Feasability of constraining the curvature parameter of the symmetry energy using elliptic flow data

A QMD type transport model supplemented by a phase-space coalescence model fitted to FOPI experimental multiplicities of free nucleons and light clusters has been used to study the density dependence of the symmetry energy above the saturation point by a comparison with experimental elliptic flow ratios measured by the FOPI-LAND and ASYEOS collaborations in $^{197}$Au+$^{197}$Au collisions at 400 MeV/nucleon impact energy. A previous calculation has proven that neutron-to-proton and neutron-to-charged particles elliptic flow ratios probe on average different densities allowing in principle the extraction of both the slope $L$ and curvature $K_{sym}$ parameters of the symmetry energy. Consequently a Gogny interaction inspired potential has been modified to allow independent changes of $L$ and $K_{sym}$. Comparing theoretical predictions with experimental data for neutron-to-proton and neutron-to-charged particles elliptic flow ratios the following constraints have been extracted: $L$=85$\pm$22(exp)$\pm$20(th)$\pm$12(sys) MeV and $K_{sym}$=96$\pm$315(exp)$\pm$170(th)$\pm$166(sys) MeV. Residual model dependence is accounted for in the magnitude of the quoted theoretical error. Systematical uncertainties are generated by the inability of the transport model to reproduce experimental light-cluster-to-proton multiplicity ratios. A value for $L$, free of systematical theoretical uncertainties, can be extracted from the neutron-to-proton elliptic flow ratio alone: $L$=84$\pm$30(exp)$\pm$19(th) MeV. It has also been demonstrated that elliptic flow ratios reach a maximum sensitivity on the $K_{sym}$ parameter in heavy-ion collisions of about 250 MeV/nucleon impact energy, allowing a reduction of the experimental component of uncertainty to about 150 MeV for this parameter.