Construction of U(4) ⊃ SU_S(2) ⊗ SU_T(2) Basis States for Nuclear Structure Studies

The U(4) supermultiplet theory was proposed in 1937 by E. Wigner [1] based on an invariance of nuclear forces between pairs of nucleons. The supermultiplet name refers to a combination of spin and isospin degrees of freedom into a single multiplet, described by the group chain U(4) ⊃ SU_S(2) ⊗ SU_T(2). This group theoretical approach has enjoyed many successes, especially in nuclear physics [2]; for instance, it provides a selection rule for the formation of α-clusters in nuclei [3], and also explains the unusually strong binding of N = Z nuclei via the Wigner energy [4]. Moreover, when constructing the model space of a nuclear shell model framework which involves the LST-coupling scheme, the U(4) ⊃ SU_S(2) ⊗ SU_T(2) quantum numbers ([n₁, n₂, n₃, n₄], S, T ) can be used to label the basis states [5]. Therefore, it is important to have a software that can be used to determine the Wigner coupling (and recoupling) coefficients of the U(4) supermultiplet. However, this task requires the canonical U(4) ⊃ U(3) ⊃ U(2) ⊃ U(1) coupling coefficients along with a transformation between these two sets of (physical and canonical) basis states. In 1970, J. Draayer provided an analytical expression for such a transformation [6] using Wigner D-matrices and the permutation operators. Recently, F. Pan et al. derived a more effective procedure [7] which utilizes solving the null space of the spin-isospin projection matrix. In this talk, these novel techniques will be presented followed by some results from a C++ program developed by the authors.