Modifications of the pion-production threshold in the nuclear medium in heavy ion collisions and the nuclear symmetry energy

Using the relativistic Vlasov-Uehling-Uhlenbeck (RVUU) equation based on mean fields from the nonlinear relativistic NLρ and NLρ δ models, which have same nuclear equation of state and symmetry energy but different symmetry energy slope parameters, we study the effect of medium modification of the pion-production threshold on the total pion yield and the π^-/π⁺ ratio in Au+Au collisions. We find that the in-medium threshold effect enhances both the total pion yield and the π^-/π⁺ ratio, compared to those without this effect. Furthermore, including the medium modification of the pion-production threshold leads to a larger π^-/π⁺ ratio for the NLρ δ model with a larger symmetry energy parameter than the NLρ model with a smaller symmetry energy parameter, opposite to that found without the in-medium threshold effect. To reproduce the total pion yield measured by the FOPI Collaboration, we introduce a density-dependent cross section for Δ baryon production from nucleon-nucleon collisions, which suppresses the total pion yield but hardly changes the π^-/π⁺ ratio. Because of the small difference in the stiffness of their symmetry energies, the π^-/π⁺ ratios obtained from both the NLρ and NLρ δ models are consistent with the FOPI data within the experimental errors.