Hot and dense matter beyond relativistic mean field theory

Properties of hot and dense matter are calculated in the framework of quantum hadrodynamics by including contributions from two-loop (TL) diagrams arising from the exchange of isoscalar and isovector mesons between nucleons. Our extension of mean field theory (MFT) employs the same five density-independent coupling strengths which are calibrated using the empirical properties at the equilibrium density of isospin-symmetric matter. Results of calculations from the MFT and TL approximations are compared for conditions of density, temperature, and proton fraction encountered in the study of core-collapse supernovae, young and old neutron stars, and mergers of compact binary stars. The TL results for the equation of state (EOS) of cold pure neutron matter at sub- and near-nuclear densities agree well with those of modern quantum Monte Carlo and effective field-theoretical approaches. Although the high-density EOS in the TL approximation for cold and $\beta$-equilibrated neutron-star matter is substantially softer than its MFT counterpart, it is able to support a $2M_{\odot }$ neutron star required by recent precise determinations. In addition, radii of $1.4M_{\odot }$ stars are smaller by $\sim 1$ km than those obtained in MFT and lie in the range indicated by analysis of astronomical data. In contrast to MFT, the TL results also give a better account of the single-particle or optical potentials extracted from analyses of medium-energy proton-nucleus and heavy-ion experiments. In degenerate conditions, the thermal variables are well reproduced by results of Landau's Fermi-liquid theory in which density-dependent effective masses feature prominently. The ratio of the thermal components of pressure and energy density expressed as ${\mathrm{\Gamma }}_{\mathrm{th}}=1+(P_{\mathrm{th}}/{\epsilon }_{\mathrm{th}})$, often used in astrophysical simulations, exhibits a stronger dependence on density than on proton fraction and temperature in both MFT and TL calculations. The prominent peak of ${\mathrm{\Gamma }}_{\mathrm{th}}$ at supranuclear density found in MFT is, however, suppressed in TL calculations. This outcome is analogous to results of nonrelativistic models when exchange contributions from finite-range interactions are included in addition to those of contact interactions.

Figure 1

Feynman diagram for the two-loop contribution to the grand-canonical potential density $\omega$. The exchange of mesons $\varphi ,V^{\mu },{\rho }^{\mu }$, and $\pi$ is indicated by the wavy line.

Figure 2

Dirac effective masses $M^{*}/M$ vs baryon density ${\rho }_B$ in SNM $\left(x=0.5\right)$ and PNM $\left(x=0\right)$ for the MFT and TL calculations.

Figure 3

Energy vs ${\rho }_B$ in SNM and PNM for the MFT and TL calculations.

Figure 4

Same as Fig. 3, but for the low-density region.

Figure 5

Panel (a) shows a comparison of the MFT and TL results for PNM energy vs ${\rho }_B$ with those of a variational calculation “APR” [79]; quantum Monte Carlo (QMC) calculations, “QMC1” [62, 80], “QMC2” based on a N3LO chiral potential with momentum cutoff 414 MeV [81], and “QMC3” [83]; and the N3LO chiral perturbation theory calculation (“Pert”) with a momentum cutoff of 500 MeV [82]. For QMC3, the two curves shown are with $S_2=32$ and 33.7 MeV. Panel (b) compares the TL results with three different DBHF calculations: “DBHF1” [84], “DBHF2” [85, 86], and “DBHF3” [87, 88].

Figure 6

Two-loop contributions to the energy vs ${\rho }_B$ in SNM and PNM.

Figure 7

The total two-loop energy and its nonrelativistic approximation in SNM and PNM.

Figure 8

The symmetry energy $S_2$ vs ${\rho }_B$. In MFT, contributions to $S_2$ arise from both the $\rho$-meson and scalar $\varphi$-meson interactions [Eq. (4.10)]. The latter piece is termed $S_2^{\text{kin}}$ in Eq. (4.17). In the TL calculation, there is an additional contribution from the exchange diagram [Eq. (4.25)].

Figure 9

The energy difference between PNM and SNM approximated by $S_2$ and $S_2+S_4$ in MFT and TL calculations. For both cases, the solid curve is the energy difference, the bottom long-dashed curve is $S_2$, and the middle short-dashed curve is $S_2+S_4$.

Figure 10

Pressure vs density in SNM and PNM.

Figure 11

Same as Fig. 10, but for PNM in the low-density region.

Figure 12

Neutron-star mass-radius diagram for the MFT and TL calculations.

Figure 13

The nucleon's optical potential in SNM at different densities. For each density, results of MFT lie above those of the TL calculations. For densities, 0.16, 0.3, and $0.4 {\mathrm{fm}}^{-3}$, the potentials are shifted by 50, 100, and 150 MeV, respectively, for clarity.

Figure 14

The neutron and proton effective Landau masses $m^{*}$ vs ${\rho }_B$ for different proton fractions. The crosses $\times$ indicate the densities at which the first derivatives become zero, whereas the asterisks $*$ mark the densities at which the Dirac effective mass $M^{*}$ equals either the neutron or the proton Fermi momentum.

Figure 15

The neutron and proton Landau effective masses $m^{*}$ vs ${\rho }_B$ for different proton fractions. Here “wo $\pi$” and “wo $\rho$” are results without including $\pi$ and $\rho$ two-loop contributions in calculating the single-particle spectra.

Figure 16

The zero-temperature neutron and proton chemical potentials (inclusive of masses) vs density for different proton fractions.

Figure 17

Comparison between the Landau masses $m_{p,n}^{*}$ [from a self-consistent (SC) calculation] and $m_{p,n}^{{*}^{\prime }}$ [from the perturbative (P) approach] for proton fractions $x=0.1$ and $0.5$.

Figure 18

Pressure isotherms at low density for SNM. The middle curve in each panel corresponds to the temperature at which the liquid-gas phase transition occurs.

Figure 19

Entropy per nucleon at $T=20$ and 50 MeV for proton fractions $x=0$ and $0.5$. The left panels compare the TL(P,SC) and MFT results. The right panels show ratios between the degenerate limits (DL) and the corresponding full results. The ratios for PNM are shifted by $-0.5$.

Figure 20

Thermal pressure at $T=20$ and 50 MeV for proton fractions $x=0$ and $0.5$. The left panels compare the TL(P,SC) and MFT results. The right panel shows ratios between the results of the full calculations and their degenerate-limit values. The ratios for PNM are shifted by $-0.5$.

Figure 21

Thermal energy per nucleon at $T=20$ and 50 MeV for proton fractions $x=0$ and $0.5$. The left panels compare the TL(P,SC) and MFT results. The right panels show ratios between the results of the full calculations and their degenerate-limit values. The ratios for PNM are shifted by $-0.5$.

Figure 22

(a) Thermal components of the neutron chemical potentials at $T=20$ MeV at the indicated proton fractions. (b) Ratios of results from full and degenerate-limit calculations.

Figure 23

Same as Fig. 22, but for $T=50$ MeV.

Figure 24

The thermal index ${\mathrm{\Gamma }}_{\mathrm{th}}$ at $\mathrm{T}=20$ and 50 MeV and proton fractions $0.1,0.3$, and $0.5$. The panels from left to right show results of MFT, TL(P), and TL(SC) calculations. The upper curves are for matter with nucleons only, whereas the lower curves are for nucleons with leptons and photons.