Calculation of the continuum contribution to hadronic current correlation functions in a chiral quark model with confinement

A powerful technique for the calculation of hadronic properties is the application of QCD sum rules in a study of correlation functions. The calculation of hadronic current correlation functions may be carried out using the operator-product expansion. The correlator may also be represented by hadronic states, which include one or more discrete levels and a continuum contribution. Comparison of the two methods of calculation of the same correlator allows for the determination of hadronic parameters after one estimates the continuum contribution and specifies the values of the vacuum condensates. In this work we consider two types of contribution to the continuum. The first involves the radially excited states of a meson. These states appear as resonances which ultimately decay into the continuum of multimeson states. In a second process, the multimeson continuum is reached in a direct process, without the excitation of a resonance. For the resonances, we calculate the contribution to the hadronic correlator by calculating the meson decay constants in the region $0l{P}^{2}l6{\mathrm{GeV}}^{2},$ making use of our generalized Nambu--Jona-Lasinio model, which includes a covariant model of confinement. [Our model provides a good fit to the decay constants of the ${a}_{0}(1450)$ and ${K}_{0}^{*}$ (1430) mesons. Our value for the ${a}_{0}(980)$ is about 60% larger than the value obtained by Maltman using QCD sum rules. Also, our values for the pion and kaon decay constants are about 35 and 10% too large, respectively. We have not performed any parameter variations to improve upon these values.] The second contribution to the correlator arises from the direct excitation of multimeson continuum states. For the ${a}_{0}(980)$ and its radial excitations, we calculate both the resonant and direct contributions. In the case of the isovector pseudoscalar states we calculate the resonance contribution and parametrize the direct multimeson contribution so as to reproduce a phenomenological expression of the quark-hadron duality model. In this case the calculated resonance contribution is about 15% of the phenomenological result after a Borel transformation is made. The resonance contribution plus a quark-hadron duality model contribution with ${s}_{0}=1.65{\mathrm{GeV}}^{2}$ yields a good fit to the phenomenological form. In this phenomenological form only the quark-hadron duality model is used with ${s}_{0}=1.53{\mathrm{GeV}}^{2}.$ We also provide values of the resonance contribution to the continuum strength in the case of the strange pseudoscalar states. In the case of the isovector scalar mesons we find significant disagreement between our microscopic calculations and the result of the quark-duality model for the continuum of the hadronic current correlator. Indeed, the contribution of the resonances to the continuum is a least an order of magnitude larger than the values obtained from the quark-hadron duality model. That result may suggest an explanation for the problems encountered by some authors who have studied the isovector scalar mesons using QCD sum rules.