Spin and pseudospin symmetries of the Dirac equation with confining central potentials

TL;DR

This paper analyzes the node structure of solutions to the Dirac equation with confining potentials under exact spin and pseudospin symmetry, revealing how potential behavior influences bound states and symmetry conditions.

Contribution

It provides a detailed derivation of the node structure for Dirac solutions under exact symmetries and highlights the impact of potential asymptotics on bound state existence.

Findings

Node structure is identical for spin symmetry with potentials vanishing at infinity.

Reversed node structure occurs under pseudospin symmetry with confining potentials.

Positive energy bound solutions are possible in pseudospin symmetry for confining potentials.

Abstract

We derive the node structure of the radial functions which are solutions of the Dirac equation with scalar $S$ and vector $V$ confining central potentials, in the conditions of exact spin or pseudospin symmetry, i.e., when one has $V=\pm S+C$, where $C$ is a constant. We show that the node structure for exact spin symmetry is the same as the one for central potentials which go to zero at infinity but for exact pseudospin symmetry the structure is reversed. We obtain the important result that it is possible to have positive energy bound solutions in exact pseudospin symmetry conditions for confining potentials of any shape, including naturally those used in hadron physics, from nuclear to quark models. Since this does not happen for potentials going to zero at large distances, used in nuclear relativistic mean-field potentials or in the atomic nucleus, this shows the decisive importance of the asymptotic behavior of the scalar and vector central potentials on the onset of pseudospin symmetry and on the node structure of the radial functions. Finally, we show that these results are still valid for negative energy bound solutions for anti-fermions.