A HADRON–QUARK VERTEX FUNCTION: INTERCONNECTION BETWEEN 3D AND 4D WAVE FUNCTIONS

Abstract

The interrelation between the 4D and 3D forms of the Bethe–Salpeter equation (BSE) with a kernel [Formula: see text] which depends on the relative four-momenta, [Formula: see text] orthogonal to Pμ is exploited to obtain a hadron–quark vertex function of the Lorentz-invariant form [Formula: see text]. The denominator function [Formula: see text] is universal and controls the 3D BSE, which provides the mass spectra with the eigenfunctions [Formula: see text]. The vertex function, directly related to the 4D wave function Ψ which satisfies a corresponding BSE, defines a natural off-shell extension over the whole of four-momentum space, and provides the basis for the evaluation of transition amplitudes via appropriate quark-loop digrams. The key role of the quantity [Formula: see text] in this formalism is clarified in relation to earlier approaches, in which the applications of this quantity had mostly been limited to the mass shell (q · P = 0). Two applications (fP values for [Formula: see text] and Fπ for π0 → γγ) are sketched as illustrations of this formalism, and attention is drawn to the problem of complex amplitudes for bigger quark loops with more hadrons, together with the role of the [Formula: see text] function in overcoming this problem.