Role of the ’t Hooft interaction in the calculation of the properties of scalar mesons

Abstract

A survey of the literature dealing with the quark-model configurations of the scalar-isoscalar light mesons suggests that a theoretical model is needed to deal with these characterizations, since various researchers have come to widely different conclusions concerning the nature of the \ensuremath{\sigma}(500--600) or the ${f}_{0}(980),$ basing their analysis on an attempt to fit experimental data with models for the wave functions of these states. In an earlier work we demonstrated that our generalized Nambu--Jona-Lasinio model may be used to provide a very good fit to the mixing angles and decay constants of the \ensuremath{\eta}(547) and ${\ensuremath{\eta}}^{\ensuremath{'}}(958)$ mesons. It was found that the \ensuremath{\eta}(547) was mainly a flavor octet state. On the other hand, when we use our model to describe the scalar-isoscalar ${f}_{0}(980)$ meson, we find that it is mainly a singlet state, in agreement with the recent suggestion of Ochs. We stress that these results for the properties of the pseudoscalar and scalar mesons arise from the different behavior of the 't Hooft interaction in the two cases. In this work we demonstrate that the singlet-octet flavor representation largely brings the interaction matrix to diagonal form, with some residual singlet-octet mixing due, in part, to the 't Hooft interaction and to the difference of the up (or down) and the strange quark constituent mass values. We find that, if we neglect the coupling to the two-meson continuum and use the same interaction that was used in our study of the \ensuremath{\eta} mesons, the ${f}_{0}(980)$ has a mass of 981 MeV and is the $q\overline{q}$ state of lowest energy. That result implies that the \ensuremath{\sigma}(500--600) is not a $q\overline{q}$ state, but is generated dynamically in \ensuremath{\pi}\ensuremath{\pi} scattering, as well have discussed in an earlier work. We suggest that the proper treatment of the 't Hooft interaction leads to a unified interpretation of the pseudoscalar and scalar meson characteristics.