U A ( 1 ) Symmetry Breaking and η , η ′ mesons in the Bethe – Salpeter Approach

U A (1) symmetry breaking is studied by introducing the flavor mixing interaction proposed by Kobayashi, Maskawa and 't Hooft. Combining the one gluon exchange interaction , the rainbow like Schwinger–Dyson equation and the ladder like Bethe–Salpeter equation are derived. The anomalous PCAC relation in the framework of this approximation is considered. The masses of the pseudoscalar mesons π, η and η ′ are calculated. It is found that the pion mass is not sensitive to the strength of the flavor mixing interaction. On the other hand, the masses of η and η ′ are reproduced by a relatively weak flavor mixing interaction, for which the chiral symmetry breaking is dominantly induced by the soft-gluon exchange interaction. The decay constants are calculated and the anomalous PCAC relation is numerically checked. It is found that the flavor structures of the η and η ′ mesons significantly depend on their masses and therefore it is quetionable to define a flavor mixing angle for η and η ′ .