Subthreshold pion production within a transport description of central Au + Au collisions

Mapping out the equation of state of nuclear matter is a long-standing problem in nuclear physics. Recent emphasis is on the density dependence of the symmetry energy, with experiments needing dedicated symmetry-energy observables. Towards the latter goal, we employ the Boltzmann-Uehling-Uhlenbeck (pBUU) transport model to simulate pion production in a heavy ion collision. We find that the net pion yield tests the momentum dependence of the nuclear mean field. In exploring the sensitivity of pion observables to the symmetry energy at higher than normal densities, we find that our calculations of pion ratios contradict, at some level, predictions from both the isospin-dependent BUU (IBUU) and improved isospin-dependent quantum molecular dynamics models. We propose to employ the pion ratio in the high-energy tail of spectra in future experiments to distinguish between different variants of high-density symmetry energy.

Figure 1

Pion multiplicity in central Au + Au collisions. Symbols represent data of the FOPI Collaboration [12]. The lines represent pBUU calculations when following either (a) the momentum-independent MF or (b) the past flow-optimized momentum-dependent MF. Solid lines are predictions for ${\pi }^{-}$, and dashed lines are predictions for ${\pi }^{+}$. The experimental error bars are about the size of symbols.

Figure 2

Pion multiplicity in central Au + Au collisions, as a function of beam energy. Symbols represent data of the FOPI Collaboration [12], while lines represent the pBUU calculations with the $N_{\pi }$-adjusted momentum-dependent MF. The experimental error bars are about the size of symbols.

Figure 3

Optical potential in nuclear matter at different indicated densities, as a function of momentum. Dashed and solid lines represent, respectively, the $v_2$-optimized and $N_{\pi }$-adjusted MFs.

Figure 4

Optical potential in nuclear matter at different indicated densities, as a function of nucleon energy. Dashed and solid lines represent, respectively, UV14 + UVII variational calculations and our $N_{\pi }$-adjusted MF.

Figure 5

Ratio of out-of-reaction plane to in-plane proton yields, as a function of transverse momentum. Symbols represent data from the measurements of the KaoS Collaboration of midperipheral Bi + Bi collisions at the beam energy of $400A$ MeV $(b\simeq 8.7$ fm) [21]. Solid line represents pBUU calculations with the $N_{\pi }$-adjusted momentum-dependent MF and dashed line represents calculations with the $v_2$-optimized momentum-dependent MF. The indicated theoretical errors are statistical, associated with the Monte Carlo sampling in the transport calculations.

Figure 6

Pion ratios in central Au + Au collisions, as a function of beam energy. (a) Comparison of predictions from IBUU and ImIQMD models to the FOPI data; the calculations employed stiff and soft symmetry energies. (b) Comparison of predictions from the pBUU model to the FOPI data; the calculations employed $v_2$-optimized and $N_{\pi }$-adjusted MFs. In our calculations here, the potential part of symmetry energy is linear in density.

Figure 7

Ratio of net charged-pion yields in central Au + Au collisions at $400A$ and $200A$ MeV, as a function of the stiffness of symmetry energy $\gamma$, from pBUU calculations using $N_{\pi }$-adjusted MF. The colored region represents the $400A$ MeV FOPI measurement. The theoretical errors are due to statistical sampling in the pBUU calculations.

Figure 8

S-wave contribution to $\pi {-}^{197}\mathrm{Au}$ optical potential. Solid line represents the work of Toki [27]. Long-dashed, dotted and dash-dotted lines represent pion potentials from pBUU parametrization for $\gamma =0.5,1.0,2.0$, respectively, in the interaction part of the symmetry energy. Short dashed line represents the lack of corresponding potentials in the IBUU and ImIQMD models.

Figure 9

Charged-pion ratio in central Au + Au collisions at $200A$ MeV, as a function of kinetic energy in the center-of-mass frame for different values of the stiffness $\gamma$ of the symmetry energy, from 0.5 to 2.0. The horizontal line represents the ratio of net charged-pion yields.

Figure 10

Charged-pion ratio in central ${}^{124}\mathrm{Sn}$ + ${}^{132}\mathrm{Sn}$ collisions at $300A$ MeV, as a function of kinetic energy in the c.m. frame for different values of the stiffness $\gamma$ of the symmetry energy, from 0.5 to 2.0.

Figure 11

Average c.m. kinetic energy of ${\pi }^{+}$ and ${\pi }^{-}$ in central Au + Au collisions at $200A$ MeV, plotted against stiffness $\gamma$ of the symmetry energy. The error bars are smaller than the symbol size.

Figure 12

Difference between average c.m. kinetic energy of ${\pi }^{+}$ and ${\pi }^{-}$ in central Au + Au collisions at $200A$ MeV, plotted against stiffness $\gamma$ of the symmetry energy. The error bars are smaller than the symbol size.

Figure 13

Ratio of neutron-to-proton numbers at net supranormal densities, $\rho >{\rho }_0$, in central Au + Au collisions at $200A$ MeV, as a function of times. At early times, the numbers in the ratio are marginal, and the ratio is thus not very meaningful.