Relativistic Brueckner-Hartree-Fock Research Articles

Saturation of nuclear matter in the relativistic Brueckner-Hatree-Fock approach with a leading order covariant chiral nuclear force

Nuclear saturation is a crucial feature in nuclear physics that plays a fundamental role in understanding various nuclear phenomena, ranging from properties of finite nuclei to those of neutron stars. However, a proper description of nuclear saturation is highly nontrivial in modern nonrelativistic ab initio studies because of the elusive three-body forces. In this letter, we calculate the equation of state for nuclear matter in the relativistic Brueckner-Hartree-Fock (RBHF) framework with the leading order covariant chiral nuclear force. We show that a simultaneous description of the nucleon-nucleon scattering data and the saturation of the symmetric nuclear matter can be achieved. In this regard, the relativistic effects nicely explain the saturation of nuclear matter. As a result, the present study based on the covariant chiral nuclear force shows that in the RBHF framework, one can achieve saturation with a leading order covariant chiral nuclear force with only two-body forces, in contrast to the vast majorities of studies in the non-relativistic framework, where the next-to-next-to-leading order two-body and three-body chiral forces are needed. This study sets the foundation for studying nuclear saturation with the covariant chiral force in the RBHF framework, which allows for a systematic understanding of one of the key features of nuclear physics more microscopically.

Properties of Pb208 predicted from the relativistic equation of state in the full Dirac space

Relativistic Brueckner-Hartree-Fock (RBHF) theory in the full Dirac space allows one to determine uniquely the momentum dependence of scalar and vector components of the single-particle potentials. In order to extend this new method from nuclear matter to finite nuclei, as a first step, properties of $^{208}\mathrm{Pb}$ are explored by using the microscopic equation of state for asymmetric nuclear matter and a liquid droplet model. The neutron and proton density distributions, the binding energies, the neutron and proton radii, and the neutron skin thickness in $^{208}\mathrm{Pb}$ are calculated. In order to further compare the charge densities predicted from the RBHF theory in the full Dirac space with the experimental charge densities, the differential cross sections and the electric charge form factors in the elastic electron-nucleus scattering are obtained by using the phase-shift analysis method. The results from the RBHF theory are in good agreement with the experimental data. In addition, the uncertainty arising from variations of the surface term parameter ${f}_{0}$ in the liquid droplet model is also discussed.

Exploring universal characteristics of neutron star matter with relativistic ab initio equations of state

Starting from the relativistic realistic nucleon-nucleon ($NN$) interactions, a newly developed relativistic \textit{ab initio} method, i.e., the relativistic Brueckner-Hartree-Fock (RBHF) theory in the full Dirac space is employed to study the neutron star properties. First, the one-to-one correspondence relation for gravitational redshift and mass is established and used to infer the mass of isolated neutron stars combining the gravitational redshift measurements. Next, the ratio of the moment of inertia $I$ to $MR^2$ as a function of the compactness $M/R$ is obtained, which is consistent with the universal relations in the literature. The moment of inertia for $1.338M_\odot$ pulsar PSR J0737-3039A $I_{1.338M_\odot}$ is predicted to be 1.356$\times10^{45}$, 1.381$\times10^{45}$, and $1.407\times10^{45}\ \mathrm{g~cm^2}$ by the RBHF theory in the full Dirac space with $NN$ interactions Bonn A, B, and C, respectively. Finally, the quadrupole moment of neutron star is calculated under the slow-rotation and small-tidal-deformation approximation. The equation of states constructed by the RBHF theory in the full Dirac space, together with those by the projection method and momentum-independence approximation, conform to universal $I$-Love-$Q$ relations as well. By combing the tidal deformability from GW170817 and the universal relations from relativistic \textit{ab initio} methods, the moment of inertia of neutron star with 1.4 solar mass is also deduced as $I_{1.4M_\odot}=1.22^{+0.40}_{-0.25}\times 10^{45}\mathrm{g\ cm^2}$.

Nuclear Matter and Neutron Stars from Relativistic Brueckner–Hartree–Fock Theory

The momentum and isospin dependence of the single-particle potential for the in-medium nucleon are the key quantities in the Relativistic Brueckner–Hartree–Fock (RBHF) theory. It depends on how to extract the scalar and the vector components of the single-particle potential inside nuclear matter. In contrast to the RBHF calculations in the Dirac space with the positive-energy states (PESs) only, the single-particle potential can be determined in a unique way by the RBHF theory together with the negative-energy states, i.e., the RBHF theory in the full Dirac space. The saturation properties of symmetric and asymmetric nuclear matter in the full Dirac space are systematically investigated based on the realistic Bonn nucleon–nucleon potentials. In order to further specify the importance of the calculations in the full Dirac space, the neutron star properties are investigated. The direct URCA process in neutron star cooling will happen at density ρ DURCA = 0.43, 0.48, 0.52 fm−3 with proton fractions of Y p,DURCA = 0.13. The radii of a 1.4M ⊙ neutron star are predicated as km, and their tidal deformabilities are for potential Bonn A, B, C. Compared with the results obtained in the Dirac space with PESs only, the full-Dirac-space RBHF calculation predicts the softest symmetry energy, which would be more favored by the gravitational wave detection of GW170817. Furthermore, the results from the full-Dirac-space RBHF theory are consistent with the recent astronomical observations of massive neutron stars and simultaneous mass–radius measurement.

The nuclear symmetry energy from relativistic Brueckner-Hartree-Fock model * *Supported by in part by the National Natural Science Foundation of China (11775119, 12175109), the Natural Science Foundation of Tianjin

The microscopic mechanisms of the symmetry energy in nuclear matter are investigated in the framework of the relativistic Brueckner-Hartree-Fock (RBHF) model with a high-precision realistic nuclear potential, pvCDBonn A. The kinetic energy and potential contributions to symmetry energy are decomposed. They are explicitly expressed by the nucleon self-energies, which are obtained through projecting the G-matrices from the RBHF model into the terms of Lorentz covariants. The nuclear medium effects on the nucleon self-energy and nucleon-nucleon interaction in symmetry energy are discussed by comparing the results from the RBHF model and those from Hartree-Fock and relativistic Hartree-Fock models. It is found that the nucleon self-energy including the nuclear medium effect on the single-nucleon wave function provides a largely positive contribution to the symmetry energy, while the nuclear medium effect on the nucleon-nucleon interaction, i.e., the effective G-matrices provides a negative contribution. The tensor force plays an essential role in the symmetry energy around the density. The scalar and vector covariant amplitudes of nucleon-nucleon interaction dominate the potential component of the symmetry energy. Furthermore, the isoscalar and isovector terms in the optical potential are extracted from the RBHF model. The isoscalar part is consistent with the results from the analysis of global optical potential, while the isovector one has obvious differences at higher incident energy due to the relativistic effect.

Nuclear matter in relativistic Brueckner-Hartree-Fock theory with Bonn potential in the full Dirac space

Starting from the Bonn potential, relativistic Brueckner-Hartree-Fock (RBHF) equations are solved for nuclear matter in the full Dirac space, which provides a unique way to determine the single-particle potentials and avoids the approximations applied in the RBHF calculations in the Dirac space with positive-energy states (PESs) only. The uncertainties of the RBHF calculations in the Dirac space with PESs only are investigated, and the importance of the RBHF calculations in the full Dirac space is demonstrated. In the RBHF calculations in the full Dirac space, the empirical saturation properties of symmetric nuclear matter are reproduced, and the obtained equation of state agrees with the results based on the relativistic Green's function approach up to the saturation density.

Relativistic Brueckner-Hartree-Fock Theory in Infinite Nuclear Matter

On the way of a microscopic derivation of covariant density functionals, the first complete solution of the relativistic Brueckner-Hartree-Fock (RBHF) equations is presented for symmetric nuclear matter. In most of the earlier investigations, theG-matrix is calculated only in the space of positive energy solutions. On the other side, for the solution of the relativistic Hartree-Fock (RHF) equations, also the elements of this matrix connecting positive and negative energy solutions are required. So far, in the literature, these matrix elements are derived in various approximations. We discuss solutions of the Thompson equation for the full Dirac space and compare the resulting equation of state with those of earlier attempts in this direction.

Origin of the evolution of spin-orbit and pseudospin-orbit splittings in neutron drops

We found the same staggering pattern in the evolution of the spin-orbit (SO) and pseudospin-orbit (PSO) splittings in neutron drops obtained by the energy-density functional (EDF) theory using SAMi-T and SLy5 parametrizations, with and without the tensor terms respectively. This staggering evolution is similar to the results recently obtained by the relativistic Brueckner-Hartree-Fock (RBHF) theory with the Bonn A potential, which was claimed to be a tensor effect. In this work, we present an intuitive but essential explanation on the origin of this staggering evolution in the Skyrme EDF theory. We found that, for both the SO and PSO partners, the energy splittings mainly originate from the SO potential. The staggering evolution of both the SO and PSO splittings is mainly due to the drastic change of density while neutrons fill different single-particle orbits. The key to determine the specific staggering pattern is the strength of the neutron density gradient term and the SO density term in the SO potential, not necessarily the tensor force.

Impact of tensor forces on spin-orbit splittings in neutron-proton drops

A systematic study of the tensor-force impact in neutron-proton drops has been reported using the relativistic Hartree-Fock (RHF) theory with the $\ensuremath{\pi}\text{\ensuremath{-}}N$ coupling strength optimized to the relativistic Brueckner-Hartree-Fock (RBHF) results for neutron drops. The evolutions of the neutron spin-orbit splittings as a function of the neutron number for neutron-proton drops with one proton behave similarly to the pure neutron drops, which show the tensor-force effect. By adding one more proton or neutron in the neutron drop with $N=20$, it is found that the tensor-force effect is more prominent between neutrons and protons than between neutrons. This can be attributed to the isospin factor in the tensor term of the $\ensuremath{\pi}\text{\ensuremath{-}}N$ interaction in the RHF density functional theory, which reflects the fact that the neutron-proton tensor force is stronger than the neutron-neutron one. Similar behavior for the spin-orbit splitting evolutions has also been found for the neutron-proton drops with 20 protons, where the tensor-force strength $\ensuremath{\lambda}$ is redetermined according to the RBHF results due to the large central densities of the systems.

Properties of nuclear matter in relativistic Brueckner–Hartree–Fock model with high-precision charge-dependent potentials

Properties of nuclear matter are investigated in the framework of relativistic Brueckner–Hartree–Fock model with the latest high-precision charge-dependent Bonn (pvCD-Bonn) potentials, where the coupling between pion and nucleon is adopted as a pseudovector form. These realistic pvCD-Bonn potentials are renormalized to effective nucleon–nucleon (NN) interactions, G matrices. They are obtained by solving the Blankenbecler-Sugar (BbS) equation in nuclear medium. Then, the saturation properties of symmetric nuclear matter are calculated with pvCD-Bonn A, B, C potentials. The energies per nucleon are around −10.72 MeV to −16.83 MeV at saturation densities, 0.139 fm−3 to 0.192 fm−3 with these three potentials, respectively. It clearly demonstrates that the pseudovector coupling between pion and nucleon can generate reasonable saturation properties comparing with pseudoscalar coupling. Furthermore, these saturation properties have strong correlations with the tensor components of NN potentials, i.e., the D-state probabilities of deuteron, PD to form a relativistic Coester band. The equations of state of pure neutron matter from pvCD-Bonn potentials are almost identical, since the prominent difference of pvCD Bonn potentials are the components of tensor force, which provides very weak contributions in the case of total isospin T = 1. In addition, the charge symmetry breaking (CSB) and charge independence breaking (CIB) effects are also discussed in nuclear matter from the partial wave contributions with these high-precision charge-dependent potentials. In general, the magnitudes of CSB from the differences between nn and pp potentials are about 0.05 MeV, while those of CIB are around 0.35 MeV from the differences between np and pp potentials.

Properties of Neutron Stars Described by a Relativistic Ab Initio Model

Abstract Properties of neutron stars are investigated by an available relativistic ab initio method, the relativistic Brueckner–Hartree–Fock (RBHF) model, with the latest high-precision, relativistic charge-dependent potentials, pvCD-Bonn A, B, C. The neutron star matter is solved within the beta equilibrium and charge neutrality conditions in the framework of the RBHF model. Compared to the conventional treatment, where the chemical potential of leptons was approximately represented by the symmetry energy of nuclear matter, the equation of state of neutron star matter in the present self-consistent calculation with pvCD-Bonn B has a striking difference above the baryon number density n b = 0.55 fm−3. However, these differences influence the global properties of neutron stars only about 1% to 2%. Then, three two-body potentials pvCD-Bonn A, B, C, with different tensor components, are systematically applied in the RBHF model to calculate the properties of neutron stars. It is found that the maximum masses of neutron stars are around 2.21–2.30 M ⊙, and the corresponding radii are R = 11.18–11.72 km. The radii of a 1.4 M ⊙ neutron star are predicated as R 1.4 = 12.34–12.91 km, and their dimensionless tidal deformabilities are Λ1.4 = 485–626. Furthermore, the direct URCA process in neutron star cooling will happen from n b = 0.414 to 0.530 fm−3 with the proton fractions Y p = 0.136–0.138. All of the results obtained from the RBHF model only with two-body pvCD-Bonn potentials completely satisfy various constraints from recent astronomical observations of massive neutron stars, gravitational wave detection (GW170817), and simultaneous mass–radius measurement.

Strength of tensor forces from neutron drops in ab initio relativistic Brueckner-Hartree-Fock theory

CITATION: Wang, S., et al. 2019. Strength of tensor forces from neutron drops in ab initio relativistic Brueckner-Hartree-Fock theory. Physical Review C, 100(6):064319, doi:10.1103/PhysRevC.100.064319.

The properties of neutron star in the framework of relativistic Hartree–Fock model with unitary correlation operator method

The properties of neutron star are studied in the framework of relativistic Hartree–Fock (RHF) model with realistic nucleon–nucleon (NN) interactions, i.e., Bonn potentials. The strong repulsion of NN interaction at short range is properly removed by the unitary correlation operator method (UCOM). Meanwhile, the tensor correlation is neglected due to the very rich neutron environment in neutron star, where the total isospin of two nucleons can be approximately regarded as [Formula: see text]. The equations of state of neutron star matter are calculated in [Formula: see text] equilibrium and charge neutrality conditions. The properties of neutron star, such as mass, radius and tidal deformability, are obtained by solving the Tolman–Oppenheimer–Volkoff equation and tidal equation. The maximum masses of neutron from Bonn A, B, C potentials are around [Formula: see text]. The radius are [Formula: see text][Formula: see text]km at [Formula: see text], respectively. The corresponding tidal deformabilities are [Formula: see text]. All of these properties are satisfied with the recent observables from the astronomical and gravitational wave devices and are consistent with the results from the relativistic Brueckner–Hartree–Fock model.

Skyrme Functional with Tensor Terms from ab initio Calculations: Results for the Spin--Orbit Splittings

A new Skyrme functional including tensor terms is presented. The tensor terms have been determined by fitting the results of relativistic Brueckner-Hartree-Fock (RBHF) studies on neutron-proton drops. Unlike all previous studies, where the tensor terms were usually determined by fitting to experimental data of single-particle levels, the pseudodata calculated by RBHF does not contain beyond mean-field effect such as the particle-vibration coupling and, therefore, can provide information on the tensor term without ambiguities. The obtained new functional, named SAMi-T, can describe well ground-state properties such as binding energies, radii, spin-orbit splittings and, at the same time, the excited state properties such as those of the Giant Monopole Resonance (GMR), Giant Dipole Resonance (GDR), Gamow-Teller Resonance (GTR), and Spin-Dipole Resonance (SDR).

Evolution of nuclear spin-orbit splittings with Skyrme functional SAMi-T

A new Skyrme functional has been developed with tensor term guided by ab initio relativistic Brueckner-Hartree-Fock (RBHF) studies on neutron-proton drops. Instead of extracting information on the tensor force from experimental single-particle energy splittings, the RBHF calculations do not contain beyond mean-field effects such as particle-vibration coupling and therefore the information on the tensor force can be obtained without ambiguities. The new functional gives a good description of nuclear ground-state properties aswell as various giant resonances. The description for the evolution of single-particle energy splittings is also improved by the new functional.

Relativistic Brueckner-Hartree-Fock theory in nuclear matter without the average momentum approximation

Brueckner-Hartree-Fock theory allows to derive the $G$-matrix as an effective interaction between nucleons in the nuclear medium. It depends on the center of mass momentum $\bm{P}$ of the two particles and on the two relative momenta $\bm{q}$ and $\bm{q'}$ before and after the scattering process. In the evaluation of the total energy per particle in nuclear matter usually the angle averaged center of mass momentum approximation has been used. We derive in detail the exact expressions of the angular integrations of the momentum $\bm{P}$ within relativistic Brueckner-Hartree-Fock (RBHF) theory, especially for the case of asymmetric nuclear matter. In order to assess the reliability of the conventional average momentum approximation for the binding energy, the saturation properties of symmetric and asymmetric nuclear matter are systematically investigated based on the realistic Bonn nucleon-nucleon potential. It is found that the exact treatment of the center of mass momentum leads to non-negligible contributions to the higher order physical quantities. The correlation between the symmetry energy $E_{\mathrm{sym}}$, the slope parameter $L$, and the curvature $K_{\mathrm{sym}}$ of the symmetry energy are investigated. The results of our RBHF calculations for the bulk parameters characterizing the equation of state are compared with recent constraints extracted from giant monopole resonance and isospin diffusion experiments.

Spin symmetry in the Dirac sea derived from the bare nucleon–nucleon interaction

The spin symmetry in the Dirac sea has been investigated with relativistic Brueckner–Hartree–Fock theory using the bare nucleon–nucleon interaction. Taking the nucleus 16O as an example and comparing the theoretical results with the data, the definition of the single-particle potential in the Dirac sea is studied in detail. It is found that if the single-particle states in the Dirac sea are treated as occupied states, the ground state properties are in better agreement with experimental data. Moreover, in this case, the spin symmetry in the Dirac sea is better conserved and it is more consistent with the findings using phenomenological relativistic density functionals.

Effects of tensor forces in nuclear spin–orbit splittings from ab initio calculations

A systematic and specific pattern due to the effects of the tensor forces is found in the evolution of spin–orbit splittings in neutron drops. This result is obtained from relativistic Brueckner–Hartree–Fock theory using the bare nucleon–nucleon interaction. It forms an important guide for future microscopic derivations of relativistic and nonrelativistic nuclear energy density functionals.

Fully self-consistent relativistic Brueckner-Hartree-Fock theory for finite nuclei

Starting from the relativistic form of the Bonn potential as a bare nucleon-nucleon interaction, the full Relativistic Brueckner-Hartree-Fock (RBHF) equations are solved for finite nuclei in a fully self-consistent basis. This provides a relativistic ab initio calculation of the ground state properties of finite nuclei without any free parameters and without three-body forces. The convergence properties for the solutions of these coupled equations are discussed in detail at the example of the nucleus $^{16}$O. The binding energies, radii, and spin-orbit splittings of the doubly magic nuclei $^{4}$He, $^{16}$O, and $^{40}$Ca are calculated and compared with the earlier RBHF calculated results in a fixed Dirac Woods-Saxon basis and other non-relativistic ab initio calculated results based on pure two-body forces.

The properties of nuclear matter with lattice NN potential in relativistic Brueckner-Hartree-Fock theory.

We study the properties of nuclear matter with lattice nucleon-nucleon (NN) potential in the relativistic Brueckner-Hartree-Fock (RBHF) theory. To use this potential in such a microscopic many-body theory, we firstly have to construct a one-boson-exchange potential (OBEP) based on the latest lattice NN potential. Three mesons, pion, σ meson, and ω meson, are considered. Their coupling constants and cut-off momenta are determined by fitting the on-shell behaviors and phase shifts of the lattice force, respectively. Therefore, we obtain two parameter sets of the OBEP potential (named as LOBEP1 and LOBEP2) with these two fitting ways. We calculate the properties of symmetric and pure neutron matter with LOBEP1 and LOBEP2. In non-relativistic Brueckner-Hartree-Fock case, the binding energies of symmetric nuclear matter are around −3 and −5 MeV at saturation density, while it becomes −8 and −12 MeV in relativistic framework with 1S0, 3S1, and 3D1 channels using our two parameter sets. For the pure neutron matter, the equations of state in non-relativistic and relativistic cases are very similar due to only consideration 1S0 channel with isospin T = 1 case.