Pure Neutron Matter Research Articles

Bayesian truncation errors in equations of state of nuclear matter with chiral nucleon-nucleon potentials

The truncation errors in equations of state (EOSs) of nuclear matter derived from the chiral nucleon-nucleon (NN) potentials at different expansion orders are analyzed by a Bayesian model. These EOSs are expanded as functions of a dimensionless parameter, Q, which is determined by Fermi momentum, kF and breakdown scale, Λb. The degree-of-belief (DoB) intervals predicted by the chiral effective field theory are calculated within the corresponding expansion coefficients and the specific prior probability distribution functions in terms of Bayes' theorem. The truncation errors of EOSs, generated by the DoB intervals, exhibit good order-by-order convergences with different chiral expansion order potentials. When DoB is considered as 1σ credibility, i.e., 68.27% confidence interval, the truncation errors of binding energy per nucleon in symmetric nuclear matter and pure neutron matter are consistent with the results given by a simple error analysis method proposed by Epelbaum, et al. Finally, the reasonable values of the breakdown scale Λb in nuclear matter are discussed through consistency checks of Bayesian method based on the calculations of success rate.

First Observation of an Attractive Interaction between a Proton and a Cascade Baryon.

This Letter presents the first experimental observation of the attractive strong interaction between a proton and a multistrange baryon (hyperon) Ξ^{-}. The result is extracted from two-particle correlations of combined p-Ξ^{-}⊕p[over ¯]-Ξ[over ¯]^{+} pairs measured in p-Pb collisions at sqrt[s_{NN}]=5.02 TeV at the LHC with ALICE. The measured correlation function is compared with the prediction obtained assuming only an attractive Coulomb interaction and a standard deviation in the range [3.6, 5.3] is found. Since the measured p-Ξ^{-}⊕p[over ¯]-Ξ[over ¯]^{+} correlation is significantly enhanced with respect to the Coulomb prediction, the presence of an additional, strong, attractive interaction is evident. The data are compatible with recent lattice calculations by the HAL-QCD Collaboration, with a standard deviation in the range [1.8, 3.7]. The lattice potential predicts a shallow repulsive Ξ^{-} interaction within pure neutron matter and this implies stiffer equations of state for neutron-rich matter including hyperons. Implications of the strong interaction for the modeling of neutron stars are discussed.

The three-body cluster energy for the [formula omitted] asymmetric [formula omitted] matter in the [formula omitted] formalism

The equation of state (EOS) of the isospin asymmetric nucleonic matter (IASM) is studied in the Lowest Order Constrained Variational (LOCV) formalism, using the AV18 two-body potential with a density-dependent three-body interaction (TBI), which was proposed by Lagaris and Pandharipande (LP) in 1981. In the LOCV framework, an adjustment mechanism is introduced for finding the appropriate parameters of the TBI. It is shown that, assuming the TBI, the corresponding LOCV symmetric nuclear (pure neutron) matter results fairly agree with those of Fermi Hypernetted Chain (FHNC). Also, the contribution of the averaged three-body cluster (TBC) energy is evaluated, at different proton to neutron ratios R. Employing the TBI, up to ρ=0.5 fm−3, the TBC energy ranges from −6 to 6 MeV per particle which considerably changes the IASMEOS, especially at low densities. Moreover, employing the TBI, the LOCVIASM state-averaged correlation functions (effective potentials) and the nucleon–nucleon distribution functions (NNDF) are reported. It is illustrated that, by increasing the density (ratio R), the impact of the TBI becomes much more evident. Finally, it should be noted that, the IASMFHNC and Monte Carlo (MC) data are not available for comparison.

Relativistic self-energy decomposition of nuclear symmetry energy and equation of state of neutron matter within QCD sum rules

Properties of the relativistic nucleon self-energy decomposition of the symmetry energy as well as the equation of state (EOS) of pure neutron matter (PNM) are explored systematically within the QCD sum rules (QCDSR). Our main conclusions are as follows: (1) The five self-energy decomposition terms of the symmetry energy according to the nucleon Lorentz structure are carefully studied, leading to the conclusion that the symmetry energy increases as the nucleon sigma term ${\ensuremath{\sigma}}_{\mathrm{N}}$ increases and the contributions to the symmetry energy due to the momentum dependence of the self-energies in symmetric nuclear matter (SNM) are very small compared with those from other decomposition terms. (2) A smaller strange quark mass is found to generate a larger symmetry energy, and this correlation is useful for understanding the origins of the uncertainties on the (nucleonic matter) symmetry energy from quark level. (3) The EOS of PNM at low densities can be effectively approximated by ${E}_{\mathrm{n}}(\ensuremath{\rho})\ensuremath{\approx}{E}_{\mathrm{n}}^{\mathrm{FFG}}(\ensuremath{\rho})+(M\ensuremath{\rho}/2{\ensuremath{\langle}\overline{q}q\ensuremath{\rangle}}_{\mathrm{vac}})[(1\ensuremath{-}\ensuremath{\xi})({\ensuremath{\sigma}}_{\mathrm{N}}/2{m}_{\mathrm{q}})\ensuremath{-}5]$ which depends only on several physical quantities such as ${m}_{\mathrm{q}},{\ensuremath{\sigma}}_{\mathrm{N}}$ and ${\ensuremath{\langle}\overline{q}q\ensuremath{\rangle}}_{\mathrm{vac}}$, and this formula already has predictive power and the results are found to be consistent with those from other celebrated microscopic many-body theories at low densities. (4) The higher order density terms in quark condensates are shown to be important to describe the empirical EOS of PNM in the density region around and above nuclear saturation density, and these higher order density terms are also found to hinder the appearance of chiral symmetry restoration in PNM at high densities. (5) The symmetry energy is shown to depend strongly on the five-dimensional condensate ${\ensuremath{\langle}{g}_{\mathrm{s}}{q}^{\ifmmode\dagger\else\textdagger\fi{}}\ensuremath{\sigma}\mathcal{G}q\ensuremath{\rangle}}_{\ensuremath{\rho},\ensuremath{\delta}}$, providing a useful approach to explore the symmetry energy through knowledge on the condensates which can be extracted from hadronic physics. (6) The twist-4 four-quark condensates are shown to have significant effects on the EOS of both SNM and PNM but have minor effects on the symmetry energy, and combined with the analyses on the effects of the higher order density terms in the chiral condensates, three parameter sets of QCDSR are constructed and they are shown to be able to describe the EOS of PNM and the symmetry energy within a wide range of densities. Our results in the present work demonstrate that the QCDSR approach can provide a useful way to understand the properties of dense nucleonic matter from nonperturbative QCD vacuum.

Analysis of nuclear structure in a converging power expansion scheme

In the framework of the KIDS generalized energy density functional (EDF), the nuclear equation of state (EoS) is expressed as an expansion in powers of the Fermi momentum or the cubic root of the density ($\rho^{1/3}$). Although an optimal number of converging terms was obtained in specific cases of fits to empirical data and pseudodata, the degree of convergence remains to be examined not only for homogeneous matter but also for finite nuclei. One goal of the present work is to validate the minimal and optimal number of EoS parameters required for the description of homogeneous nuclear matter over a wide range of densities relevant for astrophysical applications. The major goal is to examine the validity of the adopted expansion scheme for an accurate description of finite nuclei. To this end we vary the values of the high-order derivatives of the EoS, namely the skewness of the energy of symmetric nuclear matter and the kurtosis of the symmetry energy, at saturation and examine the relative importance of each term in $\rho^{1/3}$ expansion for homogeneous matter. For given sets of EoS parameters determined in this way, we define equivalent Skyrme-type functionals and examine the convergence in the description of finite nuclei focusing on the masses and charge radii of closed-shell nuclei. The EoS of symmetric nuclear matter is found to be efficiently parameterized with only 3 parameters and the symmetry energy (or the energy of pure neutron matter) with 4 parameters when the EoS is expanded in the power series of the Fermi momentum. Higher-order EoS parameters do not produce any improvement, in practice, in the description of nuclear ground-state energies and charge radii, which means that they cannot be constrained by bulk properties of nuclei.

Equation of state of dense matter in the multimessenger era

While the equation of state (EOS) of symmetric nuclear matter (SNM) at suprasaturation densities has been relatively well constrained from heavy-ion collisions, the EOS of high-density neutron-rich matter is still largely uncertain due to the poorly known high-density behavior of the symmetry energy. Using the constraints on the EOS of SNM at suprasaturation densities from heavy-ion collisions together with the data of finite nuclei and the existence of $2M_\odot$ neutron stars from electromagnetic (EM) observations, we show that the high-density symmetry energy cannot be too soft, which leads to lower bounds on dimensionless tidal deformability of $\Lambda_{1.4} \ge 193$ and radius of $R_{1.4} \ge 11.1$ km for $1.4M_\odot$ neutron star. Furthermore, we find that the recent constraint of $\Lambda_{1.4} \le 580$ from the gravitational wave signal GW170817 detected from the binary neutron star merger by the LIGO and Virgo Collaborations rules out too stiff high-density symmetry energy, leading to an upper limit of $R_{1.4} \le 13.3$ km. All these terrestrial nuclear experiments and astrophysical observations based on strong, EM and gravitational measurements together put stringent constraints on the high-density symmetry energy and the EOS of SNM, pure neutron matter and neutron star matter.

Consistent relativistic mean-field models constrained by GW170817

We have obtained the Love number and corresponding tidal deformabilites ($\Lambda$) associated with the relativistic mean-field parametrizations shown to be consistent (CRMF) with the nuclear matter, pure neutron matter, symmetry energy and its derivatives [Dutra, et al., Phys. Rev. C 90, 055203 (2014)]. Our results show that CRMF models present very good agreement with the recent data from binary neutron star merger event GW170817. They also confirm the strong correlation between $\Lambda_{1.4}$ and the radius of canonical stars ($R_{1.4}$). When a recently GW170817 constraint on $\Lambda_{1.4}$ and the corresponding radius $R_{1.4}$ is used, the majority of the models tested are shown to satisfy it.

Neutron matter in the hole-line expansion

We calculate the binding energy of pure neutron matter up to third order in the Brueckner-Bethe-Goldstone hole-line expansion, employing various modern nucleon-nucleon potentials. At that order all results appear well converged to within a few percent up to a density of $0.8\phantom{\rule{0.28em}{0ex}}{\mathrm{fm}}^{\ensuremath{-}3}$. We analyze the potentials in terms of the strength of their hard core. Comparison with results of other theoretical methods is made.

Relativistic parameterizations of neutron matter and implications for neutron stars

We construct parameter sets of the relativistic mean-field model fitted to the recent constraints on the asymmetry energy $J$ and the slope parameter $L$ for pure neutron matter. We find cases of unphysical behaviour, i.e.\ the appearance of negative pressures, for stiff parameter sets with low values of the effective mass $m^*/m$. In some cases the equation of state of pure neutron matter turns out to be outside the allowed band given by chiral effective field theory. The mass-radius relations of neutron stars for all acceptable parameter sets shows a maximum mass in excess of $2M_\odot$ being compatible with pulsar mass measurements. Given the constraints on the model in the low-density regime coming from chiral effective theory, we find that the radius of a $1.4M_\odot$ neutron star is nearly independent on the value of $L$. This is in contrast to some previous claims for a strong connection of the slope parameter with the radius of a neutron star. In fact, the mass-radius relation turns out to depend only on the isoscalar parameters of symmetric matter. The constraints of GW170817 on the tidal deformability and on the radius are also discussed.

Temperature Effects on the Equation of State and Symmetry Energy at Low and High Densities

Thermal properties of symmetric nuclear matter and pure neutron matter are studied in a selfconsistent Green’s function and Brueckner–Hartree–Fock approaches with the inclusion of the contact interaction using CDBONN potential. Also we investigate the temperature dependence of the symmetry energy. The symmetry energy at fixed density is found to generally decrease with temperature. The temperature effects on the nuclear matter symmetry energy are found to be stronger at lower densities while become much weaker at higher densities. The results of several microscopic approaches are compared. Also the results are compared with recent experimental data. There is good agreement between the experimental symmetry energy and those calculated in the Brueckner–Hartree–Fock approach.

Optimized chiral N2LO interactions in nuclear matter

We employ modern two- and three-nucleon interactions derived in chiral perturbation theory (ChPT) at next to-next-to leading order (N2LO), to calculate the energy per particle of symmetric nuclear matter and pure neutron matter in the framework of the microscopic Brueckner-Hartree-Fock approach. In particular, we present results concerning two optimized versions at N2LO of chiral potentials ( $ \mathrm{N2LO}_{opt}$ ), fitted to properties of light nuclei. We also employ the recently developed $\mathrm{N2LO}_{sat}$ interaction which has been calculated at the same order of ChPT but fitted in a very different way compared with the $\mathrm{N2LO}_{opt}$ interactions. We find that using these potentials, in general, it is not possible to reproduce all the saturation properties of nuclear matter. In particular the behaviour of the symmetry energy ( $E_{sym}$ predicted by the interactions considered is quite soft. This is shown comparing our results with the empirical constraints on $E_{sym}$ obtained from the data analysis of the excitation energies of isobaric analog states in nuclei and from experimental data of the neutron skin thickness of heavy nuclei. We finally confront our results with similar calculations performed by other research groups using nuclear chiral interactions at various order of ChPT and employing different many-body methods.

Neutron matter within QCD sum rules

Equation of state (EOS) of pure neutron matter (PNM) is studied in QCD sum rules (QCDSR). It is found that the QCDSR results on EOS of PNM are in good agreement with predictions by current advanced microscopic many-body theories. Moreover, the higher-order density terms in quark condensates are shown to be important to describe the empirical EOS of PNM in the density region around and above nuclear saturation density although they play minor role at subsaturation densities. The chiral condensates in PNM are also studied, and our results indicate that the higher-order density terms in quark condensates, which are introduced to reasonably describe the empirical EOS of PNM at suprasaturation densities, tend to hinder the appearance of chiral symmetry restoration in PNM at high densities.

New relativistic effective interaction for finite nuclei, infinite nuclear matter, and neutron stars

We carry out the study of finite nuclei, infinite nuclear matter, and neutron star properties with the newly developed relativistic force, the Institute of Physics Bhubaneswar-I (IOPB-I). Using this force, we calculate the binding energies, charge radii, and neutron-skin thickness for some selected nuclei. From the ground-state properties of superheavy nuclei ($Z=120$), it is noticed that considerable shell gaps appear at neutron numbers $N=172$, 184, and 198, manifesting the magicity at these numbers. The low-density behavior of the equation of state for pure neutron matter is compatible with other microscopic models. Along with the nuclear symmetry energy, its slope and curvature parameters at the saturation density are consistent with those extracted from various experimental data. We calculate the neutron star properties with the equation of state composed of nucleons and leptons in $\ensuremath{\beta}$-equilibrium, which are in good agreement with the x-ray observations by Steiner [Astrophys. J. 722, 33 (2010)] and N\"attil\"a [Astron. Astrophys. 591, A25 (2016)]. Based on the recent observation of GW170817 with a quasi-universal relation, Rezzolla et al. [Astrophys. J. Lett. 852, L25 (2018)] have set a limit for the maximum mass that can be supported against gravity by a nonrotating neutron star in the range $2.01\ifmmode\pm\else\textpm\fi{}0.04\ensuremath{\lesssim}M({M}_{\ensuremath{\bigodot}})\ensuremath{\lesssim}2.16\ifmmode\pm\else\textpm\fi{}0.03$. We find that the maximum mass of the neutron star for the IOPB-I parametrization is $2.15{M}_{\ensuremath{\bigodot}}$. The radius and tidal deformability of a canonical neutron star of mass $1.4{M}_{\ensuremath{\bigodot}}$ are 13.2 km and $3.9\ifmmode\times\else\texttimes\fi{}{10}^{36}\phantom{\rule{0.16em}{0ex}}\mathrm{g}\phantom{\rule{0.16em}{0ex}}{\mathrm{cm}}^{2}\phantom{\rule{4pt}{0ex}}{\mathrm{s}}^{2}$, respectively.

Minimal nuclear energy density functional

We present a minimal nuclear energy density functional (NEDF) called "SeaLL1" that has the smallest number of possible phenomenological parameters to date. SeaLL1 is defined by 7 significant phenomenological parameters, each related to a specific nuclear property. It describes the nuclear masses of even-even nuclei with a mean energy error of 0.97 MeV and a standard deviation 1.46 MeV, two-neutron and two-proton separation energies with r.m.s. errors of 0.69 MeV and 0.59 MeV respectively, and the charge radii of 345 even-even nuclei with a mean error $\epsilon_r=$0.022 fm and a standard deviation $\sigma_r=$0.025 fm. SeaLL1 incorporates constraints on the EoS of pure neutron matter from quantum Monte Carlo calculations with chiral effective field theory two-body (NN) interactions at N3LO level and three-body (NNN) interactions at the N2LO level. Two of the seven parameters are related to the saturation density and the energy per particle of the homogeneous symmetric nuclear matter, one is related to the nuclear surface tension, two are related to the symmetry energy and its density dependence, one is related to the strength of the spin-orbit interaction, and one is the coupling constant of the pairing interaction. We identify additional phenomenological parameters that have little effect on ground-state properties, but can be used to fine-tune features such as the Thomas-Reiche-Kuhn sum rule, the excitation energy of the giant dipole and Gamow-Teller resonances, the static dipole electric polarizability, and the neutron skin thickness.

Single-particle potential of the Λ hyperon in nuclear matter with chiral effective field theory NLO interactions including effects of YNN three-baryon interactions

Adopting hyperon-nucleon and hyperon-nucleon-nucleon interactions parametrized in chiral effective field theory, single-particle potentials of the $\Lambda$ and $\Sigma$ hyperons are evaluated in symmetric nuclear matter and in pure neutron matter within the framework of lowest order Bruckner theory. The chiral NLO interaction bears strong $\Lambda$N-$\Sigma$N coupling. Although the $\Lambda$ potential is repulsive if the coupling is switched off, the $\Lambda$N-$\Sigma$N correlation brings about the attraction consistent with empirical data. The $\Sigma$ potential is repulsive, which is also consistent with empirical information. The interesting result is that the $\Lambda$ potential becomes shallower beyond normal density. This provides the possibility to solve the hyperon puzzle without introducing ad hoc assumptions. The effects of the $\Lambda$NN-$\Lambda$NN and $\Lambda$NN-$\Sigma$NN three-baryon forces are considered. These three-baryon forces are first reduced to normal-ordered effective two-baryon interactions in nuclear matter and then incorporated in the $G$-matrix equation. The repulsion from the $\Lambda$NN-$\Lambda$NN interaction is of the order of 5 MeV at the normal density, and becomes larger with increasing the density. The effects of the $\Lambda$NN-$\Sigma$NN coupling compensate the repulsion at normal density. The net effect of the three-baryon interactions to the $\Lambda$ single-particle potential is repulsive at higher densities.

Nuclear matter calculations with chiral interactions

Using two-nucleon and three-nucleon interactions derived in the framework of chiral perturbation theory (ChPT) with and without the explicit Δ isobar contributions, we calculate the energy per particle of symmetric nuclear matter and pure neutron matter employing the microscopic Brueckner–Hartree–Fock approach. Specifically, we present nuclear matter calculations using the new fully local in coordinate-space two-nucleon interaction at the next-to-next-to-next-to-leading-order (N3LO) of ChPT with Δ isobar intermediate states (N3LOΔ) recently developed by Piarulli et al. [1] supplemented with a local N2LO three-nucleon interaction with explicit Δ isobar degrees of freedom. We show that for this combination of two- and three-nucleon interactions it is possible to obtain a good saturation point of symmetric nuclear matter. We also calculate the nuclear symmetry energy and compare our results with the available empirical constraints on this quantity.

Strongly Interacting Multi-component Fermions: From Ultracold Atomic Fermi Gas to Asymmetric Nuclear Matter in Neutron Stars

We investigate an asymmetric nuclear matter consisting of protons and neutrons with spin degrees of freedom (σ = ↑, ↓). By generalizing the Nozières and Schmitt-Rink theory for two-component Fermi gases to the four-component case, we analyze the critical temperature Tc of the superfluid phase transition. Although the pure neutron matter exhibits the dineutron condensation in the low-density region, the superfluid instability toward the deuteron condensation is found to take place as the proton fraction increases. We clarify the mechanism of the competition between the deuteron condensation and dineutron condensation. Our results would serve for understanding the properties of asymmetric nuclear matter realized in the interior of neutron stars.

Density dependence of the nuclear energy-density functional

The explicit density (rho) dependence in the coupling coefficients of the non-relativistic nuclear energy-density functional (EDF) encodes effects of three-nucleon forces and dynamical correlations. The necessity for a coupling coefficient in the form of a small fractional power of rho is empirical and the power often chosen arbitrarily. Consequently, precision-oriented parameterisations risk overfitting and loss of predictive power. Observing that the Fermi momentum kF~rho^1/3 is a key variable in Fermi systems, we examine if a power hierarchy in kF can be inferred from the properties of homogeneous matter in a domain of densities which is relevant for nuclear structure and neutron stars. For later applications we want to determine an EDF that is of good quality but not overtrained. We fit polynomial and other functions of rho^1/3 to existing microscopic calculations of the energy of symmetric and pure neutron matter and analyze the fits. We select a form and parameter set which we found robust and examine the parameters' naturalness and the resulting extrapolations. A statistical analysis confirms that low-order terms like rho^1/3 and rho^2/3 are the most relevant ones. It also hints at a different power hierarchy for symmetric vs. pure neutron matter, supporting the need for more than one rho^a terms in non-relativistic EDFs. The EDF we propose accommodates adopted properties of nuclear matter near saturation. Importantly, its extrapolation to dilute or asymmetric matter reproduces a range of existing microscopic results, to which it has not been fitted. It also predicts neutron-star properties consistent with observations. The coefficients display naturalness. Once determined for homogeneous matter, EDFs of the present form can be mapped onto Skyrme-type ones for use in nuclei. The statistical analysis can be extended to higher orders and for different ab initio calculations.

Equation of state of dense nuclear matter and neutron star structure from nuclear chiral interactions

Aims. We report a new microscopic equation of state (EOS) of dense symmetric nuclear matter, pure neutron matter, and asymmetric and β-stable nuclear matter at zero temperature using recent realistic two-body and three-body nuclear interactions derived in the framework of chiral perturbation theory (ChPT) and including the Δ(1232) isobar intermediate state. This EOS is provided in tabular form and in parametrized form ready for use in numerical general relativity simulations of binary neutron star merging. Here we use our new EOS for β-stable nuclear matter to compute various structural properties of non-rotating neutron stars. Methods. The EOS is derived using the Brueckner–Bethe–Goldstone quantum many-body theory in the Brueckner–Hartree–Fock approximation. Neutron star properties are next computed solving numerically the Tolman–Oppenheimer–Volkov structure equations. Results. Our EOS models are able to reproduce the empirical saturation point of symmetric nuclear matter, the symmetry energy Esym, and its slope parameter L at the empirical saturation density n0. In addition, our EOS models are compatible with experimental data from collisions between heavy nuclei at energies ranging from a few tens of MeV up to several hundreds of MeV per nucleon. These experiments provide a selective test for constraining the nuclear EOS up to ~4n0. Our EOS models are consistent with present measured neutron star masses and particularly with the mass M = 2.01 ± 0.04 M⊙ of the neutron stars in PSR J0348+0432.

Equation of state for neutron stars. Some recent developments

Calculations using the chiral effective field theory (ChEFT) indicate that the four-body force contribution to the equation of state (EOS) of pure neutron matter (PNM) at the nuclear density n0 is negligibly small. However, the overall uncertainty in the EOS of PNM at n0 remains ∼ 20%. Relativistic mean field (RMF) calculations with in-medium scaling, and including hyperons and Δ resonances, can be made consistent with recent nuclear and astrophysical constraints. Dirac-Brueckner-Hartree-Fock calculations with some medium dependence of the nuclear interaction yield neutron star (NS) models with hyperonic cores consistent with 2 M⊙ stars and agreeing with the saturation parameters of nuclear matter. Many unified EOS for the NS crust and core were calculated, and are reviewed here. The effect of the finite size of baryons on the EOS, its treatment via the excluded-volume approximation, and its relevance for the hypothetical hybrid-star twins at ∼ 2 M⊙ are dicussed.