Pion production and absorption in heavy-ion collisions
aa r X i v : . [ nu c l - t h ] S e p Pion production and absorption in heavy-ion collisions
Yuan Gao,
1, 2, ∗ Lei Zhang, Gao-Chan Yong,
2, 3
Zi-Yu Liu, and Wei Zuo
2, 3 School of Information Engineering, Hangzhou Dianzi University, Hangzhou 310018, China Institute of Modern Physics, Chinese Academy of Sciences, Lanzhou 730000, China University of Chinese Academy of Sciences, Beijing 100049, China College of Physics and Electronic Engineering, Xianyang Normal University, Xianyang 712000, China
Based on the isospin dependent transport model IBUU, the pion production and its absorptionare thoroughly studied in the central collision of Au + Au at a beam energy of 400 MeV/nucleon.It is found that the pions are firstly produced by the hard ∆ decay at the average density around1 . ρ , whereas about 18% of them are absorbed absolutely in the subsequent inelastic collisions.For the free pions observed, more than half of them have been scattered for one or more times beforethey are free from matter. And the more scattering numbers of pions, the higher the momentumthey possess. These pions, due to longer time of their existence in high density nuclear matter, carrymore information about the symmetry energy of the nuclear matter at high densities. I. INTRODUCTION
Heavy-ion collisions (HIC) offer a unique possibility tostudy bulk properties of hot and compressed nuclear mat-ter. One of the main goals of such study is to determinethe density-dependence of the symmetry energy (SE) athigh densities [1–3]. The symmetry energy plays essentialroles in understanding a number of physical phenomenaand processes. However it can not be directly measuredin experiments and has to be extracted from observableswhich are sensitive to the symmetry energy[4].Pion production is a dominant feature in heavy-ioncollisions at intermediate energies[5–8]. In the collisionsnear the pion-production threshold the nuclear mattercan be compressed up to about 2 times normal density ρ before it expands again. During the compression stagethe pions are produced mainly from the decay of the ∆resonances created[9–11]. Therefore the pion productionis considered to be important for extracting the infor-mation of the properties of the nuclear matter at highdensities[12]. Since the charged pion ratio in heavy-ioncollisions was first suggested in Ref. [13], pion productionhas attracted much attention in pinning down the den-sity dependence of the symmetry energy. During the lastdecade, a lot of pionic observables have been proposed aspromising probes, such as the collective pion flows[14–16], the cone-azimuthal emission of charged pions[17],etc. However, comparing the theoretical results of the π − /π + ratio with the experimental data, analyses cameto rather conflicting conclusions on the stiffness of thenuclear symmetry energy at high densities[18–20].In the reactions near the threshold of pion-productionthe pions are mainly the decay products of the ∆ res-onances. However most of the them can be absorbedinto ∆ resonances because of the inelastic πN collisions,and then may decay into pions frequently, i.e. scatteringprocess[21–23]. The detail of the scattering process plays ∗ Electronic address: [email protected] an important role for extracting the information of thesymmetry energy at different densities. At the same time,it is commonly known that pions observed are producedat high densities in heavy-ion collisions. The quantitativestudy of the density at which pion produces, as well asthe pion absorption or its scattering, is seldom reportedyet. To further extract the information of compressedmatter by pionic observables, it is necessary to performa detailed analysis of the pions production and its ab-sorption in HIC.However, the information of the pion absorption or itsscattering process can not be extracted in experimentsso far but can be obtained in theoretical calculations. Inthis paper, a detailed statistical investigation of all theinelastic collisions in the reaction of Au + Au at 400MeV/nucleon beam energy is performed. All the inelas-tic collisions are recorded and investigated statistically.Further more, the charged pions at the freeze out are cat-egorized by their production and re-scattering processes.The analysis shows the pions are indeed produced at highdensities. Moreover about 60% of the pions observedhave re-absorption and re-decay processes after they wereproduced first time, and due to longer time of their ex-istance in high density nuclear matter, they carry moresubstantial information of the high-density behavior ofthe symmetry energy than those without any scatteringprocess, which are produced from hard ∆ decay directlyand free from matter immediately.
II. THE THEORETICAL MODEL
In the past decade the the isospin-dependentBoltzmann-Uehling- Uhlenbeck transport model (IBUU)have been very successful in describing the dynamicalevolution of nucleons in phase space, as well as the reac-tion dynamics of heavy-ion collisions[29–32]. The presentIBUU transport model originating from the IBUU04model can describe the time evolution of the single-particle phase-space distribution function, ∂f∂t + ∇ −→ p E − ∇ −→ R f = I c , (1)where I c is the collision item and f ( ~r, ~p, t ) is the phase-space distribution function which denotes the probabilityof finding a particle at time t with momentum ~p at posi-tion ~r . E denotes the total energy, ie, E = E kin + U . U is the mean-field potential of the single particle and canbe expressed as[33] U ( ρ, δ, p , τ ) = A u ( x ) ρ τ ′ ρ + A l ( x ) ρ τ ρ + B (cid:18) ρρ (cid:19) σ (cid:0) − xδ (cid:1) − xτ Bσ + 1 ρ σ − ρ σ δρ τ ′ + 2 C τ,τ ρ Z d p ′ f τ ( r , p ′ )1 + ( p − p ′ ) / Λ + 2 C τ,τ ′ ρ Z d p ′ f τ ′ ( r , p ′ )1 + ( p − p ′ ) / Λ , (2)where τ, τ ′ = ± / x denote the stiffness of thesymmetry energy. Varying the x , one can get differentforms of the symmetry energy predicted by various many-body theories without changing any property of symmet-ric nuclear matter and the value of symmetry energy atnormal density ρ . The parameters A u ( x ) , A l ( x ) are xdependent and defined as A u ( x ) = − . − Bσ + 1 x, (3) A l ( x ) = − .
57 + 2 Bσ + 1 x. (4)The parameter values are B = 106 .
35 MeV, σ =4/3. Λ = p F is the nucleon Fermi momentum in symmetric nuclearmatter, C τ,τ ′ = − . C τ,τ = − . C τ,τ ′ and C τ,τ terms are the momentum-dependentinteractions of a nucleon with unlike and like nucleonsin the surrounding nuclear matter. With this potentialwe can get binding energy -16 MeV and incompressibility211 MeV for symmetric nuclear matter and the symmetryenergy 31.5 MeV at saturation density. The resonance ∆potential is given by U ∆ − = U n ,U ∆ = 23 U n + 13 U p ,U ∆ + = 13 U n + 23 U p ,U ∆ ++ = U p . (5)In the present work, pions are produced via the de-cay of ∆ resonance. Near the pion-production threshold, the inelastic reaction channels as follows are taken intoaccount[34], N N → N ∆( hard ∆ production ) ,N ∆ → N N (∆ absorption ) , ∆ → N π (∆ decay ) ,N π → ∆( sof t ∆ production ) . (6)The free inelastic isospin decomposition cross sections are σ pp → n ∆ ++ = σ nn → p ∆ − = σ + 12 σ ,σ pp → p ∆ + = σ nn → n ∆ = 32 σ ,σ np → p ∆ = σ np → n ∆ + = 12 σ + 14 σ (7)The σ II ′ can be parametrized by σ II ′ ( √ s ) = π ( ~ c ) p α ( p r p ) β m Γ ( q/q ) ( s ∗ − m ) + m Γ . (8)Here the I and I ′ are the initial state and final stateisospins of two nucleons, for explicit details and parame-ters, see ref.[35]. The cross section for the two-body freeinverse reaction can be described by the modified detailedbalance, σ N ∆ → NN = m ∆ p f σ NN → N ∆ δ ) p i (cid:30) Z √ s − m N m π + m N dm ∆ π P ( m ∆ ) , (9)where p f and p i are the nucleon center of mass momentain the N N and N ∆ channels, respectively. P ( m ∆ ) isthe mass function of the ∆ produced in N N collisionand can be defined according to a modified Breit-Wignerfunction[36], P ( m ∆ ) = p f m ∆ × m Γ ∆ ( m − m ) + m Γ . (10)Here m ∆0 denotes the centroid of the resonance and Γ ∆ is the width of the resonance ∆. Assuming the ∆ beproduced isotropically in the nucleon-nucleon center ofmass, and the decay of ∆ → πN with an isotropic angulardistribution in the ∆ rest frame, the width of ∆ resonancecan be given in a simplistic fashion[38],Γ ∆ = 0 . q m π [1 + 0 . q/m π ) ] . (11)The q is the pion momentum in the ∆ rest frame anddefined as q = s ( m − m n + m π m ∆ ) − m π , (12)The decay of the resonance into the nucleon and the pionis carried out by the Monte Carlo method for each timestep dt in our calculation, with the probability as P deacy = 1 − exp ( − dt Γ ∆ / ~ ) . (13)The meson-baryon interactions in our calculations aretreated via the formation of baryon resonances, and theBreit-Wigner form of resonance formation can be modi-fied as[37] σ π + N = σ max ( q q ) Γ ( m ∆ − m ∆0 ) + Γ , (14)where q represents the pion momentum at the centroid m ∆0 =1.232 GeV of the resonance mass distribution. Themaximum cross sections are given by[38–40] σ π + p → ∆ ++ max = σ π − n → ∆ − max = 200 mb,σ π − p → ∆ max = σ π + n → ∆ + max = 66 . mb,σ π p → ∆ + max = σ π n → ∆ max = 133 . mb. (15) III. RESULTS AND DISCUSSIONS
Transport theories have been very successful in de-scribing the reaction dynamics of heavy-ion collisions[24–28]. Due to their strong interaction with the nuclear en-vironment pionic observables at the freeze out are theresult of complex creation and rescattering processes. Inorder to obtain detail information of the pion productionand its absorption, we study all the inelastic collisionsin the central collision of Au + Au at a beam energy of400 MeV/nucleon within the frame work of IBUU. Firstly Au+
Au@400MeV
Hard ∆ production ∆ absorption Hard ∆ decay ∆ decay at the freeze-out d N / d ( ρ / ρ ) ρ / ρ FIG. 1: Local density distributions of the number of the in-elastic reaction in the central collision of Au + Au at a beamenergy of 400 MeV/nucleon. we investigate the numbers of different inelastic reactionsand the densities at which they take place, which is shownin Fig. 1. By comparing the solid line and the dash line,it can be seen that only 33% of hard ∆ can decay into pions and the rest of them are subsequently absorbedinto nucleon without any decay. Furthermore, due to thelow production threshold, pions are reabsorbed and re-produced quite frequently. About 18% of the pions fromhard ∆ will be absorbed thoroughly, and the rest are tobe free particles ultimately, but probably having one ormore scattering process ( πN → ∆ → πN ) before theyare detected.In Fig. 1 we can also see that the reaction N N → N ∆ takes place at the average density around 1 . ρ ,and in the almost same range of the density, the hard ∆decay into pions. Most of the pions from hard ∆ decaywill be scattered in the evolution. Due to the scatteringprocess, the ∆ decays into free pions at the freeze-outtake place in a wide density range. Nevertheless, theaverage density is up to 1.5 ρ . Therefore it is reasonablefor pion production to probe the properties of the nuclearmatter at high densities.Fig. 2 shows the evolution of the inelastic collisionnumber in the reaction. As common known, the N N inelastic collisions and the hard ∆ decay take place inthe early stage. The hard ∆ mostly produces at the timeof about 15 f m/c , and after after 30 f m/c , there are al-most no new hard ∆ produced. It can also be estimatedFig. 2 that the absorptivity of the pions is about 18%.
Hard ∆ production ∆ absorption Hard ∆ decay ∆ decay at the freeze-out Au+
Au@400MeV d N / d t ( f m / c ) - t (fm/c) FIG. 2: Time evolution of the number of the inelastic reactionin the central collision of Au + Au at a beam energy of 400MeV/nucleon. In the following, we focus on the charge pions at thefreeze out because of their advantage in the detectoracceptance. With the analysis of the complex collisionhistory, we classified the free charge pions, according tothe πN rescattering number in their history. Fraction ofcharged free pions originating from specific rescatteringprocess are plotted in Fig. 3. Here the abscissas are thecycle-index of the πN scattering of the collision historyfor the detectable pions. For example, 0 on the abscissa is Scattering Number
Free π - F r a c t i on ( % ) Free π + FIG. 3: Fraction of the different types of free pions categorizedby their scattering numbers. corresponding the pions without any absorption processafter they are produced from hard ∆ decay, 1 correspond-ing the pions have been πN scattered for one time, .ie. bythe channel N N → N ∆ → N N π → N ∆ → N N π free ,and so on.It can been seen that most of the free pions have beenabsorbed into ∆ and then re-decay into pions after theyare firstly produced by hard ∆ decay. Our calculationshows less than 40% of the detectable negative pions areseen to freeze out as soon as they are produced by hard∆ decay, without any πN scattering. It can been alsoseen that at least 5% of the negative pions have beenscattered for at least five times before they are detected.For positive pions, the fraction of those without πN scattering process is about 44%, larger than that of thenegative ones, as a result of the Coulomb potential fromprotons, i.e., negative pions are attracted to while posi-tive ones are repelled away.We next investigate the pion creation history, and fo-cus on the two processes. One is the first formation, .ie.the decay of the hard ∆ into pion, another is last for-mation, .ie. the ∆ decay into free pion at the freeze-out.Fig. 4 shows the average density at which the first andlast creation take place, for different categories of freepion which is categorised in Fig. 3. As shown the av-erage density of the first formation for all categories isapparently above the normal nuclear density. The pionswith more scattering processes are mostly produced fromhard ∆ decay at higher density. The average density ofthe last formation of course is much lower than that ofthe first formation. Nevertheless, it is still above the nor-mal nuclear density for most of categories. The averagedensities of the formations of π + are little higher thanthat of π − because of the coulomb potential.Fig. 5 shows the average time of the first and the lastformations of the pion, versus the different categories of Au+
Au@400MeV
Decay of hard ∆, for π − Decay at freeze out , for π − Decay of hard ∆, for π + Decay at freeze out , for π + < ρ / ρ > Scattering Number
FIG. 4: The average density where the pion was produced firsttime by the hard ∆ decay (first pion formation) and the decayat the freeze-out (last pion formation), versus the differenttypes of free pions categorized by their scattering numbers,in the central collision of Au + Au at 400 MeV/nucleon. Au+
Au@400MeV
Decay of hard ∆, for π − Decay at freeze out , for π − Decay of hard ∆, for π + Decay at freeze out , for π + < t > ( f m / c ) Scattering Number
FIG. 5: The average time when the pion was produced firsttime by the hard ∆ decay (first pion formation) and the decayat the freeze-out (last pion formation), versus the differenttypes of free pions categorized by their scattering numbers,in the central collision of Au + Au at 400 MeV/nucleon. pion. One can see that for the categories of the pion withless scatterings, their first formations take place later. Itis not surprising for the fact that the pions which createdfrom ∆ decay earlier have more probabilities to be scat-tered, and the whole scattering process lasts in a largertime range.In Fig. 6 we investigate the average transverse momen- ≤0.1 < P t > ( M e V ) Scattering Number π − π + Au+
Au@400MeV
FIG. 6: The average transverse momentum of the differenttypes of free pions categorized by their scattering numbers,in the central collision of Au + Au at a beam energy of 400MeV/nucleon. ≤0.1 A v e r age S c a tt e r i ng N u m be r P t (MeV) π − π + Au+
Au@400MeV
FIG. 7: The average scattering number in the history ofthe free pions as a function of the transverse momentum, inthe central collision of Au + Au at a beam energy of 400MeV/nucleon. tum p t of the different categories of free pions. One cansee that negative pions with more scattering processesin general have higher transverse momenta. It also canbeen seen the average momentum of the positive pions isobviously higher than that of the negative pions for eachcategory, due to the coulomb potential.We also calculated the average scattering number asshown in Fig. 7 as a function of pion transverse momen- Au+
Au@400MeV x=1 x=0 π − / π + Scattering Number
FIG. 8: The π − /π + ratio versus the different types of freepions categorized by their scattering numbers in the centralcollision of Au + Au at a beam energy of 400 MeV/nucleonwith different symmetry energies. tum p t . The results show the pions with higher trans-verse momentum mostly have more scattering processes,which is consistent with that shown in Fig. 6. It shouldbe noticed that in the transverse momentum spectrumat small p t ≦ M eV , the average scattering number isclosed to zero, indicating most of these pions have neverbeen scattered since they were produced from hard ∆decay.Fig. 8 shows the π − /π + ratio for different categoriesof free pions, with soft and stiff symmetry energies, re-spectively. We can find that the ratios increase with in-creasing scattering number, in both situations. Becauseof the coulomb potential, negative pions have more prob-abilities to be absorbed and scattered than positive ones.It should also be noticed the effects of the symmetry en-ergy are apparent for most categories. However for zerocategory, i.e. these pions without any scattering process,the effect appear to be negligible. While with increas-ing scattering number, the ratio shows more sensitive tothe symmetry energy. In particular for the pions withfive scattering processes, the effect reach to as much asmore than 15%. The difference of the sensitivity for dif-ferent categories is reasonable. The pions without morescattering processes, are created by hard ∆ decay andfreeze out to the detector directly. Whereas for those pi-ons with more scattering processes, their first formationstake place earlier and last formations take place later.Therefore, the time of they exit in high density matter islonger, which enhancing the sensitivity to the stiffness ofnuclear symmetry energy at high densities. This impliesthat the effect of the symmetry energy is governed notonly by the density where the pions are originated butalso by the time that they spent in high density regionduring their whole formation processes. IV. CONCLUSIONS
In conclusion, based on the framework of the trans-port model IBUU, we investigate the central collision of Au + Au at a beam energy of 400 MeV/nucleon. We ana-lyzed all the inelastic collisions to extract the informationof the pion production and its absorption. The statisti-cal investigation shows the pions are firstly produced bythe hard ∆ decay at the average density around 1 . ρ .However about 18% of them are absorbed absolutely inthe subsequent inelastic collisions, thus can not be ob-served in the experiment. From categorizing the free pi- ons by their scattering processes, it is found that most ofthe free pions have been scattered for one or more times.The pions with more scattering numbers have existed inhigh-density region for longer time. As a result those pi-ons carry more information of the high-density region andexhibit significant sensitivity to the symmetry energy. Acknowledgments
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