Structure of the Particle-Hole Amplitudes in No-core Shell Model Wave Functions

TL;DR

This study analyzes the structure of no-core shell model wave functions for $^6$Li and $^{12}$C, revealing the impact of particle-hole amplitudes and the effects of the Lee-Suzuki transformation on form factors and state properties.

Contribution

It identifies the sign discrepancy of particle-hole amplitudes and their origin from specific matrix elements, highlighting challenges in self-consistent determination and convergence in no-core shell model calculations.

Findings

Large particle-hole amplitudes have opposite signs to those needed for experimental form factors.

The Lee-Suzuki transformation introduces higher-order particle-hole excitations.

Convergence improves with larger model spaces, especially for $^6$Li, but remains slow.

Abstract

We study the structure of the no-core shell model wave functions for $^6$Li and $^{12}$C by investigating the ground state and first excited state electron scattering charge form factors. In both nuclei, large particle-hole ($ph$) amplitudes in the wave functions appear with the opposite sign to that needed to reproduce the shape of the $(e,e')$ form factors, the charge radii, and the B(E2) values for the lowest two states. The difference in sign appears to arise mainly from the monopole $\Delta\hbar\omega=2$ matrix elements of the kinetic and potential energy (T+V) that transform under the harmonic oscillator SU(3) symmetries as $(\lambda,\mu)=(2,0)$. These are difficult to determine self-consistently, but they have a strong effect on the structure of the low-lying states and on the giant monopole and quadrupole resonances. The Lee-Suzuki transformation, used to account for the restricted nature of the space in terms of an effective interaction, introduces large higher-order $\Delta\hbar\omega=n, n>$2, $ph$ amplitudes in the wave functions. The latter $ph$ excitations aggravate the disagreement between the experimental and predicted $(e,e')$ form factors with increasing model spaces, especially at high momentum transfers. For sufficiently large model spaces the situation begins to resolve itself for $^6$Li, but the convergence is slow. A prescription to constrain the $ph$ excitations would likely accelerate convergence of the calculations.