Calculation of the $\alpha$-particle ground state within the hyperspherical harmonic basis

The problem of calculating the four-nucleon bound state properties for the case of realistic two- and three-body nuclear potentials is studied using the hyperspherical harmonic (HH) approach. A careful analysis of the convergence of different classes of HH functions has been performed. A restricted basis is chosen to allow for accurate estimates of the binding energy and other properties of the ${}^4\mathrm{He}$ ground state. Results for various modern two-nucleon and two- plus three-nucleon interactions are presented. The origin of the isospin $T=1$ and $T=2$ admixtures in the ${}^4\mathrm{He}$ ground state is discussed in detail. The ${}^4\mathrm{He}$ asymptotic normalization constants for separation in $2+2$ and $1+3$ clusters are also computed.

Figure 1

Binding energy differences for the $\alpha$ particle for the classes C1 (circles), C2 (squares), and C3 (triangles) as function of the grand angular value K (see the text for more details). The potential used is the MT-V. The curves are fitted to the large K part of the energy differences.

Figure 2

Binding energy differences for the $\alpha$ particles for the classes C1 (circles), C2 (squares), C3 (up triangles), and C4 (down triangles) as function of the grand angular value K (see the text for more details). The potential used is the AV18. The curves are fitted to the large K part of the energy differences.

Figure 3

Ratios $c_{pt}(r_{pt})$ as function of the $p\text{−}{}^{3}H$ distance $r_{pt}$. The ratios obtained by the direct calculation of the overlap using Eqs. (44) and (45) with the ${}^4\mathrm{He}$ WF corresponding to the grand angular quantum numbers specified in Eq. (47) are shown by the open circles (case a), the open squares (case b), and the solid triangles (case c). The ratios obtained by the solution of the differential equation defined in Eq. (48) are shown by the dotted (case a), dashed (case b), and solid lines (case c), respectively (the dashed and solid line are almost coincident). The ${}^4\mathrm{He}$ WF were generated using the AV18 potential.

Figure 4

Functions $g(r)$ obtained for three choices of the HH basis specified in Eq. (47) are shown by the dotted (case a), dashed (case b), and solid lines (case c). The three lines are practically coincident and cannot be distinguished.