ar X iv : h ep - l at / 9 60 81 10 v 1 2 1 A ug 1 99 6 1 Quenched chiral logs , the η ′ mass , and the hairpin diagram

One of the most direct manifestations of closed quark loop effects is in the mass of the flavor singlet pseudoscalar η meson. The η propagator includes not only the valence quark-antiquark term that appears in the nonsinglet propagator, but also the contribution from diagrams in which the valence quark and antiquark annihilate. These “hairpin” diagrams probe the topological structure of the gauge field via the axial U(1) anomaly. In a large Nc approximation, the axial anomaly may be introduced perturbatively as a term which breaks the U(1)A symmetry of the massless quark Lagrangian. From the chiral Lagrangian point of view, the anomaly adds a term L1 to the U(Nf)×U(Nf ) Lagrangian L0 which gives the η ′ its mass. For low momentum, and keeping only terms quadratic in the η field, the most general form of such a term is (c.f. Ref.[1])