Medium effects on pion production in heavy ion collisions

Within the framework of the relativistic Vlasov–Uehling–Uhlenbeck transport model based on the relativistic nonlinear $\mathrm{NL}\rho$ interaction, we study pion in-medium effects on the ${\pi }^{-}/{\pi }^{+}$ ratio in $\text{Au}+\text{Au}$ collisions at the energy of $E/A=400\mathrm{MeV}$. These effects include the isospin-dependent pion $s$-wave and $p$-wave potentials, which are taken from calculations based on the chiral perturbation theory and the $\mathrm{\Delta }$-hole model, respectively. We find that the ${\pi }^{-}/{\pi }^{+}$ ratio in this collision is suppressed by the pion $s$-wave potential but enhanced by the $p$-wave potential, with a net effect of a significantly suppressed ${\pi }^{-}/{\pi }^{+}$ ratio. Including also the in-medium threshold effects on $\mathrm{\Delta }$ resonance production and decay and using a nuclear symmetry energy with a slope parameter $L=59\mathrm{MeV}$ by reducing the coupling of isovector-vector $\rho$ meson to nucleon, our result is in good agreement with measured ${\pi }^{-}/{\pi }^{+}$ ratio from the FOPI Collaboration. We further investigate the pion in-medium effects on the ratio of charged pions as a function of their kinetic energies.

Figure 1

Density dependence of the energy of a pion at fixed momentum $k=200\mathrm{MeV}$ due to its $s$-wave interaction with nucleons in asymmetric nuclear matter of isospin asymmetry $\delta =0.2$. The black line indicates the case for a free pion.

Figure 2

Density dependence of the energy of a pion at fixed momentum $k=200\mathrm{MeV}$ (left window) and momentum dependence of its energy at fixed density $\rho =2{\rho }_0$ (right window) due to its $p$-wave potential in asymmetric nuclear matter of isospin asymmetry $\delta =0.2$. The curves labeled by ${\omega }_{\mathrm{F}}$ and ${\omega }_{\mathrm{\Delta }}$ are energies of free pion [Eq. (6)] and $\mathrm{\Delta }$-hole state [Eq. (3)], respectively.

Figure 3

Same as Fig. 2 but with the inclusion of both pion $s$-wave and $p$-wave potentials.

Figure 4

Decay widths of ${\mathrm{\Delta }}^{++}$ and ${\mathrm{\Delta }}^{-}$ as functions of their masses after the inclusion of the pion $s$-wave potential (S), the $p$-wave potential (P), and both the $s$-wave and $p$-wave potentials ($\text{S}+\text{P}$) in asymmetric nuclear matter of $\rho =2{\rho }_0$ and $\delta =0.2$. The case of a $\mathrm{\Delta }$ resonance in free space is shown as black solid lines. Here nucleons and $\mathrm{\Delta }$ resonances are treated as free particles.

Figure 5

Sum of nucleon and pion energies $E_N+\omega \left(k\right)$ as a function of pion momentum $k$ in symmetric nuclear matter at different densities with the inclusion of both pion $s$-wave and $p$-wave potentials. Here the nucleon is treated as a free particle.

Figure 6

Cross sections of the reaction ${\pi }^{+}+p\to {\mathrm{\Delta }}^{++}$ and ${\pi }^{-}+n\to {\mathrm{\Delta }}^{-}$ as functions of $\sqrt{s}$ with inclusion of the pion $s$-wave potential (S), the $p$-wave potential (P) and both $s$-wave and $p$-wave potentials ($\text{S}+\text{P}$) in asymmetric nuclear matter of $\rho =2{\rho }_0$ and $\delta =0.2$. For comparison, we also show the cross section for free pions as black solid lines. Here nucleons and $\mathrm{\Delta }$ resonances are treated as if they are in free space.

Figure 7

The ${\pi }^{-}/{\pi }^{+}$ ratio in $\text{Au}+\text{Au}$ collisions at impact parameter of 1.4 fm and energy of $E/A=400\mathrm{MeV}$ from the $\mathrm{NL}\rho$ model in different cases (see text for details). Experimental data [26] from the FOPI collaboration are shown as the cyan band.

Figure 8

The ${\pi }^{-}/{\pi }^{+}$ ratio in $\text{Au}+\text{Au}$ collisions at $E/A=400\mathrm{MeV}$ and impact parameter of $1.4\mathrm{fm}$ as a function of kinetic energy in the center-of-mass frame for the three cases of without any medium effects (free), with the in-medium threshold effect (Th), and with both the threshold and pion potential effects ($\text{Th}+\text{S}+\text{P}$).