arXiv subjects

Cheng-Jun Xia

Publications and source records attributed to Cheng-Jun Xia.

At least 19 recordsLinked to original sources

arXiv2026

Pair luminosity and cooling of newborn strange star: Color-flavor-locked and two-flavor color superconducting quarks

Following the previous wor k[1] here we consider early thermal evolution of hot strange stars made of color superconducting quarks in two different pairing states: two-flavor color superconductor (2SC) and color-flavor-locked (CFL) phases, taking into account cooling by neutrinos and electron-positron pair creation due to the Schwinger process. We show that Schwinger luminosity in the electrosphere is a universal function of temperature, independent of quark matter phase for quark chemical potential $\mu_q>280$ MeV. The surface of a strange star in all the cases cools faster than its interior. This leads to a fast decrease of pair luminosity with time, so that it does not exceed $10^{46}$ erg/s at 1 second after strange star formation. Neutrino luminosity dominates over pair luminosity in all the cases except for the CFL phase with a large gap parameter $\Delta>60$ MeV, where there is a strong suppression of neutrino emission. The total energy emitted in electron-positron pairs is always smaller than the energy emitted in neutrinos, but they become comparable for the large gap parameter.

astro-ph.HE↗

arXiv2026

Asteroseismology of neutron stars with both hyperons and $\Delta$ resonances

Employing various relativistic energy density functionals for nucleon-nucleon interactions, we investigate the impact of hyperons and $\Delta$ resonances on the frequencies of non-radial oscillations in neutron stars, where the universal coupling scheme is adopted for the $\Delta$-meson couplings. It is found that the inclusion of $\Delta$ resonances is essential for neutron stars to accommodate the recent mass and radius measurements of PSR J0030+0451, PSR J0740+6620, PSR J0437-4715, PSR J0614-3329, and HESS J1731-347. As $\Delta$ resonances start to emerge in neutron stars' inner core, the $g$-mode oscillation energy becomes concentrated within the $\Delta$-admixed region, leading to a sharp increase in the $g$-mode frequency. We also examine the $f$-mode and $p$-mode oscillations and find that the impact of $\Delta$ resonances on these modes is less pronounced than the $g$-modes. The oscillation frequencies calculated in the Cowling approximation are then compared with those from full general relativity, confirming that the Cowling approximation introduces an error of approximately 20\% for the $f$-modes and within 10\% for the $g$-modes. The notable effect of $\Delta$ resonances on neutron stars' $g$-mode frequencies holds important implications for probing the internal composition of neutron stars.

astro-ph.HE↗

arXiv2026

Magnetized Coulomb crystals in neutron star crusts

We investigate the properties of magnetized Coulomb crystals in neutron star crusts using a fully three-dimensional geometry with periodic boundary conditions. The electron density profiles are fixed via the Thomas-Fermi approximation, and the nonuniform magnetic fields are treated with the equivalent magnetic charge method. The study focuses on Coulomb crystals composed of $^{12}$C at an ion number density $n_d = 10^{-9}\ \text{fm}^{-3}$, subjected to various external magnetic fields. Nuclei are described by a Gaussian wave function, where the width $b$ encapsulates the effects of zero-point ion vibrations and finite temperature. Our findings show that the crystal softens as $b$ increases. The Madelung constant $K_M$ fluctuates with the external magnetic field $B_{z0}$ at $B_{z0}\leq 3\times 10^{14}$ G. At higher field strengths, $K_M$ increases until $B_{z0} \approx 3\times 10^{15}$ G and then decreases. The body-centered cubic (BCC) lattice is slightly more stable than the face-centered cubic (FCC) lattice when $B_{z0} < 3\times 10^{15}$ G, whereas the FCC lattice may become more stable at larger $B_{z0}$. The elastic constants $c_{11}-c_{12}$ and $c_{44}$ are computed and tabulated, which grow with $B_{z0}$ for $3\times 10^{14}\ \mathrm{G}\lesssim B_{z0}\lesssim 2\times 10^{15}$ G and then decline toward zero as the field strength increases further. For $B_{z0}\gtrsim 10^{16}$ G, it becomes difficult to identify a stable lattice structure. These results provide valuable insights into the role of strong magnetic fields in shaping the properties of Coulomb crystals in compact stars.

astro-ph.HE↗

arXiv2026

Inverse-mapped density-dependent relativistic mean-field inference of the neutron-star equation of state with multi-messenger constraints

We perform a Bayesian inference of the equation of state (EOS) of cold dense matter within a density-dependent relativistic mean-field (DD-RMF) model. An explicit inverse-mapping procedure reconstructs the density-dependent couplings from a physically interpretable ten-dimensional parameter set while enforcing thermodynamic consistency together with stability and causality conditions. The EOS is constrained by complementary multi-messenger data including chiral effective field theory calculations at low density, heavy-ion collision flow information at intermediate densities, NICER mass-radius posteriors, and the existence of approximately two-solar-mass pulsars. The combined constraints strongly restrict both isoscalar and isovector sectors. In particular, the chiral effective field theory band favors a relatively soft symmetry-energy slope around 38 MeV, corresponding to a compact canonical neutron-star radius of about 11.6 km. To reconcile the intermediate-density softness suggested by heavy-ion data with the high-density stiffness required by massive pulsars, the posterior prefers a moderately large Dirac effective mass at saturation together with correlated high-density limits of the scalar and vector couplings. The resulting sound-speed profile remains causal and shows significant stiffening above the conformal limit at several times nuclear saturation density, indicating strongly interacting matter in neutron-star cores. Evidence diagnostics indicate strong compatibility among the adopted constraints within the present DD-RMF framework.

nucl-th↗

arXiv2026

Quarkyonic matter with strangeness in an extended relativistic mean-field model

Quarkyonic matter is expected to play a key role for the transition from hadronic matter to quark matter in compact stars. Within the framework of the relativistic mean-field (RMF) model and equivparticle model with density-dependent quark masses, we construct the ``quark Fermi sea" with a ``baryon Fermi surface" to characterize the properties of the quarkyonic matter. In particular, we develop a comprehensive framework to account for the strangeness degrees of freedom, incorporating $\Lambda$, $\Xi$, and $\Sigma$ hyperons as well as strange quarks in a unified quarkyonic framework. Our calculations indicate that the inevitable emergence of hyperons softens the equations of state, leading to a reduction in the equilibrium sound velocity around $n_{\rm b}\approx 2n_0$, and consequently reducing the masses and radii of neutron stars. When the quark-hadron phase transition is taken into account, the equation of state at high densities exhibits additional softening consistent with current astronomical observational constraints. This softening leads to a maximum equilibrium sound velocity of $v_{eq}^{\rm max} \approx 0.6\,c$, which is close to the ultrarelativistic limit of $0.58\,c$.

nucl-th↗

arXiv2026

The properties of strange quark matter and evolution of strange quark stars

In this work, we study the properties of strange quark matter and reveal the evolution process of strange quark stars employing a self consistent thermodynamic treatment. A comprehensive and reliable thermodynamic basis for the study of the dynamic evolution from proto-strange quark stars to stable strange stars at a zero temperature is provided. The relative abundance of particles, equation of state, temperature, and mass-radius relationship at each stage of the evolution of stars are discussed, where the cold strange quark star are consistent with the observational mass and radius of Hess J1731-347, PSR J1231-1411, PSR J0030+0451, PSR J0348+0432, and PSR J0740+6620, which could be difficult to be explained by the standard neutron star model. A schematic diagram is provided as well, illustrating the state of different stages along the evolution of stars at a fixed baryon-mass.

hep-ph↗

arXiv2025

U-spin symmetry energy and hyperon puzzle

By combining the ($u$,$d$) I-spin doublets or ($d$,$s$) U-spin doublets, the SU(3) flavor symmetry of light quarks can be decomposed into SU(2)$_I\times$U(1)$_Y$ or SU(2)$_U\times$U(1)$_Q$ subgroups, which have been widely adopted to categorize hadrons and their decay properties. The I-spin counterpart for the interactions among nucleons has been extensively investigated, i.e., the nuclear symmetry energy $E_\mathrm{sym}(n_\mathrm{b})$, which characterizes the variation of binding energy as the neutron to proton ratio in a nuclear system. In this work, we propose U-spin symmetry energy $E_\mathrm{U}(n_\mathrm{b})$ for hyperonic matter to characterize the variation of binding energy with the inclusion of hyperons. In particular, being the lightest hyperon, $\Lambda$ hyperons are included in dense matter, where the U-spin symmetry energy $E_\mathrm{U}(n_\mathrm{b})$ is fixed according to state-of-the-art constraints from nuclear physics and astrophysical observations using Bayesian inference approach. It is found that $E_\mathrm{U}(n_\mathrm{b})$ is much smaller than that of $E_\mathrm{sym}(n_\mathrm{b})$, indicating much stronger proton-neutron attraction than that of nucleon-hyperon pairs. Consequently, the $\Lambda$ hyperon potential increases significantly with density and becomes repulsive at high densities. The results indicate that there is more than 50\% probability for the emergence of $\Lambda$ hyperons in posterior EOSs, which are likely to vanish at densities $n_\mathrm{b} \gtrsim 5\,n_0$. In scenarios where $\Lambda$ hyperons do emerge, the onset density $n_{\mathrm{b}}^\Lambda$ is typically within the range of $2\,n_0$--$5\,n_0$, corresponding to neutron stars more massive than $1.0\,\rm{M_\odot}$.

hep-ph↗

arXiv2025

Nonradial oscillations of stratified neutron stars with solid crusts: Mode characterization and tidal resonances in coalescing binaries

Dynamical tides of neutron stars in the late stages of binary inspirals provide a viable probe into dense matter through gravitational waves, and potentially trigger electromagnetic precursors. We model the tidal response as a set of driven harmonic oscillators, where the natural frequencies are given by the quasinormal modes of a nonrotating neutron star. These modes are calculated in general relativity by applying linear perturbation theory to stellar models that include a solid crust and compositional stratification. For the mode spectrum, we find that the canonical interface mode associated with the crust-core boundary vanishes in stratified neutron stars and is replaced by compositional gravity modes with mixed gravity-interfacial character, driven primarily by strong buoyancy in the outer core. We also find that fluid modes such as the core gravity mode and the fundamental mode can penetrate the crust, and we establish a criterion for such penetration. Regarding the tidal interaction, we find that transfer of binding energy to oscillations is dominated by the fundamental mode despite its frequency being too high to resonate with the tidal forcing. In general, we find that lower-frequency modes induce gravitational-wave phase shifts smaller than $\sim 10^{-3},\rm rad$ for the equation of state we consider. We discover that nonresonant fundamental and crustal shear modes can trigger crust breaking already near the first gravity-mode resonance, while gravity-mode resonance concentrates strain at the base of the crust and may marginally crack it. These results suggest that both resonant and nonresonant excitations can overstress the crust and may channel energy into the magnetosphere prior to merger, potentially powering electromagnetic precursors. Our work represents an important step toward realistic modeling of dynamical tides of neutron stars in multimessenger observations.

astro-ph.HE↗

arXiv2025

Multibaryon states in the framework of an equivparticle model

Within the framework of an equivparticle model employing mean-field approximation, we investigate systematically the mass spectra of color-singlet $N$-quark configurations with $N = 3, 6, 9, 12, 15$, and 18, which are assumed to be spherically symmetric with quarks occupying the 1s$_{1/2}$ state, i.e., compact multibaryon states. At a given quark number $N$, these states collectively form a single irreducible representation under SU(6) symmetry. Our analysis yields comprehensive mass formulae that characterize these SU(6) multiplets, providing a unified description of their mass spectra. In order to effectively constrain the parameter space of the model and improve the prediction accuracy, we carry out a Bayesian parameter inference based on the experimental masses of eight baryons and $D_{03}$. The posterior probability density functions and their correlations of the model parameters are examined, based on which we further predict the masses of various multibaryon states and provide their 68$\%$ and 90$\%$ credible intervals. In our prediction, H-dibaryon, $D_{03}$, and the dibaryon with $S = -6$ are all bound states relative to $\Lambda\Lambda$, $\Delta\Delta$ and $\Omega\Omega$ thresholds, while slight probabilities of other stable dibaryons (23.64$\%$ more stable than $\Omega^- \Xi^0$ for the state wtih $I = \frac{1}{2}$, $S = -5$ and 92.49$\%$ more stable than $\Xi^0\Xi^-$ for the state wtih $I = 0$, $S = -4$) and tribaryons (0.25$\%$ more stable than $\Xi^-\Xi^0\Xi^0$ for the state wtih $S = -6$, $I = 1/2$; 2.19$\%$ more stable than $\Lambda\Xi^0\Xi^-$ for the state wtih $S = -5$, $I = 0$; and 2.21$\%$ more stable than $\Lambda\Lambda\Xi^0$ for the state wtih $S = -4$, $I = 1/2$) are observed as well. For heavier compact multibaryon states, it is unlikely for them to be stable.

hep-ph↗

arXiv2025

Light and heavy $\Lambda$ hyperclusters in nuclear matter with relativistic-mean-field models

In the framework of relativistic-mean-field (RMF) models, we investigate the properties of light and heavy $\Lambda$ hyperclusters emersed in nuclear matter at various densities $n_{\mathrm{gas}}$ and proton fractions $Y_p$. In particular, the (hyper)clusters are fixed by solving the Dirac equations imposing the Dirichlet-Neumann boundary condition, while the nuclear matter take constant densities and is treated with Thomas-Fermi approximation. The binding energies of (hyper)clusters decrease with the density of nuclear matter $n_{\mathrm{gas}}$, which eventually become unbound and melt in the presence of nuclear medium, i.e., Mott transition. For light clusters with proton numbers $N_p < 4$, with the addition of $\Lambda$ hyperons, the binding energies per baryon for $\Lambda$ hyperclusters become smaller and decrease faster with $n_{\mathrm{gas}}$ due to the weaker $N$-$\Lambda$ attraction. For heavy clusters with $N_p \geq 4$, on the contrary, the addition of $\Lambda$ hyperons increases the stability of (hyper)clusters so that the Mott transition density becomes larger as nucleons occupying higher energy states while $\Lambda$ hyperons remain in the $1s_{1/2}$ orbital. The isovector effects on (hyper)clusters in nuclear medium are also identified, where the binding energies for (hyper)clusters with $N_p> N_n$ ($N_p< N_n$) increase (decrease) with $Y_p$. For those predicted by nonlinear relativistic density functionals, light (hyper)clusters are destabilized drastically as $n_{\mathrm{gas}}$ increases, while the binding energies of heavier (hyper)clusters vary smoothly with $n_{\mathrm{gas}}$. The binding energy shifts of various (hyper)clusters due to the impact of nuclear medium are fitted to an analytical formula, which could be employed to examine the evolutions of (hyper)clusters in both heavy-ion collisions and neutron stars.

nucl-th↗

arXiv2025

Bayesian constraints on quark stars from multi-messenger observations

We perform a systematic Bayesian analysis of quark star equations of state under current multimessenger constraints, investigating the impact of prior assumptions and extreme-mass observations. Quark matter is modeled within an interacting MIT bag framework that consistently accommodates color-superconducting phases (2SC, 2SC+s, and CFL) and perturbative QCD corrections. We find that quark star models exhibit a distinct advantage in naturally accommodating the ultra-low mass object HESS J1731-347, a configuration that is challenging for standard neutron star models. In the high-mass regime, the interpretation of the secondary component of GW190814 is shown to be strongly prior-dependent: only broad priors allow for the substantial stiffness required to support such a massive object ($\sim$2.6 M$_\odot$), while more restrictive priors favor a softer equation of state consistent with standard pulsar populations. Microscopically, we demonstrate that current data tightly constrain the effective bag constant and the overall stiffness, but cannot distinguish between different color-superconducting phases. Furthermore, we validate a reduction of the model to two effective parameters without loss of information. Our results indicate that if quark stars exist, their sound speeds consistently exceeds the conformal limit ($c_s^2>1/3$) at stellar densities.

astro-ph.HE↗

arXiv2025

Constraints on $\Lambda N$ Effective Interactions from Mirror Hypernuclei in a Deformed Relativistic Hartree-Bogoliubov Model

We investigate the ground-state properties of four mirror hypernuclei pairs--$^{10}_\Lambda$Be-$^{10}_\Lambda$B, $^{12}_\Lambda$B-$^{12}_\Lambda$C, $^{16}_\Lambda$N-$^{16}_\Lambda$O, and $^{40}_\Lambda$K-$^{40}_\Lambda$Ca--within the deformed relativistic Hartree-Bogoliubov framework, analyzing their connection to $\Lambda N$ effective interactions. Systematic calculations with eight distinct effective interactions reveal linear correlations between mirror hypernuclei in $\Lambda$ separation energies and charge radii. The charge symmetry breaking effects, quantified through $\Lambda$ separation energy differences, exhibit a positive correlation with the SU(3) flavor symmetry violation. We emphasize that constraints derived from $A=10$ and $A=12$ hypernuclear pairs must explicitly incorporate rotational energy correction effects. Precision measurements of the (near) spherical $A=16$ and $A=40$ mirror systems are proposed as critical benchmarks for refining the isospin part of the hyperon-nucleon interactions.

nucl-th↗

arXiv2025

Properties and microscopic structures of dense stellar matter in RMF models

Data tables on the equation of state (EOS) and microscopic structures for cold dense stellar matter with proton fractions $Y_p =0.01$-$0.65$ and baryon number densities $n_\text{b}=10^{-8}$-$2 \ \mathrm{fm}^{-3}$ are obtained adopting 13 different relativistic density functionals, i.e., NL3, PK1, PK1r, GM1, MTVTC, DD-LZ1, PKDD, DD-ME2, TW99, DD-MEX, DD-MEX1, DD-MEX2, and DD-MEY. The EOSs of dense stellar matter inside neutron stars with baryon number densities $n_\text{b}=7.6\times 10^{-11}$-$2 \ \mathrm{fm}^{-3}$ are obtained as well fulfilling $\beta$-stability condition. In general, the dense stellar matter exhibits droplet phase at $n_\mathrm{b}\lesssim 0.015\ \mathrm{fm}^{-3}$, while more exotic structures such as rods, slabs, tubes, and bubbles appear sequentially as density increases. The critical proton fractions $Y_p^\mathrm{drip}$ ($\approx 0.26$-0.31) for neutron drip are obtained, where neutron gas emerges outside of nuclei at $Y_p< Y_p^\mathrm{drip}$. For dense stellar matter at small densities ($n_\text{b}\lesssim 10^{-5} \ \mathrm{fm}^{-3}$) or large proton fractions ($n_\text{b}\lesssim0.1 \ \mathrm{fm}^{-3}$ and $Y_p\gtrsim Y_p^\mathrm{drip}$), the EOSs and microscopic structures are generally insensitive to the adopted density functionals. With the onset of neutron drip at $Y_p\lesssim Y_p^\mathrm{drip}$, the uncertainties emerge and peak at $n_\text{b} \approx 0.02 \ \mathrm{fm}^{-3}$ within the range $10^{-5} \lesssim n_\text{b}\lesssim0.1 \ \mathrm{fm}^{-3}$. At $n_\text{b}\gtrsim0.1 \ \mathrm{fm}^{-3}$, the dense stellar matter becomes uniform and muons eventually appear, where the uncertainties in the EOSs grow significantly.

astro-ph.HE↗

arXiv2025

Unified treatment for in-medium light and heavy clusters with RMF models

It was shown that light nuclei such as $^4$He, $^8$Be, and $^{12}$C can be well described by RMF models, which enables a unified description for nuclei with baryon numbers $A\gtrsim4$. In this work, we propose a hybrid treatment for investigating the clustering phenomenon in nuclear medium, where clusters ranging from light nuclei (e.g., $^3$H, $^3$He, and $^4$He) to heavy ones (e.g., $^{12}$C, $^{16}$O, $^{40}$Ca, $^{48}$Ca, and $^{208}$Pb) can be treated in a unified manner. In particular, assuming a spherical Wigner-Seitz cell, the clusters are fixed by solving the Dirac equations imposing the Dirichlet-Neumann boundary condition, while the nuclear medium are treated with Thomas-Fermi approximation and take constant densities. In the presence of nuclear medium, the clusters eventually become unbound as density increases, while the root-mean-square charge radii increase. For clusters with different proton and neutron numbers $N_p \neq N_n$, their binding energies varies with the proton fraction of nuclear medium, which are less significant for clusters with $N_p = N_n$. The uncertainties of density functionals on the clustering phenomenon are investigated as well adopting 8 different functionals. Based on the obtained results, an analytical formula describing the binding energies of in-medium clusters is then obtained. The results presented in this work should be useful to understand the clustering phenomenon in both heavy-ion collisions and neutron stars.

nucl-th↗

arXiv2025

Effects of inner crusts on $g$-mode oscillations in neutron stars

In this work we investigate the influence of neutron stars' crusts on the non-radial $g$-mode oscillations and examine their correlations with nuclear matter properties fixed by adopting 10 different relativistic density functionals. At subsaturation densities, neutron star matter takes non-uniform structures and form the crusts. We find that the Brunt-V\"{a}is\"{a}l\"{a} (BV) frequency increases significantly at densities slightly above the neutron drip density (i.e., neutron stars' inner crusts), which leads to crust $g$-mode oscillations with their frequencies insensitive to the adopted density functional. At larger densities, BV frequency increases as well due to the core-crust transitions and emergence of muons, which lead to core $g$-mode oscillations. It is found that the obtained core $g$-mode frequencies generally increase with the slope of nuclear symmetry energy $L$, which eventually intersect with that of the crust $g$ modes adopting large enough $L$. This leads to the avoid-crossing phenomenon for the global $g$ modes that encompass contributions from both the crust and core. The correlation between the global $g_1$ mode and $L$ is identified for neutron stars with masses $M\gtrsim 1.4\ M_{\odot}$, which enables the measurements of $L$ based on gravitational wave observations. In our future study, the effects of the discontinuities in density or shear modulus should be considered, while the temperature, rotation, magnetic field, and superfluid neutron gas in neutron stars could also play important roles.

astro-ph.HE↗

arXiv2024

Elastic properties of nuclear pasta in neutron-star crusts

Based on the relativistic mean field (RMF) model with Thomas-Fermi approximation, we investigate the elastic properties of neutron star matter. The elastic constants are estimated by introducing deformations on the nuclear pasta structures in $\beta$-equilibrium, where various crystalline configurations are considered in a fully three-dimensional geometry without the Wigner-Seitz approximation. Two scenarios with different symmetry energy slope ($L = 41.34$ and 89.39 MeV) are examined, where the the elastic constants can vary by ten times. By fitting to the numerical results, we improve the analytic formulae for the elastic properties of nuclear pasta by introducing damping factors.

nucl-th↗

arXiv2024

Continuous gravitational waves from thermal mountains on accreting neutron stars: effect of the nuclear pasta phase

As density increases, the shape of nuclei transitions to non-spherical ``nuclear pasta" structures. The physical properties of the nuclear pasta, such as thermal conductivity and elasticity, have implications for detecting continuous gravitational waves from a rapidly rotating neutron star. In this work, we investigate the effect of the nuclear pasta on the quadruple moment, and find out that, compared with previous work, the quadrupole moment contributing to continuous gravitational-wave radiation can be up to two orders of magnitude larger. We also discuss the relationship between the quadruple moment and the maximum shear strain. Considering the properties of nuclear pasta, we study the detectability of known accreting neutron stars and compare predicted results to the detectable amplitude limits. These sources are well above the sensitivity curves for Cosmic Explorer and Einstein Telescope detectors. Our work advances the understanding of the properties of nuclear pasta and a possible mechanism for continuous gravitational waves.

gr-qc↗

arXiv2024

Finite nuclei in an extended Nambu-Jona-Lasinio model

We propose a new theoretical framework to investigate the properties of finite nuclei based on an extended Nambu-Jona-Lasinio (eNJL) model, where the Dirac sea, the spontaneous chiral symmetry breaking, and the quark degrees of freedom are considered by extending the SU(3) NJL model and treating baryons as clusters of quarks. The eNJL model can then be readily adopted to examine the matter states ranging from baryonic matter to quark matter in a unified manner. In this work, by assuming spherically symmetric finite nuclei and neglecting the center-of-mass or rotational corrections, we systematically investigate the properties of finite nuclei based on the eNJL model with additional pairing correlations. It is found that our model generally reproduces the binding energies of the 2495 nuclei ($A>2$) from the 2016 Atomic Mass Evaluation (AME2016) with the root-mean-square deviations $5.38$ MeV. The deviations are mainly attributed to the too large shell gaps at magic numbers $N(Z) =28$, 50, and 82 as well as the spurious shell closures at $N(Z)=34$, 58, and 92. Meanwhile, the obtained charge radii of 906 nuclei are systematically smaller than the experimental values with root-mean-square deviations $0.127$ fm. In our future study, we expect to reduce the uncertainties of our predictions by carefully calibrating the density dependence of coupling constants and considering deformations with microscopic collective corrections from the nucleons in the Fermi sea and quarks in the Dirac sea.

nucl-th↗