Sub-threshold charm production in nuclear collisions
aa r X i v : . [ nu c l - t h ] J a n Sub-threshold charm production in nuclear collisions
J. Steinheimer , A. Botvina , , and M. Bleicher , , Frankfurt Institute for Advanced Studies, Ruth-Moufang-Str. 1, 60438 Frankfurt am Main, Germany Institut f¨ur Theoretische Physik, Goethe Universit¨at Frankfurt,Max-von-Laue-Strasse 1, D-60438 Frankfurt am Main, Germany Institute for Nuclear Research, Russian Academy of Sciences, 117312 Moscow, Russia and John von Neumann Institute for Computing (NIC),Forschungszentrum J¨ulich, 52425 J¨ulich, Germany
We present the first predictions for sub-threshold open charm and charmonium production innuclear collisions. The production mechanism is driven by multi-step scatterings of nucleons andtheir resonance states, accumulating sufficient energy for the production of J/ Ψ and Λ c + D . Ourresults are of particular importance for the CBM experiment at FAIR, as they indicate that alreadyat the SIS100 accelerator one can expect a significant number of charmed hadrons to be produced.This opens new possibilities to explore charm dynamics and the formation of charmed nuclei. Charmed hadron production is considered to be an ex-cellent probe of the properties of hot and dense nuclearmatter. Early works have argued that charmonium sup-pression in central nuclear collisions may serve as signalfor the formation of a deconfined medium, the so calledquark gluon plasma (QGP) [1]. As the charm quark massis much higher than the typical scale of QCD, charm inthe traditional scenario is only produced in the very earlystages of a nuclear collision where relative momenta arestill large. In essence, it serves as a messenger of theproperties of that stage. A focus of recent investigationswas on charm production at ultra-relativistic energies,i.e. in experiments at the LHC and RHIC accelerators(see e.g. [2–12]).It is expected to be even more interesting to study charmproduction at lower energies, for example at the plannedFAIR facility. At such low energies the system created isclose to the transition between the hadronic phase andthe QGP at very high net baryon densities. Thereforethe charm quarks and hadrons will be born in a verystrongly interacting system of high baryon density, open-ing up the possibility to study charm interactions withcold and hot hadronic matter. In the physics program ofthe CBM experiment at FAIR the study of open charmand charmonium plays an essential role [13]. However,the FAIR project is planned to start with the SIS100accelerator, which will be able to accelerate a beam ofheavy ions only to an energy of E lab = 11 A GeV, anenergy which is below the charm production threshold inelementary collisions. In order to verify if the plannedCBM experiment at the FAIR facility is fit to do studieson open charm and charmonium production, it is of greatimportance to have reliable estimates on the productioncross sections these states. In this paper will provide suchestimates for sub-threshold charm production in nuclearcollisions, meaning charm production at beam energiesbelow their elementary p+p threshold.In the first part of the paper we will introduce the modelwhich we will use and the mechanism employed for charmproduction. In the second part we will show our resultson open charm, charmonium and charmed nuclei produc-tion at the SIS100 accelerator and in the final part we will discuss our results and their relevance for the planned ex-periments at FAIR. I. CHARM PRODUCTION
Estimates and predictions on the charm productioncross section in p+p and A+A collisions are usually basedon a perturbative approach of QCD (see e.g. [14]). Asone approaches the threshold energy required to producecharm in p+p collisions, as given in table I, one expectsthese perturbative methods to break down, especially inA+A collisions. Previous work on near threshold charmproduction, in A+A collisions, is based on a parametrizedp+N cross section [15, 16] or an effective interaction La-grangian [17, 18]. In such a study all charm is essentiallyproduced in the first binary collisions, leading to binaryscaling of charm production, thus the main contributionto sub-threshold production is from the Fermi momentaof the nucleons, shifting the threshold. However it wasalready shown that deep below the threshold, the role ofFermi momenta is negligible [19] and one should expecta breakdown of the strict scaling of charm productionwith the number of binary collisions because the energyper individual p+p collision is not sufficient to producea charm quark pair, i.e. that secondary baryon interac-tions are essential for particle production at and belowthe elementary production threshold. Therefore one ex-pects contributions of non-perturbative processes to thecharmed hadron cross section. One such a process couldbe the creation of charm in the decay of a heavy ex-
Process Threshold Energy [GeV] N + N → N + N + J/ψ . N + N → N + Λ c + D . N + N → N + N + D + D . N refersto any ground state nucleon. Name Mass [GeV] Width [GeV] SpinN*(2600) 2.600 0.65 11 / / / / / / / / / / / / / cited state, e.g. a baryonic resonance. Such a processhas already shown to successfully describe near thresh-old production of the φ mesons [19, 20]. Here the maincontribution to the large sub-threshold φ multiplicity arethe secondary scatterings of produced heavy baryonicstates with other incoming baryons. It was found thatthese states decay and populate the multiplicity distri-butions of produced hadronic species according to phasespace dominance. One should note that this is in essencenothing else but the application of Fermi’s ”golden rule”which states that in a decay process the final state’s pop-ulation probability is proportional to a (constant) matrixelement (which has to be fixed) and the available phasespace density for the process. It is known that such de-cays are the dominant process in associated Λ+K pro-duction (see e.g. [21]). Here, we include and explorethese processes for the first time, to make quantitativepredictions on the sub-threshold production of charmedhadrons in nuclear collisions.We will employ the UrQMD transport model [22, 23],which already includes a extensive list of baryonic reso-nances. In order to describe charm production we extendthe list of baryons to higher masses. As baryonic reso-nances act as an intermediate energy reservoir we willadditionally include all known nucleon and Delta reso-nance with masses up to 4.6 GeV [24], since only thesehigh mass states may decay into a J/ Ψ or Λ c + D or N + D + D .The direct resonance production cross section, in ele-mentary p+p collisions, implemented in UrQMD followsfrom a phenomenological parametrization of measuredexperimental cross sections and phase space considera-tions. Here the cross section has the general form: σ , → , ( √ s ) ∝ (2 S + 1)(2 S + 1) h p , ih p , i s | M ( m , m ) | (1)where S and S are the spin of the outgoing particles,and h p i,j i the average momentum of the in- and out-going p+p total inelastic p+p -> resonance OLD p+p -> resonance NEW String [ m b ] s pp [GeV] FIG. 1. [Color online] Total inelastic cross section of p+pcollisions, as implemented in UrQMD (black solid line). Thered line depicts the part of the p+p cross section which cor-responds to the excitation of at least one baryonic resonance. particles respectively. The matrix element | M ( m , m ) | is assumed to not explicitly depend on the spins but onlyon the masses of the outgoing particles. It is given as: | M ( m , m ) | = A m − m ) ( m + m ) (2)The parameter A is determined by a fit to avail-able data and is the same for any specific process, i.e. N + N → N + N *, for all N *. The total inelastic p+pcross section from resonance excitation then follows asthe sum of all possible channels (1). The mass depen-dence of the production cross section is then essentiallydetermined by phase space, and not the parameter A,which has been shown to lead to a good description ofmeasured resonance production cross sections [22]. Theresulting inelastic cross sections of p+p collisions, in themodel, are shown in figure 1. Here we compare theold resonance contribution (red dashed line) with thenew implementation which includes the above mentionedheavy states (red solid line). At low energies this crosssection essentially makes up for the total inelastic p+pcross section implemented in the model. As the resonancecontribution to the total inelastic cross section increases,the contribution of the string must decrease at the sametime in order to conserve the total inelastic cross section.All the before mentioned assumptions can also be usedto directly compute any N + N resonance contribution tothe inelastic cross section, where N can be either a protonor neutron. We can therefore directly apply our modelto collisions of heavy nuclei containing protons as well asneutrons. The heaviest baryonic states, which eventuallydecay into the open and hidden charm states, will not beexcited in the initial binary scatterings which are gov-erned by the cross section shown in figure 1 but originatefrom secondary interactions of initially produced bary-onic resonances which serve as an intermediate energy p + p - > J / + X p + p - > C + D + X Data J/UrQMD: p+p->J/ +X p+p-> C +X p+p-> D+X p+p-> D+X t o t [ nb ] s NN [GeV] SIS100 p + p - > D + D + X FIG. 2. [Color online] J/ Ψ, Λ c , D and D production crosssection, from UrQMD, in p+p collisions as function of the col-lision energy. Experimental measurements of the J/ Ψ crosssection are indicated as symbols [27–29]. The threshold ener-gies of the corresponding channels are also indicated as verti-cal lines. The grey area corresponds to the beam energy rangeexpected for heavy ion collisions at the SIS100 accelerator. reservoir.After fixing the production probability of the heavyresonances one needs to determine the branching frac-tion of the N* into the relevant charm channels, i.e. weneed to determine the probability of N* → N + J/ Ψ,N* → Λ c + D and N* → N+ D + D . If Γ tot is the total decaywidth of a given resonance and Γ F S is the partial decaywidth of that same resonance decaying into a particularfinal state FS, then the branching fraction Γ FS / Γ tot isdefined as the fraction of resonances which decay to acertain final state FS. These partial decay widths are theessential free parameters of our approach and are not cal-culated explicitly. To fix this crucial input we use exper-imental data on the measured J/ Ψ cross section in p+pcollisions at √ s pp = 6 . J/ Ψ production from protons on a Hydrogentarget. Furthermore we assume that the J/ Ψ can only becreated in a N* decay (it is forbidden in the ∆* decay).The resulting branching fraction, to describe the experi-mentally measured J/ Ψ yield is Γ N+ J/ Ψ / Γ tot = 5 · − .This branching ratio is two orders of magnitude smallerthan the corresponding decay into a φ meson. The re-sulting, energy dependent J/ Ψ cross section is shown infigure 2 as black line. We have also checked that themodel reproduces the original measurement of the crosssection of proton induced J/ Ψ production on Berylliumat a beam energy of E lab = 23 A GeV, which is of theorder of 1-2 nanobarn [25] (with very large statistical andsystematic errors).The threshold for Λ c + D creation is only slightly largerthan that of the J/ Ψ. It has been shown that the rel-ative importance of associated charm production near -4 -3 -2 -1 K - +p +p J/ +p i ne l [ m b ] s- s [GeV] FIG. 3. [Color online] Total inelastic J/ Ψ + p cross sectionas function of center of mass energy from detailed balancerelations. The J/ Ψ cross section is compared to inelastic φ and K − scattering of the proton also from detailed balancerelations. The rich structure in the K − + p scattering crosssection is due to number of hyperonic resonances which canbe excited in that channel (the peak corresponds mainly tothe excitation of the Λ(1520)). the threshold is significant [26]. In our case the relativebranching ratio, of N* → Λ c + D to N* → N+ J/ Ψ, is takenfrom the statistical approach to charmed hadron produc-tion [26]. In this approach all possible decay channelswhich may lead to a Λ c have been taken into account,for example the Σ c → Λ c + π . In a sense we there-fore take into account all possible contributions to theΛ c in this one effective parameter, the Λ c to J/ Ψ rel-ative production rate. In the statistical model, at thelowest beam energy measured √ s pp = 6 . c whilethe J/ Ψ yields is about 100 times smaller. Therefore wehave introduced the decay: N* → Λ c + D with a branchingfraction of Γ Λ C + D / Γ tot = 1 · − . Note that at the mo-ment we have not included explicitly a possible decay ofthe ∆* to Σ c → Λ c + π . Finally we have also included theN* → N + D + D branching fraction from statistical rela-tive abundances as Γ N+ D + D / Γ tot = 2 · − . The result-ing production cross sections are shown in figure 2. Notethat the associated production cross-section is smallerthan early estimates from a hadronic Lagrangian model[17]. In the mentioned model the cross section is actuallycalculated from an effective interaction Lagrangian andnot deduced from phase space considerations as in ourapproach. However, since our cross section is smaller webelieve that our results can be considered a conservativeestimate.One can clearly see that the choice of our branchingfraction fits the data point at √ s pp = 6 . Au+Au central: J/ HSD J/ C p + p - > J / + X p + p - > C + D + X p+p: J/ HSD J/ C T o t a l M u l t i p li c i t y s NN [GeV] SIS100
FIG. 4. [Color online] Production yields of J/ Ψ and Λ C inp+p and central Au+Au reactions as a function of the col-lision energy. We compare our results with previous HSDmodel predictions [16]. The threshold energies of the corre-sponding channels in p+p reactions are again indicated asvertical lines. The grey area corresponds to the beam energyrange expected for heavy ion collisions at the SIS100 acceler-ator. also the J/ Ψ cross section eventually saturates and willdrop at even larger energies, which is why we cannot de-scribe charm production at higher energies. However,since we are only interested in sub-threshold productionwe can safely ignore the contribution to charm produc-tion from even higher energy strings which would con-tribute for example to better describe J/ Ψ production atthe higher beam energy of √ s NN = 8 . D + D from the string excitation, as thecorresponding string threshold energy is larger than thatof the resonance processes. Since we are only interested inthe sub-threshold production process where string exci-tations are of minor importance. Furthermore we do nottake into account rescattering and absorption processesof the charmed hadrons in the hadronic medium. At themoment we are only interested in the total productioncross section which is not influenced much by hadronicrescatterings (charmed hadrons have a very small inelas-tic rescattering cross section).Including the production of the J/ Ψ via the excitationand decay of a heavy baryonic state has an additionaladvantage. We can use detailed balance relations to ex-tract, from the earlier determined branching ratio, thetotal inelastic cross section of J/ Ψ + p → N ∗ → X . Theresulting total inelastic cross section as a function of freeinvariant mass √ s − √ s , with √ s = m J/ Ψ + m N in caseof the J/ Ψ, is shown in figure 3. Here it is also comparedto the total inelastic cross sections of K − + p and φ + p .We find that the J/ Ψ has a very small cross section withthe nucleon, on the order of a few micro barn, as expectedfor near threshold production from short range QCD cal- p + p - > D + X p+p D D Au+Au central D D T o t a l M u l t i p li c i t y s NN [GeV] p + p - > D + X SIS100
FIG. 5. [Color online] Production yields of D and D mesonsin p+p and central Au+Au reactions as a function of thecollision energy. The threshold energies of the correspondingchannels in p+p reactions are again indicated as vertical lines.The grey area corresponds to the beam energy range expectedfor heavy ion collisions at the SIS100 accelerator. culations [30]. This result does also support the omissionof the inelastic hadronic rescattering for the J/ Ψ. II. RESULTS
Figures 4 and 5 summarize the results of the sub-threshold charm production in nuclear collisions, ob-tained with the above described model. As expectedthe production multiplicity for charmed hadrons doesnot show the strong threshold behavior, as observed inthe p+p reactions, in the Au+Au collisions. In themodel, due to secondary interactions, there essentiallyis no sharp production threshold anymore and the J/ Ψcan be produced over a wide range of beam energies evenbelow the elementary threshold. Again we want to makeclear that the heavy resonance which eventually decaysinto the charm hadrons cannot be produced in a first bi-nary p+p collision (deep) below the elementary thresh-old, even in A+A. It may however be produced in a sec-ondary collision of e.g. two already excited states whichlead to a sufficiently high √ s to produce the high massresonance. We would like to point out that the thresh-old for the D + D production is also smeared, due tosecondary interactions, in A+A collisions. However weobserve that the associated production Λ c + D dominatesthe D yield which can be observed in figures 2, 4 and 5.At a fixed target beam energy of E lab = 6 A GeV weexpect a yield of 1 × − J/ Ψ, 5 × − Λ c , 6 × − D and 1 × − D per central Au+Au event. In orderto estimate the expected numbers of measured charmedhadrons from experiment, a dedicated study on recon-struction efficiencies and experimental acceptance is inorder. This highlights the need for reliable predictions,as supplied from this paper, on expected charmed hadronyields. For the highest available beam energy for heavynuclei at the SIS100, E lab = 11 A GeV, we expect thatyield to be one order of magnitude larger. Hence we pre-dict that a significant number of charmed hadrons canbe measured at the CBM experiment, already with theSIS100 accelerator in place. This is of particular inter-est as the baryon number densities achieved at these lowbeam energies are very large, opening up the possibil-ity to study charm production and propagation in a verydense hadronic environment. Of particular interest hereis to understand and quantify the interaction of charmedhadrons with the nuclear environment and possible ef-fects of chiral symmetry restoration of charmed hadronproperties.A specific example of how the interactions of charmedhadrons can be inferred would be the measurement ofcharmed nuclei. The existence of bound states of nucle-ons and one or more Λ c was first suggested in [31, 32].Although their actual binding energy are still under dis-cussion it is generally agreed that such states should exist[33, 34]. Consequently the detection and therefore con-firmation of the existence of bound nuclear states withcharm would be an extremely important discovery. Wepropose that the CBM experiment is best suited for theproduction and detection of charmed nuclei, due to thehigh baryon density created and the expect high statis-tics measurements. Within the above discussed UrQMDmodel we can estimate the expected number of charmednuclei produced in minimum bias collisions. It has beenshown that in central nuclear collisions the coalescencemechanism, which assembles light nuclear clusters fromthe produced baryons is important and it can be used topredict the yields of fragments like nuclei and hypernu-clei with reasonable accuracy [35, 36, 38]. Previously wehave developed the Coalescence of Baryons (CB) model[37] which was successfully used to calculate the yields ofhypernuclei, i.e. nuclei with strangeness. In this work, wehave extended this approach, to include the coalescenceof charmed Λ c particles and nucleons to form charmedlight clusters. The coalescence model, as described in[37], is realized in both the coordinate and momentum(velocity) space. It is assumed that baryons containedin a cluster with mass number A can only be locatedwithin a radius of 2.0 A / fm from the center of the clus-ter. This is consistent with the nuclear density extractedfrom secondary decays of excited nuclei. The parame-ter v C represents effectively the maximum difference inrelative velocities between nucleons within the same coa-lescent cluster. The uncertainty in the binding energy ofsuch nuclei can be evaluated by the variation of the coa-lescent parameters v C , similar as it is applied for normalclusters and hyper-clusters. The largest v C =0.22 corre-sponds approximately to the Fermi motion in large nu-clei. The value around v C =0.10 was obtained previouslyfrom the coalescent description of experimental data onnormal light cluster production ( A ≤
4) in Heavy Ion col-lisions at energies around 1 −
10 A GeV (see [38]). Also, -9 -8 -7 -6 -5 -4 -3 { C NNN}{ C NN}{ C N} v C = 0.22 v C = 0.10 v C = 0.07 Y i e l d P e r E v en t Mass Number A E lab = 11 A GeV, Au+Au, min. bias C FIG. 6. [Color online] Production yields of light, singlecharmed nuclei as function of the mass number, for differ-ent values of the coalescence parameter v C . We show resultsfor min. bias collisions of Au+Au nuclei at a fixed targetbeam energy of E lab = 11 A GeV. this corresponds to internal excitation energies (calcu-lated from the relative motion of nucleons) of light clus-ters around several MeV, consistent with their bindingenergy.In Fig. 6 we show the yields (per minimum bias Au+Aucollision at a fixed target beam energy of E lab = 11 AGeV) of Λ c and light clusters, of different mass number A , containing Λ c obtained within our approach. We haveconsidered 3 different coalescence parameters for the co-alescence stage, which reflect different assumptions onthe binding energy of the Λ c . The largest v c =0.22 cor-responds to large relative velocities between the baryonsin clusters. For small clusters this can be considered avery optimistic estimate. The v c =0.1 is the most rea-sonable choice, since this parameter choice leads to agood description of the yields of normal light nuclei innuclear collisions [38]. We also show the results for aneven smaller parameter v c =0.07 in order to give an con-servative estimate on the lower limit for charmed nucleiproduction.As charmed-nuclei can be identified through theirmesonic decays (see e.g. [39, 40]), we think is it feasi-ble to investigate their decay into normal light clustersand hyperons. For example, a system with A=3 may de-cay via the production of an intermediate hyper-nucleus: Λ c np → Λ np + π + → He + π − + π + . The decay of Λ np nucleus was already reliably observed in relativistic heavyion collisions [35]. Since sophisticated correlation mea-surements and the reconstruction of the invariant massesare necessary for their identification, it is most importantto be able to provide the experiment with model simu-lations, such as presented here, which can be used forfurther feasibility studies for the CBM detector. III. SUMMARY
We provide pioneering estimates on the deep sub-threshold production of charmed hadrons in nuclear col-lisions. We find that the CBM experiment is well suitedto make successful measurements on J/ Ψ, Λ c , D and D production yields and even spectra, at fixed target beamenergies as low as E lab = 6 A GeV. This finding is veryimportant as it enables studies of charmed hadron prop-erties at very high baryon densities with the SIS100 ac-celerator. Furthermore we propose that the CBM ex-periment may be able to, for the first time, confirm the existence of bound states of charmed baryons with nu-cleons, so called charmed-nuclei. Such a finding wouldconstitute a remarkable step in understanding the inter-actions of charmed hadrons with normal nuclear matter. IV. ACKNOWLEDGMENTS
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