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 $\text{NL}\rho$ and $\text{NL}\rho \delta$ 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 ${\pi }^{-}/{\pi }^{+}$ ratio in Au+Au collisions. We find that the in-medium threshold effect enhances both the total pion yield and the ${\pi }^{-}/{\pi }^{+}$ ratio, compared to those without this effect. Furthermore, including the medium modification of the pion-production threshold leads to a larger ${\pi }^{-}/{\pi }^{+}$ ratio for the $\text{NL}\rho \delta$ model with a larger symmetry energy parameter than the $\text{NL}\rho$ 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 $\Delta$ baryon production from nucleon-nucleon collisions, which suppresses the total pion yield but hardly changes the ${\pi }^{-}/{\pi }^{+}$ ratio. Because of the small difference in the stiffness of their symmetry energies, the ${\pi }^{-}/{\pi }^{+}$ ratios obtained from both the $\text{NL}\rho$ and $\text{NL}\rho \delta$ models are consistent with the FOPI data within the experimental errors.

Figure 1

Proton and neutron effective masses in asymmetric nuclear matter of isospin asymmetry $\alpha =0.2$ and at zero temperature as functions of nuclear matter density in the $\text{NL}\rho$ and $\text{NL}\rho \delta$ models [14].

Figure 2

Binding energy per nucleon of symmetric nuclear matter at zero temperature as a function of nuclear matter density.

Figure 3

Symmetry energy as a function of nuclear density in the $\text{NL}\rho$ and $\text{NL}\rho \delta$ models [14].

Figure 4

Numbers of $\Delta$ baryons and pions (upper panel) and central density divided by the saturation density (lower panel) in the $\text{NL}\rho$ model as functions of time in Au+Au collisions at impact parameter of $1\mathrm{fm}$ and energy of $E/A=400\mathrm{MeV}$.

Figure 5

${\pi }^{-}/{\pi }^{+}$ ratio (left panel) and pion yields (right panel) as functions of the collision energy with and without the threshold effect in Au+Au collisions at impact parameter of $1.4\mathrm{fm}$ from the $\text{NL}\rho$ and $\text{NL}\rho \delta$ models. Experimental data are from the FOPI Collaboration [6].

Figure 6

${\pi }^{-}/{\pi }^{+}$ ratio (left panel) and total pion yield (right panel) as functions of the collision energy obtained with the threshold effect and the density-dependent $\Delta$ production cross section in Au+Au collisions at impact parameter of $1\mathrm{fm}$ for both the $\text{NL}\rho$ and $\text{NL}\rho \delta$ models. Experimental data are from the FOPI Collaboration [6].

Figure 7

Reduced longitudinal (left panel) and transverse (right panel) rapidity distributions of protons in Au+Au collisions at $E/A=400\mathrm{MeV}$ and $b<2.25$ fm. The experimental data are those of fragments with charge $Z\le 6$ from the FOPI Collaboration [29].

Figure 8

Reduced longitudinal (left panel) and transverse (right panel) rapidity distributions of ${\pi }^{-}$ and ${\pi }^{+}$ in Au+Au collisions at $E/A=400\mathrm{MeV}$ and $3.74<b<6.74$ fm. The experimental data are from the FOPI Collaboration [30].