Interlinked 3D and 4D 3-Quark Wave Functions in a Bethe-Salpeter Model

TL;DR

This paper develops a covariant method to reconstruct the four-dimensional Bethe-Salpeter wave function for a three-quark system from its three-dimensional form, extending previous two-body results to three-body systems.

Contribution

It generalizes the interconnection between 3D and 4D Bethe-Salpeter wave functions from two-body to three-body systems with identical relativistic spinless quarks.

Findings

Reconstruction of 4D wave function from 3D form for three-quark systems

Extension of 2-body Bethe-Salpeter formalism to 3-body case

Covariant interconnection between 3D and 4D wave functions

Abstract

Using the method of Green's functions within the framework of a Bethe-Salpeter formalism characterized by a pairwise $qq$ interaction with a 3D support to its kernel (expressed in a Lorentz-covariant manner), the 4D BS wave function for a system of three identical relativistic spinless quarks is reconstructed from the corresponding 3D form which satisfies a {\it fully connected} 3D BSE. This result is a 3-body generalization of a similar interconnection between the 3D and 4D 2-body wave functions that had been found earlier under identical conditions of a 3D support to the corresponding BS kernel, using the ansatz of Covariant Instaneity for the pairwise $q\bar q$ interaction. The generalization from spinless to fermion quarks is straightforward.