Singularity Structures in Coulomb-Type Potentials in Two Body Dirac Equations of Constraint Dynamics

TL;DR

This paper investigates the singular structures in Coulomb-type potentials within two-body Dirac equations, demonstrating that including tensor couplings yields well-behaved wave functions, applicable to QED and QCD bound states, including mesons.

Contribution

It reveals how tensor coupling inclusion in the relativistic Schrödinger equation resolves singularities, providing a consistent framework for describing bound states in QED and QCD.

Findings

Wave functions are well-behaved when full potentials and couplings are considered.

Tensor coupling is essential for regular wave functions in coupled triplet systems.

The formalism explains meson spectra and chiral symmetry breaking effects.

Abstract

Two Body Dirac Equations (TBDE) of Dirac's relativistic constraint dynamics have been successfully applied to obtain a covariant nonperturbative description of QED and QCD bound states. Coulomb-type potentials in these applications lead naively in other approaches to singular relativistic corrections at short distances that require the introduction of either perturbative treatments or smoothing parameters. We examine the corresponding singular structures in the effective potentials of the relativistic Schroedinger equation obtained from the Pauli reduction of the TBDE. We find that the relativistic Schroedinger equation lead in fact to well-behaved wave function solutions when the full potential and couplings of the system are taken into account. The most unusual case is the coupled triplet system with S=1 and L={(J-1),(J+1)}. Without the inclusion of the tensor coupling, the effective S-state potential would become attractively singular. We show how including the tensor coupling is essential in order that the wave functions be well-behaved at short distances. For example, the S-state wave function becomes simply proportional to the D-state wave function and dips sharply to zero at the origin, unlike the usual S-state wave functions. Furthermore, this behavior is similar in both QED and QCD, independent of the asymptotic freedom behavior of the assumed QCD vector potential. Light- and heavy-quark meson states can be described well by using a simplified linear-plus-Coulomb-type QCD potential apportioned appropriately between world scalar and vector potentials. We use this potential to exhibit explicitly the origin of the large pi-rho splitting and effective chiral symmetry breaking. The TBDE formalism developed here may be used to study quarkonia in quark-gluon plasma environments.