Energy Loss Versus Energy Gain of Heavy Quarks in a Hot Medium
EEnergy Loss Versus Energy Gain of Heavy Quarks in a Hot Medium
Mohammad Yousuf Jamal, Santosh K. Das, and Marco Ruggieri School of Physical Sciences, National Institute of Science Education and Research, HBNI, Jatni-752050, India School of Physical Sciences, Indian Institute of Technology Goa, Ponda-403401, Goa, India School of Nuclear Science and Technology, Lanzhou University,222 South Tianshui Road, Lanzhou 730000, China
We study the energy loss and the energy gain of heavy quarks in a hot thermal medium. Theseinclude the study of the energy change due to the polarization and to the interaction with the thermalfluctuations of the medium. The dynamics of the heavy quarks with the medium is described bythe Wong equations, that allow for the inclusion of both the backreaction on the heavy quarks dueto the polarization of the medium, and of the interaction with the thermal fluctuations of the gluonfield. Both the momentum as well as the temperature dependence of the energy loss and gain ofcharm and bottom quark are studied. We find that heavy quark energy gain dominate the energyloss at high-temperature domain achievable at the early stage of the high energy collisions. Thisfinding supports the recently observed heavy quarks results in Glasma and will have a significantimpact on heavy quark observables at RHIC and LHC energies.
I. INTRODUCTION
The medium consisting of quarks and gluons producedat various heavy-ion collision experimental facilities suchas the Relativistic Heavy Ion Collider (RHIC) and theLarge Hadron Collider (LHC) provide a unique opportu-nity to explore the Quantum Chromodynamics (QCD)matter under extreme conditions of temperature anddensity. The bulk properties of such a state of matter,called Quark-Gluon Plasma (QGP) [1, 2], are governedby the light quarks and gluons. Though the small sizeand short-lived nature of the produced medium do not al-low to observe it by the naked eyes and hence, we rely onthe signatures observed at the detector end in the formof particle spectra.Heavy quarks, namely Charm and Beauty,, are con-sidered as excellent probes of the QGP [3–10] and offerssignatures of the production of the QGP itself. In fact,one of these signatures is the suppression of high p T heavyhadrons [11–13], that is understood as a result of the lossof energy of the high-energy charm and beauty quarkswhile they propagate through the dense matter formedafter collisions. More generally, the energy change ofcharm and beauty in the QGP have two major contri-butions, namely the polarization of the medium, whichleads to energy loss, and the interaction with the back-ground thermal fluctuations fluctuations of the gluonfield that is responsible of momentum diffusion. The po-larization is responsible of energy loss [14, 15] while in-teraction with thermal fluctuations leads to energy gainand is effective in the low-velocity limit [16, 17], seealso [14, 16, 18–55]. Heavy quarks can experience dif-fusion in the early stage of high energy nuclear collisionsas well. In particular, recent studies suggest that due tothe high energy density developed in the early stage, themotion of charm and beauty is dominated by field fluc-tuations that lead to a modest energy gain of the heavyprobes and to a tilt in the spectrum [56–60], in qualita-tive agreement with previous studies on the propagationin a high temperature QGP medium [16, 17]. The purpose of the present study is to analyze thecombined effect of energy gain and energy loss of heavyquarks in a high temperature QCD medium, analyzingthe kinematic regime in which one of the two mecha-nisms dominates. In solving this problem, albeit usingseveral approximations, we will show that even when en-ergy gain and energy loss are considered consistently, theheavy quarks will experience a substantial energy gainif the temperature of the medium is large enough. Wewill address quantitatively the question which betweenenergy gain or energy loss of heavy quarks is dominantin a given kinematic regime and at a given temperature.The conclusion is easy to imagine: if the temperature isquite larger than the kinetic energy of the heavy quark,then the medium will contribute substantially to increasethe energy of the heavy probe as this propagates in thehot medium; energy loss will be important when the tem-perature of the medium is lower than the kinetic energyof the heavy quark. These qualitative statements need tobe supported by quantitative findings, aiming to identifythe kinematic regimes in which energy loss or energy gaindominate and thus giving a clearer understanding of thedynamics of heavy quarks in the QGP medium producedin collisions. This is what we want to study here.In addition to this, our results offer a case studythat supports the assumption of [56–58, 61] where heavyquarks propagate in the evolving Glasma fields, and inwhich energy loss has been ignored. In fact, althoughhere we consider a thermalized medium while the evolv-ing Glasma is out of equilibrium, the diffusion of heavyprobes in Glasma resembles that in a thermal medium,at least when an average over the heavy quark spectrumis considered (see for example Fig. 7 of [58]); the energydensity in the evolving Glasma is very large, implyingthat the effective temperature of the medium is also highand thus the loss of energy of low momentum quarks canbe neglected.The plan of the paper is as follows. In section II, weshall discuss the polarization energy loss of heavy quarksmoving in the hot QCD medium along with a brief de- a r X i v : . [ nu c l - t h ] S e p scription of the change in energy of heavy quarks due tofluctuation. In section III, we shall discuss the variousresults. Section IV, is dedicated to the summary andfuture possibilities of the present work. II. ENERGY CHANGE DUE TOPOLARIZATION AND FLUCTUATION
In this section we discuss the theoretical setup onwhich we base our analysis. We treat charm and beautyas classical color sources that obey the Wong’s equa-tions [62]; these equations describe the motion of classicalcolored particles interacting with a dynamical gluon field, F µνa , and in a Lorentz covariant form they are given by dx µ ( τ ) dτ = u µ ( τ ) , (1) dp µ ( τ ) dτ = gq a ( τ ) F µνa ( x ( τ )) u ν ( τ ) , (2) dq a ( τ ) dτ = − gf abc u µ ( τ ) A µb ( x ( τ )) q c ( τ ); (3)in these equations, q a ( τ ) is a classical charge (to not beconfused with the fundamental, quantized color chargeof the quark) that is introduced to describe the conser-vation of the color current in the classical theory, with a − , , . . . , N c − g is the coupling constant, τ , x µ ≡ X , u µ = γ (1 , v ) and p µ ( τ ) are the proper time, trajectory,4-velocity and 4-momentum of the heavy quark, respec-tively. For N c fundamental colors of quarks there areN c − f abc is thestructure constant of SU(N c ) gauge group; finally, A µa isthe gauge potential. In solving these equations we as-sume the gauge condition u µ A µa ( X ) = 0 [26, 28], namelythat the gauge potential vanishes on the trajectory of theparticle and which implies that q a is independent of τ ;moreover, we assume that in the motion of the heavyquark in the thermal medium the magnitude of the ve-locity does not change much [28].From the µ = 0 component of Eq. (2) the energychange per unit time is dEdt = g q a v · E a ( X ) , (4)where here and in the following we use E to denote theenergy of the heavy quark and E for the color-electricfield, and t = γτ is the time in the laboratory frame inwhich the heavy quark of mass M moves with velocity v = p √ p + M . The color field consists of two terms, E a = E a ind + E a fluct , (5)where E a ind denotes the field induced by the motion of theheavy quark that polarizes medium (for this reason, thisis also called the polarization contribution), hence repre-senting an energy loss and its inclusion in the equationof motion amounts to consider the backreaction on theheavy quark, while E a fluct denotes the color field induced by the thermal fluctuations in the gluon medium: theinteraction of the heavy quark with E a fluct can result inenergy loss or energy gain depending on the temperatureof the medium as well as on the heavy quark momentum,as we discuss later.For the motion of the heavy quark in a thermalmedium, the right hand side of Eq. (4) is replaced byits ensemble average, dEdt = g q a (cid:104) v ( t ) · E a ( X ( t )) (cid:105) , (6)where the electric field is given by Eq. (5). The proce-dure to evaluate right hand side of the above equation isexplained clearly in the literature, see for example [16],therefore we limit ourselves to quote the final result thatis dEdt = (cid:104) g q a v · E a (cid:105) + g q a q b E (cid:90) t dt (cid:10) E bt ( t ) · E at ( t ) (cid:11) + g q a q b E (cid:90) t dt (cid:90) t dt (cid:28) Σ j E bt,j ( t ) × ∂∂ r j v · E at ( t ) (cid:29) . (7)Equation (7) corresponds to the full energy change of theheavy quark: the first addendum on the right hand sideis the energy loss due to the work against the inducedfield that has been discussed in the previous subsection,while the remaining addenda correspond to the changeof energy due to the thermal fluctuations of the gluonfields. In the intermediate steps it has been assumedthat (cid:104) E ai B aj (cid:105) = 0 and (cid:104) ˜ E (cid:105) = 0. We rewrite Eq. (7) as dEdt = (cid:18) dEdt (cid:19) ind + (cid:18) dEdt (cid:19) fluct , (8)where (cid:18) dEdt (cid:19) ind = (cid:104) g q a v · E a (cid:105) (9)and (cid:18) dEdt (cid:19) fluct = g q a q b E (cid:90) t dt (cid:10) E bt ( t ) · E at ( t ) (cid:11) + g q a q b E (cid:90) t dt (cid:90) t dt (cid:28) Σ j E bt,j ( t ) × ∂∂ r j v · E at ( t ) (cid:29) , (10)and we discuss the two terms separately below. A. Energy loss due to the induced field
Firstly we analyze the energy loss due to the workagainst the induced field, see Eq. (9) [14, 19, 23, 63].The induced field can be obtained by solving the Yang-Mills equations for a thermalized gluon system with thesource given by the color current carried by the heavyquark, namely [15], E a ind ( X ) = − i gq a π (cid:90) d ω d k ω k (cid:20) k ( k · v ) (cid:0) (cid:15) − L − (cid:1) + (cid:16) k v − k ( k · v ) (cid:17)(cid:26) (cid:18) (cid:15) T − k ω (cid:19) − − (cid:18) − k ω (cid:19) − (cid:27)(cid:21) e i ( k · x − ωt ) ω − k · v + i + ; (11)performing the ω integration in Eq. (11) and subtitutingin Eq. (9) we get (cid:18) dEdt (cid:19) ind = − C F α s π | v |× (cid:90) k max k d k ωk (cid:26) (cid:0) k | v | − ω (cid:1) Im 1 ω (cid:15) T − k + Im 1 (cid:15) L (cid:27) ω = k · v , (12)where, k µ = ( ω, k ) with | k | = k and α s is the QCDcoupling; moreover, C F = 4 / c ), (cid:15) L and (cid:15) T are the longitudinal and transverse components of themedium dielectric permittivity, that have been computedusing the semi-classical transport theory approach in [15]. B. Interaction with the fluctuating field
The energy change due to the interaction of the heavyquark with the fluctuating field can be written as [16], (cid:18) dEdt (cid:19) fluct = C F α s π E v (cid:90) k max v dω coth βω F ( ω, k = ω/v )+ C F α s π E v (cid:90) k max dkk (cid:90) k dω coth βω G ( ω, k ) , (13)where F ( ω, k ) = 8 πω Im[ (cid:15) L ] | (cid:15) L | ,G ( ω, k ) = 16 π Im[ (cid:15) T ] | (cid:15) T − k /ω | . (14)In Eq. (13) we have put E = (cid:112) p + M and introducedan ultraviolet cutoff, k max , which is of the order of theDebye screening mass [16, 46]; in the following we willconsider two representative values of this cutoff, namely k max = m D and k max = 2 m D : while the specific valueof k max affects the results quantitatively, the qualitativepicture is almost unaffected by this choice. p = = =
10 GeV, Pol0.0 0.5 1.0 1.5 2.00.00.10.20.30.4 T ( GeV ) - d E / d x ( G e V / f m ) p = = =
10 GeV, Pol0.0 0.5 1.0 1.5 2.00.00.20.40.60.81.0 T ( GeV ) - d E / d x ( G e V / f m ) Figure 1. Energy loss of charm due to the polarization of thehot medium, − ( dE/dx ) ind , versus temperature, for three val-ues of the initial charm quark momentum. Upper and lowerpanels correspond to k max = m D and k max = 2 m D respec-tively. III. RESULTS
In this section we summarize our results: firstly wefocus on charm, then we turn on beauty. We use theset of parameters N c = 3, N f = 2 and α s = 0 .
3. Inall the figures below we show the energy change per unitlength since the latter is the most used in the literature:this can be obtained easily from the change of energy perunit time that we have computed in the previous section, dEdx = 1 | v | dEdt , (15)where v is the velocity of the heavy quark. Moreover, touniform to the existing literature we plot − dE/dx sincethis quantity is been mostly used to quantify the energyloss and is therefore positive. A. Charm
In Fig. 1 we plot − ( dE/dx ) ind versus temperature forthree values of the heavy quark momentum. In the figure,upper and lower panels correspond to k max = m D and k max = 2 m D respectively. As anticipated, the backreac-tion represented by the interaction of the heavy quark p = = =
10 GeV, Fluct0.0 0.5 1.0 1.5 2.0 - - - - - T ( GeV ) - d E / d x ( G e V / f m ) p = = =
10 GeV, Fluct0.5 1.0 1.5 2.0 - - - - - - T ( GeV ) - d E / d x ( G e V / f m ) Figure 2. Energy change of charm due to the fluctuationof the hot medium, − ( dE/dx ) fluct , versus temperature, forthree values of the initial charm quark momentum. Upperand lower panels correspond to k max = m D and k max = 2 m D respectively. with the induced field results in an energy loss. Thiscan be understood easily since the motion of the heavyquark in the thermal medium results in the polarizationof the medium itself, and for this process to happen en-ergy has to be transferred from the quark to the mediumitself. For example, for a charm quark with initial mo-mentum p = 10 GeV, at a temperature T = 1 GeV wefind − ( dE/dx ) ind ≈ . k max = m D and − ( dE/dx ) ind ≈ . k max = 2 m D .In Fig. 2 we plot − ( dE/dx ) fluct versus temperature forthree values of the initial heavy quark momentum; upperand lower panels correspond to k max = m D and k max =2 m D respectively. Differently from the cases shown inFig. 1, we find that the interaction with the thermalizedgluon field leads to energy gain rather than energy loss.For example, considering again p = 10 GeV and T = 1GeV we find − ( dE/dx ) fluct ≈ − .
02 GeV/fm for k max = m D and − ( dE/dx ) fluct ≈ − . k max = 2 m D .The results shown in Figg. 1 and 2 agree qualitativelywith those obtained within a purely classical model forthe diffusion and the energy loss in a Brownian motion[60], in which the backreaction as the source of the energyloss and the interaction with the thermal fluctuationsas resulting in energy gain and momentum broadeningappear clearly. In addition to this, comparing the resultsshown in Figg. 1 and 2 we notice that for p/T (cid:29) p = + Fluctp = + Fluctp =
10 GeV, Pol + Fluct0.0 0.5 1.0 1.5 2.0 - T ( GeV ) - d E / d x ( G e V / f m ) p = + Fluctp = + Fluctp =
10 GeV, Pol + Fluct0.0 0.5 1.0 1.5 2.0 - T ( GeV ) - d E / d x ( G e V / f m ) Figure 3. Energy change of charm quark due to fluctuationand polarization for k max = m D (top) and k max = 2 m D (bot-tom). energy loss due to polarization of the medium is largerthan the energy gain, but this situation changes when p/T < ∼ p ; this is obtained byadding the results shown in Figg. 2 and 1 . In Fig. 3the upper and lower panels correspond to k max = m D and k max = 2 m D respectively. We notice that for p = 1GeV, in the full range of temperature considered the sumof the polarization and the fluctuation contributions re-sults in an energy gain of the quark. For the other tworepresentative values of p , namely for p = 5 GeV and p = 10 GeV, we find that up to T ≈ p = 10 GeV in the figure, and k max = m D then energy loss dominates over energy gainover the whole range of temperature studied.In Fig. 4 we plot the energy change due to polarization(upper panel) and fluctuations (middle panel) of charmquarks versus the initial momentum, for two representa-tive values of temperature and for k max = 2 m D (resultsfor k max = m D are similar to those shown here). At rel-atively low temperature the energy loss dominates overenergy gain for p > ∼ T = = p ( GeV ) - d E / d x ( G e V / f m ) T = = - - - - p ( GeV ) - d E / d x ( G e V / f m ) T = + FluctT = + Fluct0 2 4 6 8 10 12 14 - - - - - p ( GeV ) - d E / d x ( G e V / f m ) Figure 4. Energy change of charm quark versus the initialmomentum, at T = 0 . T = 2 GeV (bluelines). Upper and middle panels correspond to the polar-ization and fluctuations contributions respectively, while thelower panel corresponds to the sum of the two contributions.Results correspond to k max = 2 m D . ing gluon field is more important than the energy loss. B. Beauty
In this subsection we report on the analysis of energyloss and gain of beauty quarks in the hot medium; sincethe qualitative picture is unchanged with respect to thatof the charm quark, here we limit ourselves to presentonly a few representative results. In Fig. 5 we plot en-ergy change induced by polarization (upper panel), inter-action with fluctuating medium (middle panel) and total p = = =
10 GeV, Pol0.0 0.5 1.0 1.5 2.00.00.20.40.60.81.0 T ( GeV ) - d E / d x ( G e V / f m ) p = = =
10 GeV, Fluct0.0 0.5 1.0 1.5 2.0 - - - - T ( GeV ) - d E / d x ( G e V / f m ) p = + Fluctp = + Fluctp =
10 GeV, Pol + Fluct0.0 0.5 1.0 1.5 2.0 - - T ( GeV ) - d E / d x ( G e V / f m ) Figure 5. Energy change beauty due to polarization (up-per panel) and fluctuations (middle panel), as well as thecombination of the two (lower panel). Results correspond to k max = 2 m D . (lower panel) versus temperature, for three values of theinitial beauty quark momentum; the results correspondto k max = 2 m D . Clearly, there is some quantitative dif-ference between charm and beauty, due to the differentmasses of the two quarks, e.g., for the given values of pa-rameters, beauty quark loses less energy in the case of po-larization and also gains less energy in the case of fluctua-tion as compared to charm quark. Overall, the combinedeffect of polarization and fluctuations on beauty resultsin an energy gain for p/T < ∼ p/T > ∼ IV. CONCLUSIONS
We have studied, within linear response theory, the en-ergy change of heavy quarks in a hot thermalized QCDmedium, analyzing the combined effect of energy loss dueto the polarization of the medium, and energy gain dueto interaction with the thermal fluctuations of the gluonfield of the medium.. We have considered the effects onboth charm and beauty quarks. This study has been in-spired by a series of works on the propagation of heavyprobes in the early stage of the high energy nuclear col-lisions, in which the energy gain due to the diffusion inthe evolving Glasma is crucial to bend the initial pQCDspectrum of the heavy quarks before the formation ofthe quark-gluon plasma [56, 58]. Although we do notconsider the Glasma in the present study, we think thatthe results found here support at least qualitatively thediffusion-dominated scenario found in [56, 58]: in fact,despite the fact that the evolving Glasma is a systemout of thermal equilibrium, the diffusion of heavy colorprobes (see also Ref. [64]) in it is not very different fromthe diffusion in a Brownian motion, at least when an aver-age over the full heavy quark spectrum is taken: becauseof this similarity, it is likely that the results on diffu-sion in a fluctuating medium studied here can be appliedqualitatively to the diffusion in the evolving Glasma aswell.We have found that in the kinematic regime p/T < ∼ p is the initial heavy quark momentum and T the temperature of the medium, energy gain dominates ofthe energy loss, and the situation inverts in the comple-mentary regime p/T > ∼
1. These results are consistentwith previous literature [15, 16]. If we applied these con-clusions to the early stages of high energy nuclear colli-sions, our findings would suggest a diffusion dominatedpropagation for p < ∼ p > ∼
10 GeV, while in between there wouldbe a balance between the two.The results may have a significant impact on the ex-perimental observables like the nuclear suppression fac-tor and elliptic flow [5, 7] of heavy mesons produced atRHIC and LHC energies both for the nucleus-nucleus andp-nucleus collisions. Also a thorough understanding ofthe initial stage dynamics is a timely fundamental taskand may affect observables like the triggered D − ¯ D an-gular correlation [65] and the heavy quark directed flow v [66]. Apart from this, as it is well known that thethermal systems have comparatively weaker fluctuationsthan the non equilibrated systems. Therefore, incor-porating the momentum anisotropy (which remains in-evitible throughout the medium evolution) and also vis-cosity while modelling the medium [67, 68] in the currentstudy will bring us much closer to the real picture of thehigh energy nuclear collision. Hence, it will be an im-midiate future extension to the current work. ACKNOWLEDGMENTS
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