Role of the 't Hooft interaction in the calculation of the mixing angles of the eta(547) and eta'(958) mesons
Recent work has shown that the singlet-octet mixing angles of the η(547) and η'(958) are different. That may be demonstrated either in extended chiral perturbation theory or by analysis of a large body of experimental data. The conclusion is that the η(547) is almost entirely of octet character, while the η'(958) is mainly of singlet character with about 10% octet component. It is possible to calculate the mixing angles and decay constants in our generalized Nambu-Jona-Lasinio (NJL) model, which includes a covariant model of confinement. Our model is able to give a good account of the mass values of the η(547), η'(958), η(1295), and η(1440) mesons. (We also provide predictions for the mass values of a large number of radially excited states.) It is well known that the UA(1) symmetry is broken, so that we only have eight pseudo Goldstone bosons, rather than the nine we would have otherwise. In the NJL model that feature may be introduced by including the 't Hooft interaction in the Lagrangian. That interaction reduces the energy of the octet state somewhat and significantly increases the energy of the singlet state, making it possible to fit the mass values of the η(547) and η'(958) in the NJL model when the 't Hooft interaction is included. In this work, we derive the equations of a covariant random phase approximation that may be used to study the nonet of pseudoscalar mesons. We demonstrate that a consistent treatment of the 't Hooft interaction leads to excellent results for the singlet-octet mixing angles. (The values obtained for the singlet and octet decay constants are also quite satisfactory.) It may be seen that the difference between the up (or down) constituent quark mass and the strange quark mass induces singlet-octet mixing that is too large. However, the 't Hooft interaction contains singlet-octet coupling that enters into the theory with a sign opposite to that of the term arising from the difference of the quark mass values, leading to quite satisfactory results. In this work we present the wave function amplitudes for a number of states of the eta mesons. (The inclusion of pseudoscalar-axial-vector coupling is important for our analysis and results in the need to specify eight wave function amplitudes for each state of the eta mesons.) We present the values of the various constants that parametrize our generalized NJL model and which give satisfactory values of the eta meson masses, decay constants, and mixing angles. It is found that the calculated mass values for the η(1295) and η(1440) are quite insensitive to variation of the parameters of the model whose values have largely been fixed in our earlier studies of other light mesons.