Pion production in a transport model based on mean fields from chiral effective field theory

We develop a Boltzmann-Uehling-Uhlenbeck (BUU) transport model based on the $\mathrm{Sk}\chi {\mathrm{m}}^{*}$ energy density functional, which is constructed from fitting the nuclear equation of state and nucleon effective masses in asymmetric nuclear matter predicted by the two- and three-body chiral interactions as well as the binding energies of finite nuclei. This new $\chi \mathrm{BUU}$ transport model is then used to study how baryon mean-field potentials affect the kinematics of a scattering or decay process and the equilibrium properties of a hot $N\text{−}\mathrm{\Delta }\text{−}\pi$ system in a box with periodic boundary conditions. We find that the inclusion of mean-field potentials in the energy conservation condition for scattering and decay processes is necessary to maintain the equilibrium numbers of $N,\mathrm{\Delta }$, and $\pi$. Although the baryon mean-field potentials have significant effects on the total $\mathrm{\Delta }$ and $\pi$ numbers, they only slightly affect the ratio of effective negatively to positively charged pions. We also study pion production in central ${}^{197}\mathrm{Au}+{}^{197}\mathrm{Au}$ collisions at the incident energy of $E/A=400$ MeV and compare the results with the experimental data from the Four Pi (FOPI) Collaboration at the Gesellschaft für Schwerionenforschung mbH (GSI) in Germany. We find that our model can well describe the experimental results and the threshold effect due to baryon mean-field potentials does not affect much the charged pion ratio in this model. We further make predictions on pion production for the ongoing experiments of ${}^{132}\mathrm{Sn}+{}^{124}\mathrm{Sn}$ and ${}^{108}\mathrm{Sn}+{}^{112}\mathrm{Sn}$ at the incident energy of $E/A=270$ MeV by the SAMURAI Pion-Reconstruction and Ion-Tracker ($\mathrm{S}\pi \mathrm{RIT}$) Collaboration at the Institute of Physical and Chemical Research (RIKEN) in Japan.

Figure 1

Momentum dependence of the nucleon single-particle potential in symmetric nuclear matter at saturation density from the $\mathrm{Sk}\chi {\mathrm{m}}^{*}$ interaction. Also shown are the optical potential predicted by the chiral effective theory [35] and the empirical optical potential from Hama et al. [36, 37].

Figure 2

Momentum distributions of neutron (left panel), ${\mathrm{\Delta }}^{-}$ (middle panel), and ${\pi }^{-}$ (right panel) in a box at $t=50$ fm/$c$. Solid lines are theoretical momentum distributions from the thermal model with the inclusion of baryon potentials at $T=60$ MeV, and dashed lines are those in the final state of the “free” case (see text for details). Solid circles are results from the $\chi \mathrm{BUU}$ model including the effect of baryon potentials in the energy conservation condition in elementary reactions ($\mathrm{Sk}\chi {\mathrm{m}}^{*}$), and open circles are those without including this effect (free).

Figure 3

Time evolutions of $\mathrm{\Delta }$ numbers in box calculations for the two cases of with and without mean-field potentials in the energy conservation condition of scattering and decay processes. For comparison, results from the thermal model are shown as open cycles.

Figure 4

Same as Fig. 3 for $\pi$ numbers.

Figure 5

Time evolutions of the central density ${\rho }_c$ and effective ${\pi }^{-}$ and ${\pi }^{+}$ numbers in Au + Au collisions at impact parameters of 1 fm and incident energy of $E/A=400$ MeV. The mean-field potential effects in the collision terms are not included.

Figure 6

${\pi }^{-}/{\pi }^{+}$ ratios as functions of pion kinetic energy from central ($b=3$ fm) collisions of ${}^{132}\mathrm{Sn}+{}^{124}\mathrm{Sn}$ and ${}^{108}\mathrm{Sn}+{}^{112}\mathrm{Sn}$ at the incident energy of $E/A=270$ MeV. ${\theta }_{\mathrm{cm}}$ is the polar angle of pion momentum relative to the incident beam direction.

Figure 7

Same as Fig. 6 for double $[{({\pi }^{-}/{\pi }^{+})}_{132+124}/{({\pi }^{-}/{\pi }^{+})}_{108+112}]$ and subtracted $[{({\pi }^{-}/{\pi }^{+})}_{132+124}-{({\pi }^{-}/{\pi }^{+})}_{108+112}]$ ${\pi }^{-}/{\pi }^{+}$ ratios.

Figure 8

Same as Fig. 6 for isoscaling $[{\pi }_{132+124}/{\pi }_{108+112}]$ ratios of ${\pi }^{-}$ and ${\pi }^{+}$.