Microscopic self-energy calculations and dispersive optical-model potentials

Nucleon self-energies for ${}^{40,48,60}$Ca isotopes are generated with the microscopic Faddeev-random-phase approximation (FRPA). These self-energies are compared with potentials from the dispersive optical model (DOM) that were obtained from fitting elastic-scattering and bound-state data for ${}^{40,48}$Ca. The ab initio FRPA is capable of explaining many features of the empirical DOM potentials including their nucleon asymmetry dependence. The comparison furthermore provides several suggestions to improve the functional form of the DOM potentials, including among others the exploration of parity and angular momentum dependence. The nonlocality of the FRPA imaginary self-energy, illustrated by a substantial orbital angular momentum dependence, suggests that future DOM fits should consider this feature explicitly. The roles of the nucleon-nucleon tensor force and charge-exchange component in generating the asymmetry dependence of the FRPA self-energies are explored. The global features of the FRPA self-energies are not strongly dependent on the choice of realistic nucleon-nucleon interactions.

Figure 1

Self-energy ${\Sigma }^{☆}\left(E\right)$ separates exactly into a static (mean-field) term ${\Sigma }^{\infty }$ and the polarization propagators $R^{\left(\text{2p1h/2h1p}\right)}\left(E\right)$ for the 2p1h/2h1p motion. These $R\left(E\right)$ are expanded in terms of particle-vibration couplings as depicted below in Fig. 3.

Figure 2

Expansion of the ph propagator $\Pi \left(E\right)$ in a series of ring diagrams. The second line gives examples of time-inversion patterns that are generated by the RPA. A similar expansion, in terms of ladders diagrams, applies to $g^{II}\left(E\right)$. The diagrams are time ordered, with time propagating upward.

Figure 3

Left: Example of one of the diagrams for $R^{\left(\text{2p1h}\right)}\left(E\right)$ that are summed to all orders by means of the Faddeev method. Each of the ellipses represent an infinite sum of rings [$\Pi \left(E\right)$] or ladders [$g^{II}\left(E\right)$]. The diagrams included in $\Pi \left(E\right)$ are shown in Fig. 2, and $g^{II}\left(E\right)$ is the analogous one for ladders [18]. Right: Corresponding contribution to the self-energy obtained from $R^{\left(\text{2p1h}\right)}\left(E\right)$ (compare to Fig. 1).

Figure 4

Volume ntegrals of Re ${\Sigma }_0^{\ell }$ for ineutrons in ${}^{40}$Ca. The horizontal, dashed lines are the volume integrals of ${\Sigma }_0^{\infty ,\ell }\left(E_F\right)$.

Figure 5

Angular momentum dependence for the volume integrals $J_F^{\ell }=J_V^{\ell }\left(E_F\right)$ of ${\Sigma }^{\infty ,\ell }\left(E_F\right)$ excluding the contribution of the dynamic part of the self-energy. For each $\ell$, results for protons are given by solid diamonds and neutrons by solid circles. Proton potentials are considerably less attractive due to the Coulomb energy. When the Coulomb interaction is suppressed (open diamonds), the proton results are close to the neutron results. The results shown are for ${}^{40}$Ca using the AV18 interaction.

Figure 6

Imaginary volume integrals of the volume part of a DOM self-energy with a local Woods-Saxon form factor replaced by a nonlocal form proposed by Perey and Buck. The results shown are for $\ell =0$ (solid), $\ell =1$ (long dash), $\ell =2$ (long dot dash), $\ell =3$ (short dash), and $\ell =4$ (short dot dash).

Figure 7

FRPA results for the average over all $\ell$ channels (dashed) are compared with the DOM result (solid), corrected for nonlocality.

Figure 8

Separate partial wave contributions of $J_W$ averaged over $\ell$ channels with the same number of harmonic-oscillator orbits in the model space. This plot is for neutrons in ${}^{40}$Ca. The dashed–double-dotted curve represents the DOM result.

Figure 9

$J_W$ averaged over even $\ell$ channels (solid) is compared with $J_W$ averaged over odd $\ell$ channels (dashed).

Figure 10

Asymmetry dependence of the absorption for neutrons and protons and dependence on tensor correlations. The top panels shows $J_W^{\text{avg}}$ for ${}^{40}$Ca (solid) as compared with the results for ${}^{48}$Ca (dashed) and ${}^{60}$Ca (dotted-dashed). The middle panels are obtained by suppressing the tensor component in the AV18 interaction. The difference between the top two panels is plotted in the bottom panels to provide a more detailed assessment of the correlations induced by including the tensor force.

Figure 11

Effect of the tensor force and charge exchange on correlations on the proton-${}^{48}$Ca self-energy. The solid curve is the imaginary volume integral $J_W^{\text{avg}}$ from the full FRPA calculation, while the dashed curve results from removing the tensor term in the AV18 interaction. The dashed-dotted curve is obtained by excluding charge exchange from the full calculation. Similar results are found for neutrons and the other Ca isotopes.

Figure 12

Diagonal part of FRPA imaginary self-energy for protons at $E-E_F=44$ MeV (solid) and the corresponding parametrized self-energy (dashed). The results shown are for $\ell =0$ and are offset by 5 MeV for each subsequent nucleus.

Figure 13

Left panels: FRPA imaginary self-energy integrated over $r^{\prime }$ for protons in ${}^{40}$Ca (solid), ${}^{48}$Ca (dashed), and ${}^{60}$Ca (dotted-dashed), see Eq. (21). The results shown are for $\ell =0$ and at an energy of 44 MeV above $E_F$. Right panels: For comparison, the DOM imaginary potentials at the same energy are shown for ${}^{40}$Ca and ${}^{48}$Ca. Since these are local, they have been corrected for nonlocality by the $k$ mass.