3D-4D Interlinkage Of qqq Wave Functions Under 3D Support For Pairwise Bethe-Salpeter Kernels

TL;DR

This paper develops a 3D-4D linkage for Bethe-Salpeter wave functions of three-quark systems, extending previous 2-body results, and verifies the consistency of a covariant 3D kernel approach with NJL-based models.

Contribution

It generalizes the 2-body Bethe-Salpeter results to 3-body systems with a covariant 3D support, and demonstrates the consistency of the CIA vertex with NJL model limits.

Findings

Reconstruction of 4D wave functions from 3D forms for 3-quark systems.

Verification that the CIA vertex reduces to NJL form at zero spatial range.

Extension of 2-body Bethe-Salpeter results to 3-body relativistic systems.

Abstract

Using the method of Green's functions within a Bethe-Salpeter framework characterized by a pairwise qq interaction with a Lorentz-covariant 3D support to its kernel, the 4D BS wave function for a system of 3 identical relativistic spinless quarks is reconstructed from the corresponding 3D form which satisfies a fully connected 3D BSE. This result is a 3-body generalization of a similar 2-body result found earlier under identical conditions of a 3D support to the corresponding qq-bar BS kernel under Covariant Instaneity (CIA for short). (The generalization from spinless to fermion quarks is straightforward). To set the CIA with 3D BS kernel support ansatz in the context of contemporary approaches to the qqq baryon problem, a model scalar 4D qqq BSE with pairwise contact interactions to simulate the NJL-Faddeev equations is worked out fully, and a comparison of both vertex functions shows that the CIA vertex reduces exactly to the NJL form in the limit of zero spatial range. This consistency check on the CIA vertex function is part of a fuller accounting for its mathematical structure whose physical motivation is traceable to the role of `spectroscopy' as an integral part of the dynamics.