Nuclear suppression of heavy quark production at forward rapidities in relativistic heavy ion collisions
aa r X i v : . [ nu c l - t h ] M a y Nuclear suppression of heavy quark production atforward rapidities in relativistic heavy ion collisions
Umme Jamil and Dinesh K. Srivastava
Variable Energy Cyclotron Centre, 1/AF Bidhan Nagar, Kolkata 700064, India
Abstract.
We calculate nuclear suppression R AA of heavy quarks produced from theinitial fusion of partons in nucleus-nucleus collisions at RHIC and LHC energies. Wetake the shadowing as well as the energy loss suffered by them while passing throughQuark Gluon Plasma into account. We obtain results for charm and bottom quarks atseveral rapidities using different mechanisms for energy loss, to see if we can distinguishbetween them.
1. Introduction
The heavy ion collision experiments at RHIC and LHC are designed with a hope toexplore the existence of a new form of matter known as Quark Gluon Plasma (QGP) andto explore its properties. The estimation for the energy density [1, 2, 3, 4] attained inthese collisions using the Bjorken formula [5] is well beyond the energy densities whereQGP is expected to be formed. The temperatures reached at RHIC, as revealed fromseveral studies (see e.g., [6, 7, 8] for a compilation) are also much larger than the valuesprovided by Lattice QCD calculations for the critical temperature for a transition toQGP [9]. Strong confirmation of the formation of the QGP is given by observation ofa large elliptic flow [10], jet-quenching [11], and the recombination of partons as themechanism of production of hadrons at intermediate transverse momenta [12]. Stillhigher temperatures are likely to be reached at LHC.Heavy quarks are produced from the initial fusion of gluons ( gg → QQ ) or lightquarks ( qq → QQ ). This pair would be produced at τ ≈ / M Q ≪ uclear suppression of heavy quark production.. QQ → gg , etc.) can be safely ignored. Thusheavy quarks and in turn charm and bottom hadrons will stand out in the back-groundof a multitude of light hadrons, and one can in principle track them.Along with other reasons, an interest in the study of energy loss of heavyquarks was triggered by the large back-ground that correlated charm or bottom decayprovides [13, 14] to the thermal dileptons which have been considered a signature of theformation of QGP for a long time [15, 16, 17, 18, 19]. It was pointed out by Shuryak [20],Lin et al. [21, 22], Kampfer et al. [23], and Mustafa et al. [24, 25] that the correlatedcharm and bottom decay could be suppressed if the energy loss suffered by heavy quarksbefore they form D or B mesons was accounted for. Since then several attempts havebeen made to estimate the energy loss of heavy quarks as they proceed through QGP.Of course the possibility to identify the vertex of D or B meson decay will further enrichthis study.These results have been put to a rigorous test by the measurement of single electronsfrom heavy ion collisions at RHIC which show a clear evidence for the loss of energyby the heavy quarks [26]. Their possible thermalization is also indicated by the ellipticflow that they show [27].As indicated earlier, the temperature likely to be reached at LHC in collision ofheavy nuclei could be even larger and thus this energy loss will play a more significantrole. The opening of a much wider window in rapidity at LHC is also likely to providewidely differing media at different rapidities through which the heavy quarks wouldpropagate.Thus a valuable test of various theories for energy loss suffered by heavy quarkscan be performed by studying it at RHIC and LHC and at different rapidities.We study these effects in terms of nuclear modification factor R AA for heavy quarks.In these initial studies we calculate the average energy loss suffered by them as theypass through the QGP and compare the resulting p T distribution with the same forproton-proton (pp) collisions to get R AA . Since the mass of charm or bottom quarks isquite large, the p T distribution of these quarks will closely reflect the p T distribution ofD or B mesons.We employ a local fluid approximation [13, 14, 28] in order to picture the mediumat larger rapidities. We shall come back to this later.The paper is organized as follows. As we need to compare the spectra of the heavyquarks from relativistic heavy ion collisions with those for pp collisions, as a first stepwe study the heavy quark production in LO pQCD and compare our results with aNLO pQCD calculation. We find that single quark distribution calculated using LOpQCD supplemented with a K-factor adequately reproduces the NLO results as well asthe available experimental data. Next we estimate the average energy loss suffered byheavy quarks of a given energy using various mechanisms discussed in the literature.Finally we perform a Monte Carlo calculation to obtain the average change in thetransverse momentum spectra of heavy quarks for nucleus-nucleus collisions and get R AA as a function of p T for different rapidities. We add that this work is not intended uclear suppression of heavy quark production..
2. Heavy quark production in pp collisions
At lowest order in pQCD, heavy quarks in pp collisions are produced by fusionof gluons ( gg → QQ ) or light quarks ( qq → QQ ) [31]. The so-called flavour excitationprocess ( qQ → qQ and gQ → gQ ) is now known to be suppressed when the NLOprocesses are taken into account [32, 33, 34]. In addition, Brodsky et al. [35, 36] haveshown that the total contribution of intrinsic charm in the midrapidity region is smalleven though most of the heavy quarks are produced in this region.The cross-section for the production of heavy quarks from pp collisions at lowestorder is given by [31, 37]: dσdy dy dp T = 2 x x p T X ij [ f (1) i ( x , Q ) f (2) j ( x , Q )ˆ σ ij (ˆ s, ˆ t, ˆ u ) + f (1) j ( x , Q ) f (2) i ( x , Q )ˆ σ ij (ˆ s, ˆ t, ˆ u )] / (1 + δ ij ) . (1)In the above equation, i and j are the interacting partons, f (1) i and f (2) j are the partonicstructure functions and x and x are the fractional momenta of the interacting hadronscarried by the partons i and j. The relation between p T and fractional momentum x or x through their respective rapidities can be written as x = m T √ s ( e y + e y ) , x = m T √ s ( e − y + e − y ) , (2)where m T is the transverse mass, q M + p T , of the produced heavy quark. Thefunction ˆ σ = dσ/dt , the short range subprocess for the heavy quark production isdefined as: dσdt = 116 π ˆ s |M| . (3) |M| for the heavy quark production processes gg → Q ¯ Q and q ¯ q → Q ¯ Q are expressedthrough the mass of the heavy quark and Mandelstam variables ˆ s , ˆ t , and ˆ u as (cid:12)(cid:12)(cid:12)(cid:12) M (cid:12)(cid:12)(cid:12)(cid:12) gg → Q ¯ Q ) = π α s (cid:20) s (cid:16) M − ˆ t (cid:17) (cid:16) M − ˆ u (cid:17) + 83 (cid:16) M − ˆ t (cid:17) ( M − ˆ u ) − M (cid:16) M + ˆ t (cid:17)(cid:16) M − ˆ t (cid:17) + 83 (cid:16) M − ˆ t (cid:17) ( M − ˆ u ) − M ( M + ˆ u )( M − ˆ u ) uclear suppression of heavy quark production.. − M (ˆ s − M )3 (cid:16) M − ˆ t (cid:17) ( M − ˆ u ) − (cid:16) M − ˆ t (cid:17) ( M − ˆ u ) + M (cid:16) ˆ u − ˆ t (cid:17) ˆ s (cid:16) M − ˆ t (cid:17) − (cid:16) M − ˆ t (cid:17) ( M − ˆ u ) + M (cid:16) ˆ t − ˆ u (cid:17) ˆ s ( M − ˆ u ) (4)and (cid:12)(cid:12)(cid:12)(cid:12) M (cid:12)(cid:12)(cid:12)(cid:12) q ¯ q → Q ¯ Q ) = 64 π α s (cid:16) M − ˆ t (cid:17) + ( M − ˆ u ) + 2 M ˆ s ˆ s . (5)The running coupling constant α s at lowest order is α s = 12 π (33 − N f ) ln ( Q / Λ ) , (6)where N f = 3 is the number of active flavours and Λ = Λ QCD . We use the factorizationand renormalization scales as Q = m T . We refer the readers to Vogt et al. [38] forresults on variations of these scales. We also carry out the calculation of differentialcross section for heavy quarks in pp collision at NLO in pQCD using the treatmentdeveloped by Mangano, Nason, and Ridolfi (MNR-NLO) [39]. All the calculations arecarried out by neglecting the intrinsic transverse momentum of the partons.The effect of nuclear shadowing in high energy nucleus-nucleus collisions is wellknown [40, 41, 42, 43]. With the increase of the mass number of the nucleusand increasing contribution of terms having small x, the effect becomes morepronounced. We introduce the shadowing effect in our calculations by using EKS 98parameterization [44] for nucleon structure functions. We take CTEQ4M [45] structurefunction set for nucleons.We shall see that x dependence of the shadowing function introduces interestingstructures in the nuclear modification factor as a function of p T , y, and the incidentenergy, because of the large mass of the quarks.In Fig. 1 we compare our results for heavy quark p T distribution obtained usinglowest order pQCD for pp collision with the results from NLO-MNR calculationat midrapidity for charm and bottom quarks at RHIC and LHC energies. Thesecomparisons suggest a K factor of ≈ D Ø data [46, 47]. We find a good description of these data using lowest order pQCDwith a K-factor.We calculate the total cross-section for charm quark production at lowest order forthe process pp → c ¯ c as a function of √ s considering the charm quark mass as 1.2 GeVand 1.6 GeV. We have also included results for M c (3 GeV) = 0 .
986 GeV [48], suggested uclear suppression of heavy quark production.. p T (GeV) -6 -5 -4 -3 -2 -1 d σ / d p T d y ( µ b / G e V ) Charm (M c =1.6 GeV) µ R = µ F = √ (M c2 + p T2 )pp @ 200 GeV pp @ 5.5 TeV CTEQ4M p T (GeV) -8 -7 -6 -5 -4 -3 -2 -1 d σ / d p T d y ( µ b / G e V ) Bottom (M b =4.7 GeV) µ R = µ F = √ (M b2 + p T2 )pp @ 200 GeV pp @ 5.5 TeV CTEQ4M Figure 1. [Left panel] Comparison of our lowest order pQCD results with the NLO-MNR calculation for charm quark ( M c = 1 . M b = 4 . p Tmin (GeV) -2 -1 σ ( p T > p T m i n ) ( nb ) D0 preliminary UA1 CDF preliminary 2.5 x LO √ s=630 GeV, |y| < 1.5 CTEQ4M pp bX Bottom quark (M b = 4.7 GeV) p Tmin (GeV) σ ( p T > p T m i n ) ( nb ) CDF D0 3 x LO
CTEQ4M Bottom quark (M b =4.7 GeV) pp bX √ s=1.8 TeV, |y| < 1 Figure 2.
Comparison of our result for σ (cid:0) p T > p minT (cid:1) for production of bottomquarks with experimental data. recently. Though the last value is not at the pole Q = M c , it may serve as the lower limitto the mass of the charm quark. In the left panel of Fig. 3 we compare these results andthe results from NLO-MNR calculation with the experimental data points [47, 49, 50].Here also our lowest order pQCD calculation show a good agreement with experimentaldata points for M c = 1.2 GeV. In the right panel of Fig. 3 we compare our results withthe NLO-MNR calculations up to √ s = 15000 GeV.We also calculate the total cross-section for bottom quark production for pp → b ¯ b as a function of √ s considering the bottom quark mass as 4.7 GeV and 4.163 GeV [48].In Fig. 4 we compare these results and the results obtained from NLO-MNR calculationwith the experimental data points [51]. This comparison is quite impressive as at M b = 4.7 GeV our lowest order result accurately reproduce the result of NLO-MNRcalculation with K = 2.5. uclear suppression of heavy quark production.. √ s (GeV) σ ( µ b ) SPS/FNAL p beamNA37ISR exptsUA2PHENIX single e in AuAuE743PHENIX single e in ppSTAR dAuSTAR AuAuCosmic rayM c = 0.986 GeV (LO x 2.5)M c = 1.2 GeV (LO x 2.5)M c = 1.6 GeV (LO x 2.5)M c =1.6 GeV (NLO-MNR) µ R = µ F = √ ( M c2 + p T2 ) pp cc CTEQ4M √ s (GeV) -5 -4 -3 -2 -1 σ ( µ b ) M c = 0.986 GeV (2.5 x LO)M c = 1.2 GeV (2.5 x LO)M c = 1.6 GeV (2.5 x LO)M c = 1.6 GeV (NLO-MNR) µ R = µ F = √ ( M c2 + p T2 ) pp cc CTEQ4M
Figure 3.
Total cross-section for pp → c¯ c compared with experimental data at varying √ s . Thus we see that the p T distribution and production cross-section for the heavyquarks calculated in lowest order pQCD and supplemented with a K-factor ≈ R AA . √ s (GeV) σ ( nb ) E789E771HERA-BUA1CDFM b = 4.163 GeV (2.5 x LO)M b = 4.7 GeV (2.5 x LO)M b = 4.7 GeV (NLO-MNR) µ R = µ F = √ (M b2 + p T2 ) pp bb CTEQ4M Figure 4.
Total cross-section for pp → b ¯ b compared with experimental data varying √ s .
3. The Initial conditions and the Evolution of the Plasma
The heavy quarks produced at the initial stage pass through the QGP, where they looseenergy by colliding with quarks and gluons and also by radiating gluons. The energy losswill depend upon the path-length of the heavy quarks in the plasma, the temperatureevolution of the plasma, and the energy and mass of the heavy quarks. uclear suppression of heavy quark production.. dN g dy = dN g dy ! exp (cid:16) − y / σ (cid:17) . (7)We take ( dN g dy ) ≈
900 and σ = 3 for Au+Au collisions at RHIC [52] and ≈ σ = 4 for Pb+Pb collisions at LHC [53].The Bjorken cooling is then assumed to work locally at different rapidities, andwe consider the passage of a heavy quark having rapidity y in a fluid having anidentical fluid rapidity. This approximation, which corresponds to assuming a boost-invariant expansion along with a local fluid approximation, has been used earlierin literature [13, 14, 28]. A more complete study would use a (3 + 1) dimensionalhydrodynamics [54, 55], which we plan to use in future publications. Figure 5.
The distance, L, covered by a heavy quark while passing through the QGP.For central collisions the results for h L i will not depend on Φ. We consider a heavy quark produced in a central collision, at the point (r, Φ), andmoving at an angle φ with respect to ˆr in the transverse plane. In general the distancecovered by the heavy quark before it exists the QGP, will vary from 0 to 2R, where Ris the radius of the colliding nuclei. The distance covered by the heavy quark in theplasma, L, is given by [56]: L ( φ, r) = q R − r sin φ − r cos φ. (8) uclear suppression of heavy quark production.. h L i = R R r dr π R L ( φ, r) T AA (r , b = 0) d φ R R r dr π R T AA (r , b = 0) d φ . (9)In the above the nuclear overlap function T AA (r, b = 0) provides the probability ofproduction of heavy quarks in hard binary collisions. We find that h L i is 5.78 fm forAu+Au collisions at RHIC and 6.14 fm for Pb+Pb collisions at LHC, and is about 20%smaller than the radii of the colliding nuclei, as the appearance of the nuclear overlapfunction gives a larger weight to the points having smaller r.As the heavy quarks loose most of their energy in interaction with gluons, it isenough to consider only the distribution of gluons. Their density at the time τ can bewritten as [57]: ρ ( τ ) = 1 π R τ dN g dy . (10)The corresponding temperature [57], assuming a chemically equilibrated plasma is T ( τ ) = π . ρ ( τ )(9 N f + 16) ! . (11)The rapidity dependence of the temperature of the plasma at a typical τ = h L i / Rapidity (y) T e m p er a t u re ( G e V ) Pb+Pb @ 5.5A TeVAu+Au @ 200A GeV (dN g /dy) ~ 900, τ = L/2, L ~ 5.7 fm and σ = 3 (dN g /dy) ~ 3300, τ = L/2, L ~ 6 fm and σ = 4 Figure 6.
Variation of temperature of QGP with rapidity y at a typical τ . Assuming that the QGP is formed at τ = 0.2 fm/c, we estimate the T at y = 0for RHIC as 377 MeV and LHC as 555 MeV. More detailed studies do suggest a largerformation time of ≈ uclear suppression of heavy quark production.. τ = L eff /
2, which is much larger (see thefollowing discussion). Further assuming, Bjorken’s cooling law, T τ = constant, thisprovides that the plasma would cool down to the transition or critical temperature T c ≈
160 MeV by τ c ≈ v T = p T /m T , it would take a time τ L = h L i /v T to cross the plasma. If τ c ≥ τ L the heavy quark would be inside QGP duringthe entire period, τ to τ L . However, if τ c < τ L , only while covering the distance v T × τ c , would the heavy quark be in the QGP phase. We further approximate theexpanding and cooling plasma with one at a temperature of T at τ = h L i eff /
2, where h L i eff = min [ h L i , v T × τ c ]. This procedure has been used frequently [57].
4. Mechanisms for Energy Loss
Next we discuss the energy loss mechanisms that we have included. As mentionedabove, repeatedly, the heavy quarks loose energy both by collisions as well as radiationof gluons. A number of formalisms have been proposed for the collisional as well as theradiative energy loss of heavy quarks in the literature. We shall consider the followingtreatments for the collisional energy loss.Bjorken [58] has considered the collisional energy loss of light quarks as analogousto the energy loss of a charged particle passing through a medium and losing energyby ionizing the medium. His expression for massless quarks was adapted by Braatenand Thoma to the case of heavy quarks [see Eq. A.1]. We shall continue to label thismechanism as Bjorken for clarity. Braaten and Thoma (BT) [59, 25] also modified theexpression for the energy loss suffered by muons while traversing QED plasma, to obtainthe collisional energy loss of a heavy quark as it passes through the QGP [see Eqs. A.2and A.3 in the Appendix A]. These results are valid for collisions where the momentumtransfer q << E, where E is the energy of the heavy quark. Peigne and Peshier (PP) [60]have improved this treatment by including the u-channel, which becomes important forlarge energies [see Eq. A.4 in the Appendix A].For the calculation of radiative energy loss, we consider the treatment of Djordjevic,Gyulassy, Levai, and Vitev (DGLV) [61, 57] using opacity expansion, the treatmentof Armesto, Salgado, and Wiedemann (ASW) [62] using path integral formalism formedium-induced gluon radiations off massive quarks, and the treatment of Xiang, Ding,Zhou, and Rohrich (XDZR) [63] using light cone path integral approach. Detailedexpressions for these formalisms are given in the Appendix B.
5. Results for Energy Loss
We compare the results for transverse energy loss for a heavy quark using these differentenergy loss treatments for several rapidities. We plot the transverse energy loss of charmand bottom quarks, ∆ E T as a function of transverse energy E T ( q p T + M ) in Figs. 7 uclear suppression of heavy quark production.. ASW (rad.)DGLV (rad.)XDZR (rad.)
Bjorken (coll.)PP (coll.)BT (coll.) ∆ E T ( G e V ) ( c h a r m ) y=0 y=2y=4 Pb+Pb @5.5A TeVAu+Au @200A GeV y=0y=1y=2 E T (GeV) Figure 7.
Collisional (dotted lines) and radiative (solid lines) energy loss suffered bya charm quark while passing through the QGP and 8, at RHIC and LHC energies.Several interesting features emerge. We see that the collisional energy loss for charmquarks at RHIC and LHC energies is only marginally dependent on the rapidity andthe BT formalism gives largest energy loss, as expected. In our treatment, change ofrapidity implies a change in the temperature of the plasma. Thus these results suggesta weaker dependence on the temperature and the average path length for the energyloss suffered by charm quarks due to collisions.The radiative energy loss, on the other hand, shows a much more complex behaviourand is quite different for the different formalisms under consideration. We note that theASW formalism for radiative energy loss gives largest degradation in the energy at allrapidities (except for E T < uclear suppression of heavy quark production.. ASW (rad.)DGLV (rad.)
Bjorken (coll.)PP (coll.)BT (coll.) ∆ E T ( G e V ) ( b o tt o m ) y=0y=1 E T (GeV) Au+Au @ 200A GeV y=0 y=2y=4y=2
Pb+Pb @ 5.5A TeV
Figure 8.
Same as Fig. 7 for a bottom quark loss for charm quarks at y=0 and y=2, and the corresponding results at y=4 differ byabout 10%. This, we feel, is due to a more complex dependence on the average pathlength in the ASW formulation.The collisional energy loss for bottom quarks using the PP and the Bjorken’sformulation are seen to be quite similar at RHIC and LHC energies for all rapiditiesunder consideration. The BT formulation due to the neglect of the u-channel, gives amuch smaller energy loss, both for RHIC and LHC energies and at all rapidities.We have already mentioned that due to the numerical approximations used, theXDZR formulation is not valid for evaluation of the radiative energy loss for bottomquarks. The ASW and DGLV radiative energy loss formalisms show a more complexdependence on the mass and the average path length. The ASW formulation gives alarger energy loss at RHIC energy, though the results are again comparable at LHCenergy at all rapidities. We note that while the collisional and radiative energy lossesfor bottom quarks at RHIC energy are comparable, the collisional energy loss dominatesover the radiative energy loss in the E T range under consideration at LHC energy. We uclear suppression of heavy quark production.. E T , the radiative energy loss again starts dominating.This rich structure suggests that description of energy loss for one (quark) mass atone rapidity, and one energy may not be enough to identify the most reliable treatments,for this. R AA for heavy quarks The nuclear modification factor R AA for heavy quarks can be expressed as: R AA ( b ) = dN AA /d p T dyT AA ( b ) d σ NN /d p T dy , (12)where, as mentioned earlier, T AA ( b ) is the nuclear overlap function for impact parameterb, calculated using Glauber model. We get T AA ≈
280 fm − for Au+Au collisions atRHIC and ≈
290 fm − for Pb+Pb collisions at LHC, for b = 0 fm. p T (GeV) y=0y=2y=4 y=0y=1y=2 Pb+Pb @ 5.5A TeVAu+Au @ 200A GeV R AA ( c h a r m ) Only shadowingOnly shadowing p T (GeV) y=0y=2y=4 y=0y=1y=2 Pb+Pb @ 5.5A TeVAu+Au @ 200A GeV R AA ( b o tt o m ) Only shadowingOnly shadowing
Figure 9. [Left panel] R AA of charm quarks with only nuclear shadowing effect atmore forward rapidities. [Right panel] Same as left panel for bottom quark. As a first step, we give the results for R AA with only nuclear shadowing effect forproduction of charm and bottom quarks at the rapidities considered earlier for RHICand LHC energies (see Fig. 9). We see that the fairly large masses of the charm andbottom quarks, the kinematics, and the rich behaviour of the structure function with xand Q , lends interesting features to R AA .We see that R AA for charm and bottom quarks for y = 0 and y = 1 are quite similarat RHIC energy. Similarly, the results for y = 0 and y = 2 are only marginally differentat LHC energy. The results at larger y are more strongly affected due to increasedvariation in the ’x’ values (see Eq. 2) which contribute. In order to do a full justice tothese interesting results, we now discuss them individually.For charm quarks at RHIC energy, we see a suppression at lower p T , an enhancementat intermediate p T , and again a suppression at larger p T , for y = 0 and y = 1, while fory = 2, R AA starts at about 0.8, goes up to a value slightly more than 1, then drops againto about 0.8 at p T ≈
20 GeV, and rise again to beyond 1 at p T ≈
40 GeV. Since the uclear suppression of heavy quark production.. p T , this would introduceinteresting features in R AA after this is accounted for, unless of course the p T spectrumfor the quarks drops too rapidly. We shall come back to this point again. The increasedenergy at LHC then provides a larger suppression at low p T for all the rapidities. In aninteresting development, R AA for y = 0 and y = 2 rises beyond 1 at p T ≈
20 GeV, whileit stays below 1 up to p T ≈
40 GeV, for y = 4.The results for bottom quarks are even more interesting. Due to the large mass ofthe bottom quarks, at RHIC energy, the R AA for lower p T for y = 0 and y = 1 is alreadystarts getting contributions from the region of x where anti-shadowing appears. Thus R AA starts at a value which is more than 1 at lower p T , goes up, up to p T ≈ p T ≈
20 GeV and rises again. At LHC energy, R AA for bottom quarks fory = 0 and y = 2 starts at ≈
30 % below 1 and then rises steadily to about 1.1 at p T ≈ p T . y=0y=1y=2 R AA ( c h a r m @ A G e V ) p T (GeV) Shadowing+PPShadowing+ASWShadowing+PP+ASW
Shadowing+PP+DGLVShadowing+PP+ASW R AA ( C h a r m @ A G e V ) p T (GeV) y=0y=1y=2 Figure 10. [Left panel] R AA of charm quarks with the nuclear shadowing effect aswell as the energy loss at more forward rapidities at RHIC energy. [Right panel]Comparison of the relative nuclear suppression using ASW and DGLV formalisms forcharm quarks at different rapidities at RHIC energy. Now let us discuss our results for R AA with the additional inclusion of collisionaland radiative energy losses. We shall restrict our consideration to inclusion of collisionalenergy loss using the PP formulation and the radiative energy loss using ASW or DGLVformulation.Fig. 10 gives our findings for charm quarks at RHIC energy. We see that theoutcome of shadowing and energy loss gives an interesting structure to R AA , as expected.We see that the final R AA starts at about 40 % below 1, goes up to about 0.8 at p T ≈ ≈ p T ≈
15 GeV. In an interesting development,we see that the combination of the shadowing and energy loss gives a marginally largersuppression at y = 2 compared to y = 0, even though the fractional energy loss is higherat smaller y (see Fig. 7). We have also given a comparison of R AA by replacing the uclear suppression of heavy quark production.. R AA of about 0.2 ∼ p T > y=0y=2 R AA ( c h a r m @ . A T e V ) y=4y=0 (@ 200A GeV) p T (GeV) Shadowing+PP Shadowing+DGLVShadowing+PP+DGLV
Figure 11. R AA of charm quarks with the nuclear shadowing effect as well as theenergy loss at more forward rapidities at LHC energy. The results for the R AA for charm quarks at LHC energy are shown in Fig. 11. Weshow the results only for the DGLV formulation for the radiative energy loss, as we haveseen that it is quite similar to that for the ASW formalism for charm quarks at LHCenergy. We see roughly similar behaviour, in that the R AA starts from a lower value at p T ≈ p T ≈ p T .We also find a marginally larger suppression for y = 0 compared to that for y = 4. Theresults for y = 0 for RHIC energy are also given for a comparison which suggests a muchlarger suppression at LHC, as expected.Next we discuss our findings for nuclear suppression for bottom quarks at RHICenergy (see Fig. 12). The shadowing and the large mass of the bottom quarks, withits consequences, gives an R AA ≈ p T ≈ p T ≈ ∼ p T . The shadowing results ina larger suppression for y = 2 compared to y = 0 (see Fig. 9), even though the energyloss is slightly lower for larger y (see Fig. 8). Results obtained by replacing the ASWformulation with the DGLV treatment show a smaller suppression, as for charm quarks(see Fig. 10).Finally in Fig. 13 we have given our results for R AA for bottom quarks at LHCenergy for y = 0, 2 and 4 using shadowing, collisional energy loss and radiative energyloss using the DGLV treatment. The results using ASW treatment are expected to bequite similar as seen from Fig. 8. The results for y = 0 at RHIC energy are also given uclear suppression of heavy quark production.. y=0y=1y=2 R AA ( b o tt o m @ A G e V ) p T (GeV) Shadowing+PPShadowing+ASWShadowing+PP+ASW
Shadowing+PP+DGLVShadowing+PP+ASW R AA ( B o tt o m @ A G e V ) p T (GeV) y=0y=1y=2 Figure 12.
Same as Fig. 10 for bottom quarks. y=0y=2 R AA ( b o tt o m @ . A T e V ) y=4y=0 ( @ 200A GeV) p T (GeV) Shadowing+PP Shadowing+DGLVShadowing+PP+DGLV
Figure 13.
Same as Fig. 11 for bottom quarks. for a ready reference. We again see a trend which is common to our results that R AA starts at ≈ p T ≈ ≈ p T ≈ p T ≈
20 GeV. The overall effect of shadowing and energy loss is seen tolead to very similar values for R AA from y = 0 to y = 4. Of course the R AA for RHICenergy is about twice as large, showing a much reduced suppression.
7. Summary and Discussion
We have made a detailed study of charm and bottom production from initial fusionof partons in relativistic collision of heavy nuclei. As a first step we have checked theusefulness of lowest order pQCD to reproduce the NLO results for the p T distribution ofcharm and bottom quarks in pp collisions. Next we have checked our predictions againstexperimental results of the charm and bottom quarks production from such collisions. uclear suppression of heavy quark production.. R AA by additionally incorporatingnuclear shadowing for the cases under study. A rich picture of dependence of R AA ony, p T , incident energy, and the mass of the heavy quarks emerges. We have noted thatour findings would support the suppression of single electrons seen at RHIC.Before concluding, we discuss some of the short-comings of the present work. Theseinitial calculations can be improved in several ways. We have used (1 + 1) dimensionalBjorken hydrodynamics and assumed it to apply at all y. Since the heavy quarkswill loose most of their energy at very early times, this may not be a serious short-coming. Still we are looking at the possibility of using a full fledged (3 + 1) dimensionalhydrodynamics calculations, also at b = 0. We are incorporating the single electrondecay of the resulting D and B mesons, along with the results for their back to backcorrelation. We expect this to be rewarding, especially in conjunction with NLO resultsfor pp collisions, as it may throw up an interesting detail about differences of NLOresults and the results with energy loss. These will be published shortly.Finally, we conclude that the description for energy loss for one quark mass atone rapidity for a particular incident energy may not be sufficient to identify the mostreliable energy loss treatment for either collisional or radiative energy loss valid for allcases. Acknowledgments
We gratefully acknowledge the use of MNR-NLO code [39] made available to us by theauthors. We also thank M. G. Mustafa for useful discussions.
AppendixAppendix A. Collisional energy loss
A.1. Bjorken
Bjorken argued that the elastic energy loss by partons in the QGP is very similarto the energy loss due to ionization due to passage of charged particles in ordinarymatter [58]. This treatment was adapted by Braaten and Thoma for heavy quarks [59].The fractional energy loss suffered by the heavy quark due to collisions with the quarksand gluons given as: dEdx = 8 π α s T (cid:18) N f (cid:19) " v − − v v log 1 + v − v log q max q min , (A.1)where v is the velocity of the heavy quark. As suggested by Braaten and Thoma weuse the upper limit of the momentum transfer q max as √ T E and the lower limit of themomentum transfer q min as √ m g . uclear suppression of heavy quark production.. m g can be expressed as m g = µ/ √ µ = r π α s T (cid:16) N f (cid:17) is the Debye screening mass. A.2. Braaten and Thoma
Braaten and Thoma first developed a theoretical formalism to find the collisional energyloss of a muon propagating through a plasma of electrons, positrons and photons toleading order in QED [64]. This work was further extended by them to calculate thecollisional energy loss of heavy quarks propagating through QGP [59]. The energyloss formulation is given in two energy regimes: The fractional collisional energy loss ofheavy quarks with energy
E << M /T is dEdx = 8 π α s T (cid:18) N f (cid:19) " v − − v v log 1 + v − v × log Nf Nf B ( v ) E Tm g M ! (A.2)The fractional collisional energy loss of heavy quarks with energy E >> M /T is dEdx = 8 π α s T (cid:18) N f (cid:19) log Nf ( Nf ) 0 . √ E Tm g , (A.3)where B( v ) is a smooth function of velocity having value in the range 0.6 - 0.7. Braatenand Thoma have shown the crossover energy between these energy regimes as E cross =1 . × M /T for N f = 2. A.3. Peigne and Peshier
In BT formalism, it was assumed that the momentum exchange in the elastic scatteringprocess is much less than the energy carried by the heavy quarks. Peigne and Peshierpointed out that this assumption is not reliable in the energy regime
E >> M /T ,and corrected it in the QED case while calculating the collisional energy loss of a muonin QED plasma [65]. This work in QED is then used by them to derive the collisionalenergy loss suffered by heavy quarks while passing through QGP [60].The fractional collisional energy loss suffered by heavy quarks as proposed by Peigneand Peshier is dEdx = 4 π α s T "(cid:18) N f (cid:19) log E Tµ + 29 log E TM + c ( N f ) (A.4)and c ( N f ) ≈ . N f + 0 . uclear suppression of heavy quark production.. Appendix B. Radiative energy loss
B.1. Djordjevic, Gyulassy, Levai, and Vitev
For massless quarks, Gyulassy, Levai and Vitev (GLV) calculated the induced radiationto arbitrary order in opacity χ n ( χ = L/λ ) of the plasma [66] where λ is the mean freepath of the quark. In Djordjevic, Gyulassy, Levai, and Vitev (DGLV) formulation [61],the GLV method is generalized to estimate the first order induced radiative energy lossincluding the kinematic effect for heavy quarks. Wicks et al. [57] present a simplifiedform of the DGLV formalism for the average radiative energy loss of heavy quarks:∆ E = c F α s π E Lλ g − ME + p Z mgE + p dx ∞ Z µ q dq (cid:16) E xL (cid:17) + ( q + β ) ( A log B + C ) , (B.1)where β = m g (1 − x ) + M x , λ g = ρ g σ gg + ρ q σ qg ,σ gg = 9 π α s µ ,σ qg = 49 σ gg ,ρ g = 16 T . π ,ρ q = 9 N f T . π ,A = 2 β f β (cid:16) β + q (cid:17) ,B = ( β + K ) (cid:16) β Q − µ + Q + µ Q + µ + Q + µ f β (cid:17) β (cid:16) β (cid:16) Q − µ − K (cid:17) − Q − µ K + Q + µ Q + µ + f β f µ (cid:17) ,C = 12 q f β f µ [ β µ (cid:16) q − µ (cid:17) + β (cid:16) β − µ (cid:17) K + Q + µ (cid:16) β − q Q + µ (cid:17) + f µ (cid:16) β (cid:16) − β − q + µ (cid:17) + 2 q Q + µ (cid:17) + 3 β q Q − k ] ,K = (2 p x (1 − x )) ,Q ± µ = q ± µ ,Q ± k = q ± K,f β = f (cid:16) β, Q − µ , Q + µ (cid:17) ,f µ = f (cid:16) µ, Q + k , Q − k (cid:17) and f ( x, y, z ) = q x + 2 x y + z . (B.2) uclear suppression of heavy quark production.. B.2. Armesto, Salgado, and Wiedemann
In Armesto, Salgado, and Wiedemann (ASW), formulation [62] path integral methodfor medium-induced gluon radiation is employed to calculate the radiative energy loss ofheavy quarks. This formalism provides the analysis of the double differential medium-induced gluon distribution by the heavy quarks as a function of transverse momentum.The average radiative energy loss is∆ E = α s c F π (2 n L ) E Z dω R γ Z dk ∞ Z dq × (cid:16) q + ¯ M (cid:17) − γ sin h γ (cid:16) q + ¯ M (cid:17)i(cid:16) q + ¯ M (cid:17) × q q + ¯ M × (cid:16) k + ¯ M (cid:17) + (cid:16) k − ¯ M (cid:17) ( k − q ) (cid:16) k + ¯ M (cid:17) h (1 + k + q ) − k q i . (B.3)In the above equation the gluon energy, transverse momentum and heavy quark massare expressed as dimensionless parameters. The rescaled dimensionless parameters:¯ M ≡ (cid:18) ME (cid:19) Rγ ,R = ω c L, ω c ≡ µ L, and γ ≡ ω c ω . (B.4)We use the parameter n L = 4.
B.3. Xiang, Ding, Zhou, and Rohrich
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