Vertex functions for confined quarks in momentum-space quark-hadron models

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Abstract
We consider the excitation of a quark-antiquark pair from the vacuum by a scalar-isoscalar source and define a vertex function for that process. We construct an equation for the vertex function, making use of a quark-antiquark interaction that describes confinement, and solve for the vertex function in a projected space defined using positive- and negative-energy projection operators constructed in terms of Dirac spinors. The solution of our projected equation yields a vertex function that is equal to zero when both the quark and the antiquark go on the mass shell. This result allows us to study quark dynamics in momentum space in the presence of a confining interaction. The role of our vertex function in extending the Nambu–Jona-Lasinio model to include a description of confinement is considered. It may be seen that various amplitudes are free of the discontinuities that arise when the quark and antiquark go on the mass shell and, therefore, dispersion relations may be developed that are free of unphysical singularities. (In this work, we are mainly interested in the scalar-isoscalar qq¯ channel; however, the analysis may be extended to include sources carrying other quantum numbers). The projected equations also allow for systematic approximations, such as the neglect of retardation and the neglect of pair-current effects, etc. Solutions are presented for the coupled equations obtained with the use of projection operators. We conclude that the use of projection operators, supplemented by the neglect of retardation, is a useful procedure for the calculation of vertex functions of the type considered in this work.