Quadratic momentum dependence in the nucleon-nucleon interaction

We investigate different choices for the quadratic momentum dependence required in nucleon-nucleon potentials to fit phase shifts in high partial waves. In the Argonne $v_{18}$ potential ${\mathbf{L}}^2$ and ${\left(\mathbf{L}·\mathbf{S}\right)}^2$ operators are used to represent this dependence. The $v_{18}$ potential is simple to use in many-body calculations since it has no quadratic momentum-dependent terms in $S$ waves. However, ${\mathbf{p}}^2$ rather than ${\mathbf{L}}^2$ dependence occurs naturally in meson-exchange models of nuclear forces. We construct an alternate version of the Argonne potential, designated Argonne $v_{18pq}$, in which the ${\mathbf{L}}^2$ and ${\left(\mathbf{L}·\mathbf{S}\right)}^2$ operators are replaced by ${\mathbf{p}}^2$ and $Q_{ij}$ operators, respectively. The quadratic momentum-dependent terms are smaller in the $v_{18pq}$ than in the $v_{18}$ interaction. Results for the ground-state binding energies of ${}^3\text{H}$, ${}^3\text{He}$, and ${}^4\text{He}$, obtained with the variational Monte Carlo method, are presented for both the models with and without three-nucleon interactions. We find that the nuclear wave functions obtained with $v_{18pq}$ are slightly larger than those with $v_{18}$ at interparticle distances $<1\text{fm}$. The two models provide essentially the same binding in the light nuclei, although $v_{18pq}$ gains less attraction when a fixed three-nucleon potential is added.

Figure 1

(Color online) Central potentials in even-parity waves.

Figure 2

(Color online) Central potentials in odd-parity waves.

Figure 3

(Color online) Quadratic momentum-dependent potentials in even-parity waves.

Figure 4

(Color online) Quadratic momentum-dependent potentials in odd-parity waves.

Figure 5

(Color online) Quadratic spin-orbit potentials in $S=1$ channels.

Figure 6

(Color online) Tensor potentials in $S=1$ channels.

Figure 7

(Color online) Spin-orbit potentials in $S=1$ channels.

Figure 8

(Color online) $v_{p=1,4}$ projected from $v_{ST}^c$.

Figure 9

(Color online) Deuteron $S$- and $D$-wave functions.

Figure 10

(Color online) Correlation functions for $S$-wave channels in ${}^4\mathrm{He}$.