Comparing pion production in transport simulations of heavy-ion collisions at $270A\mathrm{MeV}$ under controlled conditions

Within the Transport Model Evaluation Project (TMEP), we present a detailed study of the performance of different transport models in $\text{Sn}+\text{Sn}$ collisions at $270A\mathrm{MeV}$, which are representative reactions used to study the equation of state at suprasaturation densities. We put particular emphasis on the production of pions and $\mathrm{\Delta }$ resonances, which have been used as probes of the nuclear symmetry energy. In this paper, we aim to understand the differences in the results of different codes for a given physics model to estimate the uncertainties of transport model studies in the intermediate energy range. Thus, we prescribe a common and rather simple physics model, and follow in detail the results of four Boltzmann-Uehling-Uhlenbeck (BUU) models and six quantum molecular dynamics (QMD) models. The nucleonic evolution of the collision and the nucleonic observables in these codes do not completely converge, but the differences among the codes can be understood as being due to several reasons: the basic differences between BUU and QMD models in the representation of the phase-space distributions, computational differences in the mean-field evaluation, and differences in the adopted strategies for the Pauli blocking in the collision integrals. For pionic observables, we find that a higher maximum density leads to an enhanced pion yield and a reduced ${\pi }^{-}/{\pi }^{+}$ yield ratio, while a more effective Pauli blocking generally leads to a slightly suppressed pion yield and an enhanced ${\pi }^{-}/{\pi }^{+}$ yield ratio. We specifically investigate the effect of the Coulomb force and find that it increases the total ${\pi }^{-}/{\pi }^{+}$ yield ratio but reduces the ratio at high pion energies, although differences in its implementations do not have a dominating role in the differences among the codes. Taking into account only the results of codes that strictly follow the homework specifications, we find a convergence of the codes in the final charged-pion yield ratio to a $1\sigma$ deviation of about $5\%$. However, the uncertainty is expected to be reduced to about $1.6\%$ if the same or similar strategies and ingredients, i.e., an improved Pauli blocking and calculation of the nonlinear term in the mean-field potential, are similarly used in all codes. As a result of this work, we identify the sensitive aspects of a simulation with respect to pion observables, and suggest optimal procedures in some cases. This work provides benchmark calculations of heavy-ion collisions to be complemented in the future by simulations with more realistic physics models, which include the momentum-dependence of isoscalar and isovector mean-field potentials and pion in-medium effects.

Figure 1

(left) Distributions of centroid coordinates for neutrons and protons in the radial direction of ${}^{132}\mathrm{Sn}$ and ${}^{124}\mathrm{Sn}$. (right) Distributions of centroid rapidities for neutrons and protons in the initial stage of a ${}^{132}\mathrm{Sn}+{}^{124}\mathrm{Sn}$ collision.

Figure 2

Contours of reduced densities $\rho /{\rho }_0$ in the $x$-0-$z$ plane at different indicated times in the Full-nopb mode.

Figure 3

Time evolution of reduced densities $\rho /{\rho }_0$ (left) and asymmetries ${\rho }_n/{\rho }_p$ (right) in the central region of a volume $5\times 5\times 5{\mathrm{fm}}^3$ in the Full-nopb mode. Upper panels are for physical densities calculated from finite-size (test) particles, while lower panels are for centroid densities calculated from point (test) particles. Asymmetries are only plotted up to 40 fm/$c$, since beyond this they fluctuate strongly.

Figure 4

Final nucleon rapidity distribution (left) and transverse flow (right) in the Cascade mode, without Pauli blocking (upper) and with Pauli blocking (lower).

Figure 5

Final nucleon rapidity and transverse flow distributions as in Fig. 4 but in the Full mode, in the top and middle rows without Pauli blocking, and in the bottom row with Pauli blocking. The top row shows results for BUU models using different TP shapes and numbers, the middle row shows results for QMD models with different widths of the wave packet, and the bottom row shows results for all codes in their standard implementations and with standard parameters.

Figure 6

Successful and attempted rate of $NN\to N\mathrm{\Delta }$ reactions and the net $\mathrm{\Delta }$ production rate as a function of time and C.M. energy in the Cascade [(a)–(d)] and Full [(e)–(h)] mode with Pauli blocking.

Figure 7

C.M. energy dependence of net $\mathrm{\Delta }$ production at different time slots in the Cascade (left) and Full (right) mode for ibuu, pbuu, iqmd, and iqmd-bnu. The net production summed over time is shown as a solid line.

Figure 8

(left) Pauli blocking factor for $NN\to N\mathrm{\Delta }$ (upper) and $N\mathrm{\Delta }\to NN$ (lower) reactions as a function of C.M. energy in the Cascade [(a), (c)] and Full [(b), (d)] mode. (right) Pauli blocking factor for $nn\to p{\mathrm{\Delta }}^{-}$ (upper) and $pp\to n{\mathrm{\Delta }}^{++}$ (lower) reactions as a function of C.M. energy in the Cascade [(e), (g)] and Full [(f), (h)] mode.

Figure 9

(upper) Successful and attempted rates of $\mathrm{\Delta }\to N\pi$ decays as a function of C.M. energy in the Cascade (left) and Full (right) mode. (lower) Pauli blocking factor for $\mathrm{\Delta }\to N\pi$ decays as a function of C.M. energy in the Cascade (left) and Full (right) mode.

Figure 10

Time evolution of the multiplicities of pion-like particles in different scenarios (see legend and text).

Figure 11

Final pion multiplicities in different scenarios (see legend and text).

Figure 12

Pion kinetic spectra in the Full modes with Pauli blocking.

Figure 13

${\pi }^{-}/{\pi }^{+}$ yield ratios in different scenarios (see legend and text).

Figure 14

Correlation between the ${\pi }^{-}/{\pi }^{+}$ yield ratio and the average asymmetry ratio ${\rho }_n/{\rho }_p$ for centroid densities in the central region of ${}^{132}\mathrm{Sn}+{}^{124}\mathrm{Sn}$ collisions in the time period of $20<t<30\mathrm{fm}/c$ in the Full-nopb mode. For smf and imqmd-l, which do not calculate pions, the asymmetry ranges are indicated by symbols outside the frame.

Figure 15

Double ${\pi }^{-}/{\pi }^{+}$ yield ratios in different scenarios (see legend and text).

Figure 16

(left) ${\pi }^{-}/{\pi }^{+}$ kinetic spectra in the Full modes with Pauli blocking. (right) Double ${\pi }^{-}/{\pi }^{+}$ kinetic spectra ratio in the Full modes with Pauli blocking.

Figure 17

${\pi }^{-}/{\pi }^{+}$ yield ratios for energetic pions ($E_k>0.1$ GeV) from Full-mode calculations.