Low-momentum nucleon-nucleon interaction and its application to few-nucleon systems

Low-momentum nucleon-nucleon interactions are derived within the framework of a unitary-transformation theory, starting with realistic nucleon-nucleon interactions. A cutoff momentum $\Lambda$ is introduced to specify a border between the low- and high-momentum spaces. By Faddeev-Yakubovsky calculations the low-momentum interactions are investigated with respect to the dependence of ground-state energies of ${}^3\mathrm{H}$ and ${}^4\mathrm{He}$ on the parameter $\Lambda$. It is found that we need the momentum cutoff parameter $\Lambda ⩾5{\mathrm{fm}}^{-1}$ in order to reproduce satisfactorily the exact values of the binding energies for ${}^3\mathrm{H}$ and ${}^4\mathrm{He}$. The calculation with $\Lambda =2{\mathrm{fm}}^{-1}$ recommended by Bogner et al. leads to considerable overbinding at least for few-nucleon systems.

Figure 1

Comparison of LMNN interactions from the CD-Bonn potential in the case of $\Lambda =2.0{\mathrm{fm}}^{-1}$. The diagonal matrix elements $⟨k\mid V_{\text{low}k}\mid k⟩$ for the ${}^{1}S_0$ (a) and ${}^{3}S_1\text{−}{}^{3}D_1$ (b) partial waves are shown. The lines depict method-1 (solid) and method-2 (long-dashed) results. Because they coincide very well, one cannot distinguish both lines by the eye.

Figure 2

Comparison of LMNN interactions for the off-diagonal elements $⟨k\mid V_{\text{low}k}\mid 2k⟩$. Description is the same as in Fig. 1.

Figure 3

Comparison of the LMNN interaction in the case of $\Lambda =2.0{\mathrm{fm}}^{-1}$ and the original CD-Bonn potential for the ${}^{1}S_0$ (a) and ${}^{3}S_1\text{−}{}^{3}D_1$ (b) partial waves. The solid and dashed lines depict the diagonal matrix elements of the LMNN interaction and the original CD-Bonn potential, respectively.

Figure 4

Phase shifts for the ${}^{1}S_0$ (a) and the ${}^{3}S_1\text{−}{}^{3}D_1$ (b) channels below $E_{\text{lab}}=300\mathrm{MeV}$. Because the lines from the LMNN interactions and the original CD-Bonn potential coincide, one cannot distinguish both lines by the eye. In (b) the upper, middle, and lower lines depict the ${}^{3}S_1$ phase shift, the mixing parameter ${ϵ}_1$, and the ${}^{3}D_1$ phase shift, respectively.

Figure 5

Energy shifts $\Delta E_{\mathrm{b}}$ of ${}^3\mathrm{H}$ (a) and ${}^4\mathrm{He}$ (b) as a function of $\Lambda$ using the LMNN interaction (solid) from the CD-Bonn potential. The long-dashed lines in (a) and (b) are plotted for the case of simple momentum cutoff calculations. The short-dashed line in (a) depicts the result for the LMNN interaction from the Nijm-I potential.