Proxy-SU(3) symmetry in the shell model basis

TL;DR

This paper provides a microscopic justification for the proxy-SU(3) symmetry in the nuclear shell model, showing how it can be understood as a unitary transformation that preserves certain angular momentum projections, facilitating its use in calculations.

Contribution

It translates the proxy-SU(3) approximation into spherical coordinates and demonstrates its equivalence to a specific unitary transformation, offering a foundation for its application in shell model computations.

Findings

Proxy-SU(3) corresponds to replacing intruder orbitals with de Shalit--Goldhaber partners.

The approximation is equivalent to a unitary transformation in the z-coordinate.

The transformation preserves z-projections of angular momentum and spin.

Abstract

The proxy-SU(3) symmetry has been proposed for spin-orbit like nuclear shells using the asymptotic deformed oscillator basis for the single particle orbitals, in which the restoration of the symmetry of the harmonic oscillator shells is achieved by a change of the number of quanta in the z-direction by one unit for the intruder parity orbitals. The same definition suffices within the cartesian basis of the Elliott SU(3) model. Through a mapping of the cartesian Elliott basis onto the spherical shell model basis, we translate the proxy-SU(3) approximation into spherical coordinates, proving, that in the spherical shell model basis the proxy-SU(3) approximation corresponds to the replacement of the intruder parity orbitals by their de Shalit--Goldhaber partners. Furthermore it is shown, that the proxy-SU(3) approximation in the cartesian Elliott basis is equivalent to a unitary transformation in the z-coordinate, leaving the x-y plane intact, a result which in the asymptotic deformed oscillator coordinates implies, that the z-projections of angular momenta and spin remain unchanged. The present work offers a microscopic justification of the proxy-SU(3) approximation and in addition paves the way, for taking advantage of the proxy-SU(3) symmetry in shell model calculations.