Cooling for instantons and the wrath of Nahm

Abstract

The dynamics of instantons and anti-instantons in lattice QCD can be studied by analysing the action and topological charge of configurations as they approach a self-dual or anti-self-dual state, i.e. a state in which S/S_0=|Q|. We use cooling to reveal the semi-classical structure of the configurations we study. Improved actions which eliminate discretization errors up to and including O(a^4) are used to stabilise instantons as we cool for several thousand sweeps. An analogously improved lattice version of the continuum field-strength tensor is used to construct a topological charge free from O(a^4) discretization errors. Values of the action and topological charge obtained with these improved operators approach mutually-consistent integer values to within a few parts in 10^4 after several hundred cooling sweeps. Analysis of configurations with |Q| \approx 1 and |Q| \approx 2 supports the hypothesis that a self-dual |Q|=1 configuration cannot exist on the 4-torus.