Effects of the momentum dependence of nuclear symmetry potential on pion observables in Sn + Sn collisions at 270 MeV/nucleon

Effects of the momentum dependence of nuclear symmetry potential on pion observables in Sn + Sn collisions at 270 MeV/nucleon 1 2 corresp first-author †wei.gaofeng@gznu.edu.cn †wei.gaofeng@gznu.edu.cn wei.gaofeng@gznu.edu.cn https://orcid.org/0000-0003-0637-5500 1 1 2 1 2 3 4 School of Physics and Electronic Science, Guizhou Normal University, Guiyang 550025, China School of Physics and Electronic Science, Guizhou Normal University, Guiyang 550025, China Guizhou Provincial Key Laboratory of Radio Astronomy and Data Processing, Guizhou Normal University, Guiyang 550025, China Guizhou Provincial Key Laboratory of Radio Astronomy and Data Processing, Guizhou Normal University, Guiyang 550025, China Guizhou Provincial Key Laboratory of Public Big Data, Guizhou University, Guiyang 550025, China Guizhou Provincial Key Laboratory of Public Big Data, Guizhou University, Guiyang 550025, China College of Physics, Guizhou University, Guiyang 550025, China College of Physics, Guizhou University, Guiyang 550025, China Within a transport model, we investigated the effects of the momentum dependence of the nuclear symmetry potential on the pion observables in central Sn + Sn collisions at 270 MeV/nucleon. To this end, the quantity <math id="M1"><mrow><msubsup><mi>U</mi><mrow><mtext>sym</mtext></mrow><mi>∞</mi></msubsup><mo stretchy="false">(</mo><msub><mi>ρ</mi><mn>0</mn></msub><mo stretchy="false">)</mo></mrow></math> (i.e., the value of the nuclear symmetry potential at the saturation density ρ 0 and infinitely large nucleon momentum) was used to characterize the momentum dependence of the nuclear symmetry potential. With a certain L (i.e., the slope of the nuclear symmetry energy at ρ 0 ), the characteristic parameter <math id="M2"><mrow><msubsup><mi>U</mi><mrow><mtext>sym</mtext></mrow><mi>∞</mi></msubsup><mo stretchy="false">(</mo><msub><mi>ρ</mi><mn>0</mn></msub><mo stretchy="false">)</mo></mrow></math> of the symmetry potential significantly affects the production of π - and π + and their pion ratios. Moreover, by comparing the charged pion yields, pion ratios, and spectral pion ratios of the theoretical simulations for the reactions 108 Sn + 112 Sn and 132 Sn + 124 Sn with the corresponding data in the S π RIT experiments, we found that our results favor a constraint on <math id="M3"><mrow><msubsup><mi>U</mi><mrow><mtext>sym</mtext></mrow><mi>∞</mi></msubsup><mo stretchy="false">(</mo><msub><mi>ρ</mi><mn>0</mn></msub><mo stretchy="false">)</mo></mrow></math> (i.e., <math id="M4"><mrow><mo>−</mo><msubsup><mrow><mn>160</mn></mrow><mrow><mo>−</mo><mn>9</mn></mrow><mrow><mo>+</mo><mn>18</mn></mrow></msubsup></mrow></math> MeV), and L is also suggested within a range of 62.7 MeV < L < 93.1 MeV. In addition, the pion observable for 197 Au + 197 Au collisions at 400 MeV/nucleon also supports the extracted value for <math id="M5"><mrow><msubsup><mi>U</mi><mrow><mtext>sym</mtext></mrow><mi>∞</mi></msubsup><mo stretchy="false">(</mo><msub><mi>ρ</mi><mn>0</mn></msub><mo stretchy="false">)</mo></mrow></math> . Nuclear symmetry potential Momentum dependence Symmetry energy introduction 1 The equation of state (EoS) of asymmetric nuclear matter (ANM), especially its nuclear symmetry energy E sym ( ρ ) term, plays an essential role in studying the structure [ 1 1 - 4 4 1-4 ] and evolution of radioactive nuclei [ 5 5 - 8 8 5-8 ], as well as the synthesis of medium and heavy nuclei [ 9 9 - 12 12 9-12 ]. The E sym ( ρ ) characterizes the variation in the EoS of the symmetric nuclear matter (SNM) to that of the pure neutron matter (PNM); the latter is closely connected to the neutron star (NS) matter. Naturally, the properties of NS, such as the radius and the deformation of the NS merger, are also closely related to E sym ( ρ ), especially at densities of approximately twice the saturation density ρ 0 [ 13 13 - 20 20 13-20 ]. Nevertheless, knowledge of the E sym ( ρ ) at suprasaturation densities is still limited, although that around and below ρ 0 [ 21 21 , 22 22 ] as well as the isospin-independent part of EoS for ANM (i.e., EoS of SNM [ 23 23 - 25 25 23-25 ]) are relatively well determined. Essentially, the EoS of the ANM and its E sym ( ρ ) term are determined by the nuclear mean field, especially its isovector part (i.e., the symmetry/isovector potential [ 27 27 , 26 26 ]). However, because of the extreme challenge of relatively direct detection of the isovector potential in experiments, one extracted only using the nucleon-nucleus scattering and ( p , n ) charge-exchange reactions between isobaric analog states limited information of the isovector potential at ρ 0 and parameterized as <math id="M6"><mrow><msub><mi>U</mi><mrow><mtext>sym</mtext></mrow></msub><mo stretchy="false">(</mo><msub><mi>ρ</mi><mn>0</mn></msub><mn>,</mn><msub><mstyle mathvariant="bold"><mtext>E</mtext></mstyle><mi>k</mi></msub><mo stretchy="false">)</mo><mo>=</mo><mi>a</mi><mo>−</mo><mi>b</mi><msub><mstyle mathvariant="bold"><mtext>E</mtext></mstyle><mi>k</mi></msub></mrow></math> , where <math id="M7"><mrow><mi>a</mi><mi> </mi><mo>≈</mo><mi> </mi><mn>22</mn><mo>−</mo><mn>34</mn></mrow></math> MeV, <math id="M8"><mrow><mi>b</mi><mi> </mi><mo>≈</mo><mi> </mi><mn>0.1</mn><mo>−</mo><mn>0.2</mn></mrow></math> , and E k is limited to no more than 200 MeV [ 28 28 - 30 30 28-30 ]. Heavy-ion collision (HIC) is one of the most promising approaches for exploring the symmetry potential/energy, especially at supersaturation densities [ 3 3 , 4 4 , 13 13 , 31 31 - 33 33 31-33 ]. Recently, the S π RIT collaboration reported results from the first measurement dedicated to probing the E sym ( ρ ) at suprasaturation densities through pion production in Sn + Sn collisions at 270 MeV/nucleon carried out at RIKEN-RIBF in Japan [ 31 31 ]. Moreover, they compared the charged pion yields and their single and double pion ratios with the corresponding simulation results from seven transport models. Qualitatively, the theoretical simulations from the seven transport models reach an agreement with the data, yet quantitatively, almost all the models cannot satisfactorily reproduce both the pion yields and their single and double pion ratios of the experimental data [ 31 31 ]. In this situation, author of Ref. [ 34 34 ] claimed that by considering approximately 20% of high-momentum nucleons in colliding nuclei can reproduce both the charged pion yields and their pion ratios of the experimental data quite well because of the high momentum distribution in nuclei caused by short-range correlations (SRCs) [ 35 35 - 40 40 35-40 ]. Following this work, we focused on the momentum dependence of the symmetry potential because it plays an important role in probing the high-density behavior of E sym ( ρ ) [ 41 41 - 43 43 41-43 ]. In fact, in [ 31 31 ] and in [ 44 44 - 48 48 44-48 ] of the transport model comparison project, the possible reasons for the unsatisfactory results of the seven models quantitatively fitting the experimental data may be due to different assumptions regarding the mean field potential, pion potential, and the treatment of the Coulomb field. Therefore, exploring how the momentum dependence of the symmetry potential affects the pion production in HICs is necessary. Regarding the other aforementioned factors, we also provide detailed considerations based on sophisticated treatment methods, as discussed in Se ct. 2. In Sect. 3, we discuss the results of the present study. Finally, a summary is presented in Sect. 4. The Model 1 This study was carried out using an isospin- and momentum-dependent Boltzmann-Uehling-Uhlenbeck (IBUU) transport model. In the framework, the present model is originated from IBUU04 [ 49 49 , 50 50 ] and/or IBUU11 [ 51 51 ] models. However, the present model has been greatly improved to accurately simulate pion production, as discussed below. First, a separate density-dependent scenario for in-medium nucleon-nucleon interaction [ 52 52 - 54 54 52-54 ], which is expressed as <math display="block" id="M9"><mrow><msub><mi>v</mi><mi>D</mi></msub><mo>=</mo><msub><mi>t</mi><mn>0</mn></msub><mo stretchy="false">(</mo><mn>1</mn><mo>+</mo><msub><mi>x</mi><mn>0</mn></msub><msub><mi>P</mi><mi>σ</mi></msub><mo stretchy="false">)</mo><mo stretchy="false">[</mo><msub><mi>ρ</mi><mrow><msub><mi>τ</mi><mi>i</mi></msub></mrow></msub><mo stretchy="false">(</mo><msub><mi>r</mi><mi>i</mi></msub><mo stretchy="false">)</mo><mo>+</mo><msub><mi>ρ</mi><mrow><msub><mi>τ</mi><mi>j</mi></msub></mrow></msub><mo stretchy="false">(</mo><msub><mi>r</mi><mi>j</mi></msub><msup><mrow><mo stretchy="false">)</mo><mo stretchy="false">]</mo></mrow><mi>α</mi></msup><mi>δ</mi><mo stretchy="false">(</mo><msub><mi>r</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub><mo stretchy="false">)</mo><mn>,</mn></mrow></math> this equation is used to replace the density-dependent term of the original Gogny effective interaction [ 55 55 ], which is given by <math display="block" id="M10"><mtable><mtr><mtd><mi>v</mi><mo stretchy="false">(</mo><mi>r</mi><mo stretchy="false">)</mo><mo>=</mo><mstyle displaystyle="true"><munder><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1,2</mn></mrow></munder></mstyle><msub><mrow><mo>(</mo><mrow><mi>W</mi><mo>+</mo><mi>B</mi><msub><mi>P</mi><mi>σ</mi></msub><mo>−</mo><mi>H</mi><msub><mi>P</mi><mi>τ</mi></msub><mo>−</mo><mi>M</mi><msub><mi>P</mi><mi>σ</mi></msub><msub><mi>P</mi><mi>τ</mi></msub></mrow><mo>)</mo></mrow><mi>i</mi></msub><msup><mi>e</mi><mrow><mo>−</mo><msup><mi>r</mi><mn>2</mn></msup><mo>/</mo><msubsup><mi>μ</mi><mi>i</mi><mn>2</mn></msubsup></mrow></msup></mtd></mtr><mtr><mtd><mo>+</mo><msub><mi>t</mi><mn>0</mn></msub><mrow><mo>(</mo><mrow><mn>1</mn><mo>+</mo><msub><mi>x</mi><mn>0</mn></msub><msub><mi>P</mi><mi>σ</mi></msub></mrow><mo>)</mo></mrow><msup><mrow><mo>[</mo> <mrow><mi>ρ</mi><mrow><mo>(</mo><mrow><mfrac><mrow><msub><mstyle mathvariant="bold"><mtext>r</mtext></mstyle><mi>i</mi></msub><mo>+</mo><msub><mstyle mathvariant="bold"><mtext>r</mtext></mstyle><mi>j</mi></msub></mrow><mn>2</mn></mfrac></mrow><mo>)</mo></mrow></mrow> <mo>]</mo></mrow><mi>α</mi></msup><mi>δ</mi><mrow><mo>(</mo><mrow><msub><mstyle mathvariant="bold"><mtext>r</mtext></mstyle><mrow><mi>i</mi><mi>j</mi></mrow></msub></mrow><mo>)</mo></mrow><mn>,</mn></mtd></mtr></mtable></math> where W , B , H , M , and μ are the five parameters, P τ and P σ are the isospin and spin exchange operators, respectively, and α is the density-dependent parameter used to mimic the medium effects of many-body interactions [ 52 52 - 54 54 52-54 ]. As indicated in Ref. [ 56 56 ], the separate density dependence of the effective two-body interactions originates from the renormalization of the multibody force effects, and the latter may extend the density dependence of the effective interactions for calculations beyond the mean-field approximation. Moreover, nuclear structure studies have shown that, with the separate density-dependent scenario for the in-medium nucleon-nucleon interaction, more satisfactory results (e.g., the binding energies, single-particle energies, and electron scattering cross-sections for 16 O, 40 Ca, 48 Ca, 90 Zr, and 208 Pr [ 57 57 ]) can be achieved compared with the corresponding experiments. Correspondingly, the potential energy density for the ANM with this improved momentum-dependent interaction (IMDI) is expressed [ 53 53 ] as <math display="block" id="M11"><mtable><mtr><mtd><mi>V</mi><mo stretchy="false">(</mo><mi>ρ</mi><mn>,</mn><mi>δ</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mrow><msub><mstyle mathvariant="bold"><mtext>A</mtext></mstyle><mi>u</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><msub><mi>ρ</mi><mi>n</mi></msub><msub><mi>ρ</mi><mi>p</mi></msub></mrow><mrow><msub><mi>ρ</mi><mn>0</mn></msub></mrow></mfrac><mo>+</mo><mfrac><mrow><msub><mstyle mathvariant="bold"><mtext>A</mtext></mstyle><mi>l</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><mrow><mn>2</mn><msub><mi>ρ</mi><mn>0</mn></msub></mrow></mfrac><mrow><mo>(</mo><mrow><msubsup><mi>ρ</mi><mi>n</mi><mn>2</mn></msubsup><mo>+</mo><msubsup><mi>ρ</mi><mi>p</mi><mn>2</mn></msubsup></mrow><mo>)</mo></mrow><mo>+</mo><mfrac><mstyle mathvariant="bold"><mtext>B</mtext></mstyle><mrow><mi>σ</mi><mo>+</mo><mn>1</mn></mrow></mfrac><mfrac><mrow><msup><mi>ρ</mi><mrow><mi>σ</mi><mo>+</mo><mn>1</mn></mrow></msup></mrow><mrow><msubsup><mi>ρ</mi><mn>0</mn><mi>σ</mi></msubsup></mrow></mfrac></mtd></mtr><mtr><mtd><mo>×</mo><mrow><mo>{</mo><mrow><mfrac><mrow><mn>1</mn><mo>+</mo><mi>x</mi></mrow><mn>2</mn></mfrac><mrow><mo>(</mo><mrow><mn>1</mn><mo>−</mo><msup><mi>δ</mi><mn>2</mn></msup></mrow><mo>)</mo></mrow><mo>+</mo><mfrac><mrow><mn>1</mn><mo>−</mo><mi>x</mi></mrow><mn>4</mn></mfrac><mrow><mo>[</mo><mrow><msup><mrow><mo stretchy="false">(</mo><mn>1</mn><mo>+</mo><mi>δ</mi><mo stretchy="false">)</mo></mrow><mrow><mi>σ</mi><mo>+</mo><mn>1</mn></mrow></msup><mo>+</mo><msup><mrow><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><mi>δ</mi><mo stretchy="false">)</mo></mrow><mrow><mi>σ</mi><mo>+</mo><mn>1</mn></mrow></msup></mrow><mo>]</mo></mrow></mrow><mo>}</mo></mrow></mtd></mtr><mtr><mtd><mo>+</mo><mfrac><mn>1</mn><mrow><msub><mi>ρ</mi><mn>0</mn></msub></mrow></mfrac><mstyle displaystyle="true"><munder><mo>∑</mo><mrow><mi>τ</mi><mn>,</mn><msup><mi>τ</mi><mo>'</mo></msup></mrow></munder></mstyle><msub><mi>C</mi><mrow><mi>τ</mi><mn>,</mn><msup><mi>τ</mi><mo>'</mo></msup></mrow></msub><mstyle displaystyle="true"><mrow><mo>∫</mo></mrow></mstyle><mstyle displaystyle="true"><mrow><mo>∫</mo></mrow></mstyle><msup><mi>d</mi><mn>3</mn></msup><mi>p</mi><msup><mi>d</mi><mn>3</mn></msup><msup><mi>p</mi><mo>'</mo></msup><mfrac><mrow><msub><mi>f</mi><mi>τ</mi></msub><mo stretchy="false">(</mo><mover accent="true"><mi>r</mi><mo>→</mo></mover><mn>,</mn><mover accent="true"><mi>p</mi><mo>→</mo></mover><mo stretchy="false">)</mo><msub><mi>f</mi><mrow><msup><mi>τ</mi><mo>'</mo></msup></mrow></msub><mrow><mo>(</mo><mrow><mover accent="true"><mi>r</mi><mo>→</mo></mover><mn>,</mn><msup><mover accent="true"><mi>p</mi><mo>→</mo></mover><mo>'</mo></msup></mrow><mo>)</mo></mrow></mrow><mrow><mn>1</mn><mo>+</mo><msup><mrow><mrow><mo>(</mo><mrow><mover accent="true"><mi>p</mi><mo>→</mo></mover><mo>−</mo><msup><mover accent="true"><mi>p</mi><mo>→</mo></mover><mo>'</mo></msup></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup><mo>/</mo><msup><mi>Λ</mi><mn>2</mn></msup></mrow></mfrac><mn>.</mn></mtd></mtr></mtable></math> In the mean-field approximation, Eq. (3) leads to the following single-nucleon potential for the proposed model [ 52 52 - 54 54 52-54 ]: <math display="block" id="M12"><mtable><mtr><mtd><mi>U</mi><mrow><mo>(</mo><mrow><mi>ρ</mi><mn>,</mn><mi>δ</mi><mn>,</mn><mover accent="true"><mi>p</mi><mo>→</mo></mover><mn>,</mn><mi>τ</mi></mrow><mo>)</mo></mrow><mo>=</mo><msub><mstyle mathvariant="bold"><mtext>A</mtext></mstyle><mi>u</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mfrac><mrow><msub><mi>ρ</mi><mrow><mo>−</mo><mi>τ</mi></mrow></msub></mrow><mrow><msub><mi>ρ</mi><mn>0</mn></msub></mrow></mfrac><mo>+</mo><msub><mstyle mathvariant="bold"><mtext>A</mtext></mstyle><mi>l</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mfrac><mrow><msub><mi>ρ</mi><mi>τ</mi></msub></mrow><mrow><msub><mi>ρ</mi><mn>0</mn></msub></mrow></mfrac><mo>+</mo><mfrac><mstyle mathvariant="bold"><mtext>B</mtext></mstyle><mn>2</mn></mfrac><msup><mrow><mo>(</mo><mrow><mfrac><mrow><mn>2</mn><msub><mi>ρ</mi><mi>τ</mi></msub></mrow><mrow><msub><mi>ρ</mi><mn>0</mn></msub></mrow></mfrac></mrow><mo>)</mo></mrow><mi>σ</mi></msup><mrow><mo>(</mo><mrow><mn>1</mn><mo>−</mo><mi>x</mi></mrow><mo>)</mo></mrow></mtd></mtr><mtr><mtd><mo>+</mo><mfrac><mrow><mn>2</mn><mstyle mathvariant="bold"><mtext>B</mtext></mstyle></mrow><mrow><mi>σ</mi><mo>+</mo><mn>1</mn></mrow></mfrac><msup><mrow><mo>(</mo><mrow><mfrac><mi>ρ</mi><mrow><msub><mi>ρ</mi><mn>0</mn></msub></mrow></mfrac></mrow><mo>)</mo></mrow><mi>σ</mi></msup><mrow><mo>(</mo><mrow><mn>1</mn><mo>+</mo><mi>x</mi></mrow><mo>)</mo></mrow><mfrac><mrow><msub><mi>ρ</mi><mrow><mo>−</mo><mi>τ</mi></mrow></msub></mrow><mi>ρ</mi></mfrac><mrow><mo>[</mo><mrow><mn>1</mn><mo>+</mo><mo stretchy="false">(</mo><mi>σ</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo><mfrac><mrow><msub><mi>ρ</mi><mi>τ</mi></msub></mrow><mi>ρ</mi></mfrac></mrow><mo>]</mo></mrow></mtd></mtr><mtr><mtd><mo>+</mo><mfrac><mrow><mn>2</mn><msub><mi>C</mi><mi>l</mi></msub></mrow><mrow><msub><mi>ρ</mi><mn>0</mn></msub></mrow></mfrac><mstyle displaystyle="true"><mrow><mo>∫</mo></mrow></mstyle><msup><mi>d</mi><mn>3</mn></msup><msup><mi>p</mi><mo>'</mo></msup><mfrac><mrow><msub><mi>f</mi><mi>τ</mi></msub><mo stretchy="false">(</mo><msup><mover accent="true"><mi>p</mi><mo>→</mo></mover><mo>'</mo></msup><mo stretchy="false">)</mo></mrow><mrow><mn>1</mn><mo>+</mo><msup><mrow><mrow><mo>(</mo><mrow><mover accent="true"><mi>p</mi><mo>→</mo></mover><mo>−</mo><msup><mover accent="true"><mi>p</mi><mo>→</mo></mover><mo>'</mo></msup></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup><mo>/</mo><msup><mi>Λ</mi><mn>2</mn></msup></mrow></mfrac></mtd></mtr><mtr><mtd><mo>+</mo><mfrac><mrow><mn>2</mn><msub><mi>C</mi><mi>u</mi></msub></mrow><mrow><msub><mi>ρ</mi><mn>0</mn></msub></mrow></mfrac><mstyle displaystyle="true"><mrow><mo>∫</mo></mrow></mstyle><msup><mi>d</mi><mn>3</mn></msup><msup><mi>p</mi><mo>'</mo></msup><mfrac><mrow><msub><mi>f</mi><mrow><mo>−</mo><mi>τ</mi></mrow></msub><mo stretchy="false">(</mo><msup><mover accent="true"><mi>p</mi><mo>→</mo></mover><mo>'</mo></msup><mo stretchy="false">)</mo></mrow><mrow><mn>1</mn><mo>+</mo><msup><mrow><mrow><mo>(</mo><mrow><mover accent="true"><mi>p</mi><mo>→</mo></mover><mo>−</mo><msup><mover accent="true"><mi>p</mi><mo>→</mo></mover><mo>'</mo></msup></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup><mo>/</mo><msup><mi>Λ</mi><mn>2</mn></msup></mrow></mfrac><mn>,</mn></mtd></mtr></mtable></math> where σ = α +1, τ =1 for neutrons and -1 for protons, and the parameters A u ( x ), A l ( x ), <math id="M13"><mrow><msub><mi>C</mi><mi>u</mi></msub><mo stretchy="false">(</mo><mo>≡</mo><msub><mi>C</mi><mrow><mi>τ</mi><mn>,</mn><mo>−</mo><mi>τ</mi></mrow></msub><mo stretchy="false">)</mo></mrow></math> and <math id="M14"><mrow><msub><mi>C</mi><mi>l</mi></msub><mo stretchy="false">(</mo><mo>≡</mo><msub><mi>C</mi><mrow><mi>τ</mi><mn>,</mn><mi>τ</mi></mrow></msub><mo stretchy="false">)</mo></mrow></math> are expressed as <math display="block" id="M15"><mtable><mtr><mtd><msub><mstyle mathvariant="bold"><mtext>A</mtext></mstyle><mi>l</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mstyle mathvariant="bold"><mtext>A</mtext></mstyle><mrow><mi>l</mi><mn>0</mn></mrow></msub><mo>+</mo><msubsup><mi>U</mi><mrow><mtext>sym</mtext></mrow><mi>∞</mi></msubsup><mo stretchy="false">(</mo><msub><mi>ρ</mi><mn>0</mn></msub><mo stretchy="false">)</mo><mo>−</mo><mfrac><mrow><mn>2</mn><mstyle mathvariant="bold"><mtext>B</mtext></mstyle></mrow><mrow><mi>σ</mi><mo>+</mo><mn>1</mn></mrow></mfrac></mtd></mtr><mtr><mtd><mo>×</mo><mrow><mo>[</mo><mrow><mfrac><mrow><mrow><mo>(</mo><mrow><mn>1</mn><mo>−</mo><mi>x</mi></mrow><mo>)</mo></mrow></mrow><mn>4</mn></mfrac><mi>σ</mi><mo stretchy="false">(</mo><mi>σ</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo><mo>−</mo><mfrac><mrow><mn>1</mn><mo>+</mo><mi>x</mi></mrow><mn>2</mn></mfrac></mrow><mo>]</mo></mrow><mn>,</mn></mtd></mtr></mtable></math> <math display="block" id="M16"><mtable><mtr><mtd><msub><mstyle mathvariant="bold"><mtext>A</mtext></mstyle><mi>u</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mstyle mathvariant="bold"><mtext>A</mtext></mstyle><mrow><mi>u</mi><mn>0</mn></mrow></msub><mo>−</mo><msubsup><mi>U</mi><mrow><mtext>sym</mtext></mrow><mi>∞</mi></msubsup><mo stretchy="false">(</mo><msub><mi>ρ</mi><mn>0</mn></msub><mo stretchy="false">)</mo><mo>+</mo><mfrac><mrow><mn>2</mn><mstyle mathvariant="bold"><mtext>B</mtext></mstyle></mrow><mrow><mi>σ</mi><mo>+</mo><mn>1</mn></mrow></mfrac></mtd></mtr><mtr><mtd><mo>×</mo><mrow><mo>[</mo><mrow><mfrac><mrow><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><mn>4</mn></mfrac><mi>σ</mi><mo stretchy="false">(</mo><mi>σ</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo><mo>−</mo><mfrac><mrow><mn>1</mn><mo>+</mo><mi>x</mi></mrow><mn>2</mn></mfrac></mrow><mo>]</mo></mrow><mn>,</mn></mtd></mtr></mtable></math> <math display="block" id="M17"><mrow><msub><mi>C</mi><mi>l</mi></msub><mo>=</mo><msub><mi>C</mi><mrow><mi>l</mi><mn>0</mn></mrow></msub><mo>−</mo><mn>2</mn><msubsup><mi>U</mi><mrow><mtext>sym</mtext></mrow><mi>∞</mi></msubsup><mo stretchy="false">(</mo><msub><mi>ρ</mi><mn>0</mn></msub><mo stretchy="false">)</mo><mfrac><mrow><msubsup><mi>p</mi><mrow><mi>f</mi><mn>0</mn></mrow><mn>2</mn></msubsup></mrow><mrow><msup><mi>Λ</mi><mn>2</mn></msup><mi>ln</mi><mo stretchy="false">[</mo><mo stretchy="false">(</mo><mn>4</mn><msubsup><mi>p</mi><mrow><mi>f</mi><mn>0</mn></mrow><mn>2</mn></msubsup><mo>+</mo><msup><mi>Λ</mi><mn>2</mn></msup><mo stretchy="false">)</mo><mo>/</mo><msup><mi>Λ</mi><mn>2</mn></msup><mo stretchy="false">]</mo></mrow></mfrac><mn>,</mn></mrow></math> <math display="block" id="M18"><mrow><msub><mi>C</mi><mi>u</mi></msub><mo>=</mo><msub><mi>C</mi><mrow><mi>u</mi><mn>0</mn></mrow></msub><mo>+</mo><mn>2</mn><msubsup><mi>U</mi><mrow><mtext>sym</mtext></mrow><mi>∞</mi></msubsup><mo stretchy="false">(</mo><msub><mi>ρ</mi><mn>0</mn></msub><mo stretchy="false">)</mo><mfrac><mrow><msubsup><mi>p</mi><mrow><mi>f</mi><mn>0</mn></mrow><mn>2</mn></msubsup></mrow><mrow><msup><mi>Λ</mi><mn>2</mn></msup><mi>ln</mi><mo stretchy="false">[</mo><mo stretchy="false">(</mo><mn>4</mn><msubsup><mi>p</mi><mrow><mi>f</mi><mn>0</mn></mrow><mn>2</mn></msubsup><mo>+</mo><msup><mi>Λ</mi><mn>2</mn></msup><mo stretchy="false">)</mo><mo>/</mo><msup><mi>Λ</mi><mn>2</mn></msup><mo stretchy="false">]</mo></mrow></mfrac><mn>,</mn></mrow></math> where p f 0 is the nucleon Fermi momentum in the SNM at ρ 0 and <math id="M19"><mrow><msubsup><mi>U</mi><mrow><mtext>sym</mtext></mrow><mi>∞</mi></msubsup><mo stretchy="false">(</mo><msub><mi>ρ</mi><mn>0</mn></msub><mo stretchy="false">)</mo></mrow></math> proposed in [ 51 51 ] and is used to characterize the momentum dependence of the symmetry potential at ρ 0 . We also illustrate the derivation of Eqs. (5)-(8) and the expression of the symmetry potential and/or energy (see [ 53 53 ]). Currently, knowledge of the momentum dependence of the symmetry potential even at ρ 0 is limited [ 28 28 - 30 30 28-30 ]. Therefore, taking the parameterized symmetry potential as a reference, we treat <math id="M20"><mrow><msubsup><mi>U</mi><mrow><mtext>sym</mtext></mrow><mi>∞</mi></msubsup><mo stretchy="false">(</mo><msub><mi>ρ</mi><mn>0</mn></msub><mo stretchy="false">)</mo></mrow></math> as a free parameter similar to the x parameter, which is used to mimic the slope value <math id="M21"><mrow><mi>L</mi><mo>≡</mo><mn>3</mn><mi>ρ</mi><mo stretchy="false">(</mo><mtext>d</mtext><msub><mstyle mathvariant="bold"><mtext>E</mtext></mstyle><mrow><mtext>sym</mtext></mrow></msub><mo>/</mo><mtext>d</mtext><mi>ρ</mi><mo stretchy="false">)</mo></mrow></math> of E sym ( ρ ) at ρ 0 without changing the value of E sym ( ρ ) at ρ 0 and any properties of the SNM. A similar quantity (i.e., y parameter) in [ 58 58 - 60 60 58-60 ] has been used to describe the momentum dependence of the symmetry potential at ρ 0 ; however, its quantitative constraints have not been determined. Additionally, it should be noted that the B terms in Eqs. (3) and (4), as well as in the expressions of A u and A l are different from those in [ 58 58 - 60 60 58-60 ]. This is because the separate density-dependent scenario for in-medium nucleon-nucleon interaction has been adopted in the present model for a more delicate treatment of the in-medium many-body force effects [ 53 53 ], which significantly affects pion production in HICs [ 54 54 ]. The seven parameters A l 0 , A u 0 , B , σ , C l 0 , C u 0 , and Λ were determined by fitting seven experimental and/or empirical constraints on the properties of nuclear matter at ρ 0 =0.16 fm -3 . The first six quantities are the binding energy -16 MeV, pressure P 0 =0 MeV/fm 3 , incompressibility K 0 =230 MeV for SNM, isoscalar effective mass <math id="M22"><mrow><msubsup><mi>m</mi><mi>s</mi><mo>*</mo></msubsup><mo>=</mo><mn>0.7</mn><mi>m</mi></mrow></math> , isoscalar potential at infinitely large nucleon momentum <math id="M23"><mrow><msubsup><mi>U</mi><mn>0</mn><mi>∞</mi></msubsup><mo stretchy="false">(</mo><msub><mi>ρ</mi><mn>0</mn></msub><mo stretchy="false">)</mo><mo>=</mo><mn>75</mn></mrow></math> MeV, and symmetry energy E sym ( ρ 0 )=32.5 MeV, and the seventh quantity is considered as <math id="M24"><mrow><msubsup><mi>U</mi><mrow><mtext>sym</mtext></mrow><mi>∞</mi></msubsup><mo stretchy="false">(</mo><msub><mi>ρ</mi><mn>0</mn></msub><mo stretchy="false">)</mo></mrow></math> . The values of these parameters are A l 0 = A u 0 =-66.963 MeV, B =141.963 MeV, C l 0 =-60.486 MeV, C u 0 =-99.702 MeV, σ =1.2652, and Λ=2.424 p f 0 . The upper windows in Fig. 1 Fig. 1 show the kinetic-energy-dependent neutron and proton potentials at ρ 0 with different <math id="M26"><mrow><msubsup><mi>U</mi><mrow><mtext>sym</mtext></mrow><mi>∞</mi></msubsup><mo stretchy="false">(</mo><msub><mi>ρ</mi><mn>0</mn></msub><mo stretchy="false">)</mo></mrow></math> values calculated from the IMDI interaction. As <math id="M27"><mrow><mrow><mo>|</mo><mrow><msubsup><mi>U</mi><mrow><mtext>sym</mtext></mrow><mi>∞</mi></msubsup><mo stretchy="false">(</mo><msub><mi>ρ</mi><mn>0</mn></msub><mo stretchy="false">)</mo></mrow><mo>|</mo></mrow></mrow></math> increases, the neutron potential shows the opposite variation tendency compared with the proton potential because of the isospin effects. The middle and lower windows in Fig. 1 Fig. 1 show the isoscalar and isovector potentials at ρ 0 in comparison with the Schrödinger-equivalent isoscalar potential obtained by Hama et al. [ 61 61 , 62 62 ] and parameterized isovector potentials from experimental and/or empirical data [ 28 28 - 30 30 28-30 ]. To provide more intuitive references for <math id="M28"><mrow><msubsup><mi>U</mi><mrow><mtext>sym</mtext></mrow><mi>∞</mi></msubsup><mo stretchy="false">(</mo><msub><mi>ρ</mi><mn>0</mn></msub><mo stretchy="false">)</mo></mrow></math> , we extrapolate the experimental and/or empirical isovector potential to the nucleon kinetic energy up to 1 GeV. Evidently, good consistency can be seen for the isoscalar potential between the present model and that of Hama et al. . Moreover, the values of our symmetry potentials at the Fermi kinetic energy (i.e., approximately 36.8 MeV), are the same and within the allowed range of experimental and/or empirical data, even with different <math id="M29"><mrow><msubsup><mi>U</mi><mrow><mtext>sym</mtext></mrow><mi>∞</mi></msubsup><mo stretchy="false">(</mo><msub><mi>ρ</mi><mn>0</mn></msub><mo stretchy="false">)</mo></mrow></math> . It is based on the values of symmetry potentials at the Fermi kinetic energy and the infinite nucleon momentum that we used to determine the momentum dependence of the symmetry potential at ρ 0 . However, because the isoscalar potentials remain unchanged with different <math id="M30"><mrow><msubsup><mi>U</mi><mrow><mtext>sym</mtext></mrow><mi>∞</mi></msubsup><mo stretchy="false">(</mo><msub><mi>ρ</mi><mn>0</mn></msub><mo stretchy="false">)</mo></mrow></math> , differences in momentum dependence between symmetry potentials with different <math id="M31"><mrow><msubsup><mi>U</mi><mrow><mtext>sym</mtext></mrow><mi>∞</mi></msubsup><mo stretchy="false">(</mo><msub><mi>ρ</mi><mn>0</mn></msub><mo stretchy="false">)</mo></mrow></math> are expected to be reflected by the pion observable in HICs because the different symmetry potentials can lead to different isospin effects and thus different π - / π + ratios for neutron-rich reactions. Therefore, to clearly obtain the pion observable, in which the effects of the momentum dependence of the symmetry potentials are reflected, it is useful to map momentum-dependent symmetry potentials with different <math id="M32"><mrow><msubsup><mi>U</mi><mrow><mtext>sym</mtext></mrow><mi>∞</mi></msubsup><mo stretchy="false">(</mo><msub><mi>ρ</mi><mn>0</mn></msub><mo stretchy="false">)</mo></mrow></math> into cases with the same E sym ( ρ ). This is performed by fitting identical constraints for SNM and identical slope parameter L of E sym ( ρ ) at ρ 0 , and the corresponding results are shown in Fig. 2 Fig. 2 . Even with the same E sym ( ρ ), the corresponding symmetry potential could be considerably different because the symmetry potentials depend not only on the nucleon density but also on the nucleon momentum or energy. Second, we also considered the pion potential effects in HICs to accurately simulate pion production in HICs. Specifically, when the pionic momentum is greater than 140 MeV/ c , we use the pion potential based on the Δ -hole model of the form adopted in [ 62 62 ]. We adopt the pion potential of the form used in [ 63 63 - 65 65 63-65 ] when the pionic momentum is lower than 80 MeV/ c , whereas for the pionic momentum falling in the range of 80 to 140 MeV/ c , an interpolative pion potential constructed in [ 62 62 ] is used. The present pion potential includes the isospin- and momentum-dependent pion s - and p -wave potentials in a nuclear medium, as in [ 66 66 ] (see [ 62 62 - 65 65 62-65 ]). The in-medium isospin-dependent baryon-baryon elastic and inelastic scattering cross-sections σ medium are determined by the corresponding free-space cross-sections σ free multiplied by a factor R medium , which is expressed as <math display="block" id="M33"><mrow><msub><mi>σ</mi><mrow><mtext>medium</mtext></mrow></msub><mo>=</mo><msub><mi>σ</mi><mrow><mtext>free</mtext></mrow></msub><msub><mi>R</mi><mrow><mtext>medium</mtext></mrow></msub><mo stretchy="false">(</mo><mi>ρ</mi><mn>,</mn><mi>δ</mi><mn>,</mn><mstyle mathvariant="bold"><mtext>p</mtext></mstyle><mo stretchy="false">)</mo><mn>,</mn></mrow></math> where the reduced factor is determined as <math id="M34"><mrow><msub><mi>R</mi><mrow><mtext>medium</mtext></mrow></msub><mo>=</mo><mo stretchy="false">(</mo><msubsup><mi>μ</mi><mrow><mtext>BB</mtext></mrow><mo>*</mo></msubsup><mo>/</mo><msub><mi>μ</mi><mrow><mtext>BB</mtext></mrow></msub><msup><mo stretchy="false">)</mo><mn>2</mn></msup></mrow></math> , and μ BB and <math id="M35"><mrow><msubsup><mi>μ</mi><mrow><mtext>BB</mtext></mrow><mo>*</mo></msubsup></mrow></math> are the reduced masses of the colliding baryon pairs in free space and nuclear medium, respectively. Finally, for the treatment of the Coulomb field, we calculate the electromagnetic (EM) interactions from the Maxwell equation, that is, <math id="M38"><mrow><mstyle mathvariant="bold"><mtext>E</mtext></mstyle><mo>=</mo><mo>−</mo><mo>∇</mo><mi>φ</mi><mo>−</mo><mo>∂</mo><mstyle mathvariant="bold"><mtext>A</mtext></mstyle><mo>/</mo><mo>∂</mo><mi>t</mi></mrow></math> , <math id="M39"><mrow><mstyle mathvariant="bold"><mtext>B</mtext></mstyle><mo>=</mo><mo>∇</mo><mo>×</mo><mstyle mathvariant="bold"><mtext>A</mtext></mstyle></mrow></math> , where the scalar potential φ and vector potential A of the EM fields are calculated from the resources of charges Ze and currents Ze v. For details on EM field effects in HICs, we refer readers to [ 67 67 - 69 69 67-69 ]. Results and Discussions 1 In this section, we focus on the pion production in 108 Sn + 112 Sn and 132 Sn + 124 Sn reactions at 270 MeV/nucleon with an impact parameter of b =3 fm. To investigate the sensitivity of pion yields to the high-density behavior of E sym ( ρ ) (i.e., L ) and the momentum dependence of the symmetry potential (i.e., <math id="M40"><mrow><msubsup><mi>U</mi><mrow><mtext>sym</mtext></mrow><mi>∞</mi></msubsup><mo stretchy="false">(</mo><msub><mi>ρ</mi><mn>0</mn></msub><mo stretchy="false">)</mo></mrow></math> ), pion yields is regarded as a function of L for different <math id="M41"><mrow><msubsup><mi>U</mi><mrow><mtext>sym</mtext></mrow><mi>∞</mi></msubsup><mo stretchy="false">(</mo><msub><mi>ρ</mi><mn>0</mn></msub><mo stretchy="false">)</mo></mrow></math> values, as shown in Fig. 3 Fig. 3 . First, consistent with the findings in [ 34 34 , 70 70 ], it is observed that the multiplicities of π - are more sensitive to L than those of π + , particularly for the larger isospin asymmetry reactions 132 Sn + 124 Sn, because π - is mostly produced from the neutron-neutron inelastic collisions [ 70 70 ]. Second, with a certain L , the symmetry potential with a larger value of <math id="M42"><mrow><mrow><mo>|</mo><mrow><msubsup><mi>U</mi><mrow><mtext>sym</mtext></mrow><mi>∞</mi></msubsup><mo stretchy="false">(</mo><msub><mi>ρ</mi><mn>0</mn></msub><mo stretchy="false">)</mo></mrow><mo>|</mo></mrow></mrow></math> leads to an increased production of π - and π + . To understand this observation, we first check the evolution of the central region densities formed in the HICs. Figure 4 Figure 4 shows the evolutions of the central reduced densities ρ cent. / ρ 0 formed in 132 Sn + 124 Sn reactions with different <math id="M43"><mrow><msubsup><mi>U</mi><mrow><mtext>sym</mtext></mrow><mi>∞</mi></msubsup><mo stretchy="false">(</mo><msub><mi>ρ</mi><mn>0</mn></msub><mo stretchy="false">)</mo></mrow></math> but for a certain L of 62.7 MeV. For comparison, we also show the evolution of ρ cent. / ρ 0 for the same reaction with L =123.4 MeV and <math id="M44"><mrow><msubsup><mi>U</mi><mrow><mtext>sym</mtext></mrow><mi>∞</mi></msubsup><mo stretchy="false">(</mo><msub><mi>ρ</mi><mn>0</mn></msub><mo stretchy="false">)</mo><mo>=</mo><mo>−</mo><mn>240</mn></mrow></math> MeV. With a certain <math id="M45"><mrow><msubsup><mi>U</mi><mrow><mtext>sym</mtext></mrow><mi>∞</mi></msubsup><mo stretchy="false">(</mo><msub><mi>ρ</mi><mn>0</mn></msub><mo stretchy="false">)</mo><mo>=</mo><mo>−</mo><mn>240</mn></mrow></math> MeV, the soft symmetry energy with L =62.7 MeV leads to a higher compression compared to that with a stiff symmetry energy L =123.4 MeV, which agrees with previous observations in many studies. Interestingly, we noticed that for a certain L =62.7 MeV, <math id="M46"><mrow><msubsup><mi>U</mi><mrow><mtext>sym</mtext></mrow><mi>∞</mi></msubsup><mo stretchy="false">(</mo><msub><mi>ρ</mi><mn>0</mn></msub><mo stretchy="false">)</mo></mrow></math> also affects the evolution of the central region densities. Specifically, approximately at 13 fm/ c independent of <math id="M47"><mrow><msubsup><mi>U</mi><mrow><mtext>sym</mtext></mrow><mi>∞</mi></msubsup><mo stretchy="false">(</mo><msub><mi>ρ</mi><mn>0</mn></msub><mo stretchy="false">)</mo></mrow></math> , the reaction with a certain L =62.7 MeV approaches the maximum compression, generating a maximum compression density 1.5 ρ 0 in the central region; however, the decreasing velocity of this density is slightly faster in the case with a larger <math id="M48"><mrow><mrow><mo>|</mo><mrow><msubsup><mi>U</mi><mrow><mtext>sym</mtext></mrow><mi>∞</mi></msubsup><mo stretchy="false">(</mo><msub><mi>ρ</mi><mn>0</mn></msub><mo stretchy="false">)</mo></mrow><mo>|</mo></mrow></mrow></math> . This is because the symmetry potential with a larger <math id="M49"><mrow><mrow><mo>|</mo><mrow><msubsup><mi>U</mi><mrow><mtext>sym</mtext></mrow><mi>∞</mi></msubsup><mo stretchy="false">(</mo><msub><mi>ρ</mi><mn>0</mn></msub><mo stretchy="false">)</mo></mrow><mo>|</mo></mrow></mrow></math> causes some high-density nucleons to gain more acceleration in the subsequent reaction stages, leading to the densities of the compression region to reduce slightly faster. This can be demonstrated by checking the kinetic energy distribution of the nucleons in the compression region with local densities higher than ρ 0 at t =20 fm/ c , as shown in Fig. 5 Fig. 5 . Clearly, with a certain L =62.7 MeV but varying <math id="M50"><mrow><msubsup><mi>U</mi><mrow><mtext>sym</mtext></mrow><mi>∞</mi></msubsup><mo stretchy="false">(</mo><msub><mi>ρ</mi><mn>0</mn></msub><mo stretchy="false">)</mo></mrow></math> from -80 to -240 MeV, we observed increased high-energy nucleons but reduced low-energy nucleons. In general, because the scalar potential has the same repulsive effects on neutrons and protons, whereas the symmetry potential has repulsive (attractive) effects on the high density but low energy 1 1 neutrons (protons), these high-energy nucleons might be expected as neutrons. However, as illustrated in Fig. 6 Fig. 6 , these high-energy nucleons contain both neutrons and protons, and neutrons outnumber protons because the reaction is a neutron-rich system. Moreover, as indicated by the arrows in Fig. 5 Fig. 5 , the kinetic energies of these energetic neutrons and protons are greater than 150 MeV, whereas the threshold energy of pion production through NN inelastic collisions is not more than 300 MeV. Naturally, with a certain L but varying <math id="M51"><mrow><msubsup><mi>U</mi><mrow><mtext>sym</mtext></mrow><mi>∞</mi></msubsup><mo stretchy="false">(</mo><msub><mi>ρ</mi><mn>0</mn></msub><mo stretchy="false">)</mo></mrow></math> from -80 to -240 MeV, we can understand the increased production of both π - and π + shown in Fig. 3 Fig. 3 because π - and π + are produced mainly from inelastic <math id="M52"><mrow><mi>n</mi><mi>n</mi><mo>→</mo><mi>p</mi><mi>n</mi><msup><mi>π</mi><mo>−</mo></msup></mrow></math> and <math id="M53"><mrow><mi>p</mi><mi>p</mi><mo>→</mo><mi>p</mi><mi>n</mi><msup><mi>π</mi><mo>+</mo></msup></mrow></math> channels. Third, compared with the S π RIT data, as shown in Fig. 3 Fig. 3 , our prediction of pion multiplicities within a certain range of <math id="M54"><mrow><msubsup><mi>U</mi><mrow><mtext>sym</mtext></mrow><mi>∞</mi></msubsup><mo stretchy="false">(</mo><msub><mi>ρ</mi><mn>0</mn></msub><mo stretchy="false">)</mo></mrow></math> can fit the experimental data in general. For the predicted multiplicities, π + is not as good as π - . To date, one might wonder whether the symmetry potential at 1.5 ρ 0 (i.e., attainable maximum densities in the compression stage) with a certain L but varying <math id="M58"><mrow><msubsup><mi>U</mi><mrow><mtext>sym</mtext></mrow><mi>∞</mi></msubsup><mo stretchy="false">(</mo><msub><mi>ρ</mi><mn>0</mn></msub><mo stretchy="false">)</mo></mrow></math> from -80 to -240 MeV could cause both high-energy neutrons and protons to increase. To understand this observation, we illustrate it in the right panel of Fig. 7 Fig. 7 , which shows the symmetry potential at 1.5 ρ 0 with a certain L =62.7 MeV but different <math id="M59"><mrow><msubsup><mi>U</mi><mrow><mtext>sym</mtext></mrow><mi>∞</mi></msubsup><mo stretchy="false">(</mo><msub><mi>ρ</mi><mn>0</mn></msub><mo stretchy="false">)</mo></mrow></math> . For completeness, we also demonstrate the corresponding symmetry potential at low densities (i.e., 0.5 ρ 0 ), as shown in the left panel of Fig. 7 Fig. 7 . Similar to the symmetry potential at ρ 0 , the symmetry potentials at 1.5 ρ 0 even with different <math id="M60"><mrow><msubsup><mi>U</mi><mrow><mtext>sym</mtext></mrow><mi>∞</mi></msubsup><mo stretchy="false">(</mo><msub><mi>ρ</mi><mn>0</mn></msub><mo stretchy="false">)</mo></mrow></math> have the same value at approximately the nucleon kinetic energy of 47 MeV. In addition, the value of the symmetry potential also changes from positive to negative when the kinetic energy of the nucleon is larger than a certain value, depending on the value of <math id="M61"><mrow><msubsup><mi>U</mi><mrow><mtext>sym</mtext></mrow><mi>∞</mi></msubsup><mo stretchy="false">(</mo><msub><mi>ρ</mi><mn>0</mn></msub><mo stretchy="false">)</mo></mrow></math> . Specifically, with a certain L =62.7 MeV but varying <math id="M62"><mrow><msubsup><mi>U</mi><mrow><mtext>sym</mtext></mrow><mi>∞</mi></msubsup><mo stretchy="false">(</mo><msub><mi>ρ</mi><mn>0</mn></msub><mo stretchy="false">)</mo></mrow></math> from -80 to -240 MeV, protons (neutrons) in the high-density phase experience stronger attractive (repulsive) effects from the symmetry potentials when their kinetic energies are lower than 47 MeV. In contrast, if their kinetic energies are larger than 47 MeV but lower than approximately 81 MeV 2 2 , protons (neutrons) in the high-density phase will experience more weaker attractive (repulsive) effects from the symmetry potentials. Therefore, with a certain L but varying <math id="M64"><mrow><msubsup><mi>U</mi><mrow><mtext>sym</mtext></mrow><mi>∞</mi></msubsup><mo stretchy="false">(</mo><msub><mi>ρ</mi><mn>0</mn></msub><mo stretchy="false">)</mo></mrow></math> from -80 to -240 MeV, the repulsive scalar potential and the weakened attractive symmetry potential can cause some protons to increase their kinetic energies to 150 MeV and above. It should be emphasized that for a reaction with a certain <math id="M65"><mrow><msubsup><mi>U</mi><mrow><mtext>sym</mtext></mrow><mi>∞</mi></msubsup><mo stretchy="false">(</mo><msub><mi>ρ</mi><mn>0</mn></msub><mo stretchy="false">)</mo></mrow></math> , the effects of L on pion production can be interpreted through the density criterion (i.e., the average maximum densities formed in the reaction compression stages). Nevertheless, for the case with a certain L , interpreting the effects of <math id="M66"><mrow><msubsup><mi>U</mi><mrow><mtext>sym</mtext></mrow><mi>∞</mi></msubsup><mo stretchy="false">(</mo><msub><mi>ρ</mi><mn>0</mn></msub><mo stretchy="false">)</mo></mrow></math> on pion production requires both density and energy criteria because a small reduction in the average maximum densities formed in the reaction compression stages but a significant increase in the kinetic energy for these high-density nucleons could also lead to increased production of pions. Figure 8 Figure 8 shows the π - / π + ratios of the theoretical simulations for the same reactions in comparison with S π RIT data. First, consistent with the observations of most transport models, the upper windows of Fig. 8 Fig. 8 shows that the π - / π + ratios are indeed more sensitive to L than the pion yields, and a softer symmetry energy with a smaller L value leads to a higher π - / π + ratio. Moreover, for a more neutron-rich reaction 132 Sn + 124 Sn, the π - / π + ratios show a greater sensitivity to L . Second, from the lower windows in Fig. 8 Fig. 8 , the π - / π + ratios increase with <math id="M68"><mrow><mrow><mo>|</mo><mrow><msubsup><mi>U</mi><mrow><mtext>sym</mtext></mrow><mi>∞</mi></msubsup><mo stretchy="false">(</mo><msub><mi>ρ</mi><mn>0</mn></msub><mo stretchy="false">)</mo></mrow><mo>|</mo></mrow></mrow></math> for a certain L . Similar with the reason for more pion production, this observation can also be understood by examining the kinetic energy distribution of neutrons over protons n/p with local densities higher than ρ 0 at t =20 fm/ c in reactions 132 Sn + 124 Sn with a certain L , as shown in Fig. 9 Fig. 9 . It is observed that with a certain L =62.7 MeV, the ratio n/p increases as <math id="M69"><mrow><msubsup><mi>U</mi><mrow><mtext>sym</mtext></mrow><mi>∞</mi></msubsup><mo stretchy="false">(</mo><msub><mi>ρ</mi><mn>0</mn></msub><mo stretchy="false">)</mo></mrow></math> increases from -80 to -240 MeV, because the increment of high-energy neutrons is larger than that of protons for neutron-rich reactions. For this reason, we can observe that for a certain L the π - / π + ratios increase with the value of <math id="M70"><mrow><mrow><mo>|</mo><mrow><msubsup><mi>U</mi><mrow><mtext>sym</mtext></mrow><mi>∞</mi></msubsup><mo stretchy="false">(</mo><msub><mi>ρ</mi><mn>0</mn></msub><mo stretchy="false">)</mo></mrow><mo>|</mo></mrow></mrow></math> as shown in Fig. 8 Fig. 8 . In addition, compared with the S π RIT data, our results for π - / π + ratios also fit the experimental data quite well within a certain range for the value of <math id="M71"><mrow><msubsup><mi>U</mi><mrow><mtext>sym</mtext></mrow><mi>∞</mi></msubsup><mo stretchy="false">(</mo><msub><mi>ρ</mi><mn>0</mn></msub><mo stretchy="false">)</mo></mrow></math> . As a cleaner observable, the double ratio of the two reactions (i.e., the DR( π - / π + ) ratio of reactions 132 Sn + 124 Sn over 108 Sn + 112 Sn) has the advantage of reducing both the isoscalar potential effects and the Coulomb field effects; thus, it is expected to disentangle the effects of symmetry potential/energy from both isoscalar potentials and Coulomb fields in HICs. Therefore, in Fig. 10 Fig. 10 , we show the DR( π - / π + ) ratios of the two reactions in comparison with S π RIT data. The left panel of Fig. 10 Fig. 10 shows that the DR( π - / π + ) ratios of the two reactions are more sensitive to the high-density behavior of E sym ( ρ ). Moreover, the DR( π - / π + ) ratios are also more clearly separated by varying the value of <math id="M76"><mrow><msubsup><mi>U</mi><mrow><mtext>sym</mtext></mrow><mi>∞</mi></msubsup><mo stretchy="false">(</mo><msub><mi>ρ</mi><mn>0</mn></msub><mo stretchy="false">)</mo></mrow></math> from -80 to -240 MeV, making them more sensitive to the momentum dependence of the symmetry potential, as shown in the right panel of Fig. 10 Fig. 10 . Now, we attempted to use the above three observables (i.e., pion yields and their single π - / π + and double DR( π - / π + ) ratios) to constrain the values of <math id="M77"><mrow><msubsup><mi>U</mi><mrow><mtext>sym</mtext></mrow><mi>∞</mi></msubsup><mo stretchy="false">(</mo><msub><mi>ρ</mi><mn>0</mn></msub><mo stretchy="false">)</mo></mrow></math> and L . To this end, we perform systematic error analyses for pion yields and their single π - / π + and double DR( π - / π + ) ratios at different <math id="M78"><mrow><msubsup><mi>U</mi><mrow><mtext>sym</mtext></mrow><mi>∞</mi></msubsup><mo stretchy="false">(</mo><msub><mi>ρ</mi><mn>0</mn></msub><mo stretchy="false">)</mo></mrow></math> and L . Apart from pion yields as well as their single π - / π + and double DR( π - / π + ) ratios at <math id="M79"><mrow><msubsup><mi>U</mi><mrow><mtext>sym</mtext></mrow><mi>∞</mi></msubsup><mo stretchy="false">(</mo><msub><mi>ρ</mi><mn>0</mn></msub><mo stretchy="false">)</mo><mo>=</mo><mo>−</mo><mn>80,</mn><mo>−</mo><mn>160</mn></mrow></math> and -240 MeV and L =2.1, 32.4, 62.7, 93.1, and 123.4 MeV, the values of these observables at other <math id="M80"><mrow><msubsup><mi>U</mi><mrow><mtext>sym</mtext></mrow><mi>∞</mi></msubsup><mo stretchy="false">(</mo><msub><mi>ρ</mi><mn>0</mn></msub><mo stretchy="false">)</mo></mrow></math> and L with an interval of 10 MeV are obtained by interpolating the simulation ones. As shown in Fig. 11 Fig. 11 , the contours of the relative error are the two-dimensional function of L and <math id="M81"><mrow><msubsup><mi>U</mi><mrow><mtext>sym</mtext></mrow><mi>∞</mi></msubsup><mo stretchy="false">(</mo><msub><mi>ρ</mi><mn>0</mn></msub><mo stretchy="false">)</mo></mrow></math> . However, it is difficult to constrain the values of L and <math id="M82"><mrow><msubsup><mi>U</mi><mrow><mtext>sym</mtext></mrow><mi>∞</mi></msubsup><mo stretchy="false">(</mo><msub><mi>ρ</mi><mn>0</mn></msub><mo stretchy="false">)</mo></mrow></math> simultaneously from the relative errors between the theoretical simulations and experimental data. Therefore, we further performed systematic χ- square analyses for these observables at different L and <math id="M83"><mrow><msubsup><mi>U</mi><mrow><mtext>sym</mtext></mrow><mi>∞</mi></msubsup><mo stretchy="false">(</mo><msub><mi>ρ</mi><mn>0</mn></msub><mo stretchy="false">)</mo></mrow></math> , and the corresponding χ values as a two-dimensional function of L and <math id="M84"><mrow><msubsup><mi>U</mi><mrow><mtext>sym</mtext></mrow><mi>∞</mi></msubsup><mo stretchy="false">(</mo><msub><mi>ρ</mi><mn>0</mn></msub><mo stretchy="false">)</mo></mrow></math> are shown in Fig. 12 Fig. 12 . It is observed that there is an area with the highest confidence, in which <math id="M85"><mrow><msubsup><mi>U</mi><mrow><mtext>sym</mtext></mrow><mi>∞</mi></msubsup><mo stretchy="false">(</mo><msub><mi>ρ</mi><mn>0</mn></msub><mo stretchy="false">)</mo></mrow></math> is constrained to <math id="M86"><mrow><mo>−</mo><msubsup><mrow><mn>160</mn></mrow><mrow><mo>−</mo><mn>9</mn></mrow><mrow><mo>+</mo><mn>18</mn></mrow></msubsup></mrow></math> MeV, and the lower limit of L is approximately no less than 55 MeV. To further verify the above results, we select a value of -160 MeV for <math id="M87"><mrow><msubsup><mi>U</mi><mrow><mtext>sym</mtext></mrow><mi>∞</mi></msubsup><mo stretchy="false">(</mo><msub><mi>ρ</mi><mn>0</mn></msub><mo stretchy="false">)</mo></mrow></math> to check the spectral pion ratios of the reactions 108 Sn + 112 Sn and 132 Sn + 124 Sn because this observable, especially its high-energy parts, might be the best probe for the high-density behavior of E sym ( ρ ), as shown in [ 13 13 , 48 48 ]. Figure 13 Figure 13 shows the spectral pion ratios of the theoretical simulations in comparison with the corresponding data [ 13 13 ]. With this value for <math id="M88"><mrow><msubsup><mi>U</mi><mrow><mtext>sym</mtext></mrow><mi>∞</mi></msubsup><mo stretchy="false">(</mo><msub><mi>ρ</mi><mn>0</mn></msub><mo stretchy="false">)</mo></mrow></math> , the spectral pion ratios of our simulations, especially its high-energy parts, can fit fairly the experimental data when L ranges from 62.7 to 93.1 MeV. In addition, the pion observable in 197 Au + 197 Au collisions at 400 MeV/nucleon also supports this value for <math id="M89"><mrow><msubsup><mi>U</mi><mrow><mtext>sym</mtext></mrow><mi>∞</mi></msubsup><mo stretchy="false">(</mo><msub><mi>ρ</mi><mn>0</mn></msub><mo stretchy="false">)</mo></mrow></math> , as shown in Fig. 14 Fig. 14 . On the other hand, isospin splitting of the in-medium nucleon effective mass is resulting from the momentum dependence of the symmetry potential. Therefore, it is useful to evaluate the isospin splitting of the in-medium nucleon effective mass for <math id="M90"><mrow><msubsup><mi>U</mi><mrow><mtext>sym</mtext></mrow><mi>∞</mi></msubsup><mo stretchy="false">(</mo><msub><mi>ρ</mi><mn>0</mn></msub><mo stretchy="false">)</mo></mrow></math> . Based on the formula for the nucleon effective mass <math display="block" id="M91"><mrow><msubsup><mi>m</mi><mi>τ</mi><mo>*</mo></msubsup><mo>/</mo><mi>m</mi><mo>=</mo><mo stretchy="false">[</mo><mn>1</mn><mo>+</mo><mfrac><mi>m</mi><mrow><msub><mi>k</mi><mi>τ</mi></msub></mrow></mfrac><mfrac><mrow><mi>d</mi><msub><mi>U</mi><mi>τ</mi></msub></mrow><mrow><mi>d</mi><mi>k</mi></mrow></mfrac><msup><mo stretchy="false">]</mo><mrow><mo>−</mo><mn>1</mn></mrow></msup><mn>,</mn></mrow></math> for the used <math id="M92"><mrow><msubsup><mi>U</mi><mrow><mtext>sym</mtext></mrow><mi>∞</mi></msubsup><mo stretchy="false">(</mo><msub><mi>ρ</mi><mn>0</mn></msub><mo stretchy="false">)</mo><mo>=</mo><mo>−</mo><mn>10</mn></mrow></math> , -80, -160, and -240 MeV, the corresponding neutron-proton effective mass splittings <math id="M93"><mrow><mi>Δ</mi><msubsup><mi>m</mi><mrow><mtext>np</mtext></mrow><mo>*</mo></msubsup></mrow></math> are 0.178 δ , 0.384 δ , 0.622 δ , and 0.864 δ , respectively. So far, one can find that our results suggest a constraint on L (i.e., 62.7 < L < 93.1 MeV). This constraint on L is considerably close to the values 70 < L < 101 MeV extracted from charge exchange and elastic scattering reactions in [ 71 71 ] and within the constrained range of 42 < L < 117 MeV in [ 13 13 ]. Moreover, our results also suggest a constraint on <math id="M94"><mrow><msubsup><mi>U</mi><mrow><mtext>sym</mtext></mrow><mi>∞</mi></msubsup><mo stretchy="false">(</mo><msub><mi>ρ</mi><mn>0</mn></msub><mo stretchy="false">)</mo></mrow></math> (i.e., <math id="M95"><mrow><mo>−</mo><msubsup><mrow><mn>160</mn></mrow><mrow><mo>−</mo><mn>9</mn></mrow><mrow><mo>+</mo><mn>18</mn></mrow></msubsup></mrow></math> MeV). We also note that our extracted value for <math id="M96"><mrow><msubsup><mi>U</mi><mrow><mtext>sym</mtext></mrow><mi>∞</mi></msubsup><mo stretchy="false">(</mo><msub><mi>ρ</mi><mn>0</mn></msub><mo stretchy="false">)</mo></mrow></math> , for example, -160MeV, leads to a larger isospin splitting than the upper limit of 0.33 δ in [ 13 13 , 60 60 ]. There may be two reasons for this. First, authors of Ref. [ 13 13 ] used a more accurate criterion (i.e., E sym (2 ρ 0 /3)=25.5 MeV [ 72 72 ]), and thus considered the uncertainties of E sym ( ρ 0 ) (i.e., 32.5 MeV < E sym ( ρ 0 ) < 38.1 MeV). In this study, we use a fixed value 32.5 MeV for E sym ( ρ 0 ) as commonly used, and thus do not consider its uncertainties. Second, as the separate density-dependent scenario for in-medium nucleon-nucleon interaction has been used in this study, the corresponding potential energy density and single nucleon potential in Eqs. (3) and (4) and the corresponding expressions for A l and A u are different from those in [ 60 60 ]. These two aspects might be the reason for the difference between our extracted isospin splitting and that in [ 13 13 , 60 60 ]. As indicated in [ 73 73 , 74 74 ], the accurate inclusion of these effects might be important for transport model simulations, which may further improve our results (e.g., the deviations of the simulated π + from the corresponding data). Finally, it should be mentioned that our results in the present work are mainly based on S π RIT experiments. It will also be interesting to see how <math id="M97"><mrow><msubsup><mi>U</mi><mrow><mtext>sym</mtext></mrow><mi>∞</mi></msubsup><mo stretchy="false">(</mo><msub><mi>ρ</mi><mn>0</mn></msub><mo stretchy="false">)</mo></mrow></math> affects observables measured in other experiments, such as FOPI experiments [ 33 33 ] and ASY-EOS experiments [ 75 75 ]. Summary 1 In conclusion, we studied the effects of the momentum dependence of the symmetry potential on pion production in central Sn + Sn collisions at 270 MeV/nucleon. It was found that with a certain L , the characteristic parameter <math id="M98"><mrow><msubsup><mi>U</mi><mrow><mtext>sym</mtext></mrow><mi>∞</mi></msubsup><mo stretchy="false">(</mo><msub><mi>ρ</mi><mn>0</mn></msub><mo stretchy="false">)</mo></mrow></math> of the momentum-dependent symmetry potential significantly affected the production of π - and π + and their pion ratios. Moreover, by performing systematic analyses of these observables and comparing the spectral pion ratios of theoretical simulations with the experimental data, we found a constraint on L (i.e., 62.7 < L < 93.1 MeV). In addition, <math id="M99"><mrow><msubsup><mi>U</mi><mrow><mtext>sym</mtext></mrow><mi>∞</mi></msubsup><mo stretchy="false">(</mo><msub><mi>ρ</mi><mn>0</mn></msub><mo stretchy="false">)</mo></mrow></math> was constrained to <math id="M100"><mrow><mo>−</mo><msubsup><mrow><mn>160</mn></mrow><mrow><mo>−</mo><mn>9</mn></mrow><mrow><mo>+</mo><mn>18</mn></mrow></msubsup></mrow></math> MeV. Moreover, the pion observable for 197 Au + 197 Au collisions at 400 MeV/nucleon supported the extracted value for <math id="M101"><mrow><msubsup><mi>U</mi><mrow><mtext>sym</mtext></mrow><mi>∞</mi></msubsup><mo stretchy="false">(</mo><msub><mi>ρ</mi><mn>0</mn></msub><mo stretchy="false">)</mo></mrow></math> . Approximately at t =13 fm/ c , the reaction approaches maximum compression, the nucleons in compression region are naturally in dense but low energy phase. 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Rev. C 96 , 014612 ( 2017 ). doi: 10.1103/PhysRevC.96.014612 http://doi.org/10.1103/PhysRevC.96.014612 Proton and neutron density distributions at supranormal density in low- and medium-energy heavy-ion collisions P. Russotto , S. Gannon , S. Kupny et al. , Results of the ASY-EOS experiment at GSI: The symmetry energy at suprasaturation density . Phys. Rev. C 94 , 034608 ( 2016 ). doi: 10.1103/PhysRevC.94.034608 http://doi.org/10.1103/PhysRevC.94.034608 Results of the ASY-EOS experiment at GSI: The symmetry energy at suprasaturation density All authors contributed to the study conception and design. Material preparation, data collection and analysis were performed by Gao-Feng Wei, Xin Huang and Qi-Jun Zhi. The first draft of the manuscript was written by Gao-Feng Wei, and all authors commented on previous versions of the manuscript. All authors read and approved the final manuscript. Gao-Feng Wei would like to thank Profs. Bao-An Li and Gao-Chan Yong for their helpful discussions. 10.1007/s41365-022-01146-3 This work was supported by the National Natural Science Foundation of China (Nos. 11965008 and 11405128), the Guizhou Provincial Science and Technology Foundation (No. [2020] 1Y034), and the PhD-funded project of Guizhou Normal University (No. GZNUD [2018] 11). 2022-08-27 2022-09-27 2022-10-02 2022-12-12 2022-12-30 © China Science Publishing & Media Ltd. (Science Press), Shanghai Institute of Applied Physics, the Chinese Academy of Sciences, ChineseNuclear Society and Springer Nature Singapore Pte Ltd. 2022 This is a post-peer-review, pre-copyedit version of an article published in Nuclear Science and Techniques. The final authenticated version is available online at: http://dx.doi.org/10.1007/s41365-022-01146-3 http://dx.doi.org/10.1007/s41365-022-01146-3 . 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