Covariant Density Functionals Research Articles

New Covariant Density Functionals of Nuclear Matter for Compact Star Simulations

We generate three families of extended covariant density functionals of nuclear matter that have varying slope of symmetry energy and skewness at nuclear saturation density, but otherwise share the same basic parameters (symmetry energy, compressibility, saturation parameters, etc.) with the standard DDME2, DD2, and MPE functionals. Tables of the parameters of these new density functionals are given, which can be straightforwardly used in DDME2, DD2, and MPE parameterization-based codes. Furthermore, we provide tables of a large number of equations of state (81 for each family) that can be used in astrophysical simulations to assess the impact of variations of the not-well-known slope of symmetry energy and skewness of nuclear systems on the astrophysics of compact objects. We also provide tables of computed integral parameters (mass, radius, and tidal deformability) that can be used, e.g., for modeling gravitational waveforms. Finally, for the extended DDME2-based parameterization, we implement a first-order phase transition to quark matter to obtain a family of equations of state that accommodates a phase transition to quark matter. Analogous tables of the equations of state and integral parameters are provided for this case as well.

Density-dependent relativistic mean field approach and its application to single-Λ hypernuclei in oxygen hyperisotopes* *Supported by the Fundamental Research Funds for the Central Universities, Lanzhou University (lzujbky-2022-sp02, lzujbky-2023-stlt01), the National Natural Science Foundation of China (11875152, 12275111), and the Strategic Priority Research Program of Chinese Academy of Sciences (XDB34000000)

The in-medium feature of nuclear force, which includes both nucleon-nucleon () and hyperon-nucleon () interactions, impacts the description of single-Λ hypernuclei. With the alternated mass number or isospin of hypernuclei, such effects may be unveiled by analyzing the systematic evolution of the bulk and single-particle properties. From a density-dependent meson-nucleon/hyperon coupling perspective, a new effective interaction in the covariant density functional (CDF) theory, namely, DD-LZ1-, is obtained by fitting the experimental data of Λ separation energies for several single-Λ hypernuclei. It is then used to study the structure and transition properties of single-Λ hypernuclei in oxygen hyperisotopes, in comparison with those determined using several selected CDF Lagrangians. A discrepancy is explicitly observed in the isospin evolution of spin-orbit splitting with various effective interactions, which is attributed to the divergence of the meson-hyperon coupling strengths with increasing density. In particular, the density-dependent CDFs introduce an extra contribution to reduce the value but enhance the isospin dependence of the splitting, which originates from the rearrangement terms of Λ self-energies. In addition, the characteristics of hypernuclear radii are studied along the isotopic chain. Owing to the impurity effect of the Λ hyperon, a size shrinkage is observed in the matter radii of hypernuclei compared with the cores of normal nuclei, and its magnitude is further elucidated to correlate with the incompressibility of nuclear matter. Moreover, there is a sizable model-dependent trend in which the Λ hyperon radii evolve with neutron number, which is decided partly by the in-medium interactions and core polarization effects.

Bayesian inference of the dense matter equation of state built upon covariant density functionals

A modified version of the density dependent covariant density functional model proposed in [T. Malik, M. Ferreira, B. K. Agrawal and C. Provid\^encia, ApJ 930, 17 (2022)] is employed in a Bayesian analysis to determine the equation of state (EOS) of dense matter with nucleonic degrees of freedom. Various constraints from nuclear physics and microscopic calculations of pure neutron matter (PNM) along with a lower bound on the maximum mass of neutron stars (NSs) are imposed on the EOS models to investigate the effectiveness of progressive incorporation of the constraints, their compatibility as well as correlations among parameters of nuclear matter and properties of NSs. Our results include the different roles played by pressure and energy per particle of PNM in constraining the isovector behavior of nuclear matter; tension with the values of Dirac effective mass extracted from spin-orbit splitting; correlations between the radius of the canonical mass NS and second and third order coefficients in the Taylor expansion of energy per particle as a function of density; correlation between the central pressure of the maximum mass configuration and Dirac effective mass of the nucleon at saturation. For some of our models the tail of the NS maximum mass reaches $2.7~\mathrm{M}_{\odot}$, which means that the secondary object in GW190814 could have been a NS.

Massive relativistic compact stars from SU(3) symmetric quark models

We construct a set of hyperonic equations of state (EoS) by assuming SU(3) symmetry within the baryon octet and by using a covariant density functional (CDF) theory approach. The low-density regions of our EoS are constrained by terrestrial experiments, while the high-density regime is modeled by systematically varying the nuclear matter skewness coefficient Qsat and the symmetry energy slope Lsym. The sensitivity of the EoS predictions is explored in terms of z parameter of the SU(3) symmetric model that modifies the meson-hyperon coupling constants away from their SU(6) symmetric values. Our results show that model EoS based on our approach can support static Tolman-Oppenheimer-Volkof (TOV) masses in the range 2.3-2.5M⊙ in the large-Qsat and small-z regime, however, such stars contain only a trace amount of hyperons compared to SU(6) models. We also construct uniformly rotating Keplerian configurations for our model EoS for which the masses of stellar sequences may reach up to 3.0M⊙. These results are used to explore the systematic dependence of the ratio of maximum masses of rotating and static stars, the lower bound on the rotational frequency of the models that will allow secondary masses in the gravitational waves events to be compact stars with M2≲3.0M⊙ and the strangeness fraction on the model parameters. We conclude that very massive stellar models can be, in principle, constructed within the SU(3) symmetric model, however, they are nucleonic-like as their strangeness fraction drops below 3%.

Covariant density functional theory with localized exchange terms

A new density-dependent point-coupling covariant density functional PCF-PK1 is proposed, where the exchange terms of the four-fermion terms are local and are taken into account with the Fierz transformation. The coupling constants of the PCF-PK1 functional are determined by empirical saturation properties and ab initio equation of state and proton-neutron Dirac mass splittings for nuclear matter as well as the ground-state properties of selected spherical nuclei. The success of the PCF-PK1 is illustrated with properties of the infinite nuclear matter and finite nuclei including the ground-state properties and the Gamow-Teller resonances. In particular, the PCFPK1 eliminates the spurious shell closures at Z = 58 and Z = 92, which exist commonly in many covariant density functionals without exchange terms. Moreover, the Gamow-Teller resonances are nicely reproduced without any adjustable parameters, and this demonstrates that a self-consistent description for the Gamow-Teller resonances can be achieved with the localized exchange terms in the PCF-PK1. ?

Unified neutron star EOSs and neutron star structures in RMF models

In the framework of the Thomas-Fermi approximation, we systematically study the EOSs and microscopic structures of neutron star matter in a vast density range with n b ≈ 10−10-2 fm−3, where various covariant density functionals are adopted, i.e., those with nonlinear self couplings (NL3, PK1, TM1, GM1, MTVTC) and density-dependent couplings (DD-LZ1, DDME-X, PKDD, DD-ME2, DD2, TW99). It is found that the EOSs generally coincide with each other at n b ≲ 10−4 fm−3 and 0.1 fm−3 ≲ n b ≲ 0.3 fm−3, while in other density regions they are sensitive to the effective interactions between nucleons. By adopting functionals with a larger slope of symmetry energy L, the curvature parameter K sym and neutron drip density generally increases, while the droplet size, proton number of nucleus, core-crust transition density, and onset density of non-spherical nuclei, decrease. All functionals predict neutron stars with maximum masses exceeding the two-solar-mass limit, while those of DD2, DD-LZ1, DD-ME2, and DDME-X predict optimum neutron star radii according to the observational constraints. Nevertheless, the corresponding skewness coefficients J are much larger than expected, while only the functionals MTVTC and TW99 meet the start-of-art constraints on J. More accurate measurements on the radius of PSR J0740 + 6620 and the maximum mass of neutron stars are thus essential to identify the functional that satisfies all constraints from nuclear physics and astrophysical observations. Approximate linear correlations between neutron stars’ radii at M = 1.4M ⊙ and 2M ⊙, the slope L and curvature parameter K sym of symmetry energy are observed as well, which are mainly attributed to the curvature-slope correlations in the functionals adopted here. The results presented here are applicable for investigations of the structures and evolutions of compact stars in a unified manner.

Constraining equation of state of nuclear matter by charge-changing cross section measurements of mirror nuclei

The nuclear symmetry energy plays a key role in determining the equation of state (EoS) of dense, neutron-rich matter, which connects the atomic nuclei with the hot and dense matter in universe, thus has been the subject of intense investigations in laboratory experiments, astronomy observations and theories. Various probes have been proposed to constrain the symmetry energy and its density dependence. Currently, the extensive data yield already a good and consistent constraint to the symmetry energy (Esym(ρ)) at saturation density, but do not yet give a consistent result of one critical EoS parameter, L, the density dependence of the symmetry energy. In this work, we report a new probe of L at saturation density. A good linear correlation is found between L and the charge changing cross section difference (Δσcc) of mirror nuclei 30Si-30S for both the Skyrme-Hartree-Fock theory (SHF) and covariant (relativistic) density functionals (CDF). We found that the pairing effect for this mirror pair is essential to get a consistent correlation between L and Δσcc in both the SHF and CDF. Here, the cross sections are calculated on the same target and at the same energy using the zero-range optical-limit Glauber model. The linearity is found to be in the same precision as those found between L and neutron skin thickness or proton radius difference.

Delta-resonances and hyperons in proto-neutron stars and merger remnants

The equation of state (EoS) and composition of dense and hot varDelta -resonance admixed hypernuclear matter is studied under conditions that are characteristic of neutron star binary merger remnants and supernovas. The cold, neutrino free regime is also considered as a reference for the astrophysical constraints on the EoS of dense matter. Our formalism uses the covariant density functional (CDF) theory successfully adapted to include the full J^P=1/2^+ baryon octet and non-strange members of J^P=3/2^+ decouplet with density-dependent couplings that have been suitably adjusted to the existing laboratory and astrophysical data. The effect of varDelta -resonances at finite temperatures is to soften the EoS of hypernuclear matter at intermediate densities and stiffen it at high densities. At low temperatures, the heavy baryons varLambda , varDelta ^-, varXi ^-, varXi ^0 and varDelta ^0 appear in the given order if the varDelta -meson couplings are close to those for the nucleon-meson couplings. As is the case for hyperons, the thresholds of varDelta -resonances move to lower densities with the increase of temperature indicating a significant fraction of varDelta ’s in the low-density subnuclear regime. We find that the varDelta -resonances comprise a significant fraction of baryonic matter, of the order of 10% at temperatures of the order of several tens of MeV in the neutrino-trapped regime and, thus, may affect the supernova and binary neutron star dynamics by providing, for example, a new source for neutrino opacity or a new channel for bulk viscosity via the direct Urca processes. The mass-radius relation of isentropic static, spherically symmetric hot compact stars is discussed.

Unified nuclear matter equations of state constrained by the in-medium balance in density-dependent covariant density functionals

Considering the effects of charge screening, we propose a new numerical recipe within the framework of Thomas-Fermi approximation, where the properties of nuclear matter throughout a vast density range can be obtained self-consistently. Assuming spherical and cylindrical approximations for the Wigner-Seitz cell, typical nuclear matter structures (droplet, rod, slab, tube, bubble, and uniform) are observed. We then investigate the EOSs and microscopic structures of nuclear matter with both fixed proton fractions and $\beta$-equilibration, where two covariant density functionals DD-LZ1 and DD-ME2 are adopted. Despite the smaller slope $L$ of symmetry energy obtained with the functional DD-LZ1, the curvature parameter $K_\mathrm{sym}$ is much larger than that of DD-ME2, which is attributed to the peculiar density-dependent behavior of meson-nucleon couplings guided by the restoration of pseudo-spin symmetry around the Fermi levels in finite nuclei. Consequently, different mass-radius relations of neutron stars are predicted by the two functionals. Different microscopic structures of nonuniform nuclear matter are obtained as well, which are expected to affect various physical processes in neutron star properties and evolutions.

Global Performance of Covariant Density Functional Theory in Description of Charge Radii and Related Indicators

A short review of existing efforts to understand charge radii and related indicators on a global scale within the covariant density functional theory (CDFT) is presented. Using major classes of covariant energy density functionals (CEDFs), the global accuracy of the description of experimental absolute and differential charge radii within the CDFT framework has been established. This assessment is supplemented by an evaluation of theoretical statistical and systematic uncertainties in the description of charge radii. New results on the accuracy of the description of differential charge radii in deformed actinides and light superheavy nuclei are presented and the role of octupole deformation in their reproduction is evaluated. Novel mechanisms leading to odd-even staggering in charge radii are discussed. Finally, we analyze the role of self-consistency effects in an accurate description of differential charge radii.

Nuclear landscape in a mapped collective Hamiltonian from covariant density functional theory

The nuclear landscape is investigated within the triaxial relativistic Hartree-Bogoliubov theory with the PC-PK1 density functional, and the beyond-mean-field dynamical correlation energies are taken into account by a microscopically mapped five-dimensional collective Hamiltonian without additional free parameters. The effects of triaxial deformation and dynamical correlations on the nuclear landscape are analyzed. The present results provide a better description of the experimental binding energies of even-even nuclei, in particular for medium and heavy mass regions, in comparison with previous global calculations with the state-of-the-art covariant density functionals DD-PC1 and TMA. The inclusion of the dynamical correlation energies plays an important role in the PC-PK1 results. It is emphasized that the nuclear landscape is considerably extended by the PC-PK1 functional in comparison with the previous results obtained with the covariant density functionals DD-PC1 and TMA, which may be due to the different isovector properties in the density functionals.

Equation of State and Composition of Proto-Neutron Stars and Merger Remnants with Hyperons

Finite-temperature equation of state (EoS) and the composition of dense nuclear and hypernuclear matter under conditions characteristic of neutron star binary merger remnants and supernovas are discussed. We consider both neutrino free-streaming and trapped regimes which are separated by a temperature of a few MeV. The formalism is based on covariant density functional (CDF) theory for the full baryon octet with density-dependent couplings, suitably adjusted in the hypernuclear sector. The softening of the EoS with the introduction of the hyperons is quantified under various conditions of lepton fractions and temperatures. We find that Λ, Ξ−, and Ξ0 hyperons appear in the given order with a sharp density increase at zero temperature at the threshold being replaced by an extended increment over a wide density range at high temperatures. The Λ hyperon survives in the deep subnuclear regime. The triplet of Σs is suppressed in cold hypernuclear matter up to around seven times the nuclear saturation density, but appears in significant fractions at higher temperatures, T≥20 MeV, in both supernova and merger remnant matter. We point out that a special isospin degeneracy point exists where the baryon abundances within each of the three isospin multiplets are equal to each other as a result of (approximate) isospin symmetry. At that point, the charge chemical potential of the system vanishes. We find that under the merger remnant conditions, the fractions of electron and μ-on neutrinos are close and are about 1%, whereas in the supernova case, we only find a significant fraction (∼10%) of electron neutrinos, given that in this case, the μ-on lepton number is zero.

Hyperheavy spherical and toroidal nuclei: The role of shell structure

The properties of toroidal hyperheavy even-even nuclei and the role of toroidal shell structure are extensively studied within covariant density functional theory. The general trends in the evolution of toroidal shapes in the $Z\approx 130-180$ region of nuclear chart are established for the first time. These nuclei are stable with respect of breathing deformations. The most compact fat toroidal nuclei are located in the $Z\approx 136, N\approx 206$ region of nuclear chart, but thin toroidal nuclei become dominant with increasing proton number and on moving towards proton and neutron drip lines. The role of toroidal shell structure, its regularity, supershell structure, shell gaps as well as the role of different groups of the pairs of the orbitals in its formation are investigated in detail. The lowest in energy solutions at axial symmetry are characterized either by large shell gaps or low density of the single-particle states in the vicinity of the Fermi level in at least one of the subsystems (proton or neutron). Related quantum shell effects are expected to act against the instabilities in breathing and sausage deformations for these subsystems. The investigation with large set of covariant energy density functionals reveals that substantial proton $Z=154$ and 186 and neutron $N=228$, 308 and 406 spherical shell gaps exist in all functionals. The nuclei in the vicinity of the combination of these particle numbers form the islands of stability of spherical hyperheavy nuclei. The study suggests that the $N=210$ toroidal shell gap plays a substantial role in the stabilization of fat toroidal nuclei.

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.

Covariant density functional theory input for r -process simulations in actinides and superheavy nuclei: The ground state and fission properties

The systematic investigation of the ground state and fission properties of even-even actinides and superheavy nuclei with $Z=90-120$ from the two-proton up to two-neutron drip lines with proper assessment of systematic theoretical uncertainties has been performed for the first time in the framework of covariant density functional theory (CDFT). These results provide a necessary theoretical input for the r-process modeling in heavy nuclei and, in particular, for the study of fission recycling. Four state-of-the-art globally tested covariant energy density functionals (CEDFs), namely, DD-PC1, DD-ME2, NL3* and PC-PK1, representing the major classes of the CDFT models are employed in the present study. Ground state deformations, binding energies, two neutron separation energies, $\alpha$-decay $Q_{\alpha}$ values and half-lives and the heights of fission barriers have been calculated for all these nuclei. Theoretical uncertainties in these physical observables and their evolution as a function of proton and neutron numbers have been quantified and their major sources have been identified. Spherical shell closures at $Z=120$, $N=184$ and $N=258$ and the structure of the single-particle (especially, high-$j$) states in their vicinities as well as nuclear matter properties of employed CEDFs are two major factors contributing into theoretical uncertainties. However, different physical observables are affected in a different way by these two factors. For example, theoretical uncertainties in calculated ground state deformations are affected mostly by former factor, while theoretical uncertainties in fission barriers depend on both of these factors.

Rapidly rotating Δ-resonance-admixed hypernuclear compact stars

We use a set of hadronic equations of state derived from covariant density functional theory to study the impact of their high-density behavior on the properties of rapidly rotating Δ-resonance-admixed hyperonic compact stars. In particular, we explore systematically the effects of variations of the bulk energy isoscalar skewness, Qsat, and the symmetry energy slope, Lsym, on the masses of rapidly rotating compact stars. With models for equation of state satisfying all the modern astrophysical constraints, excessively large gravitational masses of around 2.5M⊙ are only obtained under three conditions: (a) strongly attractive Δ-resonance potential in nuclear matter, (b) maximally fast (Keplerian) rotation, and (c) parameter ranges Qsat≳500 MeV and Lsym≲50 MeV. These values of Qsat and Lsym have a rather small overlap with a large sample (total of about 260) parametrizations of covariant nucleonic density functionals. The extreme nature of requirements (a)-(c) reinforces the theoretical expectation that the secondary object involved in the GW190814 event is likely to be a low-mass black hole rather than a supramassive neutron star.

Critical parameters of the liquid-gas phase transition in thermal symmetric and asymmetric nuclear matter

The properties of critical parameters and phase diagram structure of liquid-gas phase transition are investigated in thermal symmetric and asymmetric nuclear matter with the covariant density functional (CDF) theory. Although uncertainty remains in predicting the critical parameters such as the critical temperature and pressure from various CDF functionals, several correlations are explored numerically and verified to be approximately linear between them. These correlations become worse when nuclear matter is more isospin asymmetric, resulting mainly from the effects induced by symmetry energy. By looking over the isospin dependence of the critical temperature, the role of the symmetry energy in LG transition properties of asymmetric matter is realized. The change of critical temperature with isospin asymmetry is found to be correlated well with and as a consequence could be constrained by the density slope of symmetry energy at saturation density. Then, the structure of phase diagram of thermal nuclear matter is analyzed carefully. It is revealed that the contribution from symmetry energy dominates the size of liquid-gas phase coexistence area. Moreover, the specific pattern of the phase diagram could be determined by the critical temperature at non-zero isospin asymmetry, illustrated from the correlations of the temperature with pressures at several characteristic points, paving the possible way to further explore the structure of liquid-gas phase diagram of thermal nuclear matter.

Propagation of statistical uncertainties in covariant density functional theory: Ground state observables and single-particle properties

Statistical errors in ground state observables and single-particle properties of spherical even-even nuclei and their propagation to the limits of nuclear landscape have been investigated in covariant density functional theory (CDFT) for the first time. In this study we consider only covariant energy density functionals with non-linear density dependency. Statistical errors for binding energies and neutron skins significantly increase on approaching two-neutron drip line. On the contrary, such a trend does not exist for statistical errors in charge radii and two-neutron separation energies. The absolute and relative energies of the single-particle states in the vicinity of the Fermi level are characterized by low statistical errors ($\sigma(e_i)\sim 0.1$ MeV). Statistical errors in the predictions of spin-orbit splittings are rather small. Statistical errors in physical observables are substantially smaller than related systematic uncertainties. Thus, at the present level of the development of theory, theoretical uncertainties at nuclear limits are dominated by systematic ones. Statistical errors in the description of physical observables related to the ground state and single-particle degrees of freedom are typically substantially lower in CDFT as compared with Skyrme density functional theory. The correlations between the model parameters are studied in detail. The parametric correlations are especially pronounced for the $g_2$ and $g_3$ parameters which are responsible for the density dependence of the model. The accounting of this fact potentially allows to reduce the number of free parameters of non-linear meson coupling model from six to five.

Multivariate analysis of critical parameters influencing the reliability of thermal-hydraulic passive safety system

Thermal-hydraulic passive safety systems (PSSs) are incorporated into many advanced reactor designs on the bases of simplicity, economics and inherent safety nature. Several factors among which are the critical parameters (CPs) that influence failure and reliability of thermal-hydraulic (t-h) passive systems are now being explored. For simplicity, it is assumed in most reliability analyses that the CPs are independent whereas in practice this assumption is not always valid. There is need to critically examine the dependency influence of the CPs on reliability of the t-h passive systems at design stage and in operation to guarantee safety/better performance. In this paper, two multivariate analysis methods (covariance and conditional subjective probability density function) were presented and applied to a simple PSS. The methods followed a generalized procedure for evaluating t-h reliability based on dependency consideration. A passively water-cooled steam generator was used to demonstrate the dependency of the identified key CPs using the methods. The results obtained from the methods are in agreement and justified the need to consider the dependency of CPs in t-h reliability. For dependable t-h reliability, it is advisable to adopt all possible CPs and apply suitable multivariate method in dependency consideration of CPs among other factors. • Covariance matrix dependency approach to passive system reliability. • Conditional subjective probability density approach to passive system reliability. • Dependency analysis of critical parameters influencing reliability. • Evaluation of thermal-hydraulic reliability of passive safety systems. • Application of multivariate analysis in reliability of nuclear passive systems.

Competition between delta isobars and hyperons and properties of compact stars

The Δ-isobar degrees of freedom are included in the covariant density functional (CDF) theory to study the equation of state (EoS) and composition of dense matter in compact stars. In addition to Δ's we include the full octet of baryons, which allows us to study the interplay between the onset of delta isobars and hyperonic degrees of freedom. Using both the Hartree and Hartree–Fock approximation we find that Δ's appear already at densities slightly above the saturation density of nuclear matter for a wide range of the meson–Δ coupling constants. This delays the appearance of hyperons and significantly affects the gross properties of compact stars. Specifically, Δ's soften the EoS at low densities but stiffen it at high densities. This softening reduces the radius of a canonical 1.4M⊙ star by up to 2 km for a reasonably attractive Δ potential in matter, while the stiffening results in larger maximum masses of compact stars. We conclude that the hypernuclear CDF parametrizations that satisfy the 2M⊙ maximum mass constraint remain valid when Δ isobars are included, with the important consequence that the resulting stellar radii are shifted toward lower values, which is in agreement with the analysis of neutron star radii.