The $SU(3)\supset SO(3)$ missing label problem and the analytical Bethe ansatz

TL;DR

This paper addresses the missing label problem in $SU(3)$ representations reduced to $SO(3)$, proposing a solution using the eigenvalues of specific algebraic scalars and applying the analytical Bethe ansatz for diagonalization.

Contribution

It introduces a method to determine missing labels in $SU(3) o SO(3)$ reduction via eigenvalues of algebraic scalars, employing the analytical Bethe ansatz for diagonalization.

Findings

Eigenvalues of $SO(3)$ scalars solve the missing label problem.

The degree four scalar can be diagonalized analytically.

The approach provides explicit labels for basis vectors.

Abstract

The missing label for basis vectors of $SU(3)$ representations corresponding to the reduction $SU(3) \supset SO(3)$ can be provided by the eigenvalues of $SO(3)$ scalars in the enveloping algebra of $su(3)$. There are only two such independent elements of degree three and four. It is shown how the one of degree four can be diagonalized using the analytical Bethe ansatz.