Charmonium interaction in nuclear matter at FAIR
CCharmonium interaction in nuclear matter at FAIR
Partha Pratim Bhaduri
Variable Energy Cyclotron Centre, HBNI, 1/AF Bidhan Nagar, Kolkata 700064E-mail: [email protected]
Michael Deveaux
IKF, Goethe University Frankfurt, 60438 Frankfurt/M, GermanyE-mail: [email protected]
Alberica Toia
CBM Department, GSI, 64291 Darmstadt, GermanyIKF, Goethe University Frankfurt, 60438 Frankfurt/M, GermanyE-mail: [email protected]
December 2017
Abstract.
We have studied the dissociation of
J/ψ -mesons in low energy proton-nucleus ( p + A ) collisions in the energy range of the future SIS100 accelerator atFacility for Anti-proton and Ion Research (FAIR). According to the results of ourcalculations, various scenarios of J/ψ absorption in nuclear matter show very distinctsuppression patterns in the kinematic regime to be probed at FAIR. This suggests thatthe SIS100 energies are particularly suited to shed light on the issue of interaction of
J/ψ resonance in nuclear medium.
1. Introduction
The observation of
J/ψ suppression in relativistic heavy ion collisions is considered as anevidence for the formation of quark-gluon plasma [1, 2, 3, 4]. However, a considerableamount of
J/ψ suppression is also observed in proton-nucleus ( p + A ) collisions andcommonly attributed to the dissociation of the meson in the cold nuclear matter(CNM) of the target nucleus [3, 4]. A precise understanding of this so-called “normal”suppression is crucial to establish a robust baseline, with respect to which one can isolatethe “anomalous” suppression pattern, specific to the dense QCD medium produced inheavy-ion collisions. Over a past few decades, J/ψ production in proton-nucleus ( p + A )have been studied extensively at several different fixed target experiments, in the beamenergy range of E b = 158 −
920 GeV [5, 6, 7, 8, 9, 10, 11, 12, 13] and for a variety ofnuclear targets. A common practice to estimate the cold nuclear matter effects is tofit the experimental results via the effective length traversed by the
J/ψ in the nuclear a r X i v : . [ h e p - ph ] D ec harmonium interaction in nuclear matter at FAIR σ J/ψabs extracted from thedata quantifies the overall nuclear dissociation effects [4]. The extraction of σ J/ψabs from p + A data collected at SPS by the NA50 and the NA60 Collaborations revealed asignificant beam energy dependence of the absorption cross section, with larger σ J/ψabs at lower beam energy [13]. This observation was in line with predictions discussedin [15]. Within the Glauber model framework, the authors of Ref. [15], analyzed thedata on
J/ψ production cross sections measured in p + A collisions in fixed targetexperients, with proton beam energies from 200 to 920 GeV and in d + Au collisionsat RHIC, at √ s NN = 200 GeV. Several sets of parton distributions with and withoutnuclear modifications were explicitly employed to account for the initial state effects.The magnitude of the final state absorption cross section, σ J/ψabs , is found to be sensitiveto the behavior of the opted parton distribution in the corresponding kinematic region.Moreover the results revealed a significant dependence of σ J/ψabs , on the kinematics ofthe
J/ψ and on the beam enrgy of collision, which were extraploated to estimate theexpected level of absorption in p + A collisions at 158 GeV.The production of charmonium in nuclear collisions, requires a certain formationtime. The related formation length in the target nucleus rest frame depends on therelative velocity of the c ¯ c pair and may exceed the diameter of the nucleus. Slow c ¯ c pairs would form physical resonances inside the target nucleus while fast pairs formthis resonance only in vacuum. It is likely that fully formed resonances show differentinteractions with the nuclear medium than the evoling c ¯ c pairs. The velocity of theproduced c ¯ c pairs depends on the velocity of the beam proton w.r.t the target nucleusand the kineamtic domain explored by the charmonium production. In [16], the authorsanalysed the kinematic regimes attainable for J/ψ and ψ (cid:48) production in 160 GeV p + A collisions. Selecting fully formed resonances is a prerequisite for observing differences inthe interactions of different charmonium states with the nuclear medium. Their resultsindicated that it is required to probe the phase space region x F ≤ − .
45, where x F denotes the Feynman scaling variable. In this region, the produced c ¯ c pairs at 160GeV, are slow enough to be produced inside the target nucleus. This eventually ledto distinguishably different suppresion patterns by J/ψ and ψ (cid:48) resonances due to theirdifferent binding energies. Even though NA60 Collaboration took data in 158 GeV p + A collisions, their measurements [13] were confined in the positive hemisphere over arapdity range 0 . < y cms < .
78, which corresponds to the x F domain 0 . < x F < . ψ (cid:48) data are available from NA60 Collaboration due to limitation in statistics.Up till now there is no significant measurement of the cold nuclear matter effects oncharmonium production in p + A collisions below 158 GeV. The situation changes withthe appearance of the Compressed Baryonic Matter (CBM) experiment at FAIR [17].The CBM detector set up at SIS100 is suited to measure charmonia and open charmhadrons in p + A collisions, at proton beam energies from 15 −
30 GeV, thanks toits unprecedented rate capability. These measurements will be highly interesting toinvestigate the potential issue of
J/ψ interaction in nuclear medium. In the presentarticle, we discuss the kinematics of
J/ψ production in 15 and 30 GeV p + Au collision harmonium interaction in nuclear matter at FAIR F x-1 -0.5 0 0.5 1 f o r m a t i on l e ng t h (f m ) y J / d R d = 15 GeV p E F x-1 -0.5 0 0.5 1 f o r m a t i on l e ng t h (f m ) y J / d R d = 30 GeV p E Figure 1. x F dependence of the formation length of J/ψ mesons, in the laboratoryframe, in 15 GeV (left panel) and 30 GeV (right panel) p + A collisions. At 15 GeVbeam energy, the resonance formation lengths remain less then 2 fm, even for the fastestmesons. At 30 GeV, formation length is longer than 2 fm, implying the propagationof expanding colorless c ¯ c states beyond x F (cid:39) . systems, as available at the SIS 100 accelerator at FAIR. We rely on the formulation ofproduction kinematics developed in [16]. The interesting question on charm propagationin nuclear matter that can be addressed via charmonium measurements at FAIR wasfirst qualitatively triggered in [18]. This was certainly a call for a more detailed study.We calculate in quantitaive details the differential distribution of J/ψ production crosssections in p + Au collisions. Moreover, we discuss if the J/ψ production cross-sectionobserved in p + A collisions at SIS100 is a suitable probe to distinguish different modelsfor charmonium dissociation. To do so, we estimated this cross-section for differentmodels for nuclear dissociation of J/ψ . The resultant suppression patterns are foundto be distinguishably different, giving us the opportunity to probe the mechanism ofcharmonium dissociation in nuclear medium.
2. Theoretical formulation
In the literature,
J/ψ production in hadronic collisions is usually considered as afactorizable two step process. The first step is the production of a color octet c ¯ c pairthat can be described by perturbative QCD (pQCD). This is followed by the non-perturbative formation of the color singlet resonance, which requires a finite time (seefor example Ref. [19] for an up-to-date review of the quarkonium production up tothe LHC energies). In the c ¯ c rest frame, color neutralization occurs at a time scale of τ (cid:39) .
25 fm [16]. Physical resonances with appropriate size and quantum numbersare believed to take even longer time to form. In our following calculations, we wouldadopt a value τ R (cid:39) .
35 [20] fm for intrinsic formation time. However the choice ofthe resonance formation time is to some extent arbitrary and alternative estimates areavailable in literature [21]. harmonium interaction in nuclear matter at FAIR p + A collisions, to undergo the nuclear medium effects by the resonance itself, thecharmonium states need to be formed inside the nuclear medium or even better beforehitting a nucleon, apart from the one on which it is produced. The second conditionis met if the resonance formation length in the laboratory frame, remains below theaverage distance of two nucleons in the core, which is assumed to amount ∼ d R ) = βγcτ R ) = P L M τ R ) (1)where d and d R stand for the formation lengths of color singlet c ¯ c pair and of fullydeveloped resonance state respectively. The mass and momentum of the c ¯ c resonancestate are denoted by M and P L , respectively. β is the velocity of the state in thelaboratory frame. The center-of-mass (CMS) momentum ( P CMS ) of the resonance stateamounts: P CMS = γ CMS P L − γ CMS β CMS (cid:113) P L + M (2)where β CMS is the velocity of the CMS in the laboratory. The maximum momentum ofa
J/ψ meson in the CMS frame in an elementary reaction, like p + N → J/ψ + p + N ,occurs if the two nucleons travel both in the one direction opposite to that of J/ψ . Ifthe two nucleons travel in forward direction and
J/ψ in backward direction, then in thelaboratory frame
J/ψ would be emitted with the minimum possible momentum (slowest
J/ψ ). The maximum CMS momentum ( P max ) can be obtained from the relation: s = (cid:16)(cid:112) P max + M + (cid:112) P max + 4 m (cid:17) (3)where s and m denote the square of CMS energy and the nucleon mass respectively.Solving Eq.3 one obtains P max = (cid:115)(cid:18) s + 4 m − M √ s (cid:19) − m (4)Concerning the interaction between the charmonium and the nuclear medium, threedifferent kinematical regimes can be distinguished. In a first case, called “color octet”region, the c ¯ c pair penetrates the nuclear core before forming a resonance and theresonance is formed in vacuum. In the so-called “resonance region” the resonance is fullydeveloped before the c ¯ c pair hits a nucleon. In the “transition region”, the formationoccurs while the c ¯ c penetrates the core.The inelastic J/ψ − N dissociation cross section ( σ J/ψ ) can be measured bycomparing the yield in p + A with the one observed in p + p collisions. The survivalprobability S J/ψ for fully formed
J/ψ mesons depend on the average path length insidethe target nucleus ( L A ) as follows: S J/ψ = exp( − n σ J/ψ L A ) (5)where n = 0 .
15 fm − is the saturation nuclear density. Following prescription givenin [22], we have used L A (cid:39) R A ( R A being nuclear radius) for heavy nuclei. In the“transition region”, the dissociation cross section changes while the pair is being formed. harmonium interaction in nuclear matter at FAIR F x-1 -0.5 0 0.5 1 E ( G e V ) (GeV) N ψ s (GeV) ψ J/ E = 15 GeV p E F x-1 -0.5 0 0.5 1 E ( G e V ) (GeV) N ψ s (GeV) ψ J/ E = 30 GeV p E Figure 2. x F dependence of the energy of the J/ψ mesons in the laboratory frame,and the corresponding CMS energy of
J/ψ + N interaction, in 15 GeV (left panel) and30 GeV (right panel) p + A collisions. Based on a classical color dipolar approximation [16], the dissociation cross section ofan evolving color neutral c ¯ c pair can be parametrized as: σ ( d ) = σ (cid:18) d ¯ L (cid:19) (6)Here, d denotes the instantaneous size of an expanding c ¯ c pair and σ is thedissociation cross section for fully developed resonances by nucleons inside the target.¯ L stands for the effective distance travelled until full resonance formation. Using theparameterization would modify the survival probability given in Eq. 5 as S J/ψ = exp (cid:18) − n σ (cid:20) L A −
23 ¯ L (cid:21)(cid:19) (7)which remains valid until the resonance is fully formed. Note that there are otherparametrizations available for modelling the dissociation of an expanding c ¯ c pair [23].As the extent of the transition region (where this scenario would be operative) is notlarge at CBM energies, we refrain from using them. The kinematic threshold for charmoniumproduction in p + p collisions amounts E J/ψth (cid:39) . J/ψ , E ψ (cid:48) th (cid:39) . ψ (cid:48) , and E χ c th (cid:39) . χ c . The SIS100 synchrotron at FAIR will provide protonbeams up to 30 GeV. This allows to produce all three particles near but above threshold.However, experiments will be presumably restricted to J/ψ due to the low yield of theother mesons.At a beam energy of 15 GeV, the maximum momentum of a
J/ψ in the CMS frameof the initial collision amounts | P max | = 1 . P x F = − = 5 .
58 GeV to P x F =1 = 13 .
34 GeV. This corresponds to a rapiditycoverage of 1 . < y LabJψ < .
17. A
J/ψ produced at rest in the CMS frame would flywith a monemtum of 8 .
73 GeV in the laboratory frame. Singlet as well as resonance harmonium interaction in nuclear matter at FAIR x F . Therefore, all the J/ψ are formed in the “resonanceregion”. At a beam energy of 30 GeV, we find | P max | = 2 .
94 GeV. In the laboratoyframe, the momentum range spans from P x F = − = 4 .
94 GeV to P x F =1 = 29 . . < y LabJψ < .
94. At this energy, the momentumof a
J/ψ produced at rest in the CMS frame would be 12 . x F (cid:39) . x F . Once formed, the J/ψ mesons may be absorbed (dissociated) by processes like
J/ψ + N → Λ C + ¯ D or like J/ψ + N → D + ¯ D + N , while travelling inside the nuclear medium. As this dissociationis endothermic, its cross section depends most likely on the CMS energy ( √ s ψN ) of the J/ψ + N collision. This energy is plotted in Fig. 2 along with the energy of the J/ψ mesons ( E J/ψ ) in the laboratory frame. √ s ψN spans from 4.76 to 5.9 GeV for 15 GeVbeam energy, whereas at 30 GeV, it covers a range from 4.6 to 8 GeV. The thresholdCMS energy required for the Λ c ¯ D production is 4 . D ¯ DN channelis 4 . (GeV) N y s5 6 7 8 ( m b ) Q CD a b s s Figure 3.
Variation of
J/ψ dissociation cross section by nucleons as obtained frompQCD in p + A collisions in the FAIR energy domain. Measuring the absorption cross section σ abs with the above mentioned approach requires that the absorption is sufficiently strongto modify the survival probability of J/ψ significantly. As there is no direct experimentalmeasurements on σ abs , we study this question based on different theoretical estimatesavailable in literature.Within geometric approach, σ abs is proportional to the square of the radius of theparticular charmonium state. Even though validity of asymptotic cross sections nearthreshold is not free from doubt, in our calculations we use σ Geoabs (cid:39) . J/ψ following [16].The theoretical estimates of the dynamical dissociation cross sections can be broadly harmonium interaction in nuclear matter at FAIR (GeV) N y s4.5 5 5.5 6 6.5 7 7.5 ( m b ) H adab s s (sum) Hadabs s )D C L ( Hadabs s N)D (D
Hadabs s Figure 4.
Variation of
J/ψ dissociation cross section by nucleons, as obtained fromhadronic models in 15 GeV p + A collisions. The effect of Fermi motion of the nucleonsinside the nucleus is taken into account. divided into two categories. The first approach is based on pQCD. The nucleardissociation cross section within this framework using QCD sum rules was found tobe approximately parametrized as [24] σ J/ψ ≈ . × (cid:34) − (cid:32) M J/ψ ( m N + (cid:15) J/ψ )( s ψN − M J/ψ ) (cid:33)(cid:35) . (8)where, m N = 0 .
94 GeV is the mass of the target nucleon, (cid:15)
J/ψ = 0 .
64 GeV the bindingenergy of the
J/ψ . Since the bound nucleons do not contain a sufficient number of hardgluons, this cross section shows a large threshold damping. Over the energy range tobe probed at FAIR, this cross section is around an order of magnitude smaller than thecorresponding asymptotic values as evident from the Fig. 3. A more rigorous discussionof the pQCD inspired dissociation cross sections can be found in [25, 26]. The maincaveat in this theory is that the
J/ψ is considered as a Coulombic bound state, whichcould only be applicable at very large charm quark mass limit of m c >
25 GeV. Howeverresults from vector meson dominance (VDM) model were found to be in agreement withthe short distance QCD calculations [27].The other approach is non perturbative and uses hadronic models based on quarkexchange [28, 29] or meson exchange [30, 31, 32]. Due to a lack to experimental data,the total inelastic
J/ψ + N cross sections as predicted by these models spreads over asizeable range. But a common feature of all these model is that the dissociation crosssection peaks close to threshold, which stands in contrast to the pQCD predictions.To the best of our knowledge, the most recent effective theory calculations on J/ψ interaction with nuclei appear in [32]. The dominant contribution to
J/ψ dissociationcomes from Λ C ¯ D channel, due to the lowest threshold. The total inelastic cross sectionpeaks around √ s ψN = 4 .
46 GeV which gets diluted once the cross section is averagedover the momenta distributions of the nucleons inside the nucleus. However the crosssections available in these calculations are restricted to a
J/ψ − N cms energy of up to 5GeV. To extend this data to the full FAIR energy range of √ s ψN (cid:39) . − harmonium interaction in nuclear matter at FAIR F x-0.5 0 0.5 y J / S QCDabs s Geoabs s Hadabs s p+Au @ 15 GeV F x-0.5 0 0.5 y J / S QCDabs s Geoabs s Hadabs s p+Au @ 30 geV Figure 5. x F dependence of the J/ψ survival probability in p + Au collisions at FAIR.The effect of feed down from excited states is not taken into account. (fm) t ) t P ( Figure 6.
Distribution of
J/ψ formation time. Black circles denote the data pointsfrom the original calculations and the red line corresponds to the fitted curve requiredfor our modelling. to complement the
J/ψ dissociation via D ¯ DN channel as well, which is not includedin [32]. An estimate of the dissociation cross section via both, the Λ C ¯ D and the D ¯ DN channel, are available in [33]. Including the D ¯ DN channel leads to a monotonic riseof the total inelastic cross section with energy, which would otherwise descend beyondthe Λ C ¯ D threshold. However the total cross section computed for the Λ C ¯ D channelamounts more than a factor of two less than the ones calculated by other authors [31, 32].This is plausibly because the contact interaction is ignored, which seems to provide thedominating contribution at low energies. In the absence of suitable calculations thatinclude all relevant processes with appropriate interactions at FAIR, we estimate theenergy dependence of the total inelastic cross section as follows: Up to D ¯ DN threshold,we take the energy dependence from [32], where as beyond this the total cross section isthe sum of the extrapolated cross section from [32] and the inelastic cross section from D ¯ DN channel as reported in [33]. Since the choice of form factors in such calculationsis no way unique, we took the cross section without form factor corrections. The energydependence of the cross sections for the two processes and their sum as included in ourcalculations is shown in Fig. 4. For consistency, we used the raw inelastic cross sections harmonium interaction in nuclear matter at FAIR (fm) R d0 1 2 3 4 ) R P ( d =0.0 F x =-0.5 F x =0.5 F x = 15 GeV p E (fm) R d0 2 4 6 8 ) R P ( d =0.0 F x =-0.5 F x =0.5 F x = 30 GeV p E Figure 7.
Distribution of
J/ψ formation lengths in 15 (left panel) and 30 (right panel)GeV p + A collisions at x F = − . . . and made suitable extrapolations with linear functions where necessary. Hereafter, wecalculated the average cross section accounting for the Fermi motion of the nucleons.The total average cross section shows a small bump around Λ C ¯ D and then increasesmonotonically with energy due to growing contribution from the D ¯ DN channel. In thekinematic domain probed by FAIR, the dissociation cross sections from hadronic modelsis orders of magnitude larger than the pQCD inspired values. With increase in energyof the J/ψ + N collisions, the difference in the cross sections from the two approachesgradually decreases [33]. We now have all the ingredients neededto calculate the
J/ψ survival probability ( S J/ψ ) at FAIR. Fig. 5 displays the
J/ψ suppression pattern as a function of x F for p + Au collisions of 15 and 30 GeV protonenergy. For 15 GeV, the constant geometric cross sections ( σ Geoabs ) reduces S J/ψ by 20%and remains mostly independent of x F . This is because the absorption depends onlyon the path length of the J/ Ψ in the frame of the target nucleus. For the dynamicalcross sections, S J/ψ varies with x F . However, the tiny cross sections obtained from theperturbative models ( σ QCDabs ) create a negligible absorption and S J/ψ remains nearlyunity. The hadronic models ( σ Hadabs ) predict a much larger suppression at both theenergies. At 15 GeV, the suppression shows a slow increase with x F due to opening ofthe D ¯ DN channel, which grows with energy. At 30 GeV a step appears in the survivalprobability, around x F = 0 . J/ψ suppressionpattern on the different input parameters. Let us first investigate the impact of theresonance formation time ( τ J/ψ ) on the evaluation of S J/ψ . For a given energy, τ J/ψ determines the fate of the produced c ¯ c pairs during their evolution inside the nuclearmedium. So far, a constant value of τ J/ψ (cid:39) .
35 fm is considered. But the choice of τ J/ψ is not unique and highly model dependent. In [34], for the first time, dispersion harmonium interaction in nuclear matter at FAIR F x-1 -0.5 0 0.5 1 y J / S QCDabs s Geoabs s Hadabs s p+Au @ 15 GeV F x-1 -0.5 0 0.5 1 y J / S QCDabs s Geoabs s Hadabs s p+Au @ 30 GeV Figure 8. x F dependence of J/ψ survival probability within the variable formationtime approach, for 15 and 30 GeV p+Au collisions. relations are used to reconstruct, in a model independent way, the formation dynamics ofquarkonium states from the experimental data on e + e → Q ¯ Q annihilation. In contrastto a universal formation time, those calculations lead to a distribution of τ J/ψ with amean value of < τ
J/ψ > = 0 .
44 fm and a width of δ J/ψ = 0 .
31 fm. This certainly calls foran investigation of the effect of such variable formation time to the observed survivalprobabilities. For this purpose we model the extracted distribution of τ J/ψ ( P ( τ )) asshown in Fig. 6. Hereafter, we generate the J/ψ mesons randomly according to thisdistribution of τ J/ψ for a given collision energy and x F . The corresponding resonanceformation lengths in the laboratory frame are given in Fig. 7 for three typical x F valuesat backward, central and forward rapidities. At both the beam energies, most of the J/ψ mesons are found to be formed within a spatial range of 2 fm. For a given x F , thesurvival probability now depends on the corresponding formation time. The averagesurvival probability < S ( x F ) > can then be obtained from the instantaneous survivalprobability ( S ( x F , τ J/ψ ) weighted with the distribution P ( τ ). The resulting suppressionpatterns are depicted in Fig. 8. In contrast to the situation of fixed τ J/ψ , the suddenjump in the survival probability around x F (cid:39) . p + Au collisions is nowwashed out due to averaging effects. For any x F , the resonance region is more populatedthan the transition region. Otherwise the differences in S J/ψ between the two cases offixed and variable τ J/ψ are too meagre to be observed in the experimental data.Next we would like to explore the influence of the alternative estimations of σ QCDabs onthe resulting suppression pattern. In the previous calculations, σ QCDabs has been extractedfrom the parametric form given in Eq. 8. Full QCD calculations of
J/ψ + N dissociationusing short distance QCD methods based on operator product expansion are availablein literature [25]. Within a first order approximation neglecting the correction termsdue to finite mass of the nucleons, one obtains σ QCDabs = 2 π α s m c (cid:90) ξ ( ξx − ( ξx ) g ( x ) x dx (9)with ξ = λ(cid:15) J/ψ , where λ = s ψN − M J/ψ − m N M J/ψ . The charm quark mass and the strong harmonium interaction in nuclear matter at FAIR (GeV) N y s5 6 7 8 ( m b ) Q CD a b s s parametrized QCD order st QCD 1 order: MSTW 2008 st QCD 1 order: EPS09 st QCD 1
Figure 9.
Comparison of
J/ψ + N dissociation cross sections from short distanceQCD calculations for various parameterization of gluon density distributions. F x-1 -0.5 0 0.5 1 y J / S :parametric form QCDabs s order st : 1 QCDabs s Hadabs s p+A @ 15GeV F x-1 -0.5 0 0.5 1 y J / S :parametric form QCDabs s order st : 1 QCDabs s Hadabs s p+Au @ 30 GeV Figure 10.
Comparison of
J/ψ survival probabilities for different estimations of QCDabsorption cross sections. For completeness, the suppression from hadronic modelcalculation is also shown. coupling constant are respectively denoted by m c and α s . g ( x ) denotes the gluondistribution function in the nucleons inside the nucleus. We have evaluated σ QCDabs within FAIR energy domain, for four different cases namely the parametric form givenin Eq. 8, and estimations using Eq. 9 with parameterization of gluon distribution g ( x ) = 2 . − x ) , and two more realistic gluon densities, namely MSTW 2008 leadingorder (LO) free proton parton distribution function (PDF) [35] and EPS09 LO nuclearparton distribution function (nPDF) [36]. Results are shown in Fig. 9. The firstorder QCD estimations with parameterized gluon distribution generates much largerdissociation compared to the parameterized dissociation cross section. The differencegrows with increasing energy of the J/ψ + N collisions. Results with the realistic gluondistributions lie in between but closer to the estimations with parameterized gluondistributions.The resulting effect of the amplified σ QCDabs to the
J/ψ suppression scenario is shownin Fig. 10 which includes the
J/ψ survival probability for two different estimationsof σ QCDabs producing smallest and largest absorptions. For meaningful comparison, the harmonium interaction in nuclear matter at FAIR F x-0.5 0 0.5 p A R no absorption QCDabs s Geoabs s Hadabs s p+Au @ 15 GeV F x-0.5 0 0.5 p A R no absorption QCDabs s Geoabs s Hadabs s p+Au @ 30 GeV Figure 11. x F dependence of J/ψ R pAu in 15 and 30 GeV p + Au collisions, withvariable formation time approach. suppression within hadronic picture is also added. Calculations are done within thevariable formation time approach as this possibly simulates the more realistic formationtime dynamics. Maximum difference between two QCD estimates is (cid:39)
13% at close to x F = 1. Otherwise the difference is minimal and a distinct separation from the hadroniccase is still prominent. R pA In experiments, the survival probability S J/ψ is hard tomeasure as this requires to isolate the final state absorption of the particles fromthe initial state effects determining the particle production. Therefore, experimentalcollaborations report rather the
J/ψ production cross sections for different target nuclei.From those measured cross sections, one can construct the ratio R pA defined as ratioof the production cross section in p + A to that in p + p collisions. This ratio encodesall the possible CNM effects that modify the production during different stages of the J/ψ evolution. Here, we give predictions for the x F dependence of J/ψ R pA in p + Au collisions at FAIR, within variable formation time approach. Besides the final statedissociation discussed above, we include the initial state modification of the partondensities inside the target nucleus. This effect modifies the overall c ¯ c production in p + A collisions.The J/ψ production is calculated using Color Evaporation Model (CEM) [37]. Thetotal c ¯ c production cross sections in p + p are estimated for two leading order partonicsub processes namely gg fusion and q ¯ q annihilation, with MSTW 2008 [35] LO freeproton PDF. Higher order corrections are accounted by a phenomenological K factor.In case of p + Au collisions, the shadowing effects inside the Au nucleus, are incorporatedusing the EPS09 [36] LO nPDF set. For both parton densities, we have used the centralsets having minimum uncertainties. It might be interesting to note here that at thekinematic domian probed by the charmonium production at SPS corresponded to anantishadowing region, where the parton densities in nuclei are enhanced with respect tothose at free protons. However at FAIR energies, close to mid-rapidity nuclear partondensities are depleted leading to a reduction of overall c ¯ c production cross sections, even harmonium interaction in nuclear matter at FAIR x F dependence of R pAu for J/ψ in 15 and 30 GeV p + Au collisions areshown in Fig. 11, for the three absorption scenarios discussed above. For completeness,we also added a curve without any final state dissociation. The production crosssections are found to be negligible beyond | x F | > .
5, and thus ignored. As evident,the suppression curves for different absorption mechanisms are clearly distinguishable.Given the unprecedented beam intensities aimed at FAIR accelerators, the collecteddata are expected to suffer from much smaller statistical uncertainties. This wouldmake an experimental distinction amongst different dissociation patterns feasible.At this juncture, we would like to remind our readers that subsequent to thepredictions in [16], E866/NuSea Collaboration [11] at Fermilab observed for thefirst time, substantial differences in suppression pattern between the ψ (cid:48) and J/ψ in p + A collisions in the backward hemisphere. Subsequent measurements by NA50Collaboration at SPS [9, 10] were also in qualitative agreement with this observation.Stronger absorption of ψ (cid:48) compared to J/ψ was seen while measuring inclusivecharmonium production cross sections around mid-rapidity in p + A collisions fora variety of nuclear targets. Assuming resonance formation within nuclear core,Fermilab data and preliminary NA50 data collected at 450 GeV, were explained usingmodels that employ radial expansion of the color transparent tiny c ¯ c pair to the fullresonance [23, 40, 41]. However the formation time as well as the inelastic reaction crosssections were generally kept as free parameters among the others which were fixed fromthe data and shadowing corrections at the initial stage of production were not takeninto account. In [42], the absorption cross sections were extracted over a broad range ofcollision energies starting from d + Au collisions at √ s NN = 200 GeV at RHIC, downto p + A collisions at 158 GeV at SPS, within the same model [23] of expanding c ¯ c pairsbut with shadowing corrections explicitly taken into account.As mentioned earlier, at SIS 100 measurements of ψ (cid:48) would not seem to be feasible,and charmonium measurements would possibly be confined only to the study of J/ψ mesons, which are always formed inside the target nucleus. At SIS 100 energies
J/ψ production would occur close to the kinematic thershold, where the differencebetween the dissociation cross sections obtained from perturbative and non-perturbativeapproaches show maximum difference, which is evident from the x F dependence of theabsorption profile. This makes these upcoming measurements very relevant and uniquecompared to the existing studies. Before we close, it is important to take a note on the limitations of our present analysis.We employ pQCD to calculate
J/ψ production cross sections, at SIS100 energies.This assumption is debatable, see Refs. [43, 44, 45], for alternative approaches of
J/ψ production at near threshold beam energies. However, the validity of QCD factorization,in the near threshold quarkonium production can only be tested with data for FAIR harmonium interaction in nuclear matter at FAIR χ c , ψ (cid:48) ) to the overall J/ψ production has not been taken into account assuming their rare occurrence due tohigher kinematic threshold.In the evaluation of R pA , the parton energy loss, which might affect the momentumdistribution of the produced c ¯ c pairs, is not accounted for. The transverse momentum( p T ) of the J/ψ mesons is artificially set to zero. At FAIR, the
J/ψ mesons are expectedto be produced with a very small p T , which justifies this approximation.For extracting the J/ψ dissociation cross sections from the hadronic models, weextrapolated the results of [32] to higher energies and combined with calculationsobtained from a different article [33]. Combining parameters obtained from two differentcalculations as much as extrapolating bears its associated uncertainties, which had to beaccepted due to a lack of better suited calculations. A consistent calculation within thesame model framework including all known contributions to
J/ψ + N inelastic interactionat FAIR energies would be highly welcome.
3. Summary
In this work, we estimated the
J/ψ suppression pattern in p + Au collisions in thekinematic domain suitable for the experiments planned at FAIR. Our calculationssuggest that the slow J/ψ mesons produced in those low energy collisions will propagatethrough the nuclear matter as fully developed physical resonances.In p + A interactions the final state absorption effects are limited to the targetnucleons alone. The possibility of additional suppression due to co-moving secondaryhadrons is drastically reduced in p + A relative to A + B collisions at these low energies.The experimental examination of the x F (or rapidity) dependence in p + A reactions is avery promising test of J/ψ propagation in baryonic matter, because of the very distinctbehaviour of the corresponding survival probability.The world data collected so far on
J/ψ production from different experiments, cannot shed much light on this issue. Most of the data are collected at higher energies,where propagation of the pre-resonance c ¯ c states through the nuclear matter dominatesthe experimentally explored phase space domain. Hence neither a direct extrapolationof those results (dissociation cross sections) to lower energies nor an application of thesedissociation mechanisms at higher energies seems to be viable. At FAIR energies, thevarious scenarios of J/ψ absorption give a very distinct survival probability leading todistinguishably different pattern of R pA .Such measurements are certainly feasible with the CBM detector set up at SIS100energies which would also be capable to experimentally discriminate among variousabsorption scenarios. These results will also prove useful for interpretation of heavy-iondata to be collected in future. harmonium interaction in nuclear matter at FAIR
4. Acknowledgements
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