Renormalized relativistic Hartree-Bogoliubov equations with a zero-range pairing interaction

A recently introduced scheme for the renormalization of the Hartree-Fock-Bogoliubov equations in the case of zero-range pairing interaction is extended to the relativistic Hartree-Bogoliubov model. A density-dependent strength parameter of the zero-range pairing is adjusted in such a way that the renormalization procedure reproduces the empirical ${}^1{S}_0$ pairing gap in isospin-symmetric nuclear matter. The model is applied to the calculation of ground-state pairing properties of finite spherical nuclei.

Figure 1

Pairing gap in symmetric nuclear matter as a function of density for the zero-range interaction with constant coupling $g=-330\text{MeV}\hspace{1em}{\mathrm{fm}}^3$ (dashed) and the density-dependent coupling Eq. (23) (solid). The corresponding density-dependent curves $g_{\mathrm{eff}}$ [Eq. (19)] are plotted in the panel on the right. The dots in the left panel denote the pairing gap calculated with the Gogny D1S interaction.

Figure 2

Pairing gap in symmetric nuclear matter as a function of the Fermi momentum, calculated with the zero-range interaction and the density-dependent coupling Eq. (23), for a series of cutoff energies.

Figure 3

Average neutron pairing gaps (left) and pairing energies (right) for ${}^{114}\mathrm{Sn}$ calculated with a zero-range interaction as functions of the cutoff energy. The calculations are performed with the renormalization of the constant coupling $g=-330\text{MeV}$ ${\mathrm{fm}}^3$ (squares), and with the renormalization of the density-dependent coupling Eq. (23) (dots).

Figure 4

The pairing fields as functions of the radial coordinate (left), and the curves $g_{\mathrm{eff}}[r(\rho )]$ [Eq. (19)] as functions of the corresponding density in ${}^{114}\mathrm{Sn}$ (right), for a series of energy cutoffs. The upper panels display results of the renormalization procedure for the zero-range force with constant coupling $g=-330\text{MeV}\hspace{1em}{\mathrm{fm}}^3$ (RCC), and the lower ones correspond to the density-dependent coupling Eq. (23) (RDDC).

Figure 5

The effective single-nucleon potential in ${}^{114}\mathrm{Sn}$ as a function of the radial coordinate (left) and the correction to the strength parameter of the zero-range effective force (right) as a function of the square of the Fermi momentum. μ denotes the position of the chemical potential.

Figure 6

Average neutron pairing gaps (left) and pairing energies (right) for the chain of even-even Sn isotopes with $110\hspace{1em}\le \hspace{1em}A\hspace{1em}\le \hspace{1em}160$. The calculations are performed with the renormalization of the constant coupling $g=-330\text{MeV}\hspace{1em}{\mathrm{fm}}^3$ (dashed) and with the renormalization of the density-dependent coupling Eq. (23) (solid).

Figure 7

The anomalous densities in ${}^{114}\mathrm{Sn}$, ${}^{124}\mathrm{Sn}$, ${}^{150}\mathrm{Sn}$ isotopes as functions of the radial coordinate (left) and the density-dependent curves $g_{\mathrm{eff}}$ [Eq. (19)] (right). The calculations are performed with the renormalization of the density-dependent coupling Eq. (23) (RDDC).

Figure 8

Same as in Fig. 7 but for the RCC case with the constant coupling $g=-330\text{MeV}\hspace{1em}{\mathrm{fm}}^3$.