Nucleon-nucleon potentials in phase-space representation

A phase-space representation of nuclear interactions, which depends on the distance $\vec{r}$ and relative momentum $\vec{p}$ of the nucleons, is presented. A method is developed that permits us to extract the interaction $V(\vec{r},\vec{p})$ from antisymmetrized matrix elements given in a spherical basis with angular momentum quantum numbers, either in momentum- or coordinate-space representation. This representation visualizes in an intuitive way the nonlocal behavior introduced by cutoffs in momentum space or renormalization procedures that are used to adapt the interaction to low-momentum many-body Hilbert spaces, as done in the unitary correlation operator method (UCOM) or with the similarity renormalization group (SRG). It allows us to develop intuition about the various interactions and illustrates how the softened interactions reduce the short-range repulsion in favor of nonlocality or momentum dependence while keeping the scattering phase shifts invariant. It also reveals that these effective interactions can have undesired complicated momentum dependencies at momenta around and above the Fermi momentum. Properties, similarities, and differences of the phase-space representations of the Argonne and the N3LO chiral potential and their UCOM and SRG derivatives are discussed.

Figure 1

Phase-space representation $V_{ps}^{\Lambda }(r,p)$ in arbitrary units for (a) ${\mathit{V}}^{\mathrm{loc}}=V\left(\mathit{r}\right)$, (b) ${\mathit{V}}^{p2}=\frac{1}{2}[{\vec{\mathit{p}}}^{2}V\left(\mathit{r}\right)+V\left(\mathit{r}\right){\vec{\mathit{p}}}^2]$, and (c) ${\mathit{V}}^{L2}=V\left(\mathit{r}\right){\vec{\mathit{L}}}^2$. The radial function is taken to be a sum of Gaussians: $V\left(r\right)=2e^{-\frac{r^2}{1{\mathrm{fm}}^2}}-e^{-\frac{r^2}{2{\mathrm{fm}}^2}}$.

Figure 2

Deviations $D(r,p)$ in the phase-space representation of the Argonne potential (a) and its SRG $\left(\alpha =0.2{\mathrm{fm}}^4\right)$ transformation (b) for various $r$ and $p$ as a function of ${\Lambda }_V$.

Figure 3

Matrix elements ${}_a\langle r;L,1\left|\mathit{V}\right|p;L,1\rangle {}_a$ (in MeV) for (a) the bare Argonne potential, (b) the UCOM transformed Argonne potential, and (c) the SRG transformed Argonne potential with a flow parameter of $\alpha =0.04{\mathrm{fm}}^4$ for $L=0$,  2, and  4.

Figure 4

Phase-space representation $V_{\mathrm{ps}}^{T=1,\Lambda }(r,p)$ (in MeV) for (a) the bare Argonne potential, (b) the UCOM transformed Argonne potential, (c) the SRG transformed Argonne potential with a flow parameter of $\alpha =0.04{\mathrm{fm}}^4$, and (d) the SRG transformed Argonne potential with a flow parameter of $\alpha =0.2{\mathrm{fm}}^4$.

Figure 5

Phase-space representation $V_{\mathrm{ps}}^{T=0,\Lambda }(r,p)$ (in MeV) for (a) the bare Argonne potential, (b) the UCOM transformed Argonne potential, (c) the SRG transformed Argonne potential with a flow parameter of $\alpha =0.04{\mathrm{fm}}^4$, and (d) the SRG transformed Argonne potential with a flow parameter of $\alpha =0.2{\mathrm{fm}}^4$.

Figure 6

Phase-space representation $V_{\mathrm{ps}}^{T=1,\Lambda }(r,p)$ (in MeV) for (a) the bare chiral N3LO potential, (b) the SRG transformed potential with a flow parameter of $\alpha =0.04{\mathrm{fm}}^4$, and (c) the SRG transformed potential with a flow parameter $\alpha =0.2{\mathrm{fm}}^4$.

Figure 7

Phase-space representation for $p=0,V_{\mathrm{ps}}^{T=1,\Lambda =0}(r,0)$ (in MeV) for (a) the Argonne potential and various effective potentials based on it and (b) the N3LO potential and its SRG transformed version with flow parameter $\mathrm{s}\alpha =0.04{\mathrm{fm}}^4$ and $\alpha =0.2{\mathrm{fm}}^4$ compared with the SRG transformed Argonne potential with $\alpha =0.2{\mathrm{fm}}^4$.