In the 80’s of last century, pion production has been used as a useful tool to probe the nuclear equation of state (EOS).^[1–4]From then on, numerous experimental measurements^[5–20]and theoretical studies^[21–48]have been devoted to understanding pion production mechanisms. In particular, the ratio of negatively to positively charged pions π⁻/π⁺ in heavy-ion collisions induced by neutron-rich nuclei at energies near pion threshold was proposed to extract the information of the nuclear symmetry energy E_sym(ρ).^[28] However, unfortunately, the predicted form for the E_sym(ρ) by using the ratios of π⁻/π⁺ is highly controversial.^[30–34,37]

The EOS of nuclear matter at density ρ can be written as E(ρ,δ) = E(ρ,δ = 0) + E_sym(ρ)δ² + O(δ⁴) in terms of δ = (ρ_n − ρ_p/ρ) with ρ, ρ_n, and ρ_p representing the total, neutron, and proton densities, respectively. Until now the EOS of the isospin symmetric part E(ρ,δ = 0) is constrained quite acceptably,^[49] but the EOS of isospin asymmetric nuclear matter, namely the E_sym(ρ), is not yet constrained. In the past decades, a lot of probes to the E_sym(ρ) have been found. Among them, the ratios of π⁻/π⁺ is one of the most controversial probes. Many inconsistent results have been obtained by comparing the theoretical simulations to the experimental data taken by the FOPI collaboration for pion production.^[31–34]The in-depth understanding about the pion production mechanisms and the more precise experimental data on the pion production are required.

In theory, there are many aspects to influence the simulated results of π⁻/π⁺, such as the pion in-medium potentials,^{[34,38,42–46]} the isosvector potentials for resonances Δ,^[35] threshold effects,^[39] nucleon short-range correlation,^[36,48] and so on. The absorption reaction NΔ → NN and resorption reaction πN → Δ play a significant role in discussing the dependence of the π⁻/π⁺ on the E_sym(ρ). However, this point is hardly stressed in above theoretical studies. In heavy-ion collisions induced by neutron-rich nuclei at intermediate energies, the different cross sections for the two channels will correspond to the completely different ratios of π⁻/π⁺. For the cross section of NΔ → NN, the simplest form is derived from the detailed balance principle, which is considered to be correct only for resonances with zero width.^[27] Thus a modified detailed balance formula was proposed in order to take the finite width of resonances into account. It is found that it enhances the reabsorption of pions, particularly near pion threshold value.^[13,21] For the cross section of πN → Δ, there are several different cases used in theoretical approaches, e.g., for reaction π⁻p → Δ⁰, where p denotes protons, the maximum value of the cross section σ_max = 70 mb is used in Refs. [26, 50], σ_max = 30 mb is used in Refs. [51--52], and σ_max = 66.7 mb is used in Ref. [53]. It is necessary to study the impact of the reactions of NΔ → NN and πN → Δ on the final simulated results for pion production. It is also important for sheding light on the sensitivity of the ratios of π⁻/π⁺ on the E_sym(ρ).

In this work, within the isospin-dependent quantum molecular model developed in the Beijing Normal University (IQMD-BNU),^[54–55]the roles of NΔ → NN and πN → Δ reactions in heavy-ion collisions at intermediate energies are investigated. The article is organized as follows: In Sec. 2, we give a description of the recent version of the IQMD-BNU model. The calculated results are shown and discussed in Sec. 3. In Sec. 4, conclusions are given.

In the QMD approach, the nucleons are specified as particles with finite widths of Gaussian shape instead of point particles. The width of the wave packet is fixed in the evolution and increases as the mass number of the nuclide increases. The final N-body wave function is considered as the direct product of the coherent states and the antisymmetrization is not taken into account. The time evolution of the baryons, including nucleons and resonances, and mesons in the system obeys the Hamiltonian equations of motion:
where the Hamiltonian H consists of the relativistic energy, the Coulomb, the local, and the non-local potential energies. That is,
Here . The Coulomb interaction potential energy is
where e_i is the charged number of the i-th baryon or meson. r_ij = |r_i − r_j| denotes the relative distance between two charged particles. The local potential energy is written as
with

The parameters α, β, γ, and ρ₀ are taken as −390 MeV, 320 MeV, 1.14, and 0.16 fm⁻³, respectively, and the corresponding compressibility is 200 MeV. g_sur is taken as 130 MeV⋅fm⁵ and δ is the isospin asymmetry. represents the local (density-dependent) part of the symmetry energy and is expressed as
with γ_s = 0.5, 1.0, and 2.0 corresponding to the soft, linear, and hard symmetry energies, respectively. The linear symmetry energy is used in the present work. The non-local (momentum-dependent) potential energy is written as^[56]
where t₅ = 0.0005 MeV⁻², C_τ,τ = t₄(1 − x), and C_τ,τ′ = t₄(1 + x) with t₄ = 1.76 MeV. The parameters x, C_sym are taken as the values of −0.65, 52.5 MeV, and 0.65, 23.52 MeV, corresponding to the positive and negative mass splittings, respectively.^[56] In this work, as the studies in Refs. [31, 57], the positive mass splitting is taken since the sign of nucleon effective mass splitting is still uncertain so far.^[58]

We use the same method as in the cascade model to treat two-particle collisions.^[59] The elastic cross sections in free space, which is taken from Ref. [60], and the inelastic cross section, which is derived from the one-boson exchange model,^[61] with an in-medium factor of C = 1 − 0.2 ρ/ρ₀,^[62] are used in the present work.

The pion is generated through the decay of resonances Δ and N*(1440). The spectral function of resonances is obtained by a modified Breit-Wigner function:^[25]
with m_r and Γ_r being respectively the centroid and width of the resonance. p_f is the final-state momentum for NN → NΔ channel in center-of-mass system. The decay width for resonances is expressed as
Here p is the pion momentum in the resonance rest system and is given in units of GeV. The parameters A, B, and C is taken as 22.48 (17.22), 39.69 and 0.04 (0.09) for the Δ (N*) with the free decay width Γ₀ = 0.12 GeV (0.2 GeV). For the resonance absorption, a modified detailed balance expression is used and written as^[25]
where p_NN and p_NΔ are the initial and final momenta, respectively, for the channel NN → NΔ in center-of-mass system, and f(M) is calculated according to the following expression^[25]
The resonance decay is performed by using a Monte Carlo sampling method according to the probability^[35]
with dt = 0.5 fm/c representing the temporal interval of each step and ℏ the Planck constant. Both the elastic and inelastic pion-nucleon channels are incorporated into the IQMD-BNU model. The cross sections for pion-nucleon inelastic channels are written as^[52]
where p₀ is the pion momentum at the centroid m_r of the resonance mass distribution and m is the mass of produced resonance. The maximum cross sections σ_max are taken as follows^[52]
The pion-nucleon potential is not considered in the present work and the force suffered by pions in the whole stage is just the Coulomb force.

To test our model, firstly, we compare the calculated results with the corresponding experimental data taken by the FOPI collaboration.^[15,18] Figure 1 shows the rapidity distributions of π⁺ produced in central Au+Au collisions at 400 and 1500 MeV/nucleon. The FOPI data and the IQMD SM results are taken from Ref. [18], where the SM denotes the soft EOS plus the momentum-dependent interactions. It is found that our results overestimate the experimental data but are close to the IQMD SM results. Shown in Fig. 2 is the transverse momentum distributions of π⁻ produced in Au+Au collisions at 400 and 1500 MeV/nucleon at midrapidity with the corresponding results of FOPI, which is taken from Ref. [15]. Our results describe well the experimental data at 400 MeV/nucleon but overestimate the data in larger transverse momentum region at 1500 MeV/nucleon.

Fig. 1

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	Fig. 1 Rapidity distributions of π+ produced in Au+Au collisions at 400 (a) and 1500 MeV/nucleon (b). The FOPI data and the IQMD SM results are taken from Ref. [18].

Fig. 2

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	Fig. 2 The same as in Fig. 1, but for the transverse momentum distributions of π−. The FOPI data are taken from Ref. [15].

In order to shed light on the impact of the reactions NΔ → NN and πN → Δ on the pion production dynamics, the relative differences of the kinetic energy and rapidity spectra are used, that is,

Here (dN/dE_k)_{(with NΔ → NN)} − (dN/dE_k)_{(w/o NΔ → NN)} means the difference between the kinetic energy distribution with NΔ → NN and those without NΔ → NN. The similar expressions are taken for the channel of πN → Δ. For the reaction of NΔ → NN, we do that through neglecting the channel NΔ → NN. Figure 3 represents the impacts of NΔ → NN on the kinetic energy and rapidity spectra of π⁰ produced in ¹³²Sn + ¹²⁴Sn collisions at 1000 MeV/nucleon with impact factors of b = (1, 3, 5, and 7) fm. One can see that the impacts of NΔ → NN on the kinetic energy and rapidity spectra of π⁰ become weaker as the impact parameter increases. One can neglect the influence of the reaction of NΔ → NN on pion observables in semicentra1 or peripheral collisions.

Fig. 3

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	Fig. 3 Impacts of the reaction NΔ → NN on the kinetic energy (a) and rapidity (b) spectra of π0 produced in 132Sn + 124Sn collisions at 1000 MeV/nucleon with b = (1, 3, 5, and 7) fm.

To further explore the impacts of the reaction NΔ → NN on pion observables, in Fig. 4, the influences of NΔ → NN on the kinetic energy and rapidity spectra of π⁰ from different reaction systems as indicated are shown. The results are obtained by simulating head-on ⁹⁶Ru+⁹⁶Ru and ¹⁹⁷Au+¹⁹⁷Au collisions at 1000 MeV/nucleon. It is found that the impact is stronger for the system of ¹⁹⁷Au+¹⁹⁷Au than that of ⁹⁶Ru+⁹⁶Ru as indicated. One may neglect the influence the reaction of NΔ → NN on pion observables in lighter system, e.g. in Ca+Ca collisions.

Fig. 4

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	Fig. 4 Impacts of the reaction NΔ → NN on the kinetic energy (a) and rapidity (b) spectra of π0 produced in different reaction systems as indicated.

Shown in Fig. 5 is the impacts of the resorption reaction of πN → Δ in head-on ⁹⁶Ru+⁹⁶Ru, ¹⁰⁸Sn+¹⁰⁸Sn, ¹³²Sn+¹³²Sn and ¹⁹⁷Au+¹⁹⁷Au collisions at 1000 MeV/nucleon. It is found, the same as that of NΔ → NN, its influences become stronger as the reaction system becomes heavier. However, as shown in Figs. 6 and 7, contrary to the case of the reaction NΔ → NN, the influences of the reaction πN → Δ on the kinetic energy and rapidity distributions, and on the rapidity distributions of the directed flow v₁(y₀), which is given by v₁ = p_x/p_t with , become larger when the collision parameter becomes larger. From Fig. 7, one can see that the pion flow shows the opposite sign as the nucleon flow. The anticorrelation of pions and nucleons can be explained by the shadowing effect.^[22,24] Instead of the reaction of NΔ → NN, it is the reaction of πN → Δ that causes the shadowing effect, as shown in Fig. 8.

Fig. 5

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	Fig. 5 The same as in Fig. 4, but for the impacts of the resorption reaction of πN → Δ.

Fig. 6

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	Fig. 6 The same as in Fig. 3, but for the impacts of the resorption reaction of πN → Δ at b = 1 fm and 3 fm.

Fig. 7

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	Fig. 7 Impacts of the reaction πN → Δ on the rapidity spectra of the directed flow for π0 produced in 132Sn + 124Sn collisions at 1000 MeV/nucleon with b = 5 fm and 7 fm.

Fig. 8

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	Fig. 8 Impacts of the reactions NΔ → NN and πN → Δ on the rapidity spectra of the directed flow for π0 produced in 132Sn + 124Sn collisions at 1000 MeV/nucleon with b = 5 fm.

In summary, within the IQMD-BNU model, in which the inelastic channels to generate pions are incorporated, the impacts of absorption and resorption processes on pion production are investigated through simulating heavy-ion collisions with various impact parameters. It is found that our model overestimate the experimental data on the rapidity distribution of π⁺ and the transverse momentum distribution of π⁻ produced in Au+Au collisions at 400 and 1500 MeV/nucleon. Moreover, our results are identical with those from other theoretical calculations. The absorption process NΔ → NN is more important for central nucleus-nucleus collisions and heavy collision systems than for semicentral or peripheral nucleus-nucleus collisions and light systems. The resorption process πN → Δ is of lesser significance for light systems and nucleus-nucleus collisions with small impact parameters.