The quenching of jets (particles with $p_T>>T, \Lambda_{QCD}$) in
ultra-relativistic heavy-ion collisions has been one of the main prediction and
discovery at RHIC. We have studied, by a simple jet quenching modeling, the
correlation between different observables like the nuclear modification factor
$\Rapt$, the elliptic flow $v_2$ and the ratio of quark to gluon suppression
$R_{AA}(quark)/R_{AA}(gluon)$. We show that the relation among these
observables is strongly affected by the temperature dependence of the energy
loss. In particular the large $v_2$ and and the nearly equal $\Rapt$ of quarks
and gluons can be accounted for only if the energy loss occurs mainly around
the temperature $T_c$ and the flavour conversion is significant.Finally we
point out that the efficency in the conversion of the space eccentricity into
the momentum one ($v_2$) results to be quite smaller respect to the one coming
from elastic scatterings in a fluid with a viscosity to entropy density ratio
$4\pi\eta/s=1$.Comment: 7 pages, 8 figures, Workshop WISH 201

Di Toro, M.

Greco, V.

Scardina, F.

English

arXiv

The quenching of jets (particles with pT � T,ΛQCD) in ultrarelativistic heavy-ion collisions has been one of the main prediction and discovery at RHIC.We have studied, by a simple jet quenching modeling, the correlation between

different observables like the nuclear modification factor RAA(pT ), the elliptic flow v2 and the ratio of quark to gluon suppression RAA(quark)/RAA(gluon). We show that the relation among these observables is strongly affected by the temperature dependence of the energy loss. In particular the large v2 and and the nearly equal RAA(pT) of quarks and gluons can be accounted for only if the energy loss occurs mainly around the temperature Tc and the flavour conversion is significant. Finally we point out that the efficency in the conversion of the space eccentricity into the momentum one (v2) results to be quite smaller with respect to the one coming from elastic scatterings in a fluid with a viscosity-to-entropy density ratio 4πη/s = 1

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DOI 10.1393/ncc/i2011-10835-8Colloquia: WISH2010IL NUOVO CIMENTO Vol. 34 C, N. 2 Marzo-Aprile 2011Impact of temperature dependence of the energy losson jet quenching observablesF. Scardina(1)(3), M. Di Toro(2)(3) and V. Greco(2)(3)(1) Dipartimento di Fisica, Universita` di Messina - I-98166 Messina, Italy(2) Dipartimento di Fisica e Astronomia, Universita` di CataniaVia S. Sofia 64, I-95125 Catania, Italy(3) INFN, Laboratori Nazionali del Sud - Via S. Sofia 62, I-95125 Catania, Italy(ricevuto l’ 8 Novembre 2010; approvato il 12 Gennaio 2011; pubblicato online il 22 Marzo 2011)Summary. — The quenching of jets (particles with pT  T,ΛQCD) in ultra-relativistic heavy-ion collisions has been one of the main prediction and discovery atRHIC. We have studied, by a simple jet quenching modeling, the correlation betweendifferent observables like the nuclear modification factor RAA(pT ), the elliptic flowv2 and the ratio of quark to gluon suppression RAA(quark)/RAA(gluon). We showthat the relation among these observables is strongly affected by the temperaturedependence of the energy loss. In particular the large v2 and and the nearly equalRAA(pT ) of quarks and gluons can be accounted for only if the energy loss occursmainly around the temperature Tc and the flavour conversion is significant. Finallywe point out that the efficency in the conversion of the space eccentricity into themomentum one (v2) results to be quite smaller with respect to the one coming fromelastic scatterings in a fluid with a viscosity-to-entropy density ratio 4πη/s = 1.PACS 12.38.Mh – Quark-gluon plasma.PACS 25.75.-q – Relativistic heavy-ion collisions.PACS 25.75.Ld – Collective flow.1. – IntroductionThe experiments at the Relativistic Heavy-Ion Collider (RHIC) have given clear in-dications of the formation of Quark-Gluon Plasma (QGP). One way to probe the QGPis to exploit the high-energy jets (pT  T,ΛQCD) produced by the hard collisions atthe initial stage. They are internal probes propagating through the fireball and inter-acting with the medium losing energy, hence carrying information on its properties asproposed long ago in ref. [1]. This energy loss can be quantified by the suppression ofobserved hadron spectra at high transverse momenta pT , namely RAA(pT ) [2]. Althoughthe observation of the jet suppression cannot be questioned there are several fundamentalquestions that still remain open. To investigate them we have constructed a model tostudy two observables beyond the RAA(pT ). One is the elliptic flow v2(pT ), that gives ac© Societa` Italiana di Fisica 6768 F. SCARDINA, M. DI TORO and V. GRECOmeasure of the angular dependence of quenching, and the other is RAA(q)/RAA(g) thatdetermines the flavor dependence of the suppression. We suggest that the study of thecorrelation between v2(pT ) and RAA(q)/RAA(g) carry information on the temperaturedependence of the quenching and on the mechanism of parton flavor conversion. We findthat an energy loss that increase as T → Tc, the q ↔ g in-medium conversion and anexpansion-cooling of the fireball according to a lattice QCD EoS improve the agreementwith the experimental data. Even if the T -dependence that has to be considered in thepresent modeling including only path-lenth energy loss appear to be too extreme.2. – Modeling the jet quenchingIn the model the density profile of the bulk is given by the standard Glauber modelwhile the hard-parton distributions in momenta space are calculated in the next-to-leading order (NLO) pQCD scheme. For the hadronization the Albino-Kramer-Kniehl(AKK) fragmentation functions have been employed. For further details see ref. [3].With regard to the jets energy loss we have employed various schemes, however to makea connection to the large amount of effort to evaluate gluon radiation in a pQCD frame,we have used also the Gyulassy-Levai-Vitev (GLV) formula at first order in the opacityexpansion [4, 5]:(1)ΔE(ρ, τ, μ)Δτ=9π4CRα3sρ(x, y, τ) τ log(2Eμ2τ),where CR is the Casimir factor equal to 4/3 for quarks and 3 for gluons, αs is the strongcoupling, ρ(x, y, z) is the local density, τ the proper time, E is the energy of the jetand μ = gT is the screening mass. There are corrections to eq. (1) coming from higherorder that can be approximately accounted for by a rescaling Z factor of the energyloss. However this is not really relevant for the objectives of the present work becausewe will renormalize the energy loss in order to have the measured amount of suppressionRAA(pT ) for central collisions. Usually in the GLV, as well as in other approaches, thetemperature evolution of the strong coupling αs is discarded. We will consider the impactof such a dependence to understand the amount of T -dependence coming simply fromthe asymptotic freedom. In the right panel of fig. 1 we show by dot-dashed and dashedlines the temperature dependence of the energy loss for the GLV with a dependence ofthe coupling (GLV-αs(T )) and with a constant coupling αs = 0.27 (GLVc). Furthermorewe show two other opposite cases for ΔE/Δτ : the thick line that shifts the energy lossto lower temperature (hence low density ρ or entropy density s) as suggested in [6,7] andthe other (thin line) that gives a dominance of quenching at high T , considered here justfor comparison with respect to the opposite case. We have applied our modelling of thejet quenching to Au + Au collisions at 200AGeV and in the left panel of fig. 1 we cansee that, once the RAA(pT ) at 0–5% is fixed, the dependence on centrality is correctlypredicted with a GLV formula for both constant and T -dependent αs. Hence looking atRAA one is not able to clearly discriminate the temperature dependence of the quenchingand not even the details of the density profile [3].3. – Angular and flavor dependence of the quenchingGenerally, jet quenching modeling has not been able to simultaneously describe RAAand the elliptic flow v2. In particular, experimental data relative to v2 are considerablyIMPACT OF T -DEPENDENCE OF THE ENERGY LOSS ON JET QUENCHING OBSERVABLES 690.15 0.2 0.25 0.3Temperature [GeV]00.511.52ΔE/Δτ [GeV/fm]GLV (α=cost)GLV α(T)Energy loss at high TEnergy loss at low T0 100 200 300 400Npart00.10.20.30.40.50.60.70.80.91RAA [6<p T<10 GeV]α=costα(T)Fig. 1. – Left panel: nuclear modification factor as a function of the number of participantsin Au + Au at 200 AGeV [8]. Right panel: temperature dependence of the energy loss for aparton with transverse momentum pT equal to 10GeV. Dashed and dot-dashed lines representthe GLV energy loss with constant and T -dependent αs coupling (see text). The thick line isthe case in which the energy loss takes place only closer to the phase transition and the thin linerepresents an opposite case in which the energy loss takes place only at high temperature T .larger than theoretical prediction. We have explored the relation between the temper-ature dependence of quenching and the value of the elliptic flow, and in the left panelof fig. 2 we can see that even if the amount of total quenching has been fixed to theexperimental value of RAA(pT ) the amount of elliptic flow is strongly dependent on thetemperature dependence of the Eloss.This correlation is due to the fact that at variance with RAA(pT ) the v2(pT ) hasa longer formation time because the jets have to explore the shape of the fireball torealize its asymmetry. Therefore it seems to be quite likely that experiments are tellingus that quenching does not take place mainly at the very early time (temperature) ofthe collision but mainly at later times close to the phase transition. This would meanFig. 2. – Left panel: elliptic flow for pions (b = 7.5 fm) coming from quark and gluon fragmenta-tion for the different T -dependence of the energy loss as shown in the right panel of fig. 1. Theshaded area shows the experimental data [8,9]. Right panel: ratio of quark to gluon RAA for thedifferent T -dependence of the energy loss as shown in the right panel of fig. 1. The shaded areaapproximatively shows the value expected for the ratio according to experimental observationsand using the AKK fragmentation function.70 F. SCARDINA, M. DI TORO and V. GRECOthat the quenching is not proportional to the density (or entropy density) but it isa decreasing function of it with a maximum at T ∼ Tc. This is what is essentiallydiscussed also in refs. [6, 7] where however it was implicitly assumed that the amountsof quenching of quarks and gluons are equal to each other as well as to the hadronicone. Here we have modified such assumptions showing that the temperature (or entropydensity) dependence of Eloss modifies not only the v2(pT ) but also the relative amountof quenching of quarks and gluons.3.1. Quark to gluon modification factor . – Due to its SU(3) Lie algebra the energy lossof gluons is 9/4 larger than the quark-one. For this reason sometimes it is assumed thatthe ratio between the quark and gluon suppression (RAA(q)/RAA(g)) is equal to 9/4.From this, one would think that the (anti-)protons are more suppressed with respect topions because they come more from gluon fragmentation than from quarks fragmentationwith respect to pions, at least according to the AKK fragmentation function we employ.The data at RHIC, however, have shown that even outside the region where coalescenceshould be dominant [10,11] the protons and the antiprotons appear to be less suppressedthan the pions and ρ0 [12,13]. Again we can see that going beyond the simple amount ofquenching given by RAA(pT ) both the azimuthal dependence and the flavor dependenceof the quenching appear to be in disagreement with the data. We call this open issuesthe “azimuthal” and the “flavor” puzzle, respectively. We will show that even if RAA forcentral collisions is fixed to be ∼ 0.2 the RAA(q)/RAA(g) is significantly affected by thetemperature dependence of Eloss.In the right panel of fig. 2 we show the ratio of the RAA(q)/RAA(g) for four differenttemperature dependences of the energy loss Eloss, as in fig. 1 (right). We can see that thestandard GLVc energy loss does not give the expected ratio 9/4 for RAA(q)/RAA(g) buta lower value, around 1.8, which represents already a non-negligible deviation from 2.25.We can however see that if the energy loss would be strongly T -dependent and dominantin the T ∼ Tc region RAA(q)/RAA(g) can increase up to about 2.2, on the contrary, ifit is dominant in the high-temperature region (thin solid line) the RAA(q)/RAA(g) canbecome as small as 1.5.To understand this behavior it is useful to consider the left panel of fig. 3 wherethe transverse momentum distribution of initial parton and those obtained if all partonslose the same amount of energy (3GeV and 4GeV in the dotted and dot-dashed line,respectively) are shown. The effect of the quenching in this oversimplified case is to shiftthe spectra by a quantity equal to the amount of energy loss in a way indicated by theblack arrow. Because of the rapid falling distribution, the spectra after quenchig are inthese cases one order of magnitude smaller with respect to the initial one. Therefore the10% of the spectrum without quenching, indicated by the thin line, is comparable to thespectrum in the case of Eloss = 3–4GeV. This means that if there are particles that losea very small quantity of energy, like in the case of quenching at high temperature, theystrongly influence the final spectra for both quarks and gluons and damping the differencebetween the RAA of quarks and gluons. For energy loss dominated by low temperatureall particles lose energy and this increases the difference between the respective RAA. Asimilar effect could come not only from the T -dependence of Eloss but from a core-coronaeffect [14]. For further explanations see ref. [3].4. – Correlation between RAA(q)/RAA(g) and elliptic flowWe have seen (fig. 2, left and right panel) that an energy loss predominant at lowT moves v2 toward observed data but moves also the ratio RAA(q)/RAA(g) away fromIMPACT OF T -DEPENDENCE OF THE ENERGY LOSS ON JET QUENCHING OBSERVABLES 71Fig. 3. – Left panel: spectra for partons that lose a fixed amount of energy (see text). Rightpanel: correlation between the RAA(q)/RAA(g) and the elliptic flow of pions with transversemomentum in the range 6 < pT (GeV) < 10. The squares refer to calculation without jetconversion while the circles are the values obtained including jet conversion with a Kc = 6factor.experimental indications. This is evident if one looks at the upper symbols of fig. 3(right) where RAA(q)/RAA(g) vs. v2 is shown. To solve the “flavor puzzle” inelasticcollisions that cause a change of the flavor has been invoked [15-17]. Such a processwould at the end produce a net conversion of quarks into gluons. Hence a decrease ofgluon suppression with respect to the original suppression and an increase of the quarkone. In ref. [15] it has been calculated the conversion rate of a quark jet to a gluonjet and vice versa due to two-body scatterings. An enhancement factor Kc = 4–6 thataccounts for non-perturbative effect is needed to produce a nearly equal suppression ofquarks and gluons. We have included such a mechanism in our model. The results arethe lower symbols in the right panel of fig. 3. We can see that the q ↔ g conversiondoes not affect the v2 and allows to get closer to the experimental observed value, i.e. av2 ∼ 0.1 and an RAA(q)/RAA(g) ≤ 1 (to account for the RAA(p + p¯) > RAA(π+ + π−)with AKK fragmentation function).5. – Impact of the equation of stateAs a last point, we have observed that if the quenching is predominant near thephase transition the question of the correct equation of state arises. In fact the free gasapproximation is no longer a reasonable approximation just close to Tc. We have madesome explorative studies on the impact that a more correct equation of state (EoS) canhave on the correlation between RAA(q)/RAA(g) and v2(pT ). Making a fit to the latticeQCD data [18] we have obtained the following relation between density and temperature:(2)TT0=(ρρ0)β(T ),where β(T ) = 1/3 − a(Tc/T )n with T ≥ Tc, a = 0.15 and n = 1.89 and of coursefor T  Tc one gets β ∼ 1/3. In order to estimate the impact of this correction wehave performed a simulation for the ΔEloss(T ) behavior represented by the thick solidline in the right panel of fig. 1, which is similar to the delayed energy loss proposed byPantuev [6] as a solution for the observed large elliptic flow. We consider only this case72 F. SCARDINA, M. DI TORO and V. GRECO4 6 8 10 12 14 16pT[GeV]00.050.10.150.20.250.30.350.4v 2/εWS-GlauberGLVcGLV - α(T)Eloss at high TEloss at low T0 0.5 1 1.5 2 2.5 3pT [GeV]00.20.40.60.8v 2/ε    |y|<0.35b=9 fmb=7 fmb=5 fmb=3 fm4πη/sQGP=1Fig. 4. – Left panel: ratio between the elliptic flow and the eccentricity v2/ as a function of thetransverse momentum pT of pions produced in collision with impact parameter equal to 7.5 fmfor different temperature dependence of the energy loss, as shown in the right panel of fig. 1.Right panel: v2/ as a transverse momentum function estimated in a parton cascade approachref. [19] at different impact parameters.because it is of course the one that is much more affected by the modification implied byeq. (2). The results are given by open symbols in the right panel of fig. 3. With respectto the free gas expansion cooling the system spends more time at T ∼ Tc. This reflectsin a further enhancement of both v2(pT ) and efficency of the q ↔ g conversion movingthe two observables closer to experimental data (shaded area in fig. 3 (right)).As a last point we want to mention the issue of the efficiency of conversion of initialspatial asimmetry  into a v2, namely the v2/. In the left panel of fig. 4 we showthe different values of v2/ that one obtains through the simple path-length mechanismstudied with our modeling, noticing that it is at maximum v2/ ≤ 0.25. In the rightpanel of fig. 4 we show the v2/ coming from elastic scattering in a cascade approachfor a fluid at finite shear viscosity to entropy density 4πη/s = 1 [19]. Even if at slightlylower pT the v2/ in this case is about a factor of two larger. This seems to indicate thata proper treatment of elastic energy loss can give an important contribution at least atintermediate pT ∼ 4–6GeV where presently the experimental data on v2 are available.6. – ConclusionsWe have pointed out the impact of peculiar temperature dependences of the energyloss on the elliptic flow and on the ratio between the quark and gluon suppression andtheir correlation. Moreover, we have spotted the relevance that the EoS may have incase of Eloss dominant in the T ∼ Tc region. In any case our study, although alreadyrevealing several interesting indications, is mainly explorative and it can be considered asa benchmark. A more quantitative analysis should be performed with more sophisticatedmodels that include the energy loss fluctuations, realistic gain and loss processes, elasticenergy loss and a more accurate description of the bulk. Furthermore, it should beexplored if the mass-dependent formation time can affect the correlation between v2 andRAA(q)/RAA(g) [20]. Obviously, it is also important to study how the longer lifetime andhigher temperatures, which will be reached at LHC energies, could affect the observedcorrelations.IMPACT OF T -DEPENDENCE OF THE ENERGY LOSS ON JET QUENCHING OBSERVABLES 73REFERENCES[1] Gyulassy M. and Plumer M., Phys. Lett. B, 243 (1990) 432; Wang X. N. andGyulassy M., Phys. Rev. Lett., 68 (1992) 1480.[2] Adams J. et al., Nucl. Phys. A, 757 (2005) 102; Adcox K. et al., Nucl. Phys. A, 757(2005) 184.[3] Scardina F., Di Toro M. and Greco V., Phys. Rev. C, 82 (2010) 054901.[4] Gyulassy M., Vitev I. and Wang X. N., Phys. Rev. Lett., 86 (2001) 2537.[5] Gyulassy M., Vitev I. and Zhang B. W., arXiv:nucl-th/0302077; Wang X. N., Nucl.Phys. A, 750 (2005) 98.[6] Pantuev V. S., JETP Lett., 85 (2007) 104.[7] Liao J. and Shuryak E., Phys. Rev. Lett., 102 (2009) 202302.[8] Adler S. S. et al. 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Impact of temperature dependence of the energy loss on jet quenching observables

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Impact of temperature dependence of the energy loss on jet quenching observables

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