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 K(q^{^},q^{^}′) which depends on the relative four-momenta,q^{^}_μ = q_μ - P . q^P_μ/P² orthogonal to P_µ is exploited to obtain a hadron–quark vertex function of the Lorentz-invariant form .Γ(q^{^}) = D(q^{^2}) ⊗ φ (q^{^}). The denominator function D (q^{^2}) is universal and controls the 3D BSE, which provides the mass spectra with the eigenfunctions φ(q^{^}). 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 q^{^2} 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 (f_P values for P→ γγ̅) and F_n 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 D(q^{^}) function in overcoming this problem.