Nonrelativistic nucleon effective masses in nuclear matter: Brueckner-Hartree-Fock model versus relativistic Hartree-Fock model

The density and isospin dependencies of nonrelativistic nucleon effective mass $\left(m_N^{*}\right)$ are studied, which is a measure of the nonlocality of the single particle (s.p.) potential. It can be decoupled as the so-called $k$ mass ($m_k^{*}$, i.e., the nonlocality in space) and $E$ mass ($m_E^{*}$, i.e., the nonlocality in time). Both $k$ mass and $E$ mass are determined and compared by using the latest versions of the nonrelativistic Brueckner-Hartree-Fock (BHF) model and the relativistic Hartree-Fock (RHF) model. The latter is achieved based on the corresponding Schrödinger equivalent s.p. potential in a relativistic framework. We demonstrate the origins of different effective masses and discuss also their neutron-proton splitting in the asymmetric matter in different models. We find that the neutron-proton splittings of both the $k$ mass and the $E$ mass have the same asymmetry dependencies at the densities considered; namely, $m_{k,n}^{*}>m_{k,p}^{*}$ and $m_{E,p}^{*}>m_{E,n}^{*}$. However, the resulting splittings of nucleon effective masses could have different asymmetry dependencies in these two models because they could be dominated either by the $k$ mass (then we have $m_n^{*}>m_p^{*}$ in the BHF model), or by the $E$ mass (then we have $m_p^{*}>m_n^{*}$ in the RHF model). The isospin splitting in the BHF model is more consistent with the recent analysis from the nucleon-nucleus-scattering data, while the small $E$ mass $m_E^{*}$ in the RHF case as a result of the missing ladder summation finally leads to an opposite splitting behavior.

Figure 1

Nucleon effective mass $m_N^{*}/m_N$ as a function of the density $\rho$ for isospin symmetric matter, with the BHF model with (dashed line) or without TBF (solid line), the RHF model (dash-dotted line), to be compared with the DBHF model (dotted line).

Figure 2

Density dependence of both $k$ mass and $E$ mass are compared in the $\mathrm{BHF}+\mathrm{TBF}$ model and the RHF model for isospin symmetric matter.

Figure 3

Proton effective mass $m_p^{*}/m_N$ as a function of the proton fraction $x_p$ at two densities: $\rho ={\rho }_0$ (shown as lines) and $\rho =2{\rho }_0$ (shown as symbolled lines), for both the BHF model (with TBF in dashed lines, without TBF in solid lines) and the RHF model (dotted lines).

Figure 4

Asymmetry dependencies of the effective mass $m_N^{*}/m_N$ are compared in the $\mathrm{BHF}+\mathrm{TBF}$ model and the RHF model for two densities: $\rho ={\rho }_0$ (shown in lines with squares) and $\rho =2{\rho }_0$ (shown in lines with dots). Results of neutrons (protons) are displayed with solid (dashed) lines.

Figure 5

Mass splittings of both $k$ mass (left panel) and $E$ mass (right panel) with the asymmetry parameter are compared in the $\mathrm{BHF}+\mathrm{TBF}$ model and the RHF model for two densities: $\rho ={\rho }_0$ (shown in lines with squares) and $\rho =2{\rho }_0$ (shown in lines with dots).