Isospin-dependent pion in-medium effects on the charged-pion ratio in heavy ion collisions

Using results from the chiral perturbation theory for the $s$-wave interaction and the $\Delta$-resonance model for the $p$-wave interaction of pions with nucleons, we evaluated the spectral functions of pions in asymmetric nuclear matter with unequal proton and neutron densities. We find that in hot dense neutron-rich matter the strength of the spectral function of positively charged pions at low energies is somewhat larger than that of negatively charged pions. In a thermal model, this isospin-dependent effect slightly reduces the ratio of negatively charged to positively charged pions that are produced in heavy ion collisions induced by radioactive beams. The relevance of our results to the determination of the nuclear symmetry energy from the measured ratio of negatively to positively charged pions produced in heavy ion collisions is discussed.

Figure 1

Nuclear symmetry energies as functions of nuclear density from the MID interaction for different values of the symmetry energy parameter $x$.

Figure 2

Spectral functions of pions in asymmetric nuclear matter of density $2{\rho }_0$ and isospin asymmetry ${\delta }_{\mathrm{like}}=0.135$ as functions of pion energy for different pion momenta of (a) $m_{\pi }$, (b) $2m_{\pi }$, (c) $3m_{\pi }$, and (d) $4m_{\pi }$. All are calculated with the Migdal parameter $g^{\prime }=1/3$.

Figure 3

Mass distributions of $\Delta$ resonances at rest in asymmetric nuclear matter of density $2{\rho }_0$ and isospin asymmetry ${\delta }_{\mathrm{like}}=0.135$. The solid line corresponds to that in free space. The distributions near the threshold and at the peak are enlarged in the insets.

Figure 4

The ${\pi }^{-}/{\pi }^{+}$ ratio in $\mathrm{Au}+\mathrm{Au}$ collisions at the beam energy of $0.4\hspace{0.5em}A\hspace{0.5em}\mathrm{GeV}$ for different values of nuclear symmetry energy parameter ($x=0$, $0.5$, and $1$) and the Migdal parameter $g^{\prime }=1/3$, $0.4$, $0.5$, and $0.6$. Results for $g^{\prime }=\infty$ correspond to the case without the pion in-medium effects.