Transport properties from Charm to Bottom: p T suppression, anisotropic flow v n and their correlations to the bulk dynamics
aa r X i v : . [ h e p - ph ] J a n Nuclear Physics A 00 (2019) 1–4
NuclearPhysics A / locate / procedia XXVIIth International Conference on Ultrarelativistic Nucleus-Nucleus Collisions(Quark Matter 2018)
Transport properties from Charm to Bottom: p T suppression,anisotropic flow v n and their correlations to the bulk dynamics S. Plumari a , G. Coci a,b , S.K. Das c,a , V. Minissale a,b , V. Greco a,b a Department of Physics and Astronomy, University of Catania, Via S. Sofia 64, 1-95125 Catania, Italy b Laboratori Nazionali del Sud, INFN-LNS, Via S. Sofia 62, I-95123 Catania, Italy c School of Nuclear Science and Technology, Lanzhou University, 730000 Lanzhou, China
Abstract
We study the propagation of heavy quarks (HQs) in the quark-gluon plasma (QGP) by means of a relativistic Boltzmanntransport (RBT) approach. The non-perturbative interaction between heavy quarks and light quarks is described bymeans of a quasi-particle approach able to describe simultaneously the experimental data for the nuclear suppressionfactor R AA and the elliptic flow v ( p T ) of D mesons from RHIC to LHC energies. In the same framework we predict theB meson nuclear modification factor at LHC. Finally, we discuss the relevance of initial state fluctuations that allowsto extend the analysis to high order anisotropic flows v n ( p T ) as well as to investigate the role of QCD interaction indeveloping correlations between the light and the heavy flavour anisotropic flows. Keywords:
Quark-Gluon Plasma, Heavy Quarks
1. Introduction
Heavy quarks (HQs), charm and bottom quarks, represent excellent probes of the system created atultra-Relativistic Heavy Ion Collisions (uRHIC). Their formation time is very small compared to the lightquarks one, and due to their large masses they are expected to thermalize slower in the Quark-Gluon Plasma(QGP). Therefore, HQ can probe both for the initial stages of uRHIC and the thermalized QGP evolution. Intheir final state the charm quarks appear as constituent of charmed and bottom hadrons mainly D , B mesonsand Λ c , Λ b baryons. Two key observables in HQ sector are the nuclear suppression factor R AA (the ratiobetween the spectra of heavy flavour hadrons in nucleus-nucleus collisions and the one in proton-protoncollisions) and the elliptic flow v ( p T ) (a measure of the anisotropy in momentum space). Experimentalmeasurements, both at RHIC and LHC, have shown many interesting observations for heavy mesons R AA and v ( p T ). In particular it was observed a small R AA value and large value of v ( p T ), which are almostcomparable to those of light hadrons. Several theoretical e ff orts have been made to study the R AA and the v within di ff erent models [1, 2, 3, 4, 5, 6, 7, 8, 9, 10]. Besides the well studied v ( p T ), it has been recentlyshown that also the triangular flow v ( p T ) of D mesons is not vanishing [11, 12]. In recent years, it hasbeen recognized that a strong Electromagnetic (EM) field is created at early times of uRHIC. Since HQs are / Nuclear Physics A 00 (2019) 1–4 produced in the very early stages of uRHICs they will be directly a ff ected by such a strong EM field andthis results in a rapidity-odd directed flow v for D and ¯ D [13].
2. Transport equation for charm quarks in the QGP
We describe the charm quarks evolution in the QGP by solving the RBT equations where charm quarksinteracts with a bulk medium of quarks and gluons as described by the following eq.s p µ ∂ µ f Q ( x , p ) = C [ f q , f g , f Q ]( x , p ) (1) p µ ∂ µ f j ( x , p ) = C [ f q , f g ]( x q , p q ) j = q , g (2)where f j ( x , p ) is the phase-space one-body distribution function of the j parton (quark, anti-quark or gluon)while C [ f q , f g , f Q ]( x , p ) refers to the relativistic Boltzmann-like collision integral. As shown in Eq.(1) thephase-space distribution function of the bulk medium (quarks and gluons) enters in the evolution equationfor charm quarks as an external quantities with C [ f q , f g , f Q ], and the evolution of f q and f g have been as-sumed to be independent of f Q ( x , p ). We discard collisions between heavy quarks which is by far a solidapproximation. The evolution of the bulk is given by the two equations Eqs.(2), where in C [ f q , f g ] the totalcross section is determined in order to keep fixed the ratio η/ s = / (4 π ) during the evolution of the QGP, fora detailed discussion see ref.s [14, 15, 16]. The non-perturbative interaction between heavy quarks and lightquarks is described by means of a quasi-particle approach [17]. This provides a softening of the equation ofstate, with a decreasing speed of sound approaching the cross-over region. In this approach we describe theevolution of a system that dynamically has approximatively the lQCD equation of state [18]. The hadroniza-tion process plays a crucial role in determining the final spectra, R AA ( p T ) and v ( p T ). When the temperatureof the QGP phase goes below the quark-hadron transition temperature, T c =
155 MeV, we hadronize thecharm quark to D-meson. We have considered a hybrid model of coalescence plus fragmentation (for adetailed discussion of the hadronization model see [19]). We have studied Au + Au collisions at √ s NN = Pb + Pb collisions at √ s NN = .
76 TeV at LHC. The initial conditions for the bulk in thecoordinate space are given by the standard Glauber model assuming boost invariance along the longitudinaldirection. In momentum space are given by a Boltzmann-Juttner distribution function up to a transverse mo-mentum p T = T =
365 MeV with the initial time forthe simulations τ ≃ / T = . f m / c for RHIC and T =
490 MeV with τ ≃ / T = . f m / c at LHC.In coordinate space we initialize the charm quark distribution according to the number of binary nucleon-nucleon collisions, N coll . In momentum space we use charm quark production according to the Fixed Order + Next - to - Leading Log (FONLL) calculation (from Ref. [21]) which describes the D-mesons spectra inproton-proton collisions after fragmentation Ref. [10].
3. Results
We have calculated the R AA and v ( p T ) of D meson at di ff erent centrality class from RHIC to LHCenergies, using the same interaction. Our model gives a good description for D mesons R AA and v ( p T ) atboth energies. In the left panel of Fig. 1 it is shown the comparison of our results for the R AA ( p T ) withthe experimental data for both systems created at RHIC and LHC energies for (30 − ff ect of coalescence is to increasethe R AA for momenta larger than 1 GeV. For the hadronization mechanism by coalescence, D mesons,which are composed by one light quark and a charm quark, get a larger momentum with respect to the Dmesons obtained from fragmentation. On the other hand, at larger momenta, the coalescence contributionto hadron formation becomes very small and fragmentation becomes, anyway, the dominant mechanismof hadronization. From the comparison at di ff erent energies, the e ff ect of coalescence is less significant atLHC than at RHIC. This is because the e ff ect of coalescence depends on the slope of HQ spectrum. For anharder charm quark distribution, like at LHC energy, the impact of coalescence is therefore less pronounced(for details see [10, 19]). We note that, if Λ c is included the D meson R AA will be substantially modified at Nuclear Physics A 00 (2019) 1–4 p T (GeV) R AA p T (GeV) D (Frag+Coal)D (Only Frag)B (Frag+Coal)B (Only Frag)
[email protected] TeV (0-10)%Au+Au@200 GeV (0-10)% p T (GeV) v ( p T ) p T (GeV) Only Coal.Frag+CoalOnly Frag
Pb+Pb@LHC (30-50)%Au+Au@RHIC (0-80)%
Fig. 1. Left panel: D meson R AA for (0 − Au + Au collisions at √ s NN =
200 GeV and LHC in Pb + Pb collisionsat √ s NN = .
76 TeV compared to the experimental data (Ref.s [22, 23]). The orange dashed and solid lines refer to the B meson R AA with only fragmentation and with both coalescence plus fragmentation respectively. Right panel: v at RHIC energy for (0 − − T (GeV)-0.0500.050.1 v n ( p T ) T (GeV) n=2 n=3 LHC: [email protected] TeV (0-10)% C ( v n ( li gh t ) , v n ( h ea vy ) ) QP ModelpQCD (k factor)LHC: [email protected] TeV |y|<1
Fig. 2. Left panel: v n ( p T ) with n = , Pb + Pb at √ s NN = .
02 TeV for (0 − ffi cient C ( v lightn , v heavym ) for (0 − n . Black solid line refers toQPM model while red dashed line to the case with pQCD interaction. low p T . We have also shown in Fig. 1, by the orange dashed and solid lines, the B meson R AA with onlyfragmentation and with both fragmentation plus coalescence. Due to their larger masses, bottoms quarkslose a smaller amount of energy than charm quark and therefore the B meson displays larger R AA than Dmesons. Moreover, due to the fact that for bottom quarks is easier to combine with light quarks, the e ff ectof coalescence in B meson R AA is more evident than for D meson R AA . In the right panel of Fig. 1, weshow the corresponding results for the final v ( p T ) of D mesons for both RHIC and LHC in mid-peripheralcollisions. As shown, the v developed via only coalescence (red solid lines) is larger than the v dueto fragmentation (black dashed lines). This di ff erence cames from the fact that the D meson elliptic flowformed via coalescence reflects both the heavy quark and light quark anisotropies in momentum space and itcan even lead to an increase of about a factor two at p T > v of the D-mesons increaseswith respect to the v of D meson from only fragmentation by about a 30%. The coalescence play a key roleto get a good description of the experimental data and moreover the extracted di ff usion coe ffi cient 2 π T D s is in agreement with lattice QCD results at least within the current uncertainties (for a detailed discussionsee [10]). Recently, we have developed an event-by-event transport approach for the bulk in order to studythe role of finite η/ s on the anisotropic flows v n ( p T ) (see [16]). We have used the event-by-event transportapproach to extend our analysis to D meson anisotropic flow harmonics. In the left panel of Fig.2 we presentthe D mesons v ( p T ) and v ( p T ) for central collisions in Pb + Pb at √ s NN = .
02 TeV. The anisotropic flows v n ( p T ) have been calculated using the event plane method, v n = h cos [ n ( φ − Ψ n )] i , with the momentumspace angles Ψ n = (1 / n ) arctan ( h sin ( n φ ) i / h cos ( n φ ) i ). As shown in Fig.2 we obtain a finite v ( p T ) whichis in good agreement with recent experimental data [26]. The results shown have been obtained including / Nuclear Physics A 00 (2019) 1–4 only fragmentation, the inclusion of coalescence will give an enhancement of the final anisotropic flows v n .In the right panel of Fig.2 it is shown the comparison of event-by-event correlations between light flavourand heavy mesons flow harmonics at LHC energies for QPM model (black line) and pQCD interaction (reddashed line) that has a T dependence of the drag while the QPM one has a quite weak T dependence.The measure of the linear correlation is given by the correlation coe ffi cient between light flavour and heavymesons flow harmonics C ( v lightn , v heavym ) = P i ( v i ( light ) n − h v ( light ) n i )( v i ( heavy ) m − h v ( heavy ) m i ) qP i ( v i ( light ) n − h v ( light ) n i ) P i ( v i ( heavy ) m − h v ( heavy ) m i ) (3)We observe that the second and third harmonics of HQs are strongly correlated to the corresponding har-monics of light quarks with C ( v lightn , v heavyn ) ∼ . n = ,
3. Moreover, to highlight the impact of theinteraction on v lightn − v heavyn correlation, we also consider pQCD interaction (red line). The C ( v lightn , v heavym ) issensitive to the heavy quarks - bulk interaction and in particular we observe a smaller correlation for pQCDinteraction, suggesting that it could give information about the T-dependence of transport coe ffi cients.
4. Conclusions
We have studied the HQ propagation in QGP at RHIC and LHC energies within a relativistic Boltzmanntransport approach where the interaction between HQs and light quarks of the bulk is described withinquasi-particle model. Within our approach we have a good description for D meson R AA and v both atRHIC and LHC energies within the experimental uncertainties. We observe that the e ff ect of the hadroniza-tion by coalescence is to increase the R AA and v ( p T ) for p T > R AA and v ( p T ). We have also studied the heavy flavour anisotropic flowswithin a event-by-event transport approach. Initial state fluctuations allows to extend the analysis to highorder harmonics v n ( p T ) as well as to study the role of QCD interaction in developing correlations betweenthe light and the heavy flavour anisotropic flows. This shows how the temperature dependence of the heavyquarks - bulk interaction a ff ect the heavy-light event by event v lightn − v heavyn correlations. References [1] H. van Hees, M. Mannarelli, V. Greco, R. Rapp, Phys. Rev. Lett. 100 (2008) 192301.[2] W. M. Alberico, A. Beraudo, A. De Pace, A. Molinari, M. Monteno, M. Nardi, F. Prino, Eur. Phys. J. C71 (2011) 1666.[3] S. Cao, G.-Y. Qin, S. A. Bass, Phys. Rev. C92 (2) (2015) 024907.[4] J. Upho ffff