Discrete wave-packet representation in nuclear matter calculations

The Lippmann-Schwinger equation for the nucleon-nucleon $t$ matrix as well as the corresponding Bethe-Goldstone equation to determine the Brueckner reaction matrix in nuclear matter are reformulated in terms of the resolvents for the total two-nucleon Hamiltonians defined in free space and in medium correspondingly. This allows one to find solutions at many energies simultaneously by using the respective Hamiltonian matrix diagonalization in the stationary wave-packet basis. Among other important advantages, this approach simplifies greatly the whole computation procedures both for the coupled-channel $t$ matrix and the Brueckner reaction matrix. Therefore this principally novel scheme is expected to be especially useful for self-consistent nuclear matter calculations because it allows one to accelerate in a high degree single-particle potential iterations. Furthermore the method provides direct access to the properties of possible two-nucleon bound states in the nuclear medium. The comparison between reaction matrices found via the numerical solution of the Bethe-Goldstone integral equation and the straightforward Hamiltonian diagonalization shows a high accuracy of the method suggested. The proposed fully discrete approach opens a new way to an accurate treatment of two- and three-particle correlations in nuclear matter on the basis of the three-particle Bethe-Faddeev equation by an effective Hamiltonian diagonalization procedure.

Figure 1

The partial phase shift $\delta$ for uncoupled spin-singlet (left panel) and spin-triplet (right panel) channels derived from a total Hamiltonian diagonalization in the WP basis (solid curves) and from the direct numerical solution of the one-channel Lippmann-Schwinger equation (dashed curves) for the CD Bonn $NN$ potential. The dash-dotted curves correspond to partial phase shifts evaluated for the Nijmegen $\mathit{NN}$ potential.

Figure 2

The partial phase shifts ${\delta }_1$ (a), ${\delta }_2$ (b), and the mixing parameter $\varepsilon$ (c) for the coupled spin-triplet channels ${}^3{S}_1-{}^3{D}_1$ (left panel) and ${}^3{F}_2-{}^3{P}_2$ (right panel) of the $NN$ scattering found for the CD-Bonn and Nijmegen $NN$ potentials. The notations of curves are the same as in Fig. 1.

Figure 3

The energy of the two-nucleon bound states in nuclear matter in the deuteron channel as a function of c.m. momentum $K$ and Fermi momentum $k_F$.

Figure 4

The density of the quasideuteron as a function of relative distance $r$ for various Fermi momenta $k_F$. The results for $k_F$ = 0.5 ${\mathrm{fm}}^{-1}$ (blue dashed line) and 0.8 ${\mathrm{fm}}^{-1}$ (red dashed dotted line) have been determined for a c.m. momentum $K=0$ and are compared to the corresponding density profile of the deuteron in free space (black solid line). For $k_F$ = 0.5 ${\mathrm{fm}}^{-1}$, a profile for c.m. momentum $K$ = 0.3 $\mathrm{fm}{}^{-1}$ (green line) is presented also. Note that the density $\rho \left(r\right)$ is multiplied by $r^2$ to enhance the density at large $r$.

Figure 5

The energy of the bound two-neutron states in the ${}^{1}S_0$ channel as a function of c.m. momentum and the Fermi momentum.

Figure 6

Real part of nucleon self-energy as a function of momentum and energy, calculated for a Fermi momentum $k_F$ = 1.3 ${\mathrm{fm}}^{-1}$.

Figure 7

Imaginary part of nucleon self-energy as a function of momentum and energy, calculated for a Fermi momentum $k_F$ = 1.3 ${\mathrm{fm}}^{-1}$.

Figure 8

Real part of the self-energy, calculated at fixed energies $\omega =-50$ MeV (dashed curve) and $\omega =0$ MeV (dash-dotted curve) and found self-consistently following Eq. (24) (solid curve). Example for a Fermi momentum $k_F$ =1.3 ${\mathrm{fm}}^{-1}$.

Figure 9

Real (a) and imaginary (b) parts of the sp potential calculated self-consistently by using the effective mass approximations according to calculations 1 (dash-dotted curve) and 2 (solid curve).

Figure 10

Binding energy per nucleon calculated via the wave-packet diagonalization technique for the effective mass approximations in calculations 1 (dashed curve) and 2 (solid curve) in comparison with the results for the conventional sp spectrum choice (dotted curve). The dash-dotted curve corresponds to the results of the fully self-consistent calculations for the continuous choice from Refs. [6, 30].