Passage of heavy quarks through the fluctuating hot QCD medium
NNoname manuscript No. (will be inserted by the editor)
Passage of heavy quarks through the fluctuating hot QCD medium
Mohammad Yousuf Jamal a,1 , Bedangadas Mohanty School of Physical Sciences, National Institute of Science Education and Research, HBNI, Jatni-752050, Indiathe date of receipt and acceptance should be inserted later
Abstract
The change in the energy of the moving heavy(charm and bottom) quarks due to field fluctuations presentin the hot QCD medium has been studied. A finite quarkchemical potential has been considered while modeling thehot QCD medium counting the fact that the upcoming ex-perimental facilities such as Anti-proton and Ion Research(FAIR) and Nuclotron-based Ion Collider fAcility (NICA)are expected to operate at finite baryon density and moderatetemperature. The effective kinetic theory approach has beenadopted where the collisions have been incorporated usingthe well defined collisional kernel, known as Bhatnagar-Gross-Krook (BGK). To incorporate the non-ideal equations ofstate (EoSs) effects/ medium interaction effects, an extendedeffective fugacity model has been adopted. The momentumdependence of the energy change due to fluctuation for thecharm and bottom quark has been investigated at differentvalues of collision frequency and chemical potential. Theresults are exciting as the heavy quarks are found to gain en-ergy due to fluctuations while moving through the producedmedium at finite chemical potential and collision frequency.
Keywords : Fluctuations, Energy gain, Energy loss, Debyemass, Quasi-Parton, Chemical potential, Effective fugacityand BGK-kernel.
The hot QCD medium produces at various experimental fa-cilities provides a basis that helps in understanding the dif-ferent phases of the early Universe. Especially, it helps instudying the Universe of the age of a few microseconds wherethe Quark-Gluon Plasma (QGP) phase is expected to ex-ist. This medium is created by colliding the heavy-ions atultra-relativistic speed in the laboratory that mimics the Big-Bang (MiniBang). The major hindrance in its study is its a [email protected] small size and short-lived nature. Therefore, to study sucha medium, one depends on the observed signatures at thedetector end. One of the most prominent signatures is thesuppression/enhancement in the yield of high p T hadronsmainly caused due to the energy loss/gain of heavy quarkswhile passing through the hot QCD medium created in heavy-ion collision (HIC). Moreover, the upcoming experimentalfacilities such as FAIR, NICA, etc , are expected to explorethe QCD phases at finite baryon density and moderate tem-perature. Hence, one needs to incorporate the finite baryonchemical potential to analyze this medium.The study regarding the change in the energy of the heavyquarks started back in the 1980s [1, 2]. Several articles arepresent in the literature that contain invaluable informationabout the energy loss/gain of heavy quarks moving in the hotQCD medium through several processes and also by differ-ent approaches [3–19].In the current manuscript, the mainaim is to study the change in the energy of moving heavyquarks (charm and bottom) due to fluctuations in the pres-ence of small but finite quark chemical potential within thehot QCD medium. Moreover, we intend to see how the pres-ence of finite quark chemical potential in the medium, thecollisional frequency of the medium constituents, and thenon-ideal medium effects influence their energies. For that,we employ the effective kinetic theory approach, consid-ering the BGK- collisional kernel along with the extendedEQPM [20–23].The manuscript is organized as follows. In section 2, theformalism for the change in the energy of moving heavyquarks due to fluctuations at finite chemical potential is pro-vided. Sections 3, contains various observations and resultsof the energy change at different values of quark chemicalpotential, collision frequency, and the presence of the non-ideal medium effect. Section 4, is dedicated to the summaryand future possibilities of the present work. a r X i v : . [ nu c l - t h ] J a n The heavy quarks produced at the early stages after the HICtraverse through the produced medium almost as the inde-pendent degrees of freedom. Treating them as a classicalparticle, their motion within the medium can be describedusing Wong’s equations [24]. These equations are a set ofclassical equations of motion for a particle interacting witha chromo-dynamical field, F µν a , given in the Lorentz covari-ant form as, dx µ ( τ ) d τ = u µ ( τ ) , d p µ ( τ ) d τ = gq a ( τ ) F µν a ( x ( τ )) u ν ( τ ) , dq a ( τ ) d τ = − g f abc u µ ( τ ) A µ b ( x ( τ )) q c ( τ ) , (1)where, q a ( τ ) is the quark’s color charge and g , is the cou-pling constant. τ , x µ ≡ X , u µ = γ ( , v ) and p µ ( τ ) are theproper time, trajectory, four velocity and four momentum ofthe heavy quark, respectively. Here, we have N c − f abc belongto the SU ( N c ) gauge group and A µ a is the four potential. Theexpression of the energy change of heavy quarks can be ob-tained from Eq. 1 considering the following two assump-tions [9 ? ]. First, chosing the gauge condition u µ A µ a = q a is independent of τ . Second, the quark’smomentum and energy evolve in time without changing muchthe magnitude of its velocity while interacting with the chro-modynamic field. Solving Eq. 1 considering only the zerothcomponent, µ = t = γτ , we obtained,dEdt = g q a (cid:104) v ( t ) · E a ( X ( t )) (cid:105) , (2)where, (cid:104) ..... (cid:105) denotes the ensemble average. The chromo-electric field in Eq. (2) consists of the induced field as wellas the spontaneously generated microscopic field due to therandom function of position and time. Next, solving µ (cid:54) = p dt = g q a [ E at ( X ) + v × B at ( X )] . (3)Using Eq. (3) while considering (cid:68) E ai B aj (cid:69) = i.e., (cid:10) ˜ E (cid:11) =
0, Eq. (2) further reduces as[16],dEdt = (cid:104) g q a v · E a (cid:105) + g q a q b E (cid:90) t dt (cid:68) E bt ( t ) · E at ( t ) (cid:69) + 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) , (4)where, v = p √ p + M is the velocity of the heavy quark ofmass M and E = (cid:112) p + M is the initial heavy quark en-ergy. The Eq. (4) represents the full expression of the change in energy of heavy quarks where the first term correspondsto the change due to polarization. The other two terms cor-respond to the statistical change in the energy of the mov-ing heavy quark in the medium due to the fluctuations ofthe chromo-electromagnetic fields as well as the velocityof the particle under the influence of this field. Precisely,the second term in Eq. (4) corresponds to the statistical partof the dynamic friction due to the space-time correlation inthe fluctuations in the chromo-electrical field. Whereas, thethird term corresponds to the average change in the energy ofthe moving heavy quark due to the correlation between thefluctuation in the velocity of the particle and the fluctuationin the chromo-electrical field in the plasma. The change inenergy due to polarization has been studied earlier [20] andfound that both charm and bottom quarks lose their energywhile passing through the medium. Here, the main focus isto understand the effects on the change of energy of movingheavy quarks due to fluctuations in the medium. Therefore,to get the total contribution of energy exchange due to fluc-tuations, we shall only concentrate on the plus of the sec-ond and the third terms of Eq. 4 that can be further writtenas [16], dEd x = C F α s π E v (cid:90) k ∞ v0 ω d ω coth (cid:18) β ω (cid:19) Im [ ε L ( ω , k = ω / v )] | ε L ( ω , k = ω / v ) | + C F α s π E v (cid:90) k ∞ k dkk (cid:90) k d ω coth (cid:18) β ω (cid:19) Im [ ε T ( ω , k )] | ε T ( ω , k ) − k / ω | , (5) where, C F = / ( N c ) . The QCD running couplingconstant, α s ( µ q , T ) at finite chemical potential and temper-ature [21, 25] given as, α s ( µ q , T ) = g s ( µ q , T ) π = π (cid:0) − N f (cid:1) ln (cid:18) T Λ T (cid:113) + µ q π T (cid:19) × (cid:18) − (cid:0) − N f (cid:1)(cid:0) − N f (cid:1) ln (cid:18) (cid:18) T Λ T (cid:113) + µ q π T (cid:19)(cid:19) ln (cid:18) T Λ T (cid:113) + µ q π T (cid:19) (cid:19) . (6) The ε L and ε T are the longitudinal and transverse compo-nents of the medium dielectric permittivity discussed earlierin Ref. [18]. We shall briefly the major steps in the next sub-section.2.1 Dieletric permittivity at finite quark chemical potentialFor the isotropic hot QCD medium, the dielectric permittiv-ity can be expanded in terms of its longitudinal and trans-verse projections as, ε ij ( K , ν , T , µ q ) = A i j ε T ( K , ν , T , µ q ) + B i j ε L ( K , ν , T , µ q ) , (7) µ= ν = m D LO µ= ν = m D HTL µ= ν = m D LB - - - - p ( GeV ) - d E / d x ( G e V / f m ) µ= ν = m D LO µ= ν = m D HTL µ= ν = m D LB - - - - p ( GeV ) - d E / d x ( G e V / f m ) Fig. 1
Energy change of charm quark due to fluctuation with various EoSs. In the left panel, µ = µ = . µ= ν = m D LO µ= ν = m D HTL µ= ν = m D LB - - - - p ( GeV ) - d E / d x ( G e V / f m ) µ= ν = m D LO µ= ν = m D HTL µ= ν = m D LB - - - - p ( GeV ) - d E / d x ( G e V / f m ) Fig. 2
Energy change of bottom quark due to fluctuation with various EoSs. In the left panel, µ = µ = . where, K ≡ K µ = ( ω , k ) , µ q represents the quark chemi-cal potential and ν , shows the collision frequency of themedium particles. The projection tensors are defined as, A ij = δ ij − k i k j k , and B ij = k i k j k . The permittivity tensor, ε i j can beobtained from the polarisation tenser, Π i j as, ε i j = δ i j − ω Π ij . (8)where, Π i j can be derived from the induced current as, Π i j = δ J ia , ind δ A j , a . (9)The current induced, J iind , a ( X ) inside the QCD medium dueto the change in the particle distribution function is givenas [26–29], J iind , a ( X ) = g (cid:90) d p ( π ) u i { N c δ f ga ( p , X ) + N f [ δ f qa ( p , X ) − δ f ¯ qa ( p , X )] } . (10)Next, the change in the medium particles distribution func-tions, δ f i can be obtained by solving the Boltzman-Vlasovtransport equation that can be written for each species as, u ¯ µ ∂ ¯ µ δ f ia ( p , X ) + g θ i u ¯ µ F ¯ µ ¯ ν a ( X ) ∂ ( p ) ¯ ν f i ( p ) = C ia ( ν , p , X ) , (11)where, ¯ µ and ¯ ν are the Lorentz four indices (not to confusewith chemical potential and collision frequency). Index, i represents the particle species ( i ∈ { quarks, anti-quarks andgluons } ) and θ i ∈ { θ g , θ q , θ ¯ q } have the values θ g = θ q = θ ¯ q = −
1. The partial four derivatives, ∂ µ , ∂ ( p ) ν corre-spond to the space and momentum, respectively. The colli-sional kernel, C ia ( p , X ) is considered here to be the BGK-type [30] given as follows, C ia ( ν , p , X ) = − ν (cid:34) f ia ( p , X ) − N ia ( X ) N i eq f i eq ( | p | ) (cid:35) , (12)where, f ia ( p , X ) = f i ( p ) + δ f ia ( p , X ) , (13)are the distribution functions of quarks, anti-quarks and glu-ons, f i ( p ) is equilibrium part while, δ f ia ( p , X ) is the per-turbed part of the medium particle distribution functions suchthat δ f ia ( p , X ) (cid:28) f i ( p ) . The collisions frequency, ν is con-sidered here to be independent of momentum and particlespecies. The particle number, N ia ( X ) and its equilibrium value, N i eq are defined as follows, N ia ( X ) = (cid:90) d p ( π ) f ia ( p , X ) , (14) N i eq = (cid:90) d p ( π ) f i ( p ) . (15) µ= ν = m D LO , charm µ= ν = m D LO , bottom0 5 10 15 20 - - - - - p ( GeV ) - d E / d x ( G e V / f m ) Fig. 3
Energy change of charm and bottom quarks at fixed µ = . ν = . m LOD considering only the leading order case.
Solving Eq. (10) along with Eq. (11) and Eq. (9), in theFourier space, we obtained, Π i j ( K , ν , µ q , T ) = m D ( T , µ q ) (cid:90) d Ω π u i u l (cid:110) u j k l + ( ω − k · v ) δ l j (cid:111) D − ( K , ν ) , (16)where, D ( K , ν ) = ω + i ν − k · v . The squared Debye massgiven as, m D ( T , µ q ) = πα s ( T , µ q ) (cid:18) − N c (cid:90) d p ( π ) ∂ p f g ( p ) − N f (cid:90) d p ( π ) ∂ p ( f q ( p ) + f ¯ q ( p )) (cid:19) . (17)From Eq. (7), (8) and (16) we have, ε L ( K , ν , T , µ q ) = + m D ( T , µ q ) k − ( ω + i ν ) ln (cid:0) k + i ν + ω − k + i ν + ω (cid:1) k (cid:0) k − i ν ln (cid:0) k + i ν + ω − k + i ν + ω (cid:1)(cid:1) ε T ( K , ν , T , µ q ) = − m D ( T , µ q ) ω k (cid:20) ω + i ν k + (cid:16) − ( ω + i ν ) k (cid:17) × ln (cid:18) k + i ν + ω − k + i ν + ω (cid:19) (cid:21) . (18)In the next subsection, we shall discuss the inclusion of non-ideal hot QCD medium effects using lattice and HTL equa-tions of states (EoSs) along with the finite quark chemicalpotential.2.2 Extended EQPM to incorporate the medium interactionand the finite quark chemical potentialThe extended EQPM has been employed to incorporate thefinite quark chemical potential along with the medium in-teraction effects (extended with the consideration of finitequark chemical potential, for details see Ref. [21]). The modelmaps the hot QCD medium interaction effects present in thehot QCD EoSs either computed within perturbative QCD(pQCD) or lattice QCD simulations into the effective equi-librium distribution functions for the quasi-partons, f ieq ≡ { f g , f q , f ¯ q } , that in turn, describes the strong interaction ef-fects in terms of effective fugacities, z g , q [22, 23]. The hotQCD EoSs described here are the recent (2 + et, al. [32, 33] which agrees reasonably well with the lattice re-sults [31, 34]. The EQPM has already been successfully ap-plied to study the various aspects of hot QCD medium [35–41]. In the present case, at finite quark chemical potential,the momentum distributions of gluon, quark, and anti-quarkare given as, f g = z g exp [ − β E g ] (cid:18) − z g exp [ − β E g ] (cid:19) , f q = z q exp [ − β ( E q − µ q )] (cid:18) + z q exp [ − β ( E q − µ q )] (cid:19) , f ¯ q = z q exp [ − β ( E q + µ q )] (cid:18) + z q exp [ − β ( E q + µ q )] (cid:19) , (19)where, E g = | p g | for the gluons and, (cid:113) | p q | + m q for thequark ( m q , denotes the mass of the light quarks). The fu-gacity parameter, z g / q → T → ∞ . Since themodel is valid only in the deconfined phase of QCD ( i.e., be-yond T c , T c being the critical temperature), the masses of thelight quarks can be neglected. Next, solving Eq. (17), con-sidering the distribution functions given in Eq. (19) withinthe limit, µ q T (cid:28) m , EoSD ( T , µ q ) = πα s ( T , µ q ) T (cid:20)(cid:16) N c π PolyLog [ , z EoSg ] − N f π PolyLog [ , − z EoSq ] (cid:17) + µ q T N f π z EoSq + z EoSq (cid:21) , (20)In the high temperature limit, z g , q → m D ( T , µ q ) = πα s ( T , µ q ) T (cid:20) N c + N f + µ q T N f π (cid:21) . (21)From Eq. (20) and Eq. (21), one can obtain the effectivecoupling due to medium interaction effects through variousEoSs at finite chemical potential as, α EoSs ( T , µ q ) = α s ( T , µ q ) (cid:16) N c + N f (cid:17) + µ q T N f π (cid:20)(cid:16) N c π PolyLog [ , z EoSg ] − N f π PolyLog [ , − z EoSq ] (cid:17) + µ q T N f π z EoSq + z EoSq (cid:21) . (22)Let us summarize here the inclusion of various observablesin this investigation: µ= ν = µ= ν = m D LO µ= ν = m D LO - - - - p ( GeV ) - d E / d x ( G e V / f m ) µ= ν = µ= ν = m D LO µ= ν = m D LO - - - - p ( GeV ) - d E / d x ( G e V / f m ) Fig. 4
Energy change of charm quark (left panel) and bottom quark (right panel) at fixed µ = . (i) The frequency, ν of the medium particle collision entersusing the BGK-collisional kernel that in turn appeared in thedielectric permittivity.(ii) The finite chemical potential is introduced in the analysisin two different ways, that is, the strong coupling and themedium quark/anti-quark distribution functions.(iii) The non-ideal medium interaction effects incorporatedusing the fugacity parameter in the gluon, quark and anti-quark distribution functions.Next, we shall discuss the various results regarding thechange in energy of the moving charm and bottom quarksdue to fluctuations inside the medium considering the abovementioned observables. The change in the energy of moving charm and bottom dueto fluctuation has been plotted against their momenta con-sidering the presence of finite quark chemical potential, µ q ,collision frequency, ν , and non-ideal medium effects (EoSs).To do so, Eq.(5) has been solved numerically. To perform thenumerical integration, the lower limit is taken as k EoSs = k ∞ has an ultraviolet cutoff of the orderof the Debye mass i.e., k EoSs ∞ ∼ m EoSsD ( T ) for each EoS. Itis to note that the results obtained here have a contributionto the change in the heavy-quark energy arising from soft-momentum exchange processes. The hard collisions and medium-induced gluon radiation are not considered here as these arebeyond the scope of the current analysis. Here, we worked attemperature, T = T c , T c = .
17 GeV, N c = N f =
3. Thedifferent values of collision frequency have been taken in theorder of screening mass that could be taken maximum up to ν = . m D as shown in Ref. [27]. The quark chemical po-tential is taken within the limit µ q T <
1. In all the figures, they-axis of the plots show the energy loss heavy quark per unitlength ( i.e., − dEdx in GeV/fm) and the x-axis show their cor-responding momenta (GeV). The results for the ideal case orthe leading order denoted as ‘LO’ and the non-ideal cases, viz., ( + ) − lattice EoS and 3-loop HTL EoS denoted as‘LB’ and ‘HTL PT ’, respectively in the plots.In Fig. 1, the plots are shown for the energy change ofcharm quark at ν = . m EoSsD and µ q = µ q = . ( + ) - Lattice and 3-loop HTLpt are al-most overlapping. But as compared to the LO case, the re-sult using non-ideal EoSs are found to be suppressed upto 50%. Furthermore, in the presence of chemical poten-tial, µ = . µ =
0. Although, the presence of finite but smallquark chemical potential does not much affect the results asit reduces the gain maximum up to 5%.The EoSs effects are quite clear in Fig. 1 and 2, thusto avoid the bulk we will now only focusing on LO case.Fig. 3 represents a comparison between change of energy ofcharm and bottom quark at µ = . ν = . m LOD . Ithas been observed that the charm quark gain more energy ascompared to the bottom quark although they both reachesto the same saturation values at high momentum. This isbecause, at very high momentum, v →
1, and hence, theirmasses do not affect the results.In Fig. 4, only the LO results have been shown for charm(left panel) and bottom (right panel) at fixed µ but differ-ent ν , ν = , . m L OD , . m L OD to make a visible effectsof collision frequencies. It has been noticed that both thequarks gain less energy when the medium is considered to becollisionless i.e., ν = ν (cid:54) =
0. Although the effect of collision at ν = . m L OD and ν = . m L OD do not differ much. But at low momentum boththe quarks are gaining more energy at ν = . m L OD than at ν = . m L OD . Next, we shall summarise the analysis and discuss thefuture extension of the current project.
The formalism for the change in the energy of the mov-ing heavy quarks due to fluctuations inside the hot QCDmedium has been presented. The impact of small but fi-nite quark chemical potential, collision frequency, and non-ideal medium effects has also been studied. Effective semi-classical transport theory has been adopted throughout. Thecollision effects have been incorporated using the BGK-collisionalkernel. For the inclusion of non-ideal interaction effects, aswell as the finite quark chemical potential, extended EQPMhas been employed. It has been observed that both the heavyquarks gain energy due to fluctuations while traversing throughthe hot QGP medium even in the presence of chemical po-tential and collisions in the medium. Although, the gain sat-urates at high momentum. The non-ideal medium interac-tion effects are found to suppress the gain whereas with theincrease in collision frequency the gain increases. For thesame values to chemical potential and collision frequency,the charm quark is found to gain more energy than the bot-tom quark. Moreover, the small quark chemical potentialcaused some reduction in the gain but does not affect the re-sults much. The higher values of chemical potential may af-fect the observed results appreciably but is beyond the scopeof the current analysis. Therefore, in the near future, the aimis to develop a formalism for µ q T > R AA of heavy mesons for the nucleus-nucleusand/or p-nucleus collisions. Therefore, an immediate futureextension includes the investigation of R AA in the presentcontext. M. Y. Jamal acknowledges NISER Bhubaneswar for provid-ing a postdoctoral position. We would like to acknowledgethe people of INDIA for their generous support for the re-search in fundamental sciences in the country.
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