3D-4D interlinkage of qqq wave functions under 3D support for pairwise Bethe-Salpeter kernels

Abstract

Using the method of Green functions within the framework of a Bethe–Salpeter formalism characterized by a pairwise qq interaction under the Markov–Yukawa transversality principle (MYTP) which gives Lorentz-covariant 3D support to the BS kernel, the 4D BS amplitude for three identical spinless quarks is reconstructed from the corresponding 3D quantities which satisfy a fully connected 3D BSE. This result is a three-body generalization of a similar 3D–4D interlinkage for the corresponding two-body wave functions found earlier under identical conditions of a Lorentz-covariant 3D support to the corresponding BS kernel, ("CIA" for short), for the qq interaction. (The generalization from spinless to fermion quarks is straightforward). To set "CIA" for 3D kernel support in the context of contemporary approaches to the qqq baryon problem, a model scalar 4D qqq BSE with pairwise contact interactions simulating the NJL–Faddeev equations, is worked out fully and compared with the "CIA" vertex function which reduces exactly to the 4D NJL-Faddeev form in the limit of zero spatial range. This consistency check is part of a detailed accounting of the CIA vertex structure whose physical motivation stems from the role of spectroscopy when considered as an integral part of any QCD-motivated dynamical investigation.