Noncompact Lattice Simulations of SU(2) Gauge Theory

In 1980 Creutz [1] displayed quark confinement at moderate coupling in lattice simulations [2] of both abelian and nonabelian gauge theories. Whether nonabelian confinement is as much an artifact of Wilson’s action as is abelian confinement remains unclear. The basic variables of Wilson’s formulation are elements of a compact group and enter the action only through traces of their products. Wilson’s action has extra minima [3]. Mack and Pietarinen [4] and Grady [5] have shown that these false vacua affect the string tension. In their simulations of SU(2), they placed gauge-invariant infinite potential barriers between the true vacuum and the false vacua. Mack and Pietarinen saw a sharp drop in the string tension; Grady found that it vanished. To avoid using an action that has confinement built in, some physicists have introduced lattice actions that are noncompact discretizations of the continuum action with fields as the basic variables [3, 6–11]. Patrascioiu, Seiler, Stamatescu, Wolff, and Zwanziger [6] performed the first noncompact simulations of SU(2) by using simple discretizations of the classical action. They fixed the gauge and saw a force rather like Coulomb’s. It is possible to use the exact classical action in a noncompact simulation if one interpolates the gauge fields from their values on the vertices of simplices [3, 7,8] or hypercubes. For U(1) these noncompact formulations are accurate for general coupling strengths [8]. But in four dimensions and without gauge fixing, such formulations are implementable only in code that is quite slow. One may develop faster code by interpolating across plaquettes rather than throughout simplices. For SU(2) such noncompact simulations agree well with perturbation theory at very weak coupling [9]. This report relates the results of measuring Wilson loops at strong coupling, β ≡ 4/g = 0.5, on a 12 lattice in a noncompact simulation of SU(2) gauge theory without gauge fixing or fermions. Creutz ratios of large Wilson loops provide a lattice estimate of the qq̄-force for heavy quarks. There is no sign of quark confinement.