U(1) problem and topological excitations on a lattice

We investigate in lattice QCD the U(1) problem that the η’ meson (960 MeV) is much heavier than the π meson (140 MeV), taking Wilson fermions as quarks. We first derive the spectral representation of the quark propagator and investigate the Atiyah-Singer index theorem on a lattice. Then we calculate the η propagators as well as the π propagators for 10 configurations on an $8^3$×16 lattice and show the following: The large splitting between the flavor-singlet pseudoscalar meson and the π meson is caused both by the existence of topologically nontrivial configurations and by the fact that the u and d bare-quark masses are very small. We further obtain that the mass of ‘‘η’’=(ū${\gamma }_5$u+d${\gamma }_5$d)/ √2 and the mass of ${\eta }_s$=s¯${\gamma }_5$s are both about 750 MeV. This is in accord with experiment, because if we assume the transition mass matrix between the two states is about 200 MeV, we obtain the correct masses for the η’ and η mesons and the mixing angle φ=10° (η’=${\eta }_1$ ×cosφ+${\eta }_8$ sinφ). On the other hand there is no noticeable splitting between the ρ and ω mesons. We clarify the reason for the difference between the π-η splitting and the ρ-ω splitting. We point out that if the U(1) problem should be resolved both for Wilson quarks and for Kogut-Susskind quarks, the mechanisms would be completely different from each other. We also compare our results with the prediction of 1/N-expanded QCD.