Production of Light Nuclei in Heavy Ion Collisions via Hagedorn Resonances
EEPJ manuscript No. (will be inserted by the editor)
Production of Light Nuclei in Heavy Ion Collisions via HagedornResonances
K. Gallmeister and C. Greiner
Institut f¨ur Theoretische Physik, Goethe-Universit¨at Frankfurt am Main, Max-von-Laue-Str. 1, 60438 Frankfurt am Main,Germany Received: date / Revised version: date
Abstract.
The physical processes behind the production of light nuclei in heavy ion collisions are unclear.The nice theoretical description of experimental yields by thermal models conflicts with the very smallbinding energies of the observed states, being fragile in such a hot and dense environment. Other availableideas are delayed production via coalescence, or a cooling of the system after the chemical freeze-outaccording a Saha equation, or a ‘quench’ instead of a thermal freeze-out. A recently derived prescriptionof an (interacting) Hagedorn gas is applied to consolidate the above pictures. The tabulation of decayrates of Hagedorn states into light nuclei allows to calculate yields usually inaccessible due to very poorMonte Carlo statistics. Decay yields of stable hadrons and light nuclei are calculated. While the scale-freedecays of Hagedorn states alone are not compatible with the experimental data, a thermalized hadron andHagedorn state gas is able to describe the experimental data. Applying a cooling of the system according aSaha-equation with conservation of nucleons and anti-nucleons in number leads to (nearly) temperatureindependent yields, thus a production of the light nuclei at temperatures much lower than the chemicalfreeze-out temperature is possible.
PACS.
In recent years, the production of light nuclei in (ultra-)relativistic heavy ion collisions has gained new interest.Experimental measurements of the production of deuteron,triton, helium-3 and helium-4, their anti-particles, andalso hyper-triton in high-energetic collisions by the ALICEcollaboration at the LHC or some subset of these nuclei inlow-energetic collisions by the HADES collaboration at GSIintroduce a fundamental question onto their productionmechanism. It is unclear, why the experimental yields canbe described so well by thermal models as e.g. shown in[1,2,3].Under the assumption, that a thermalized system hasbeen built up, the binding energies of the observed statesare so small, that a survival in such a virulent system ofsuch fragile states at the chemical freeze-out temperaturesof O (150 MeV) is improbable. Therefore a later productionof these nuclei in the time evolution of the collision maybe some explanation.Here the first ansatz is, that in the framework of coales-cence, the production of high-mass resonances is governedby the yields of the lower mass states [4,5,6,7], while stillenergy conservation is not given in this picture.Another explanation relies on the assumption of de-tailed balance, resp. the law of mass action, resp. a kindof Saha-equation, which dictates the yields at later stages already by the chemical-freeze-out conditions of the stablehadrons [8]. Adjusting chemical potentials have also beenintroduced in [9].Recently, the additional idea has been discussed, thatall these observed yields do not originate from a thermal-ized gas after a phase transition, but are generated by a‘quench’ into a state described by Hagedorn states andtheir decays [10]. Here the underlying picture is a so-called‘self organized criticality’ (SOC). Thus, instead being ina thermalized and stable state, the system is assumed tobe in a critical state, where modifications in all extensionsare possible, but keeping the system in its (critical) state,and it just looks like it would be in a stable state.In refs. [11,12] the authors have developed a prescrip-tion of a microcanonical bootstrap of Hagedorn states withthe explicitly conserved baryon number B , strangeness S and electric charge Q , which has been augmented bythe consideration of B , S and isospin I in [13]. It is areformulation of the original concept by Hagedorn him-self [14] according to Frautschi [15], where the covariantformulation is analogous to [16,17].We are thus in the favorable situation to test the aboveassumptions against experimental data. We will thereforefirst show, that the Hagedorn states defined in our pre-scription indeed (nearly) produce a scale independence con-cerning their decay branching ratios. Nevertheless, thesedecays modestly fail to describe the experimental yields. a r X i v : . [ h e p - ph ] J u l K. Gallmeister, C. Greiner: Production of Light Nuclei in Heavy Ion Collisions via Hagedorn Resonances
On the other hand the assumption of a thermalized sys-tem of hadrons together with Hagedorn states leads to asatisfactory description of the experimental data. Whetherit was really a thermal system at the freeze-out tempera-ture, or a much cooler system following a Saha equation,which finally produced the observed particles, is not distin-guishable within our framework. Thus the criticism againstthermal models by confronting low binding energies withlarge temperatures is not legitimate in our approach.The paper is organized as follows. We start to recapit-ulate the basics of the present Hagedorn state prescriptionand elaborate on the extensions needed for the inclusionof light nuclei. Then we first show the decay multiplicitiesassuming a fixed mass Hagedorn state and second, afterdecays of a thermal Hagedorn state gas. Finally, we discussthe effect of cooling the Hagedorn state gas under the as-sumption of holding yields constant according to the Sahaequation.
We use the microscopic prescription developed in [11,12]in its improved formulation described in [13]. In order topursue the extensions needed for the light nuclei, we willhere first repeat the basic equations as given in [13], whichare implemented into the transport framework GiBUU[18].Under the basic assumption, that only subsequent two-particle decay participate, the bootstrap equation to beused is τ C ( m ) = τ C ( m ) + V ( m )(2 π ) m (cid:88) ∗ C C (cid:90) (cid:90) d m d m × τ C ( m ) τ C ( m ) m m p cm ( m, m , m ) , (1)which describes, how the mass degeneration spectrum ofthe Hagedorn states τ C ( m ) is built up from a low massinput τ C ( m ) and the combination of two lower lying Hage-dorn states. Here, as usual, 4 m p = ( m − m − m ) − m m , and the special notation (cid:80) ∗ indicates, that thesum only runs over ‘allowed’ quantum number combina-tions; τ C ( m ) stands for the inhomogeneity, i.e. the hadronicinput, while the volume V ( m ) ≡ V is just a constant atthe moment. The quantum number vector C may stand for( BSQ ) or (
BSI ) with B , S , Q , I indicating baryon number,strangeness, electrical charge, isospin. As elaborated in[13], the combination ( BSI ) is fully equivalent to (
BSQ ),but preferable internally.Selecting different values for the radius, R , and thus, via V = 4 π/ R , also for the volume V of the Hagedorn statesin the bootstrap equation eq. (1) yields different slopesand thus different values of the Hagedorn temperature asan intrinsic parameter; larger radii yield steeper increaseof the spectrum, thus smaller values of the Hagedorn tem-perature. The default value R = 1 . T H = 167 MeV, while R = 1 . T H = 152 MeV(cf. also [11]).We extend the prescription by the inclusion of lightnuclei as stable particles in the input to the bootstrap. mass B J I S [GeV]d= H 1.876 2 1 0 0t = H , He 2.809 3 1/2 1/2 0 Λ H 2.992 3 1/2 0 -1 α = He 3.728 4 0 0 0
Table 1.
Properties of light nuclei. Listed are baryon number B , spin J , isospin I , and strangeness S . Details of the particles are listed in table 1. The resultingHagedorn spectrum is only very slightly influenced by thisaddition and the differences are hardly visible. Nevertheless,decays of high mass Hagedorn states now may end in lightnuclei as final particles.In the spirit of ref. [19], also the inclusion of non-stableresonances could be in order. For this, one would firstinclude these resonances into the Hagedorn bootstrap as ifthey would also be stable particles. In a second step onethen would extend the transport code to implement theirdecays into stable nuclei and hadrons, as also the decaysof hadronic resonances are treated. At the moment, thisimplies deeper modifications of the algorithm itself and isleft for future studies.It is favorable for the (
BSI ) prescription, that all ofthe light nuclei are realized in their lowest isospin level, i.e. I = 0 or I = 1 /
2. The fact, that H and He aretwo different charge states in a I = 1 / e.g. their charge states. Like the assumptionof a common volume of all Hagedorn state specific detailsbetween different particle yields are not accessible withinour prescription.The second basic equation is the connection of thedecay width Γ of some Hagedorn state with its productioncross section σ , which is given by [13] Γ C ( m ) = σ ( m )(2 π ) τ C ( m ) − τ C ( m ) (cid:88) ∗ C C (cid:90) (cid:90) d m d m × τ C ( m ) τ C ( m ) p ( m, m , m ) . (2)At the moment, the cross section is assumed to show nomass dependence or some other details and is assumedto be a constant. In the actual prescription, it is alsodirectly connected with the radius of the Hagedorn stateby σ ( m ) ≡ σ = πR , i.e. σ = 31 . R = 1 . . Gallmeister, C. Greiner: Production of Light Nuclei in Heavy Ion Collisions via Hagedorn Resonances 3 ratios may be defined by dividing every summand of thisexpression by its total,d B C ; C , C ( m, m , m ) .. = d m d m τ C ( m ) τ C ( m ) p ( m, m , m ) (cid:88) ∗ C C (cid:82)(cid:82) d m d m τ C ( m ) τ C ( m ) p ( . . . ) , (3)such that (cid:88) ∗ C C (cid:82)(cid:82) d B ≡
1. It is interesting to observe,that here for the relative branching ratios, contrary to thedecay width eq. (2), the cross section σ ( m ) completelydrops out. The number of a specific light nucleus (A =d , t , . . . , cf. table 1) a given Hagedorn state finally decaysinto is calculated as n (A) C ( m ) = (cid:88) ∗ C C (cid:90) (cid:90) d B C ; C , C ( m, m , m ) × (cid:16) n (A) C ( m ) + n (A) C ( m ) (cid:17) . (4)For this purpose, one has to initialize the input correctly,as e.g. n (d)(2 , , ( m ) .. = δ ( m − .
876 GeV) ,n (t)(3 , , . ( m ) .. = δ ( m − .
809 GeV) ,. . . . (5)This quantity n (A) C ( m ) gives the total fraction for the decayinto the light nucleus, i.e. the direct decay and also theindirect decay chain via intermediate Hagedorn states.The tabulation has to be done for all quantum numbersand masses of the mother particle. As long as the ’finalstate’ A has fixed quantum numbers and mass, as e.g. thestates listed in table 1, the tabulation is manageable. Whenlooking for an extension of this tabulation to more states,the major problem will be a mass distribution of the finalstates. In this case, the tabulation will very soon exceedactual memory setups of the HPC computer clusters. Thusa na¨ıve extension of eq. (4), especially in the spirit ofref. [8], is not possible. The actual implementation of the Hagedorn bootstrapexplicitly respects conservation of the quantum numbers.It is obvious, that the quantum numbers of the initial(mother) state directly influences the yields of the different(daughter and grandchild) states with different quantumnumbers. (As an example, starting with a Hagedorn statewith B = 2 yields obviously and considerable more nucle-ons than starting with B = 0.)Calculating ‘stochastic’ averages (contrary to ‘statis-tical’ averages, which have thermal weights and chemicalpotentials), one has to average over all possible quantumnumbers given by τ C alone. While averaging over all quan-tum numbers which are accessible for a given Hagedornstate mass m , one observes two general features: – The overall yield grows linearly with the Hagedorn statemass according (cid:104) N tot (cid:105) (cid:39) .
27 + 1 .
44 GeV − m (see also[11]) – The relative yields are rather independent on the Hage-dorn state mass, but obey mass thresholds.The latter is illustrated in fig. 1. Here all the results for -9 -8 -7 -6 -5 -4 -3 -2 -1
2 3 4 5 6 7 8 9 10BSI(Q)=average 〈 N 〉 / 〈 N t o t 〉 m [GeV] π KN ΛΣΞΩ + − H He Λ H He Fig. 1.
The relative multiplicity of several species as functionof the Hagedorn state mass m for T H = 167 MeV. Different linecolors indicate different daughter particles, while line styles asindicated in the plot show different (electrical) charge states. hadronic states are calculated by MC runs, while the yieldsfor the light nuclei are generated by the tabulation de-scribed above. For d= H, results from both approachesare available, match identically, and prove the correctnessof the tabulation approach eq. (4). Albeit fig. 1 also showsthe different electric-charge states separately, only for pi-ons a slight difference between the charge states is visible.This is due to non-isospin symmetric decay channels of thehadronic resonances.Of course one has to take the previous statements aboutthe scaling bahavior with some grain of salt, since theyrely on figures with logarithmic axis scaling. Nevertheless,for large masses it seems hard to deduce the mass of themother particle just from relative yields.It is now worth comparing these relative yields withexperimental yields. We take here the high-energy LHCdata measured by ALICE [20,21,22,23,24,25]. A compari-son of the relative yields of the decays of a Hagedorn statewith m = 10 GeV with the experimental data is shown infig. 2. Here, and also in all following figures, the absolutenormalisation will be fixed to the experimental protonyield. As can be seen, a single Hagedorn state with largemass is not able to describe the experimental multiplici-ties; the distribution is too hard. Even with a bootstrapwith R = 1 . T H = 152 MeV, as described above, higher massstates, especially the light nuclei, are overestimated. Onlya reduction of the Hagedorn temperature further down K. Gallmeister, C. Greiner: Production of Light Nuclei in Heavy Ion Collisions via Hagedorn Resonances -7 -6 -5 -4 -3 -2 -1
0 0.5 1 1.5 2 2.5 3 3.5 4 π + K + p Λ Ξ − Ω − H He Λ H He m u l t i p li c i t y m [GeV]dataT H = 167 MeVT H = 152 MeV Fig. 2.
The multiplicities of given stable particles after Hage-dorn and hadronic decay cascades for a potential Hagedorn statewith m = 10 GeV. The overall normalization is arbitrary fixedto the experimental proton value. Experimental data by ALICE[20,21,22,23,24,25]. Results are shown for two different values T H = 167 MeV and 152 MeV for the Hagedorn temperature. to even lower values could yield a satisfactory description.Anyhow, this can only be achieved by further increasingthe Hagedorn state size [11].While here the mass distribution looks thermal, it isonly governed by the Hagedorn temperature. Thus a SOCprescription may lead to thermal (looking) yields. Con-cluding from fig. 2, the intrinsic Hagedorn temperatureleads to a mass dependence, which is too hard. This state-ment relies on the results of the ad-hoc mass choice of m = 10 GeV. Since the branching rations are only massindependent on a logarithmic scale, the yields could changeby looking into them with some detail and some changesof the mass. Nevertheless, a qualitative change of the pic-ture is not expected. Therefore, we conclude this sectionwith the statement, that within our Hagedorn state decayscenario, a scale invariant decay of Hagedorn state resultsin particle yields which are too hard, i.e. show a slopeparameter, which is too large. Turning to the picture of a thermalized gas of hadronicand Hagedorn resonances, an additional degree of freedomis introduced by the temperature of the system. An inte-gration of the Hagedorn state mass spectra weighted bythe Boltzmann factor for a given mass m , n ( m, T ) = 4 π (2 π ) m T K ( m/T ) , (6)with K n indicating modified Bessel functions, is necessary.Then the hadronic feed down of these thermal averagedHagedorn states has to be calculated. A fitting procedure applied after the decays to fit the experimental data of pro-tons and light nuclei yields a temperature of T = 149 MeVfor T H = 167 MeV and T = 144 MeV for T H = 152 MeV.The results for the first setup are shown in fig. 3; thedifferences to the second setup are nearly invisible. -7 -6 -5 -4 -3 -2 -1
0 0.5 1 1.5 2 2.5 3 3.5 4T H = 167 MeV T = 149 MeV π + K + p Λ Ξ − Ω − H He Λ H He ϕ m u l t i p li c i t y m [GeV]dataafter decay Fig. 3.
Multiplicities as in fig. 2, but now for a gas of thermal-ized ( T = 149 MeV) hadron resonances and Hagedorn states( T H = 167 MeV). Some comments are in order. First, the number ofmesons included in GiBUU is quite low compared to e.g.
UrQMD [26,27], or PDG [28]. Therefore also the mesonicyields in these Hagedorn state decays may be underrepre-sented. This is why we only used the baryonic sector to fixthe temperature. Second, the yields of the strange mesonsand baryons are shown as is; no strangeness suppressionfactor has been applied. Third, these fits are meant topresent the overall success. These fits are not intendedto be high precision fits; therefore we just provide theresulting temperatures and abstain to give χ values.It is obvious, that due to the additional degree of free-dom the agreement of the model is much better than in theprevious section. But also the production channels of thedifferent particles is qualitatively different. In the picture ofa thermalized gas, one has a thermal contribution of stableparticles, while also the feed down from decays of higher ly-ing resonances contributes. So only approx. half of the final H may be claimed to be (directly) thermal, while the otherhalf stems from decays of Hagedorn states. In the case ofthe higher masses of the nuclei, the situation is even moreextreme: only approx. 20 % of He are thermal, while 80 %stem from feed down. (Interestingly, this finding seems todepend on the underlying Hagedorn temperature; for thelower Hagedorn temperature T H = 152 MeV, the relativecontribution of the feed-down decays is much larger.)Therefore it would be worthwhile to study the influenceof higher lying resonances of the nuclei, as e.g. in ref. [19].There the importance of these higher resonances was lim-ited to a level of 5 % at high energetic collisions at LHC and . Gallmeister, C. Greiner: Production of Light Nuclei in Heavy Ion Collisions via Hagedorn Resonances 5 had a sizable effect for low energetic collisions with largebaryochemical potentials. If one would expand the modelpresented here by all these higher resonances, one wouldalso expect a large occupation of these states, which wouldthen lead to sizable contribution to the yields of stablenuclei after hadronic feed down. Anyhow, as mentionedabove, this may be left for a future study.In order to illustrate, that the final yield of stableparticle is far from the spectrum of Hagedorn states beforedecay, we indicate in fig. 4 this spectrum in comparisonto the final yields. It is worth to emphasize, that here the -7 -6 -5 -4 -3 -2 -1
0 0.5 1 1.5 2 2.5 3 3.5 4T H = 167 MeV T = 149 MeV π + K + p Λ Ξ − Ω − H He Λ H He m u l t i p li c i t y m [GeV]dataafter decaybefore decay Fig. 4.
As fig. 3, but here is also shown the mass distributionof all potential Hagedorn states before decays. normalization both of the spectra ‘before decays’ and ‘afterdecays’ are the same and the number of Hagedorn statewith masses comparable to that of e.g. α = He are indeedseven orders of magnitude larger.
One may apply the same criticism to the Hagedorn gaspicture as to a thermal model relying on a hadron resonancegas alone: how could these loosely bound states survive atthese temperatures?We thus will follow the arguments in [8], where theSaha equation is the natural explanation how thermalyields behave under the cooling of the system. The as-sumption that during the cooling of the system the yieldsof stable particles are frozen at the ‘chemical freeze-out’(most important for nucleons and antinucleons), chemicalpotentials for all resonances are fixed in their temperaturedependence. While in [8] it was possible to calculate a full‘decay matrix’, this is more involved for the prescriptionpresented here, since one would have to tabulate the de-cays of all quantum number states (
BSI ) with mass m into stable hadrons. This asks for the extension of eq. (4) from light nuclei to all stable hadrons. While possible inprinciple, it is a challenging task due to computer memoryconstraints and not yet feasible.Instead we apply a simplified setup, where we adjustthe chemical potentials before feed down and restrict tobaryon number. We introduce a chemical potential forthe absolute value of the particles baryon and anti-baryonnumber, µ | B | ( T ), i.e. both the number of protons and an-tiprotons are anchored, while the yields of the other stableparticles are not considered. In this case, with particlenumbers given by eq. (6), the chemical potential is fixedby exp( µ | B | ( T ) /T ) = T cfo K ( m N /T cfo ) / ( T K ( m N /T )),where m N = 0 .
938 GeV stands for the nucleon mass and T cfo indicates the chemical freeze-out temperature. Theresulting yields of light nuclei are displayed in fig. 5. Within -7 -6 -5 -4 -3 -2 -1
70 80 90 100 110 120 130 140 150 T H = 167 MeV H He Λ H He m u l t i p li c i t y T [MeV] thermalthermal + decay
Fig. 5.
The yields of light nuclei, when the yields of stablenucleons (protons and neutrons) and anti-nucleons are fixed to T cfo = 149 MeV with T H = 167 MeV. Solid lines indicate thetotal yields, while dashed lines show the contribution of thermalparticles only. The colored bands indicate the experimental errorbars of the data by the ALICE collaboration. this picture, the yields of the light nuclei are (nearly) con-stant as function of the final temperature within somecertain range. With decreasing temperature all yields startto increase. This behavior is not so pronounced for hyper-triton as for the other nuclei. Here the lack of introducinga chemical potential for the strange sector is visible.This overall behavior has to be confronted with thatof the Boltzmann factors eq. (6), which would govern thetemperature behavior otherwise and lead to a nearly expo-nential dependence of the yields as function of temperature(see the discussion in [8]).Also shown in fig. 5 is the relative contribution ofthermal particles to the overall yield. With decreasingtemperatures, the relative importance of feed-down par-ticles vanishes. This may be of interest, since it is wellknown since the results of [29], that the decay products ofthermally distributed particles are not thermal, but look K. Gallmeister, C. Greiner: Production of Light Nuclei in Heavy Ion Collisions via Hagedorn Resonances effectively cooler (the slope is steeper). Therefore a deeperinspection of the slopes of the decay products could leadto new insights about the production mechanism. Anyhow,this is beyond the possibilities of our approach, where onlythe absolute numbers of the light nuclei are accessible bythe method relying on the tabulation according eq. (4).In order to justify the Saha equation picture also withinthe Hagedorn state prescription, we show in fig. 6 theinteraction rate of specific particles within a Hagedornstate gas. Mass differences show up in slightly different -3 -2 -1
70 80 90 100 110 120 130 140 150 160 170 Γ = n σ 〈 v 〉 [ G e V ] T [MeV] π H He Fig. 6.
The collision rate of pions, d = H, and α = He asfunction of temperature of the Hagedorn state gas ( σ = 30 mb). curves. It is now worth to realize some numbers. A valuefor the rate of Γ = 0 . T = 150-160 MeV, directly translates in lifetimes τ =1 /Γ (cid:39) µ | B | ( T ) according the Saha pictureonly slightly changes the total density.On the other hand, the binding energies of the lightnuclei are in the region 2 . . Using directly a Hagedorn state prescription devolopedduring the recent years can not allow to calculate decayrates into very rare channels as e.g. into light nuclei. Rele-vant relative decay branchings may go down to 10 − , which is below any usual statistics available in Monte Carlo cal-culations by a factor O (10 ). For the decays of Hagedornstates with given quantum numbers and masses, a tabula-tion according to the usual bootstrap has been developedand allows to access these low yields. Since this tabulationonly covers the number of particles, no other observablethan the yields may be calculated in this way; quantitieslike energy spectra or flow still stay beyond reach.The (relative) branching of Hagedorn states into stablehadrons and light nuclei shows up to be nearly independentof the mass of the parent particle. Still, mass thresholdsinfluence the yields and the above statement holds only trueon a level, where the yields are depicted with a logarithmicscaling. The most general scaling behavior is reached foran averaging over all possible quantum numbers withoutany chemical potentials. Only this case is covered in thiswork.The relative branchings are comparable with the exper-imental yields of the ALICE experiment. It shows that theHagedorn state decays lead to an over-prediction of heavymass states. Even lowering the Hagedorn temperaturewithin reasonable ranges does not allow for a successfulagreement. Therefore the assumption of a scale-free systemof Hagedorn states is not sustained by our prescription,since the Hagedorn temperature is still too high comparedto experimental data.On the other hand, the introduction of an additionaldegree of freedom by assuming a thermalized system ofHagedorn states, where in addition to the Hagedorn tem-perature also the temperature of the gas sets a scale, asatisfactory description of the experimental yields is achiev-able. With different values of the Hagedorn temperature,different temperatures yield the same level of accuracy ofagreement.As in a hadron resonance gas picture, a production ofthe light nuclei at chemical freeze-out temperature withinthe Hagedorn state gas suffers the same argument of havingtoo small binding energies compared to the temperature.Taking the notion of ‘chemical freeze-out’ seriously, allyields of stable particles are fixed at this point. Thereforea cooling below this temperature has to be considered akinto the Saha equation; chemical potentials of the stablehadrons influence those of the unstable once. In the presentwork, a simplified prescription of using a chemical potentialfor the absolute value of baryon and anti-baryon numberhas been shown. Even in such a exploratory picture, thefinal yields of the light nuclei do only depend marginallyon the final temperature, when staying within some range(as proposed in [8]).A temperature dependence may be observed when look-ing at the ratio of ‘thermal’ over ‘all particles’; if the finaltemperature is higher, the contribution of feed down par-ticles may be larger. This could maybe be attacked bylooking theoretically at the energy spectra of the particles.Anyhow, these spectra are beyond the given analysis. Also,only a description of experimental spectra using a realisticflow profile could really pin down that point.In the present work, only high energetic heavy ioncollisions have been covered. Here only the the thermody- . Gallmeister, C. Greiner: Production of Light Nuclei in Heavy Ion Collisions via Hagedorn Resonances 7 namical properties of the Hagedorn state gas developed inour prescription are used. Looking at the (very) low energyside, as e.g. HADES at GSI, the full dynamical machineryimplemented in the transport code may be used and there,also spectra of light nuclei may be calculated, maybe evenwith respect to the centrality of the collisions. This is leftfor future studies.
This work was supported by the Bundesministerium f¨ur Bildungund Forschung (BMBF), grant No. 3313040033.
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