Isospin splitting of the Dirac mass probed by the relativistic Brueckner-Hartree-Fock theory in the full Dirac space

The isospin splitting of the Dirac mass obtained with the relativistic Brueckner-Hartree-Fock (RBHF) theory is thoroughly investigated. From the perspective in the full Dirac space, the long-standing controversy between the momentum-independence approximation (MIA) method and the projection method on the isospin splitting of the Dirac mass in asymmetric nuclear matter (ANM) is analyzed in detail. We find that, the $\mathit{\text{assumption}}\mathit{\text{procedure}}$ of the MIA method, which assumes that the single-particle potentials are momentum independent, is not a sufficient condition that directly leads to the wrong sign of the isospin splitting of the Dirac mass, while the $\mathit{\text{extraction}}\mathit{\text{procedure}}$ of the MIA method, which extracts the single-particle potentials from the single-particle potential energy, leads to the wrong sign. By approximately solving the set of equations involved in the $\mathit{\text{extraction}}\mathit{\text{procedure}}$, a formal expression of the Dirac mass is obtained. The wrong isospin splitting of the Dirac mass is mainly caused by that the $\mathit{\text{extraction}}\mathit{\text{procedure}}$ forcely assumes the momentum dependence of the single-particle potential energy to be a quadratic form where the strength is solely determined by the constant scalar potential.