Generalized isobaric multiplet mass equation and its application to the Nolen-Schiffer anomaly

The Wigner isobaric multiplet mass equation (IMME) is the most fundamental prediction in nuclear physics with the concept of isospin. However, it was deduced based on the Wigner-Eckart theorem with the assumption that all charge-violating interactions can be written as tensors of rank two. In the present work, the charge-symmetry breaking (CSB) and charge-independent breaking (CIB) components of the nucleon-nucleon force, which contribute to the effective interaction in nuclear medium, are established in the framework of Brueckner theory with AV18 and AV14 bare interactions. Because such charge-violating components can no longer be expressed as an irreducible tensor due to density dependence, its matrix element cannot be analytically reduced by the Wigner-Eckart theorem. With an alternative approach, we derive a generalized IMME (GIMME) that modifies the coefficients of the original IMME. As the first application of GIMME, we study the long-standing question of the origin of the Nolen-Schiffer anomaly (NSA) found in the Coulomb displacement energy of mirror nuclei. We find that the naturally emerged CSB term in GIMME is largely responsible for explaining the NSA.

Figure 1

Density-dependent $S_1^{\text{(CSB)}}\left(\rho \right)$ (dots) and $S_2^{\text{(CIB)}}\left(\rho \right)$ (squares) of nuclear matter obtained with the Brueckner-Hartree-Fock approach adopting the AV18 along with AV14 bare interactions. The curves represent the fittings with Eqs. (5) and (6).

Figure 2

Coefficients of (a) $T_z$ and (b) $T_z^2$ in Eq. (1) extracted from the experimental data [45] for the $T=3/2$ quartets. The calculated coefficients of the Coulomb contribution (plus ${\mathrm{\Delta }}_{\text{np}}$) from a simple nonuniformly charged sphere with Eq. (17) (dashed curves) are shown for comparison. The contributions of the CSB and CIB effects in Eq. (15), taking $T_z=T$ nuclei as examples, are calculated by using Eqs. (11) and (12) with the SLy4 interaction.

Figure 3

Comparison of calculated CDE of the $T=1$ mirror pairs with experimental data [56]. The calculated results with and without the second term in Eq. (20) are plotted as triangles and dots, respectively. The first term in Eq. (20) is calculated using Eq. (17); the second, using Eq. (11) with the SLy4 interaction.