Effects of density-dependent scenarios of in-medium nucleon-nucleon interactions in heavy-ion collisions
Gao-Feng Wei, Chang Xu, Wei Xie, Qi-Jun Zhi, Shi-Guo Chen, Zheng-Wen Long
aa r X i v : . [ nu c l - t h ] A ug Effects of density-dependent scenarios of in-medium nucleon-nucleon interactions inheavy-ion collisions
Gao-Feng Wei,
1, 2, ∗ Chang Xu, † Wei Xie,
1, 2
Qi-Jun Zhi,
1, 2
Shi-Guo Chen, and Zheng-Wen Long 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 School of Physics, Nanjing University, Nanjing 210008, China College of Physics, Guizhou University, Guiyang 550025, China
Using a more reasonable separate density-dependent scenario instead of the total density-dependent scenario for in-medium nn , pp and np interactions, we examine effects of differencesof in-medium nucleon-nucleon interactions in two density-dependent scenarios on isospin-sensitiveobservables in central Au+
Au collisions at 400 MeV/nucleon. It is shown that the symme-try potentials and resulting symmetry energies in two density-dependent scenarios indeed becometo deviate at nonsaturation densities, especially at suprasaturation densities. Naturally, severaltypical isospin-sensitive observables such as the free neutron-proton ratios and the π − /π + ratiosin heavy-ion collisions are affected significantly. Moreover, to more physically detect the differ-ences between the nucleon-nucleon interactions in two density-dependent scenarios, we also mapthe nucleon-nucleon interaction in the separate density-dependent scenario into that in the totaldensity-dependent scenario through fitting the identical constraints for symmetric nuclear matteras well as the identical slope parameter of nuclear symmetry energy at the saturation density. It isshown that two density-dependent scenarios also lead to essentially different symmetry potentialsespecially at high densities although they can lead to the identical equation of state for the sym-metry nuclear matter as well as the identical symmetry energy for the isospin asymmetric nuclearmatter. Consequently, these isospin-sensitive observables are also appreciably affected by the differ-ent density-dependent scenarios of in-medium nucleon-nucleon interactions. Therefore, according tothese findings, it is suggested that effects of the separate density-dependent scenario of in-mediumnucleon-nucleon interactions should be taken into account when probing the high-density symmetryenergy using these isospin-sensitive observables in heavy-ion collisions. I. INTRODUCTION
Simulations of heavy-ion collisions (HICs) as well ascomparisons with the corresponding experiments providean important tool to explore the properties of strong in-teracting nucleonic matter at extreme conditions. As theimportant inputs in simulations of HICs, the density-dependent nucleon-nucleon interactions as well as the re-sulting nuclear mean field have been paid much atten-tion in the past few decades [1–12]. However, the nuclearmean field especially its isovector part, i.e., the symme-try potential, is still incompletely understood at present.Essentially, the symmetry potential is determined bythe competition between the isospin singlet and isospintriplet channels of nucleon-nucleon interactions [13–15].The symmetry potential is also found to be sensitiveto the in-medium effects such as the in-medium nucleareffective many-body force and the tensor force due tothe in-medium ρ -meson exchange [16]. In nonrelativisticmodels, the in-medium many-body forces effects are usu-ally taken into account by a density-dependent term inthe two-body effective interactions [17–19], and the rel-ativistic models generate the similar density-dependentterm in the two-body effective interactions as in the non- ∗ E-mail: [email protected] † E-mail: [email protected] relativistic models through dressing of the in-mediumspinors [20]. Nevertheless, how exactly does the resultingtwo-body effective interaction depend on the in-mediumnucleon densities remains an open question. For exam-ple, a total density-dependent scenario without distin-guishing the density dependence for in-medium nn , pp and np interactions is usually assumed in the Skyrme,M3Y and Gogny forces and then adopted in some the-oretical simulations of HICs [21–27]. However, withinthe Brueckner theory [28–30], Brueckner and Dabrowskipointed out that the G-matrix of nucleon-nucleon inter-actions depends strongly on the respective Fermi mo-menta of neutrons and protons in isospin asymmetric nu-clear matter. Actually, the separate density-dependentscenario for in-medium nucleon-nucleon interactions hasbeen used in studying the structure of finite nuclei aswell as properties of infinite nuclear matter by some au-thors such as Negele [31], Sprung and Banerjee [32] aswell as Brueckner and Dabrowski [28–30]. Of partic-ular interest, authors in Refs. [33] and [34] employingthe Gogny effective interactions, respectively, studied ef-fects of the separate density dependence of in-mediumnucleon-nucleon interactions on the symmetry potentialand energy, they found consistently that the resultingsymmetry potential and energy at nonsaturation densi-ties in the separate density-dependent scenario indeedbecome to deviate significantly from those in the totaldensity-dependent scenario. Stimulated by these studies,we examine effects of differences of in-medium nucleon-nucleon interactions in these two density-dependent sce-narios on isospin-sensitive observables in HICs at in-termediate energies. The main purpose of this articleis to answer whether one needs to consider the sepa-rate density-dependent scenario for in-medium nucleon-nucleon interactions in HICs, especially for probing thedensity-dependent nuclear symmetry energy using theseisospin-sensitive observables in HICs, which is seldomconsidered in simulations of HICs to our best knowledge. II. THE MODEL
For completeness, we first recall the total density-dependent scenario for in-medium nucleon-nucleon inter-actions according to the original Gogny effective interac-tion [35], v ( r ) = X i =1 , ( W + BP σ − HP τ − M P σ P τ ) i e − r /µ i + t (1 + x P σ ) (cid:2) ρ (cid:0) r i + r j (cid:1)(cid:3) α δ ( r ij ) , (1)where W , B , H , M , and µ are five parameters, and P τ and P σ are the isospin and spin exchange operators,respectively; while α is the density-dependent parame-ter used to mimic in-medium effects of the many-bodyinteractions, particularly, the case with α = 1 corre-sponds to an effective density-dependent two-body in-teraction deduced from a three-body contact interactionin spin-saturated nuclear matter [17, 34]. Based on theHatree-Fock approximation using the original Gogny ef-fective interaction, i.e., Eq. (1), Das et al . derived amomentum-dependent interaction (MDI) single-nucleonpotential for the Boltzmann-Uehling-Uhlenbeck (BUU)transport model expressed as [6, 36] U ( ρ, δ, ~p, τ ) = A u ( x ) ρ − τ ρ + A l ( x ) ρ τ ρ + B ( ρρ ) σ (1 − xδ ) − τ x Bσ + 1 ρ σ − ρ σ δρ − τ + 2 C l ρ Z d p ′ f τ ( ~p ′ )1 + ( ~p − ~p ′ ) / Λ + 2 C u ρ Z d p ′ f − τ ( ~p ′ )1 + ( ~p − ~p ′ ) / Λ , (2)where τ = 1 / − / A u ( x ) and A l ( x ) are determined as A u ( x ) = − . − x Bσ + 1 , A l ( x ) = − .
57 + x Bσ + 1 . (3)Here, the parameter x is related to the spin(isospin)-dependent parameter x via x = (1 + 2 x ) / ∝ (1 + x ) ρ α +1 ] and triplet chan-nel [ ∝ (1 − x ) ρ α +1 ] [35]. Therefore, varying the x pa-rameter can cover uncertainties of the spin(isospin) de-pendence of in-medium many-body forces which are re-sponsible for the divergent density dependence of nu-clear symmetry energy in the Gogny Hatree-Fock calcu-lations [7, 15, 34]. However, it should be emphasizedthat the x parameter does not affect the equation ofstate of symmetric nuclear matter as well as the sym-metry energy at the saturation density due to the con-tributions of different channels are cancelled out exactly,i.e., ∝ (1 + x ) ρ α +1 + (1 − x ) ρ α +1 = 2 ρ α +1 . The pa-rameters B = 106 .
35 MeV and σ = 4 / t and α in theoriginal Gogny effective interactions via t = Bσ +1 1 ρ σ and σ = α + 1 [15, 33]. While C u = − . C l = − . τ interacting, respectively,with unlike and like nucleons in the nuclear matter, andthus account for the momentum dependence of the single-nucleon potential. These parameters are all obtained byfitting the reached consensuses on properties of nuclearmatter at the saturation density ρ = 0 .
16 fm − includ-ing the binding energy E ( ρ ) = −
16 MeV, the incom-pressibility K = 212 MeV for symmetric nuclear matter,as well as the symmetry energy E sym ( ρ ) = 30 . V D = t (1 + x P σ )[ ρ τ i ( r i )+ ρ τ j ( r j )] α δ ( r ij ) . (4)Here, the interaction explicitly depends on densities oftwo nucleons at positions r i and r j instead of the to-tal density of two-nucleon central position ( r i + r j ) / U ′ ( ρ, δ, ~p, τ ) = A ′ u ( x ) ρ − τ ρ + A ′ l ( x ) ρ τ ρ + B (cid:0) ρ τ ρ (cid:1) σ (1 − x )+ 2 Bσ + 1 (cid:0) ρρ (cid:1) σ (1 + x ) ρ − τ ρ (cid:2) σ − ρ τ ρ (cid:3) + 2 C l ρ Z d p ′ f τ ( ~p ′ )1 + ( ~p − ~p ′ ) / Λ + 2 C u ρ Z d p ′ f − τ ( ~p ′ )1 + ( ~p − ~p ′ ) / Λ , (5)and the corresponding parameters A u ( x ) and A l ( x ) are E sy m ( M e V ) MDI(-1) & IMDI(-0.5874) MDI(0) & IMDI(0.2063) MDI(1) & IMDI(1) I M D I ( ) I M D I (- ) FIG. 1. (Color online) The density dependencies of nuclearsymmetry energies calculated from the MDI and IMDI single-nucleon potentials. changed as A ′ u ( x ) = − . − Bσ + 1 (cid:2) − σ − (1 − x ) (cid:3) , (6) A ′ l ( x ) = − .
57 + 2 Bσ + 1 (cid:2) − σ − (1 − x ) (cid:3) . (7)It should be mentioned that the properties of symmetricnuclear matter are not changed from the MDI interactionto the IMDI interaction due to the isospin scalar poten-tials U ( ρ, , ~p, τ ) = U ′ ( ρ, , ~p, τ ) by setting δ = 0 and ρ n = ρ p = ρ . While for the isospin asymmetric nuclearmatter, the properties are expected to change from theMDI interaction to the IMDI interaction.Shown in Fig. 1 are the density dependencies of nuclearsymmetry energy calculated from the MDI and IMDIsingle-nucleon potentials. For parameter x = 1, it isseen that the symmetry energy calculated from the IMDIsingle-nucleon potential is the same as that calculatedfrom the MDI single-nucleon potential. This is becausethe fourth term in the IMDI single-nucleon potential iszero with x = 1 while other terms are unchanged as in theMDI single-nucleon potential. However, for parametersof x = − ρ , the correspondingresults are also shown in Fig. 1. It is seen that the symme-try energy calculated from the IMDI single-nucleon po-tential with mapped parameters x = − . . x = − ρ = 0 . ρ , ρ , 1 . ρ and 2 ρ calcu-lated from the MDI and IMDI single-nucleon potentialswith parameters x = −
1, 0 and 1 as well as the mappedparameters x = − . . x = 1,the symmetry potentials are completely identical to eachother in calculations using the MDI and IMDI single-nucleon potentials at either low densities or high densi-ties. While for parameter x = −
1, the symmetry po-tentials are significantly stronger especially at high den-sities in calculations using the IMDI single-nucleon po-tential compared to those in calculations using the MDIsingle-nucleon potential. However, for the case of pa-rameter x = 0, the symmetry potentials calculated fromthe IMDI single-nucleon potential change from strong toweak (weak to strong) at suprasaturation (subsaturation)densities with the increase of nucleons momenta, com-pared to those calculated from the MDI single-nucleonpotential. As to the mapped symmetry potentials cal-culated from the IMDI single-nucleon potential with pa-rameters x = − . . x = − E sym ( ρ ) ≈ t ( k F ) + 16 ∂U ∂k | k F k F + 12 U sym ( k F ) , (8)with t ( k ) denotes the nucleon kinetic energy and k F represents the Fermi momentum of nucleons in sym-metry nuclear matter, one can know that these aremainly due to the differences between the symmetrypotential calculated from the MDI single-nucleon po-tential and the mapped symmetry potential calculatedfrom the IMDI single-nucleon potential. Therefore, ef-fects of two density-dependent scenarios for in-mediumnucleon-nucleon interactions on isospin-sensitive observ-ables in HICs can be reflected through examining ef-fects of the symmetry potential calculated from the MDIsingle-nucleon potential and the mapped symmetry po-tential calculated from the IMDI single-nucleon potentialon these observables in HICs. Nevertheless, it should beemphasized that the differences between the MDI sym-metry potential and the IMDI mapped symmetry po-tential are essentially resulting from different density-dependent scenarios because the momentum-dependent -20-10010203040 0 1 2 3 4-40-200204060 0 1 2 3 4 IMDI(0)IMDI(-1)
MDI(-1) IMDI(-0.5874) MDI(0) IMDI(0.2063) MDI(1) IMDI(1) (a) U sy m ( M e V ) IMDI(0) IMDI(-1) (b) -100102030
IMDI(0) IMDI(-1) (c) p (GeV/c)
IMDI(0)IMDI(-1) (d) p (GeV/c) -80-4004080
FIG. 2. (Color online) The momentum dependencies of symmetry potentials calculated from the MDI and IMDI single-nucleonpotentials. but the x parameter independent C terms in Eq. (5) arecompletely identical to that in Eq. (2). As a result, ac-cording to the formula of nucleon effective mass, i.e., m ∗ τ /m = (cid:2) mk τ dU τ dk (cid:3) − , (9)which is only related to the C terms in Eq. (2) andEq. (5), one can know that the nucleon effective massas well as its isospin splitting are not changed from theMDI interaction to the IMDI interaction. III. RESULTS AND DISCUSSIONS
Now, we compare effects of the symmetry potential cal-culated from the MDI single-nucleon potential and themapped symmetry potential calculated from the IMDIsingle-nucleon potential on isospin-sensitive observablesin HICs. As comparisons, we also include the correspond-ing results calculated from the IMDI single-nucleon po-tential with parameters x = − ( n / p ) f r ee t (fm/c) MDI(-1) MDI(0) MDI(1) IMDI(-0.5874) IMDI(0.2063) IMDI(1)IMDI(-1)IMDI(0)
FIG. 3. (Color online) Evolutions of free neutron-proton ra-tios in calculations with the MDI and IMDI single-nucleonpotentials.
Show in Fig. 3 are the free neutron-proton ratiosgenerated in central
Au +
Au collisions at 400MeV/nucleon where the free neutrons and protons are de-fined as those with local densities less than ρ /
8. First,as expected, with parameter x = 1, the free neutron-proton ratio generated in simulations using the IMDIsingle-nucleon potential is completely identical with thatin simulations using the MDI single-nucleon potential.Second, with parameter x = − x = 0,the competition of symmetry potentials at high densitiesbetween low nucleon momentum and high nucleon mo-mentum as aforementioned causes the observed effectsto be not so obvious as those in the case of parameter x = −
1. However, it should be emphasized that, besidesthe different density-dependent scenarios of in-mediumnucleon-nucleon interactions, these effects are also result-ing from the different symmetry energy settings becausethe slope values L of nuclear symmetry energy at ρ arecompletely different although the identical parameters x are used in the MDI and IMDI single-nucleon poten-tials. Therefore, to more physically detect the effectsof differences of in-medium nucleon-nucleon interactionsin two density-dependent scenarios on the free neutron-proton ratios, we compare the free neutron-proton ratiosgenerated in calculations using the MDI single-nucleonpotential with parameters x = − x = − . . x = − x = − . . x = − x = − . . x = − x parameter set-tings. Naturally, these different single-nucleon potentialsdominate the different reaction dynamics as well as dif-ferent free neutron-proton ratios which thus indirectlyreflect different symmetry energy settings. This can beconfirmed by comparing the reduced maximum densities ρ max /ρ reached at the maximum compression stages incollisions with the IMDI single-nucleon potential under
10 15 20 25 302.202.222.242.26 m a x f m - t (fm/c) MDI(-1) IMDI(-0.5874) IMDI(-1)
FIG. 4. (Color online) Evolutions of reduced maximum den-sities ρ max /ρ reached at the maximum compression stages incollisions with the MDI and IMDI single-nucleon potentials. setting two different parameters x = − x = − . x = − x = − . ρ max /ρ are also differ-ent although their corresponding symmetry energies arecompletely identical as shown in Fig. 1. Actually, ac-cording to the formula (8) as well as the Fig. 2, this isexactly the difference of in-medium nucleon-nucleon in-teractions in two density-dependent scenarios that leadsto the different ρ max /ρ in collisions, and thus gener-ates different free neutron-proton ratios. Correspond-ingly, the kinetic energy distributions of free neutron-proton ratios at the end of reactions are also affected bythe density-dependent scenarios of in-medium nucleon-nucleon interactions as shown in Fig. 5. Therefore, effectsof the separate density-dependent scenario of in-mediumnucleon-nucleon interactions should be carefully consid-ered in studies of using the free neutron-proton ratio asa probe of nuclear symmetry energy especially at highdensities.On the other hand, according to the production mecha-nism of pions, i.e., pions are produced mainly at the com-pression stages during collisions and π − is mainly from nn inelastic collisions but π + mainly from pp inelastic col-lisions, one naturally expects that the effects of density-dependent scenarios of in-medium nucleon-nucleon inter-actions also hold for the π − /π + ratio, which has beenindicated to be very sensitive to the symmetry energyand potential at high densities [37–41] but still affectedby some incompletely known uncertainties [42–50]. InHICs at intermediate energies, pions are produced duringcollisions mostly from the decay of ∆(1232), therefore,it is useful to examine the effects of density-dependentscenarios of in-medium nucleon-nucleon interactions on MDI(-1) MDI(0) MDI(1) IMDI(-0.5874) IMDI(0.2063) IMDI(1) ( n / p ) f r ee E kin (MeV) IMDI(-1)IMDI(0)
FIG. 5. (Color online) Kinetic energy distributions of freeneutron-proton ratios at the end of reactions with the MDIand IMDI single-nucleon potentials.
IMDI(0)IMDI(-1) () li k e t (fm/c) MDI(-1) IMDI(-0.5874) MDI(0) IMDI(0.2063) MDI(1) IMDI(1)
FIG. 6. (Color online) Evolutions of π − /π + ratios in calcula-tions with the MDI and IMDI single-nucleon potentials. dynamic pion ratio ( π − /π + ) like , i.e.,( π − /π + ) like = π − + ∆ − + ∆ π + + ∆ ++ + ∆ + . (10)Certainly, because all the ∆ resonances will eventuallydecay into nucleons and pions, the ratio ( π − /π + ) like willnaturally become to the free π − /π + ratio at the end of re-actions. Shown in Fig. 6 are the evolutions of ( π − /π + ) like ratios generated in central Au +
Au collisions at400 MeV/nucleon. First, it is obvious to see that the π − /π + ratio is indeed sensitive to the density depen-dence of nuclear symmetry energy regardless of using theIMDI single-nucleon potential or the MDI single-nucleonpotential, and a softer symmetry energy usually leads toa higher π − /π + ratio, reflecting a more neutron-rich par-ticipant region formed in the reaction [37–41]. Second,with parameter x = 1, the π − /π + ratio calculated fromthe IMDI single-nucleon potential is completely identicalwith that calculated from the MDI single-nucleon poten-tial. Third, for the case of identical parameter x = − π − /π + ratio in calculations us-ing the IMDI single-nucleon potential to be significantlysmaller than that in calculations using the MDI single-nucleon potential. Actually, due to the stronger posi-tive symmetry potential with parameter x = − π − is produced mainly from the channel n + n → π − + p but π + from the channel p + p → π + + p ,we can observe a smaller π − /π + ratios in calculations us-ing the IMDI single-nucleon potential. Certainly, due tothe symmetry potential calculated from the IMDI single-nucleon potential with parameter x = 0 changes fromstrong to weak at suprasaturation densities with the in-crease of nucleons momenta compared to that calculatedfrom the MDI single-nucleon potential with parameter x = 0, we can also see that the differences of π − /π + ratiosin this case are not as larger as those in the case of param-eter x = −
1. Again, these effects are resulting from bothdifferent density-dependent scenarios and different sym-metry energy settings. While comparing the π − /π + ra-tios in calculations using the IMDI single-nucleon poten-tial with mapped parameters x = − . . x = − x = − . x = − π − /π + ratios are relative appre-ciable. Certainly, it should be emphasized that this effectis independent of nuclear symmetry energy but is exactlyresulting from different density-dependent scenarios of in-medium nucleon-nucleon interactions. Therefore, effectsof the separate density-dependent scenario of in-mediumnucleon-nucleon interactions should also be carefully con-sidered in studies of using the π − /π + ratio as a probe ofnuclear symmetry energy especially at high densities.Before ending this part, we give two useful remarks.First, although there are currently no physical stud-ies based on first principles to illustrate more accuracyof the separate density-dependent scenario, some resultsrelevant to nuclear structure studies have been shown toyield very satisfactory agreement with the correspondingexperiments such as the binding energies, single-particleenergies, and electron scattering cross sections for O, Ca, Ca, Zr, and
Pr [31, 51]. Moreover, as in-dicated in Ref. [51], the separate density dependenceof effective two-body interactions is originated from therenormalization of multibody force effects, and the lattermay extend the density dependence of effective interac-tions for calculations beyond the mean-field approxima-tion and open a new freedom in the effective interactions.Second, it is well known that the double neutron-proton and/or π − /π + ratios from two reaction systemshave the advantage of reducing both systematic errorsand the influences of isoscalar potentials in HICs [52, 53].This could enlarge the contribution of the isovector po-tentials and better discriminate between the two sce-narios. Therefore, the double ratios of these observ-ables from two reactions as well as the cross examina-tions of these observables using various experimental datasuch as the FOPI data [40] and that from the symme-try energy measurement experiment at RIBF-RIKEN inJapan [54] could be good candidates in probing the ef-fects of density-dependent scenarios in HICs in future. IV. SUMMARY
In conclusion, we have studied effects of differencesof in-medium nucleon-nucleon interactions in the sepa-rate and total density-dependent scenarios on isospin-sensitive observables in HICs within a transport model.Consistent with the previous studies, the nuclear sym-metry energy and potential at nonsaturation densities inthe separate density-dependent scenario indeed becometo deviate significantly from those in the total density-dependent scenario for the identical x parameters exceptfor the parameter x = 1. Two typical isospin-sensitiveobservables including the free neutron-proton ratios andthe π − /π + ratios in HICs are affected significantly. Nev-ertheless, it should be emphasized that these effectsare resulting from both the different symmetry energy settings and the different density-dependent scenariosof in-medium nucleon-nucleon interactions. Therefore,to more physically detect the differences of in-mediumnucleon-nucleon interactions as well as the resulting sym-metry potential in two density-dependent scenarios, wehave also mapped the nucleon-nucleon interaction in theseparate density-dependent scenario into that in the totaldensity-dependent scenario through fitting the identicalconstraints for symmetric nuclear matter as well as theidentical slope parameter of symmetry energy at the sat-uration density. It is shown that the mapped symmetrypotentials calculated from the IMDI single-nucleon po-tential indeed deviate from those in calculations using theMDI single-nucleon potential especially at high densities.Consequently, these isospin-sensitive observables in HICscould also be appreciably affected. Therefore, accordingto these findings as well as the Brueckner theory andprevious findings in nuclear structure studies, we con-clude that effects of the separate density-dependent sce-nario for in-medium nucleon-nucleon interactions mightbe very important and thus should be taken into accountwhen probing the high-density symmetry energy usingthese isospin-sensitive observables in HICs. ACKNOWLEDGMENTS
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