Intrinsic and dynamically generated scalar meson states

Recent work by Maltman has given us confidence that our assignment of scalar meson states to various nonets based upon our generalized Nambu–Jona-Lasinio (NJL) model is correct. [For example, in our model the $a_0\mathrm{}\left(980\right)\mathrm{}$ and the $f_0\mathrm{}\left(980\right)\mathrm{}$ are in the same nonet as the $K_0^{*}\mathrm{}\left(1430\right).]$ In this work we make use of our model to provide a precise definition of “preexisting” resonances and “dynamically generated” resonances when considering various scalar mesons. [This distinction has been noted by Meissner in his characterization of the $f_0(400\mathrm{}–1200)$ as nonpreexisting.] We define preexisting (or intrinsic) resonances as those that appear as singularities of the $q\overline{q}\hspace{0.5em}T$ matrix and are in correspondence with $q\overline{q}$ states bound in the confining field. [Additional singularities may be found when studying the T matrices describing π-π or $\pi -K$ scattering, for example. Such features may be seen to arise, in part, from t-channel and u-channel ρ exchange in the case of π-π scattering, leading to the introduction of the σ(500–600). In addition, threshold effects in the $q\overline{q}\hspace{0.5em}T$ matrix can give rise to significant broad cross section enhancements. The latter is, in part, responsible for the introduction of the κ(900) in a study of $\pi -K$ scattering, for example.] We suggest that it is only the intrinsic resonances which correspond to $q\overline{q}$ quark-model states, and it is only the intrinsic states that are to be used to form quark-model $q\overline{q}$ nonets of states. [While the κ(900) and σ(500–600) could be placed in a nonet of dynamically generated states, it is unclear whether there is evidence that requires the introduction of other members of such a nonet.] In this work we show how the phenomena related to the introduction of the σ(500–600) and the κ(900) are generated in studies of π-π and $\pi -K$ scattering, making use of our generalized Nambu–Jona-Lasinio model. We also calculate the decay constants for the $a_0$ and $K_0^{*}$ mesons and compare our results with those obtained by Maltman. We find that the value obtained using QCD sum-rule techniques for the $a_0\mathrm{}\left(980\right)\mathrm{}$ decay constant is smaller than the decay constant calculated using our generalized NJL model, which suggests that the $a_0\mathrm{}\left(980\right)\mathrm{}$ may have a significant $K\overline{K}$ component. We find rather good agreement with Maltman’s values for the decay constants of the $a_0\mathrm{}\left(1450\right)\mathrm{}$ and $K_0^{*}\mathrm{}\left(1430\right).$ Maltman suggests that the $a_0\mathrm{}\left(980\right)\mathrm{}$ and $K_0^{*}\mathrm{}\left(1430\right)\mathrm{}$ should be in the same nonet, a result in agreement with our analysis.