A New Lattice Action for Studying Topological Charge

We propose a new lattice action for non-abelian gauge theories, which will reduce short-range lattice artifacts in the computation of the topological susceptibility. The standard Wilson action is replaced by the Wilson action of a gauge covariant interpolation of the original fields to a finer lattice. If the latter is fine enough, the action of all configurations with non-zero topological charge will satisfy the continuum bound. As a simpler example we consider the O(3) σ-model in two dimensions, where a numerical analysis of discretized continuum instantons indicates that a finer lattice with half the lattice spacing of the original is enough to satisfy the continuum bound. e-mail: [email protected]. e-mail: [email protected]. Field configurations with non-zero topological charge are expected to have a strong influence on the dynamics of asymptotically free theories. In QCD, such configurations are responsible for breaking axial symmetry and resolving the U(1) problem [1]. The study of these effects however requires non-perturbative techniques and one would expect that ultimately Monte Carlo methods on the lattice would be best suited to it. The observable to consider is inspired by the classic large-Nc analyses of Witten and of Veneziano [2], which showed that, mη′ +m 2 η − 2 mK = 6 χt f2 π , (1) where χt is the topological susceptibility. In the continuum it is given by, χt ≡ ∫ dx < q(x)q(0) > |no quarks (2) with q(x) being the topological charge density, q(x) = 1 32π2 ǫμνρσTr[ Fμν Fρσ]. (3) The topological charge, Q ≡ ∫ q(x), is an integer if the field strength vanishes at infinity or if (euclidean) space-time is compact. A continuum analysis also shows that the action of any configuration with non-zero topological charge must satisfy the following bound, S ≥ 8π 2|Q| g2 0 . (4) A big effort has been devoted to the study of the topological susceptibility on the lattice. There are several choices for the operator q(x). The naive discretization of (3) does not yield integer values for Q and requires renormalization factors [3]. On the other hand, the cleaner geometrical definition due to Lüscher [4] gives an integer-valued topological charge and does not require renormalization. Here we will deal only with a geometrical definition very similar to Lüscher’s. The geometrical topological susceptibility is then obtained by computing,