Symmetry energy of nucleonic matter with tensor correlations

The nuclear symmetry energy $\left(E_{\mathrm{sym}}\left(\rho \right)\right)$ is a vital ingredient of our understanding of many processes, from heavy-ion collisions to neutron stars structure. While the total nuclear symmetry energy at nuclear saturation density $\left({\rho }_0\right)$ is relatively well determined, its value at supranuclear densities is not. The latter can be better constrained by separately examining its kinetic and potential terms and their density dependencies. The kinetic term of the symmetry energy, $E_{\mathrm{sym}}^{\mathrm{kin}}\left({\rho }_0\right)$, equals the difference in the per-nucleon kinetic energy between pure neutron matter (PNM) and symmetric nuclear matter (SNM), often calculated using a simple Fermi gas model. However, experiments show that tensor force induced short-range correlations (SRC) between proton-neutron pairs shift nucleons to high momentum in SNM, where there are equal numbers of neutrons and protons, but have almost no effect in PNM. We present an approximate analytical expression for $E_{\mathrm{sym}}^{\mathrm{kin}}\left({\rho }_0\right)$ of correlated nucleonic matter. In our model, $E_{\mathrm{sym}}^{\mathrm{kin}}\left({\rho }_0\right)=-10$ MeV, which differs significantly from +12.5 MeV for the widely-used free Fermi gas model. This result is consistent with our analysis of recent data on the free proton-to-neutron ratios measured in intermediate energy nucleus-nucleus collisions as well as with microscopic many-body calculations, and previous phenomenological extractions. We then use our calculated $E_{\mathrm{sym}}^{\mathrm{kin}}\left(\rho \right)$ in combination with the known total symmetry energy and its density dependence at saturation density to constrain the value and density dependence of the potential part and to extrapolate the total symmetry energy to supranuclear densities.

Figure 1

The per-nucleon kinetic energy calculated using the correlated Fermi gas (CFG) model (diagonal red [gray] band) for atomic nuclei from ${}^{12}\mathrm{C}$ to ${}^{208}\mathrm{Pb}$. The calculated kinetic energy is shown as a function of $\lambda$, the high-momentum tail cutoff parameter. The vertical blue (gray) band shows the constraints on $\lambda$ from the deuteron momentum distribution. The red band reflects the model uncertainties. Also shown are the results from the uncorrelated Fermi gas model (dashed purple [gray] line) and a horizontal black band spanning the results from many-body nuclear calculations for various nuclei from ${}^{12}\mathrm{C}$ to ${}^{208}\mathrm{Pb}$ [42] and from exact variational Monte Carlo (VMC) calculations for ${}^{12}\mathrm{C}$ [41].

Figure 2

The per-nucleon kinetic energy for symmetric nuclear matter calculated using the correlated Fermi gas (CFG) model (diagonal red [gray] band). The calculated kinetic energy is shown as a function of $\lambda$, the high-momentum tail cutoff parameter. The vertical blue (gray) band shows the constraints on $\lambda$ from the deuteron momentum distribution. The red band reflects the model uncertainties. Also shown are the results from the uncorrelated Fermi gas model (dashed purple [gray, bottom] line), the Brueckner-Hartree-Fock (BHF) model using the AV-18 interaction [28], and the self-consistent Green's function (SCGF) approach using the CDBonn, N3LO, and AV18 nucleon-nucleon interactions [30, 31].

Figure 3

The per-nucleon kinetic symmetry energy at saturation density, $E_{\mathrm{sym}}^{\mathrm{kin}}\left({\rho }_0\right)$, calculated using the correlated Fermi gas model (diagonal red [gray] band) as a function of $\lambda$, the high-momentum tail cutoff parameter. The dashed purple (gray, top) line shows the results of the uncorrelated Fermi gas model. The horizontal green (gray) band shows the results from transport model analyses of Sn+Sn collisions described in the text. Also shown for comparison are the results from microscopic calculations: Brueckner-Hartree-Fock (BHF) [28], Fermi hypernetted chain (FHNC) [29], and the self-consistent Green's gunction (SCGF) using the CDBonn, N3LO, Nij1, and AV18 nucleon-nucleon interactions [30, 31].

Figure 4

The calculated double ratio of free neutron-protons in the two reactions in comparison with the MSU data for transversely emitted nucleons in the angular range of ${70}^{\circ }\le {\theta }_{\mathrm{cms}}\le {110}^{\circ }$ [40]. The bands represent $1\sigma$ uncertainty of the calculations.

Figure 5

The density dependence of the kinetic, potential and total symmetry energy extracted using the CFG and FG models. See text for details.