Intrinsic and dynamically generated scalar meson states

Abstract

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 ${a}_{0}(980)$ and the ${f}_{0}(980)$ are in the same nonet as the ${K}_{0}^{*}(1430).]$ 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 ${f}_{0}(400--1200)$ as nonpreexisting.] We define preexisting (or intrinsic) resonances as those that appear as singularities of the $q\overline{q} T$ 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 \ensuremath{\pi}-\ensuremath{\pi} or $\ensuremath{\pi}\ensuremath{-}K$ scattering, for example. Such features may be seen to arise, in part, from t-channel and u-channel \ensuremath{\rho} exchange in the case of \ensuremath{\pi}-\ensuremath{\pi} scattering, leading to the introduction of the \ensuremath{\sigma}(500--600). In addition, threshold effects in the $q\overline{q} T$ matrix can give rise to significant broad cross section enhancements. The latter is, in part, responsible for the introduction of the \ensuremath{\kappa}(900) in a study of $\ensuremath{\pi}\ensuremath{-}K$ 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 \ensuremath{\kappa}(900) and \ensuremath{\sigma}(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 \ensuremath{\sigma}(500--600) and the \ensuremath{\kappa}(900) are generated in studies of \ensuremath{\pi}-\ensuremath{\pi} and $\ensuremath{\pi}\ensuremath{-}K$ 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 ${a}_{0}(980)$ decay constant is smaller than the decay constant calculated using our generalized NJL model, which suggests that the ${a}_{0}(980)$ may have a significant $K\overline{K}$ component. We find rather good agreement with Maltman's values for the decay constants of the ${a}_{0}(1450)$ and ${K}_{0}^{*}(1430).$ Maltman suggests that the ${a}_{0}(980)$ and ${K}_{0}^{*}(1430)$ should be in the same nonet, a result in agreement with our analysis.