Toward a model-independent constraint of the high-density dependence of the symmetry energy

Neutron-proton elliptic flow difference and ratio have been shown to be promising observables in the attempt to constrain the density dependence of the symmetry energy above the saturation point from heavy-ion collision data. Their dependence on model parameters such as microscopic nucleon-nucleon cross sections, compressibility of nuclear matter, optical potential, and symmetry energy parametrization is thoroughly studied. By using a parametrization of the symmetry energy derived from the momentum-dependent Gogny force in conjunction with the Tübingen quantum molecular dynamics (QMD) model and comparing the results with the experimental FOPI-LAND data for ${}^{197}$Au +${}^{197}$Au collisions at 400 MeV/nucleon, a moderately stiff ($L_{\mathrm{sym}}=122\pm 57$ MeV and $K_{\mathrm{sym}}=229\pm 363$ MeV) symmetry energy is extracted, a result that agrees with that of a similar study that employed the UrQMD transport model and a power-law parametrization of the symmetry energy. This contrasts with diverging results extracted from the FOPI ${\pi }^{-}/{\pi }^{+}$ ratio available in the literature.

Figure 1

Variations in the values of the impact parameter integrated ($b⩽7.5$ fm) npEFD (a) and npEFR (b) due to different choices for the optical potential ($V_{\mathrm{opt}}$), parametrization of symmetry-energy ($S$) as well as the combined, quadratically added, uncertainty.

Figure 2

Model dependence of npEFD (a) and npEFR (b) and comparison with FOPI-LAND experimental data, integrated over impact parameter $b⩽7.5$ fm. Sensitivity to the different model parameters, compressibility modulus $K$, width of nucleon wave function $L$, optical potential $V_{\mathrm{opt}}$, and parametrization of the symmetry energy $S$ are displayed. The total model dependence is obtained by adding, in quadrature, individual sensitivities.

Figure 3

Constraints on the density dependence of the symmetry energy obtained from comparing theoretical predictions for npEFD and npEFR to FOPI-LAND experimental data. The result of Russotto et al. [15] is also shown together with the Gogny-inspired SE parametrization for three values of the stiffness parameter: $x=-1$ (stiff), $x=0$, and $x=1$ (soft).