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

A QMD transport model that employs a modified momentum dependent interaction (MDI2) potential, supplemented by a phase-space coalescence model fitted to FOPI experimental multiplicities of free nucleons and light clusters is 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 Au-197 + Au-197 collisions at 400 MeV/nucleon impact energy. A previous calculation using the same model 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. To make use of this result a Gogny interaction inspired potential is modified by the addition of a density dependent, momentum independent term, while enforcing a close description of the empirical nucleon optical potential, allowing independent modifications 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 constraint is extracted: L = 85 +/- 22(exp) +/- 20(th) +/- 12(sys) MeV and K-sym = 96 +/- 315(exp) +/- 170(th) +/- 166(sys) MeV. Theoretical errors include effects due to uncertainties in the isoscalar part of the equation of state, value of the isovector neutron-proton effective mass splitting, in-medium effects on the elastic nucleon-nucleon cross-sections, Pauli blocking algorithm variants and scenario considered for the conservation of the total energy of the system. 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 +/- 30(exp) +/- 19(th) MeV. It is 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 its experimental component of uncertainty to about 150 MeV.