Introduction

The equation of state (EOS) of nuclear matter, which originates from the interactions between nuclear matter, plays an important role in nuclear physics and astrophysics. Heavy-ion collisions, the properties of nuclei, and neutron stars (NSs) have been widely studied to extract the nuclear EOS. Because nuclear many-body problems are highly nonlinear and the EOS is not a directly observable quantity in experiments, there are still some uncertainties in the EOS despite great efforts [1-6]. For instance, the EOS extracted from the GW170817 event has uncertainties at high nuclear densities [2]. Although the EOS can be extracted from the properties of NSs, the internal composition of NSs is still poorly understood. The core of an NS may contain exotic materials such as hyperons, kaons, pions, and deconfined quark matter [7-12]. Heavy-ion collisions in terrestrial laboratories provide a unique opportunity to study both the EOS and exotic materials.

Collective flows of heavy-ion collisions were proposed in the 1970s and first detected in experiments at Bevalac [13-16]. Because collective flows are associated with nucleon-nucleon interactions, nucleon-nucleon scattering, etc., collective flows have been used to extract the nuclear EOS [1]. Collective flows are also helpful for understanding the phase transition between hadronic and quark matter. Generally, when a phase transition between hadronic and quark matter occurs, the collective flows of heavy-ion collisions indicate a soft EOS [17-21]. In addition, the ratios of the isospin particles in heavy-ion collisions, such as π^-/π⁺, K⁰/K⁺, and ∑^-/∑⁺, are thought to be sensitive to the stiffness of the EOS [22-28]. The production of pions and kaons has been experimentally measured in ¹⁹⁷Au+¹⁹⁷Au collisions. The K⁺ production predicted by various transport models favors a soft EOS of isospin-symmetric nuclear matter at high baryon densities [29-33]. The π^-/π⁺ ratio predicted by various transport models is still model-dependent [34-37]. Based on the FOPI data for the π^-/π⁺ ratio [38], some results favor a stiff symmetry energy (isospin asymmetric part of the EOS) [34, 35], whereas others imply a soft symmetry energy [36, 37]. Recently, by analyzing the ratios of charged pions in ¹³²Sn + ¹²⁴Sn, ¹¹²Sn +¹²⁴Sn, and ¹⁰⁸Sn + ¹¹²Sn collisions [39], a slope of the symmetry energy ranging from 42 to 117 MeV was suggested [40, 41]. The collective flows and ratios of charged pions are still worth studying to find the sources of the difference in various transport models and to extract information about the EOS from heavy-ion collisions.

Quantum molecular dynamics (QMD) is a popular transport model that has been developed into many versions and used to successfully describe the properties of nuclear matter, nuclei, mesons, and hyperons [33, 34, 42-55]. In high-energy heavy-ion collisions, the relativistic effects should be considered in QMD because they become significant. The RQMD approach has been proposed for this purpose [42, 43]. Recently, relativistic mean field theory (RMF) was implemented in RQMD (RQMD.RMF) [44-46]. The RQMD.RMF has been successfully applied to investigate the collective flows of hadrons [44-46]. In this study, we implemented RMF theory with isovector-vector and isovector-scalar fields in the Lanzhou quantum molecular dynamics model (LQMD.RMF). The channels for the generation and decay of resonances (Δ(1232), N*(1440), N*(1535), etc.), hyperons, and mesons were included in the previous LQMD model [33, 34, 47-50]. Using the LQMD.RMF, we explored the relationship between the symmetry energy and the properties of the collective flow and pion production.

The remainder of this paper is organized as follows. In Sect. 2, we briefly introduce the formulas and approaches used in this study. The formulas include RMF theory, the dispersion relation, and the production of pions. The results and discussion are presented in Sect. 3. Finally, a summary is presented in Sect. 4.

Formalism

2.1

Relativistic mean field approach

The RMF interaction is achieved by exchanging mesons. Scalar and vector mesons provide medium-range attraction and short-range repulsion between nucleons, respectively [56]. The nonlinear self-interaction of the σ meson is introduced to reduce the incompressibility to a reasonable domain [57]. To investigate the properties of symmetry energy, we also consider the isovector-vector ρ [58] and isovector-scalar δ mesons [59]. The Lagrangian density is expressed as [59, 60] $\begin{array}{c}\mathcal{L}=\overline{\psi }[{\gamma }_{\mu }\left(i{\partial }^{\mu }-g_{\omega }{\omega }^{\mu }-g_{\rho }\text{τ}\cdot {\text{b}}^{\mu }\right)-(M_{\text{N}}-g_{\sigma }\sigma \\ -g_{\delta }\text{τ}\cdot \delta )]\psi +\frac{1}{2}\left({\partial }_{\mu }\sigma {\partial }^{\mu }\sigma -m_{\sigma }^2{\sigma }^2\right)-\frac{1}{3}{g}_2{\sigma }^3\\ -\frac{1}{4}{g}_3{\sigma }^4+\frac{1}{2}{m}_{\omega }^2{\omega }_{\mu }{\omega }^{\mu }-\frac{1}{4}{F}_{\mu \nu }{F}^{\mu \nu }+\frac{1}{2}{m}_{\rho }^2{\text{b}}_{\mu }{\text{b}}^{\mu }\\ -\frac{1}{4}{\text{B}}_{\mu \nu }{\text{B}}^{\mu \nu }+\frac{1}{2}\left({\partial }_{\mu }\delta \cdot {\partial }^{\mu }\delta -m_{\delta }^2{\delta }^2\right)\mathrm{,}\end{array}$ (1) where M_N=938 MeV is the nucleon mass in free space, gi with $i=\sigma \mathrm{,}\omega \mathrm{,}\rho \mathrm{,}\delta$ is the coupling constant between nucleons, mi with $i=\sigma \mathrm{,}\omega \mathrm{,}\rho \mathrm{,}\delta$ denotes the meson mass, g₂ and g₃ are the coupling constants for the nonlinear self-interaction of the σ meson, and $F_{\mu \nu }={\partial }_{\mu }{\omega }_{\nu }-{\partial }_{\nu }{\omega }_{\mu }$ and ${\text{B}}_{\mu \nu }={\partial }_{\mu }{\text{b}}_{\nu }-{\partial }_{\nu }{\text{b}}_{\mu }$ are the strength tensors of the ω and ρ mesons, respectively. The equations of motion for the nucleon and meson are obtained from the Euler–Lagrange equations and are written as: $\begin{array}{l}[i{\gamma }^{\mu }{\partial }_{\mu }-g_{\omega }{\gamma }^0{\omega }_0-g_{\rho }{\gamma }^0{b}_0{\tau }_3\\ -\left(M_N-g_{\sigma }\sigma -g_{\delta }{\tau }_3{\delta }_3\right)]\psi =0,\end{array}$ (2) $m_{\sigma }^2\sigma +g_2{\sigma }^2+g_3{\sigma }^3=g_{\sigma }\overline{\psi }\psi =g_{\sigma }{\rho }_S,$ (3) $m_{\omega }^2{\omega }_0=g_{\omega }\overline{\psi }{\gamma }^0\psi =g_{\omega }\rho ,$ (4) $m_{\rho }^2{b}_0=g_{\rho }\overline{\psi }{\gamma }^0{\tau }_3\psi =g_{\rho }{\rho }_3\mathrm{,}$ (5) $m_{\delta }^2{\delta }_3=g_{\delta }\overline{\psi }{\tau }_3\psi =g_{\delta }{\rho }_{S3}\mathrm{,}$ (6) where ρ and ρ_S are the baryon and scalar densities, respectively, ${\rho }_3={\rho }_{\text{p}}-{\rho }_{\text{n}}$ is the difference between the proton and neutron densities, and ${\rho }_{S3}={\rho }_S{}_{\text{p}}-{\rho }_S{}_{\text{n}}$ is the difference between the proton and neutron scalar densities.

In the RMF approximation, the energy density is given by $\begin{array}{c}ϵ={\displaystyle \sum _{i=\text{n}\mathrm{,}\text{p}}2}{\displaystyle {\int }_0^{p_{\text{F}}}}\frac{d^{3}p}{{\left(2\pi \right)}^3}\sqrt{p^2+M_i^{*2}}+\frac{1}{2}{m}_{\sigma }^2{\sigma }^2+\frac{1}{3}{g}_2{\sigma }^3\\ +\frac{1}{4}{g}_3{\sigma }^4+\frac{1}{2}{m}_{\omega }^2{\omega }_0^2+\frac{1}{2}{m}_{\rho }^2{b}_0^2+\frac{1}{2}{m}_{\delta }^2{\delta }_3^2\mathrm{,}\end{array}$ (7) where p_F denotes the nucleon Fermi momentum, and $M_i^{*}=M_{\text{N}}-g_{\sigma }\sigma \mp g_{\delta }{\delta }_3$ (- proton, + neutron) denotes the effective nucleon mass. The isospin splitting of the effective nucleon mass $M_{\text{n}}^{*}-M_{\text{p}}^{*}$ still has a large uncertainty at this point. Analyses of nucleon-nucleus scattering data based on the optical model favor $M_{\text{n}}^{*}-M_{\text{p}}^{*}>0$ [61, 62]. Calculations based on Brueckner theory [63-65] and density-dependent relativistic Hartree-Fock [66] also indicate $M_{\text{n}}^{*}-M_{\text{p}}^{*}>0$. However, $M_{\text{n}}^{*}-M_{\text{p}}^{*}<0$ is predicted by the transport model for heavy-ion collisions [67, 68] and nonlinear RMF models [59, 60, 69]. In addition, both $M_{\text{n}}^{*}-M_{\text{p}}^{*}<0$ and $M_{\text{n}}^{*}-M_{\text{p}}^{*}>0$ can be predicted by the point-coupling RMF [69] and Skyrme and Gogny forces [70-76]. Because the Lagrangian density in this study is the same as that in Ref. [59] and [60] except for the parameter settings, as shown in Fig. 1, the negative isospin splitting of the effective nucleon mass $M_{\text{n}}^{*}-M_{\text{p}}^{*}<0$ is consistent with that in Ref. [59] and [60]. In the nonlinear RMF model, the isospin splitting of the effective nucleon mass is primarily determined by the coupling strength of the δ meson. The large coupling strength of the δ meson results in large isospin splitting of the effective nucleon mass. When the coupling strength of the δ meson is zero (set1), there is no isospin splitting of the effective nucleon mass.

Fig. 1Full-screen View

(Color online) Difference between neutron and proton effective masses as a function of the baryon density

Using the isospin asymmetry parameter $\alpha =\left({\rho }_{\text{n}}-{\rho }_{\text{p}}\right)/\left({\rho }_{\text{n}}+{\rho }_{\text{p}}\right)$, the symmetry energy can be written as [59] $\begin{array}{c}{E}_{\text{sym}}=\frac{1}{2}\frac{{\partial }^{2}E\left(\rho \mathrm{,}\alpha \right)}{\partial {\alpha }^2}{|}_{\alpha =0}=\frac{1}{2}\frac{{\partial }^2\left[ϵ\left(\rho \mathrm{,}\alpha \right)/\rho \right]}{\partial {\alpha }^2}{|}_{\alpha =0}\\ =\frac{1}{6}\frac{p_F^2}{E_{\text{F}}^{*}}+\frac{1}{2}{f}_{\rho }\rho -\frac{f_{\delta }}{2}\frac{M^{*2}\rho }{E_{\text{F}}^{*2}\left[1+f_{\delta }A\left(p_{\text{F}}\mathrm{,}{M}^{*}\right)\right]}\mathrm{,}\end{array}$ (8) where $f_i\equiv \frac{g_i^2}{m_i^2}$, i=ρ,δ, and $E_{\text{F}}^{*}=\sqrt{p_{\text{F}}^2+M^{*2}}$ and $M^{*}=M_{\text{N}}-g_{\sigma }\sigma$ are the effective nucleon masses of the symmetric nuclear matter. The integral $A\left(p_{\text{F}}\mathrm{,}{M}^{*}\right)$ is defined as $\begin{array}{c}A\left(p_{\text{F}}\mathrm{,}{M}^{*}\right)=\frac{4}{{\left(2\pi \right)}^3}{\displaystyle \int }{d}^{3}p\frac{p^2}{{\left(p^2+M^{*2}\right)}^{3/2}}\\ =3\left(\frac{{\rho }_S}{M^{*}}-\frac{\rho }{E_{\text{F}}^{*}}\right)\mathrm{.}\end{array}$ (9) Based on the symmetry energy E_sym, the slope L and curvature K_sym of the symmetry energy at the saturation density ρ₀ can be obtained as $\begin{array}{l}L=3{\rho }_0\left(\frac{\partial E_{\text{sym}}}{\partial \rho }\right){|}_{\rho ={\rho }_0}\\ K_{\text{sym}}=9{\rho }_0^2\left(\frac{{\partial }^2{E}_{\text{sym}}}{\partial {\rho }^2}\right){|}_{\rho ={\rho }_0}\mathrm{.}\end{array}$ (10) In this study, we set the saturation density to ${\rho }_0=0.16{\text{fm}}^{-3}$, and the binding energy per particle of the symmetric nuclear matter was set to E/A-M_N=-16 MeV. For symmetric nuclear matter, we set set1, set2, and set3 models to be the same as the result of vanishing isospin asymmetry. As shown in Table 1 and Fig. 2, the symmetry energy of set1, set2, and set3 was set to be 31.6 MeV at the saturation density. Set1 contained only ρ mesons; however, set2 and set3 contained both the ρ and δ mesons. For set1, when the symmetry energy was set to be 31.6 MeV at the saturation density, the coupling parameter gρ was fixed and the slope of symmetry was fixed at L=85.3 MeV. For set2 and set3, when the symmetry energy was fixed at 31.6 MeV, the slope of the symmetry energy was obtained by varying the coupling parameters gρ and gδ. Because the effective mass M^* of the above models for symmetric nuclear matter was the same, the symmetry energy with both the ρ and δ mesons could not be softer than that of set1 containing only ρ mesons. To broaden the range of the slope parameters, we set the slope parameter of set2 and set3 to be 109.3 and 145.0 MeV by varying the coupling parameters gρ and gδ, respectively. A broader range of slope parameters would be helpful for understanding the relationship between the properties of symmetry energy and the observables of heavy-ion collisions. The curvature of the symmetry energy K_sym, which is a higher-order expansion coefficient of the symmetry energy compared to the slope parameter L, may also affect the properties of the symmetry energy and the observables of heavy-ion collisions at densities far beyond the saturation density. K_sym of set1, set2, and set3 is obtained as -15, 141, and 391 MeV, respectively.

Table 1Full-screen View

Parameter sets for RMF

Model	$g_{\sigma }$	gω	g2 (fm-1)	g3	gρ	gδ	K (MeV)	Esym(ρ0) (MeV)	L (ρ0) (MeV)	$M^{*}\left({\rho }_0\right)/M_{\text{N}}$	Ksym (MeV)
Set1	8.145	7.570	31.820	28.100	4.049	-	230	31.6	85.3	0.81	-15
Set2	8.145	7.570	31.820	28.100	8.673	5.347	230	31.6	109.3	0.81	141
Set3	8.145	7.570	31.820	28.100	11.768	7.752	230	31.6	145.0	0.81	391

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The saturation density ρ₀ is set to 0.16 fm^-3. The binding energy of the saturation density is E/A-M_N=-16 MeV. The isoscalar-vector ω and isovector-vector ρ masses are fixed at their physical values, mω = 783 MeV and mρ = 763 MeV, respectively. The remaining meson masses, $m_{\sigma }$ and mδ, are set to be 550 and 500 MeV, respectively

Fig. 2Full-screen View

(Color online) Symmetry energy as a function of the baryon density

Results and discussions

The directed and elliptic flows were derived from the Fourier expansion of the azimuthal distribution: $\begin{array}{c}\frac{\text{d}N}{\text{d}\varphi }\left(y\mathrm{,}{p}_{\text{T}}\right)=N_0[1+2V_1\left(y\mathrm{,}{p}_{\text{T}}\right)\text{cos}\left(\varphi \right)\\ +2V_2\left(y\mathrm{,}{p}_{\text{T}}\right)\text{cos}\left(2\varphi \right)]\mathrm{,}\end{array}$ (36) where the azimuthal angle of the emitted particle ϕ was measured from the reaction plane. $p_{\text{T}}=\sqrt{p_x^2+p_y^2}$ denotes the transverse momentum, and the directed flow V₁ and elliptic flow V₂ are expressed as follows: $\begin{array}{l}{V}_1\equiv \langle \text{cos}(\varphi )\rangle =\langle \frac{p_x}{p_T}\rangle \mathrm{,}\\ V_2\equiv \langle \text{cos}(2\varphi )\rangle =\langle \frac{p_x^2-p_y^2}{p_T^2}\rangle \mathrm{.}\end{array}$ (37) The directed flow provides information on the azimuthal anisotropy of the transverse emission. The elliptic flow describes the competition between the in-plane (V_2>0) and out-of-plane (V_2<0) emissions.

Firstly, the collective flows of LQMD.RMF in the ¹⁹⁷Au + ¹⁹⁷Au collisions have been investigated at an incident energy of 2.92 GeV/nucleon (the corresponding nucleon-nucleon center-of-mass energy is $\sqrt{S_{NN}}=3$ GeV). The collective flows of LQMD with Skyrme interaction and without the momentum-dependent interaction have also been investigated. The Skyrme interaction of symmetric nuclear matter is taken to be the same as that of SLy6, with an incompressibility of K=230 MeV at ${\rho }_0=0.16{\text{ fm}}^{-3}$[92, 93]. The symmetry energy of Skyrme interaction is defined as $E_{\text{sym}}=\frac{1}{3}\frac{{\hslash }^2}{2M_{\text{N}}}{\left(\frac{3{\pi }^2\rho }{2}\right)}^{2/3}+\frac{C_{\text{sym}}}{3}{\left(\rho /{\rho }_0\right)}^{{\gamma }_{\text{s}}}$. When the C_sym and γ_s are set to be 38.6 MeV and 1.049, respectively, the symmetry energy and the slope parameter of symmetry energy are 31.6 MeV and 85.3 MeV, respectively. The symmetry energy and the slope parameter of symmetry energy of the Skyrme interaction are as same as those of set1. As shown in Fig. 3, we have compared the collective flows of LQMD.RMF and LQMD with Skyrme interaction with recent experimental data from STAR Collaboration [94]. The collective follows of LQMD.RMF and LQMD with Skyrme interaction can describe the STAR data at an impact parameter b=4 fm and b=7 fm, respectively. The directed flows of LQMD.RMF are almost consistent with the STAR data across the entire rapidity. However, The directed flows of LQMD with Skyrme interaction are weaker than the STAR data across the entire rapidity. This result may be due to the fact that the value of directed transverse momentum with the Lorentz effect is larger than that without the Lorentz effect [43]. The elliptic flows of both LQMD.RMF and LQMD with Skyrme interaction are consistent with the STAR data at rapidities smaller than 0.5. However, the elliptic flows of both LQMD.RMF and LQMD with Skyrme interaction are weaker than the STAR data at large rapidity. At an incident energy of 2.92 GeV/nucleon, the LQMD.RMF can better describe the experimental data compared to the LQMD with Skyrme interaction and without the momentum-dependent interaction. Based on the above analyses, the RMF models have been implemented into the LQMD model successfully.

Fig. 3Full-screen View

(Color online) Rapidity distribution of the collective flows of free protons in the ¹⁹⁷Au + ¹⁹⁷Au reaction at an incident energy of 2.92 GeV/nucleon. The open circles correspond to the STAR data [94]

With this LQMD.RMF model, the ¹⁰⁸Sn + ¹¹²Sn and ¹³²Sn + ¹²⁴Sn collisions in this study were investigated at an incident energy of 270 A MeV and an impact parameter b=3 fm. At an incident energy of 270 A MeV, the nuclear matter of the collision center can be compressed to densities approaching 2ρ₀. In this dense region, collective flows, which reflect nucleon-nucleon interactions, can be used to extract the high-density behavior of the EOS [1, 44, 79, 86, 95]. The collective flows of free protons in the ¹⁰⁸Sn + ¹¹²Sn and ¹³²Sn + ¹²⁴Sn collisions are shown in Fig. 4 and Fig. 5, respectively. It is reasonable that the maximum value of the directed flow V₁ was significantly larger than that of the elliptic flow V₂. In the same reaction system, the difference in the directed flows with various slopes of symmetry energy (set1, set2, and set3) was small. The difference in the elliptic flows with various slopes of symmetry energy was also small. To determine the relationship between the slope of the symmetry energy and the collective flow, we must process the collective flow data.

Fig. 4Full-screen View

(Color online) Rapidity distribution of the collective flows of free protons in the ¹⁰⁸Sn + ¹¹²Sn reaction at an incident energy of 270 MeV/nucleon

Fig. 5Full-screen View

(Color online) Rapidity distribution of the collective flows of free protons in the ¹³²Sn + ¹²⁴Sn reaction at an incident energy of 270 MeV/nucleon

The difference between neutron and proton directed flows emitted from heavy-ion collisions can be used to extract the density dependence of the symmetry energy [86, 95]. The difference between the neutron and proton directed flows is defined as V_1n-V_1p, as shown in Fig. 6. The trend and shape of the difference between the neutron and proton directed flows were similar to those of nonrelativistic LQMD [86]. For a given reaction system (nearly symmetric ¹⁰⁸Sn + ¹¹²Sn system or neutron-rich ¹³²Sn + ¹²⁴Sn system), the absolute value of the difference between the neutron and proton directed flows with a soft symmetry energy was higher than that with a stiff symmetry energy. Interestingly, the stiffness of the symmetry energy can be reflected through the difference between the neutron and proton directed flows. The relationship between the curvature of the symmetry energy K_sym and the collective flows was then investigated. The difference between K_sym of set1 and K_sym of set3 was 406 MeV, which was significantly larger than the 59.7 MeV difference between L of set1 and L of set3. Although the curvature of the symmetry energy K_sym also affected the difference between the neutron and proton directed flows, because the nuclear matter of the collision center could only approach 2ρ₀ at an incident energy of 270 A MeV, the effects of K_sym were not significant compared to the effects of the slope parameter L.

Fig. 6Full-screen View

(Color online) Difference between neutron and proton directed flows in the ¹⁰⁸Sn + ¹¹²Sn and ¹³²Sn + ¹²⁴Sn reactions at an incident energy of 270 MeV/nucleon

In addition to collective flows, the production of isospin exotic particles such as hyperons, kaons, and pions can also be used to extract the symmetry energy [22-28]. Because the thresholds of hyperons and kaons were not reached at the incident energies in this study, the isospin exotic particles were pions. First, we calculated the ratio between the π^- and π⁺ yields of the neutron-rich ¹³²Sn + ¹²⁴Sn system as a function of the collision energy at the impact parameter b=3fm and θ_cm<90°. Because set1 had the softest symmetry energy, it had the highest neutron density. Consequently, there were more neutron-neutron scatterings in set1, resulting in the production of more Δ^- and π^-. As shown in Fig. 7, the π^-/π⁺ ratio of set1 was the highest, and the π^-/π⁺ ratio of set2 was higher than that of set3. Specifically, at a collision energy of 270 MeV/nucleon, the π^-/π⁺ ratio without (with) the π potential changed from 2.71 (2.54) to 2.23 (2.06) when the slope parameter was varied from L=85.3 to 145.0 MeV, that is, from set1 to set3. In other words, the π^-/π⁺ ratio as a function of collision energy depends on the stiffness of the symmetry energy. This result was consistent with the predictions of most transport models [28, 39, 90]. When the π potential was considered, the interaction between π and the nucleon became attractive, resulting in a decrease in both π^- and π⁺ via the absorbed channels $\pi N\to \Delta \left(1232\right)$ and $\Delta (1232)N\to NN$. However, with the π potential, because there were more neutrons in the neutron-rich ¹³²Sn + ¹²⁴Sn system, π^- was more easily absorbed than π⁺. Consequently, the π^-/π⁺ ratio without the π potential was higher than that with the π potential in the same RMF model. Moreover, it is worth mentioning that the threshold effect neglected in this study may cause the π^-/π⁺ ratio to be reversed. In other words, with the threshold effect [81, 90, 91], π^-/π⁺ of set3 may be the highest, and the π^-/π⁺ ratio of set2 may be higher than that of set1.

Fig. 7Full-screen View

(Color online) Ratio between the π^- and π⁺ yields as a function of the incident energy in the ¹³²Sn + ¹²⁴Sn reaction

Next, the properties of π were predicted as a function of the transverse momentum. As shown in Fig. 8, the left and right panels are the transverse momentum spectra of pions for the nearly symmetric ¹⁰⁸Sn + ¹¹²Sn and neutron rich ¹³²Sn + ¹²⁴Sn reactions at θ_cm<90°, respectively. In collisions between isotopes, π⁺ is mainly generated from collisions between protons and π^- is mainly generated from collisions between neutrons. Theoretically, a stiffer symmetry energy would have a stronger repulsive force to push out neutrons and a stronger attractive force to squeeze protons, resulting in a decrease in the π^- yield and an increase in the π⁺ yield, respectively. As shown in panels (b) and (d) of Fig. 8, the stiffer symmetry energy indeed led to larger transverse momentum spectra for π⁺. However, the relationship between the transverse momentum spectra of π^- and the stiffness of the symmetry energy could not be directly explained. Compared with the stiffness of the symmetry energy, the π potential had a more significant impact on the transverse momentum spectrum of π. For the neutron-rich ¹³²Sn + ¹²⁴Sn system, the transverse momentum spectra of both π⁺ and π^- without the π potential were lower than those of the SπRIT data at p_T≲200 MeV. When the π potential was considered, the transverse momentum spectra of both π⁺ and π^- increased at p_T≲200 MeV but decreased at p_T≳200 MeV. Consequently, the transverse momentum spectra of π⁺ with the π potential were almost consistent with the SπRIT data [40]; however, the transverse momentum spectra of π^- were still lower than the SπRIT data for the entire p_T domain. The lower transverse momentum spectra of π^- may be due to the absence of a threshold effect. The threshold effect, which was not considered in this study, may enhance the production of π^- [81, 90, 91].

Fig. 8Full-screen View

(Color online) Transverse momentum spectra of pions as functions of transverse momentum at an incident energy of 270 MeV/nucleon. The left two panels [(a) and (b)] are the results of the ¹⁰⁸Sn + ¹¹²Sn reaction, and the right two panels [(c) and (d)] are the results of the ¹³²Sn + ¹²⁴Sn reaction. The full circles correspond to the SπRIT data [40]

Because a stiffer symmetry energy would have a stronger repulsive force to push out neutrons and a stronger attractive force to squeeze protons, resulting in a decrease in the π^- yield and an increase in the π⁺ yield, respectively, the single ratio $\text{SR}\left({\pi }^{-}/{\pi }^{+}\right)=\left[\text{d}M\left({\pi }^{-}\right)/\text{d}{p}_{\text{T}}\right]/\left[\text{d}M\left({\pi }^{+}\right)/\text{d}{p}_{\text{T}}\right]$ may depend on the stiffness of the symmetry energy and the reaction system. As shown in Fig. 9, for both the nearly symmetric system and neutron-rich system, a softer symmetry energy led to a larger single ratio. In addition, for the same stiffness of the symmetry energy, because the neutron-neutron scattering of the neutron-rich ¹³²Sn + ¹²⁴Sn system was greater than that of the nearly symmetric ¹⁰⁸Sn + ¹¹²Sn system, the single ratio of the neutron-rich system was higher than that of the nearly symmetric system. It is worth mentioning that the single ratio of ¹⁰⁸Sn + ¹¹²Sn was almost consistent with the experimental data. However, the single ratio of ¹³²Sn + ¹²⁴Sn was lower than that of the experimental data for the entire pT domain. This was because the transverse momentum spectra π^- of ¹³²Sn + ¹²⁴Sn were lower than the experimental data.

Fig. 9Full-screen View

(Color online) Single spectral ratios of pions as functions of transverse momentum for the ¹⁰⁸Sn + ¹¹²Sn and ¹³²Sn + ¹²⁴Sn reactions at an incident energy of 270 MeV/nucleon. The full circles correspond to the SπRIT data [40]

The double ratio between the neutron rich system and the nearly symmetric system $\text{DR}\left({\pi }^{-}/{\pi }^{+}\right)=\text{SR}{\left({\pi }^{-}/{\pi }^{+}\right)}_{132+124}/\text{SR}{\left({\pi }^{-}/{\pi }^{+}\right)}_{108+112}$, which can cancel out most of the systematic errors caused by Coulomb and isoscalar interactions, was suggested to extract the properties of the symmetry energy [40]. However, because a lower symmetry energy will lead to a larger single ratio for both the nearly symmetric system and neutron-rich system, as shown in Fig. 10, it is still difficult to understand the dependence of the double ratio on the stiffness of symmetry energy. In addition, the double ratio without the π potential decreased slightly as a function of the transverse momentum, whereas it increased slightly as the transverse momentum increased when the π potential was considered. However, the increasing trend was still considerably weaker than that of the experimental results. The lower double ratio originated from the lower single ratio of the neutron-rich ¹³²Sn + ¹²⁴Sn system compared with the experimental data. The threshold effect may enhance the production of π^- [81, 90, 91] and the single ratio of the neutron-rich system, resulting in a larger double ratio.

Fig. 10Full-screen View

(Color online) Double ratio of pions as a function of transverse momentum at an incident energy of 270 MeV/nucleon. The full circles correspond to the SπRIT data [40]

Conclusion

An RMF with various symmetry energies, namely set1, set2, and set3, was implemented in LQMD. The collective flows of the nearly symmetric ¹⁰⁸Sn + ¹¹²Sn and neutron-rich ¹³²Sn + ¹²⁴Sn systems were successfully generated from the LQMD.RMF. It has been observed that the directed flow V₁ was an order of magnitude larger than the elliptic flow V₂. However, the difference between the directed flows V₁ with various slopes of symmetry energy was small. The difference in the directed flows V₂ with various slopes of symmetry energy was also small. To explore the relationship between the collective flow and the stiffness of the symmetry energy, we defined the difference between neutron and proton directed flows V_1n-V_1p. For a given nearly symmetric system or neutron-rich system, the absolute value of V_1n-V_1p increased with decreasing slope of the symmetry energy.

We also investigated the relationship between isospin exotic particles and the stiffness of the symmetry energy. The ratio between π^- yield and π⁺ yield of the neutron-rich ¹³²Sn + ¹²⁴Sn system as a function of the collision energy increased with a decrease in the slope parameter of the symmetry energy. At an incident energy of 270 MeV/nucleon, the properties of π were predicted as a function of the transverse momentum. For the nearly symmetric ¹⁰⁸Sn + ¹¹²Sn system, the single ratio of the nearly symmetric system was consistent with the experimental data. However, for the neutron-rich ¹³²Sn + ¹²⁴Sn system, the single ratio was lower than the experimental data, resulting in a double ratio lower than the experimental data. The π potential did not explain the lower transverse momentum spectra of π^- in the neutron-rich system. Considering the π potential, the double ratio increased slightly with increasing transverse momentum. However, the increasing trend was still considerably weaker than that observed in the experimental results. The single ratio of the neutron-rich system and the double ratio may be lower than the experimental data because of the lack of a threshold effect. The threshold effect, which can enhance the production of π^-, could be a candidate for enhancing the single ratio of the neutron-rich system to a double ratio. Moreover, because a softer symmetry energy led to a larger single ratio for both nearly symmetric and neutron-rich systems, the dependence of the double ratio on the stiffness of the symmetry energy was not significant. The sensitivity of the double ratio to the stiffness of the symmetry energy may also be due to the lack of a threshold effect. When the threshold effect is considered, the production of π^- in a neutron-rich system may be more significant than that in a nearly symmetric system. In the future, when collective flows, the single ratio of the neutron-rich system and the double ratio of the LQMD.RMF are consistent with the experimental data, $V_{1n}-V_{1p}$ and the single ratio of the neutron-rich system π^-/π⁺, which are sensitive to the stiffness of the symmetry energy, may be used to extract the slope parameter of the symmetry energy.