Fixed point action and topological charge for SU(2) gauge theory

Instantons in the QCD vacuum can explain the UA(1) problem, and they could be responsible for the low energy hadron and glueball spectrum and the chiral phase transition as well[1]. Instantons have been studied on the lattice for over a decade but these studies have produced about as much controversy as physical results. The problems are three-fold: 1) Instantons are not conserved by topology on the lattice small instantons can ”fall through” the lattice. 2) Most lattice actions are not scale invariant the lattice action of an instanton depends on its size. 3) The definition of the topological charge is problematic the geometric definition erroneously identifies dislocations as instantons, while the algebraic definition relies on some sort of smoothing algorithm like cooling or APE-smearing and the topological susceptibility has both additive and multiplicative renormalization. In figure 1 the action of a single smooth instanton calculated with the Wilson action is plotted as a function of its size ρ. (SI = 8π 2 is the continuum instanton action.) The dotted vertical line indicates where the instanton disappears from the lattice according to the geometric definition and the dashed line is the value of the topological charge evaluated with the simplest algebraic FF̃ operator, the twisted plaquette. The scale violation of the Wilson action is well known but it is interesting to note that the algebraic definition gives Q < 0.8 if ρ < 2.0 even for very smooth instantons. Fixed point actions are, by construction, scale