Gao?Feng Wei · Xin Huang · Qi?Jun Zhi · Ai?Jun Dong · Chang?Gen Peng · Zheng?Wen Long

Abstract 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 U∞sym(ρ0) (i.e., the value of the nuclear symmetry potential at the saturation density ρ0 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 ρ0 ), the characteristic parameter U∞sym(ρ0) 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 πRIT experiments, we found that our results favor a constraint on U∞sym(ρ0) (i.e., ?160+18?9 MeV), and L is also suggested within a range of 62.7 MeV

Keywords Nuclear symmetry potential · Momentum dependence · Symmetry energy

1 introduction

The equation of state (EoS) of asymmetric nuclear matter (ANM), especially its nuclear symmetry energyEsym(ρ) term, plays an essential role in studying the structure [1—4] and evolution of radioactive nuclei [5—8], as well as the synthesis of medium and heavy nuclei [9—12]. TheEsym(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 toEsym(ρ) , especially at densities of approximately twice the saturation densityρ0[13—20]. Nevertheless,knowledge of theEsym(ρ) at suprasaturation densities is still limited, although that around and belowρ0[21, 22] as well as the isospin-independent part of EoS for ANM (i.e., EoS of SNM [23—25]) are relatively well determined. Essentially,the EoS of the ANM and itsEsym(ρ) term are determined by the nuclear mean field, especially its isovector part (i.e., the symmetry/isovector potential [26, 27]). 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) chargeexchange reactions between isobaric analog states limited information of the isovector potential atρ0and parameterized asUsym(ρ0,Ek)=a?bEk, wherea≈22 ?34 MeV,b≈0.1 ?0.2 , andEkis limited to no more than 200 MeV[28—30].

Heavy-ion collision (HIC) is one of the most promising approaches for exploring the symmetry potential/energy, especially at supersaturation densities [3, 4, 13, 31—33]. Recently,the SπRIT collaboration reported results from the first measurement dedicated to probing theEsym(ρ) at suprasaturation densities through pion production in Sn + Sn collisions at 270 MeV/nucleon carried out at RIKEN-RIBF in Japan [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]. In this situation, author of Ref. [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—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 ofEsym(ρ) [41—43]. In fact, in [31]and in [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 Sect. 2. In Sect. 3, we discuss the results of the present study. Finally, a summary is presented in Sect. 4.

2 The model

This study was carried out using an isospin- and momentumdependent Boltzmann-Uehling-Uhlenbeck (IBUU) transport model. In the framework, the present model is originated from IBUU04 [49, 50] and/or IBUU11 [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—54] is expressed as

this equation is used to replace the density-dependent term of the original Gogny effective interaction [55], which is given by

whereW,B,H,M, andμare the five parameters,PτandPσare the isospin and spin exchange operators, respectively,andαis the density-dependent parameter used to mimic the medium effects of many-body interactions [52—54].As indicated in Ref. [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 for16O,40Ca,48Ca,90Zr, and208Pr[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] as

The upper windows in Fig. 1 show the kinetic energydependent neutron and proton potentials atρ0with

Fig. 1 (Color online) Kinetic energy-dependent neutron a and proton b potentials as well the isoscalar c and isovector d potentials at ρ0 calculated from the IMDI interaction. The Schr?dinger-equivalent isoscalar potential obtained by Hama et al. and the parameterized isovector potential from the experimental and/or empirical data are also shown to compare with the isoscalar and isovector potentials calculated from the IMDI interaction

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]. We adopt the pion potential of the form used in [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] is used. The present pion potential includes the isospin- and momentum-dependent pions- andp-wave potentials in a nuclear medium, as in[66] (see [62—65]).

Fig. 2 (Color online) Density dependence of the Esym(ρ) with different U∞sym(ρ0) calculated from the IMDI interaction

The in-medium isospin-dependent baryon-baryon elastic and inelastic scattering cross sectionsσmediumare determined by the corresponding free space cross sectionsσfreemultiplied by a factorRmedium, which is expressed as

where the reduced factor is determined asRmedium=(μ?BB∕μBB)2, andμBBandμ?BBare 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, E=??φ??A∕?t, B=?×A , where the scalar potentialφand vector potential A of the EM fields are calculated from the resources of chargesZeand currentsZev . For details on EM field effects in HICs, we refer readers to [67—69].

3 Results and discussion

Fig. 3 (Color online) Upper: Multiplicities of π? generated in reactions 108 Sn + 112 Sn a and 132 Sn + 124 Sn b with different U∞sym(ρ0) as a function of L in comparison with the corresponding S πRIT data.Lower: Multiplicities of π+ generated in reactions 108 Sn + 112 Sn c and 132 Sn + 124 Sn d with different U∞sym(ρ0) as a function of L in comparison with the corresponding S πRIT data

Fig. 4 (Color online) Evolution of the reduced average densities in central region ( ρcent.∕ρ0 ) produced in 132 Sn + 124 Sn reactions at 270 MeV/nucleon

mostly produced from the neutron—neutron inelastic collisions [70]. Second, with a certainL, the symmetry

Fig. 5 (Color online) Kinetic energy distribution of nucleons in compression region at t=20 fm/c in 132 Sn + 124 Sn reactions at 270 MeV/nucleon

Fig. 6 (Color online) Kinetic energy distribution of neutrons a and protons b in compression region at t=20 fm/c in 132 Sn + 124 Sn reactions at 270 MeV/nucleon

Fig. 7 (Color online) Kinetic energy-dependent symmetry potentials at ρ=0.5ρ0 a and ρ=1.5ρ0 b with different U∞sym(ρ0) calculated from the IMDI interaction. The values of symmetry potential at ρ=0.5ρ0 are multiplied by a factor of 2.5

Fig. 8 (Color online) Upper: Ratios of π?∕π+ generated in reactions 108 Sn + 112 Sn a and 132 Sn + 124 Sn b with different U∞sym(ρ0) as a function of L in comparison with the corresponding S πRIT data. Lower:Ratios of π?∕π+ generated in reactions 108 Sn + 112 Sn c and 132 Sn +124 Sn d with different L as a function of U∞sym(ρ0) in comparison with the corresponding S πRIT data

Fig. 9 (Color online) Kinetic energy distribution of neutrons over protons n/p with local densities higher than ρ0 produced at t=20 fm/c in the reaction 132 Sn + 124 Sn with different U∞sym(ρ0) and a certain L

Fig. 10 (Color online) The double π?∕π+ ratios [i.e., DR(π?∕π+ )] of the reactions 132 Sn + 124 Sn over 108 Sn + 112 Sn with different U∞sym(ρ0)as a function of L a and different L as a function of U∞sym(ρ0) b in comparison with the corresponding S πRIT data

Fig. 11 (Color online) Contours of the relative errors for pion yields and their single and double pion ratios as a function of L and(ρ0) in reactions 108 Sn+ 112 Sn and 132 Sn + 124Sn

Fig. 12 (Color online) The value of χ as a two-dimensional function of (ρ0) and L in reactions 108 Sn + 112 Sn and 132 Sn + 124Sn

Fig. 13 (Color online) The spectral pion ratios of theoretical simulations for the reactions 108 Sn + 112 Sn a and 132 Sn + 124 Sn b as a function of transverse momentum in comparison with the corresponding data

Fig. 14 (Color online) Multiplicities of charged pions a and their pion ratios b in 197 Au + 197 Au collisions at 400 MeV/nucleonin comparison with the corresponding data

4 Summary

AcknowledgementsGao-Feng Wei would like to thank Profs. Bao-An Li and Gao-Chan Yong for their helpful discussions.

Author ContributionsAll 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.