The U(1) problem and topological charge on a lattice

Abstract

The U(1) problem is investigated by taking Wilson fermions as quarks. First we show that the π − η splitting is caused both by the existence of topologically non-trivial gauge configurations and by the fact that the u and d bare quark masses are very small: This is based on the fact that quark propagator has a pole at some hopping parameter close to K C for a topologically non-trivial configuration because of the Atiyah-Singer theorem. Then we calculate the propagators of the pion and the η meson in the quenched approximation. When the quark masses are small, the π − η splitting is clearly seen. We also estimate the masses of flavor singlet mesons and find that the estimated masses are in accord with the real world.