HHeavy Flavours in Quark-Gluon Plasma
Seyong Kim ∗ Department of Physics, Sejong University, Seoul 05000, Republic of KoreaE-mail: [email protected]
Recent progresses in lattice studies of heavy quark and quarkonium at non-zero temperature arediscussed. Formulating a tail of spectral functions as a transport coefficient allows lattice deter-mination of momentum diffusion coefficient ( κ ) for charm quark in the heavy quark mass limitand lattice determination of heavy quark/heavy anti-quark chemical equilibration rate in NRQCD.Quenched lattice study on a large volume gives κ / T = . · · · . N f = + Γ chem . At T =
400 MeV with M ∼ . Γ − ∼
150 fm/c. Earlier results from the twostudies (with different lattice setups and with different Bayesian priors) which calculate bottomo-nium correlators using NRQCD and employ Bayesian method to calculate spectral functions aresummarized: ϒ ( S ) survives upto T ∼ . T c and excited states of ϒ are sequentially suppressed.The spectral functions of χ b channel shows a Bayesian prior dependence of its thermal behavior:the χ b spectral function with MEM prior shows melting above T c but that with a new Bayesianprior hints survival of χ b upto ∼ . T c . Preliminary results from the efforts to understand thedifference in the behavior of χ b spectral function is given. ∗ Speaker. c (cid:13) Copyright owned by the author(s) under the terms of the Creative CommonsAttribution-NonCommercial-NoDerivatives 4.0 International License (CC BY-NC-ND 4.0). http://pos.sissa.it/ a r X i v : . [ h e p - l a t ] F e b eavy Flavors in QGP Seyong Kim
1. Introduction
Understanding the properties of Quark-Gluon Plasma (QGP) and the transition from the hadronicgas phase to the QGP phase quantitatively is one of the most important subjects in the studies ofQCD thermodynamics. Heavy flavours (charm and bottom quark) open an important window forthe lattice community to this question together with accompanying difficulties.Experimentally, investigations of QGP properties usually require comparisons between resultsof proton-proton collisions and those of relativistic heavy ion collisions (see e.g., the right figurein Fig. 1 where ϒ production in p − p collisions is compared with that in Pb − Pb collisions)and having better grasps on the baseline properties of p − p collisions help us to better distinguishthe differences between quarkonium in non-interacting collections of hadron-hadron collisions andthat in nucleus-nucleus collisions. In this regard, hadronic processes which involve heavy quark(s)can be advantageous since these processes can be understood more precisely due to “factorization”of the corresponding processes in terms of the short distance perturbative interactions and the longdistance non-perturbative matrix elements through effective field theories such as Non-RelativisticQCD [3], potential NRQCD [4]. The same effective field theories can be applied in non-zerotemperature environment as long as T / M << energy-loss models describe the measured v reasonably well within uncertainties. However, asimultaneous description of both v and R AA still represents a challenge for the models (for adiscussion see e.g.[11]). (GeV/c) T p v -0.200.20.4 |>2} η∆ {EP,| vCharged particles, {EP} v average, |y|<0.8, *+ , D + ,D Prompt DSyst. from dataSyst. from B feed-down = 2.76 TeV NN sPb-Pb, Centrality 30-50%
ALICE
ALI − PUB − Figure 3: Average v of D , D + and D ⇤ + mesons as afunction of p T compared with v of charged particles[11]. (GeV/c) T p v -0.100.10.20.30.4 average *+ , D + ,D ALICE DSyst. from dataSyst. from B feed-downWHDG rad+collPOWLANGCao, Qin, BassMC@sHQ+EPOS, Coll+Rad(LPM) TAMU elasticBAMPS UrQMD = 2.76 TeV NN sPb-Pb, Centrality 30-50%
ALI − PUB − Figure 4: Average v of D , D + and D ⇤ + mesons as afunction of p T compared with model calculations [11]. Fig. 5 shows the measurement of the average R pPb of D , D + and D ⇤ + as a function of p T for p-Pb collisions at p s NN =5.02 TeV. The R pPb is compatible with unity for p T > /c , which suggests that CNM e↵ects are small and that the suppression observed in Pb–Pbcollisions is dominated by the influence of the hot and dense medium. Models including CNMe↵ects describe the data within uncertainties (for a more detailed discussion see e.g.[12]). ) c (GeV/ T p p P b R *+ , D + , D Average D<0.04 cms y -0.96< CGC (Fujii-Watanabe)pQCD NLO (MNR) with CTEQ6M+EPS09 PDF broad + CNM Eloss T Vitev: power corr. + k
ALICE =5.02 TeV NN s p-Pb, ALI − PUB − Figure 5: Average R pPb of D , D + and D ⇤ + as a function of p T [12], compared with model calculations [13, 14, 15, 16, 17].
3. ALICE upgrade studies
An upgrade of the ALICE experiment is planned for the next long shutdown of the LHC (2018-2019). The heavy-flavour measurements will benefit in particular from the upgrade of theInner Tracking System (ITS) and the Time Projection Chamber (TPC). As one of the mainobjectives, the resolution of the track impact-parameter measurement will be improved by afactor of 3 in the plane transverse to the beam direction. This will largely enhance the rejectionof the combinatorial background in the heavy-flavour reconstruction. The upgrade of the readoutcapabilities of the TPC and several other detectors will allow to record minimum bias Pb–Pbcollisions – which are used for open charm measurements at low momentum – at hundred timesthe rate compared to the current detector.The expected performance for the open charm and beauty measurements with the upgradedALICE detector was studied with simulations of Pb-Pb collisions at p s NN = 5 . XI International Conference on Hyperons, Charm and Beauty Hadrons (BEACH 2014) IOP PublishingJournal of Physics: Conference Series (2014) 012024 doi:10.1088/1742-6596/556/1/012024 Figure 1:
Elliptic flow ( v ) of light charged hadron and charmed meson ( D , D + , D ∗ + ) from ALICE collab-oration [1] (left) and sequential suppression of ϒ ( S ) , ϒ ( S ) and ϒ ( S ) from CMS collaboration [2]. Of course, relating experiments which includes complex time evolution of highly relativisticscattering of nuclei to the quantities which lattice gauge theory can calculate is highly nontrivialand is involved since lattice gauge theory mostly describes physics in thermal equilibrium. Further-1 eavy Flavors in QGP
Seyong Kim more, studying properties of heavy quarks and quarkonia at T (cid:54) = T (cid:54) = T (cid:54) =
2. Heavy Quarks
Hydrodynamic flow of charmed meson in relativistic Pb − Pb collisions is comparable to thatof light hadron (Fig. 1), which suggests that an effective thermalization of charm quark is similarto that of light quarks [1]. This is in conflict with a perturbative consideration and calls for a latticeQCD determination of transport properties of heavy quarks in QGP. Transport properties are non-equilibrium characteristics and are usually accessed by the “transport peak” of the spectral functionfor the corresponding correlators in thermal equilibrium through Kubo relations [15]. However,obtaining the transport coefficient directly from the “transport peak” of the spectral function forheavy quark current-current correlator is hard because the width of transport peak in the spectralfunction scales as ∼ α s T / M and the peak is narrow for the heavy quark case. A HQET basedformulation by [6] allows one to extract a transport coefficient from the “power-law frequency tail”of the spectral function, instead of “zero frequency limit peak” of the spectral function. A quenchedlattice determination of the transport coefficient related to the kinetic equilibriation of heavy quarkin thermal environment following this formulation is reported recently in [16]One of other interesting questions concerning the behavior of heavy quarks in QGP is whetherthe abundance of heavy quarks/anti-heavy quarks in QGP follows thermal Boltzmann distribution.Experimentally, it is related to whether the chemical equilibriation rate of heavy quarks is largecompared to the lifetime of QGP achieved in relativistic heavy ion collisions. Similarly to the caseof kinetic equilibriation of heavy quark in thermal medium [6], chemical equilibriation of heavyquark/heavy anti-quark in thermal equilibrium can be nonperturbatively defined through density-density correlators in NRQCD limit [7, 8]. A lattice determination (in full N f = + eavy Flavors in QGP Seyong Kim
Let us briefly describe the extraction of a transport coefficient from power-law tail of a spectralfunction. Hydrodynamic property of heavy quark moving through a thermal medium is character-ized by the diffusion coefficient, D , defined as D = χ lim ω → ∑ i = ,.. ρ iiV ( ω ) ω , ρ iiV ( ω ) = (cid:90) ∞ − ∞ dt e i ω t (cid:90) d x (cid:104) (cid:2) J i ( t , x ) , J i ( , ) (cid:3) (cid:105) (2.1)where J i = ψγ i ψ ( ψ is the relativistic heavy quark field) through fluctuation-dissipation theorem. χ is a susceptibility related to the 0-th component of the four-current by χ = T (cid:90) d x (cid:104) J ( t , x ) J ( t , ) (cid:105) . (2.2)The transport peak, ω → ρ iiV ( ω ) , becomes narrower as M → ∞ and is difficult to access.Instead, from the observation that in a large heavy quark mass limit, the momentum diffusionconstant( κ ), the drag coefficient ( η D ) and the kinetic equilibriation time ( τ kin ) are related to D by D = T κ , η D = κ MT , τ kin = η D (2.3)with an assumption of “narrow transport peak” and a Lorentzian width around the peak [6], authorsof [16] focus on the momentum diffusion coefficient, κ . The mass dependent momentum diffusioncoefficient, κ M , is defined as κ M = M ω T χ ∑ i T ρ iiV ( ω ) ω | η D << | ω |≤ ω UV , (2.4)where M kin is the kinetic mass of heavy quark and ω UV is a cut-off isolating the narrow transportpeak. It can then be written as κ = T ∑ i = , lim M → ∞ χ (cid:90) dtd x (cid:104) (cid:0)(cid:2) φ † gE i φ − θ † gE i θ (cid:3) ( t , x ) , (cid:2) φ † gE i φ − θ † gE i θ (cid:3) ( , ) (cid:1) (cid:105) , (2.5)where φ is the non-relativistic spinor for heavy quark, θ is that for heavy anti-quark, and E i is thecolor electric field. One can express Eq. 2.5 as G E ( τ ) = − ∑ = , (cid:104) ReTr U ( β , τ ) gE i ( τ , ) U ( τ , ) gE i ( , ) (cid:105)(cid:104) ReTr [ U ( β , )] (cid:105) , (2.6)where U is the product of the time directional links. This is amenable to a lattice calculation.It is still a difficult lattice measurement because the observable is gluonic and suffers from largestatistical noise. [16] announced result of their multi-year effort. On large quenched lattices (64 × ∼ × T = . T c ), they performed a continuum extrapolationof lattice-measured correlators (Eq. 2.6) obtained with multi-level (actually two-level) algorithmas the first step. Then, instead of applying a general Bayesian reconstruction of the full spectralfunction for Eq. 2.6, the IR limit and the UV limit of the spectral function is argued to be a specificfunctional form. These two limits are interpolated using various fitting forms and strategies. Fitted3 eavy Flavors in QGP Seyong Kim κ / T α a1 α b1 β a1 β b2 α a2 α b2 β a2 β b3a BGM m od e l strategy (i)strategy (ii) T ~ c Figure 2:
The fitted results from various fitting strategies and fitting forms [16], where the gray band is theirfinal estimate (refer to [16] for the various symbols). results are also tested against standard Maximum Entropy Method. Fig. 2 shows the fitted resultsof κ / T . The gray band corresponds to κ / T = . − .
4, which gives an estimate for the kineticequilibriation time scale, τ kin = η D = ( . · · · . ) (cid:18) T c T (cid:19) (cid:18) M . (cid:19) fm / c . (2.7)This suggests that near T c , charm quark kinetic equilibriation is as fast as light parton kineticequilibriation which is ∼ Similar to the kinetic equilibriation of heavy quarks in QGP, one can define the chemicalequilibriation rate as a transport coefficient from a spectral function of a density-density correlator[7, 8]. The chemical equilibriation is related to how the number density ( n ) is adjusted, where ( ∂ t + h ) n = − c ( n − n ) (2.8)in Boltzmann equation approach and ( ∂ t + h ) n = − Γ chem ( n − n eq ) + O ( n − n eq ) (2.9)in a linearized form near the equilibrium where h is a kind of “Hubble expansion constant” if ex-panding QGP is considered. Γ chem = cn eq is called the chemical equilibriation rate. The linearizedform can be described in terms of a Langevin equation ∂∂ t δ n ( t ) = − Γ chem δ n ( t ) + ξ ( t ) , (cid:104)(cid:104) ξ ( t ) ξ ( t (cid:48) ) (cid:105)(cid:105) = Ω chem δ ( t − t (cid:48) ) , (cid:104)(cid:104) ξ ( t ) (cid:105)(cid:105) = δ n ( t ) = n − n eq and (cid:104)(cid:104)· · ·(cid:105)(cid:105) denotes the average over the Langevin noise. A transport propertycan be studied through the spectral function of a two-point correlator ∆ ( t , t (cid:48) ) = (cid:104) { δ n ( t ) , δ n ( t (cid:48) ) }(cid:105) . (2.11)4 eavy Flavors in QGP Seyong Kim Ω chem and Γ chem is related to the tail of the spectral function for ∆ ( t , t (cid:48) ) , Ω chem = lim Γ chem (cid:28) ω (cid:28) ω UV ω (cid:90) ∞ − ∞ dt e i ω ( t − t (cid:48) ) (cid:104) { δ n ( t ) , δ n ( t (cid:48) ) }(cid:105) , Γ chem = Ω chem χ f M , (2.12)where χ f is the quark-number susceptibility related to the heavy flavour. For a lattice calculationof the chemical equilibriation of heavy quark, we need the imaginary-time formulation of Eq. 2.12.Since in non-relativistic limit, H = M ( θ † θ − φ † φ ) , (2.13)is related to the heavy quark number operator, the density-density correlator in the imaginary timeformalism, ∆ ( τ ) = (cid:90) d x (cid:104) H ( τ , x ) H ( , ) (cid:105) , < τ < T = β , (2.14)is considered. Within NRQCD framework, the first order perturbation due to pair annihilation forS-wave (Im f ( S ) θ † φ θ φ † ) gives ∆ ( τ ) ≈ Im f ( S ) π M (cid:90) d x (cid:90) d y (cid:90) β d τ (cid:90) β d τ (cid:104) H ( τ , x ) H ( , )( θ † φ )( τ , )( φ † θ ( τ , ) (cid:105)| τ − τ | . (2.15)Then, from the tail of the spectral function, Ω chem = f ( S ) Z ∑ m , n e − β E m (cid:104) qqm | θ † φ | n (cid:105)(cid:104) n | φ † θ | qqm (cid:105) = f ( S ) Z Tr (cid:104) e − β H ( θ † φ )( + , )( φ † θ )( , ) (cid:105) , (2.16)where q , q are heavy quark and heavy anti-quark, and m , n denote the other degrees of freedom.Note that1 Z Tr (cid:104) e − β H ( θ † φ )( + , )( φ † θ )( , ) (cid:105) = (cid:104) ( θ † φ )( + , )( φ † θ )( , ) (cid:105) = Tr (cid:104) G θ ( β , ; 0 , ) G θ † ( β , ; 0 , ) (cid:105) (2.17)and χ f = (cid:90) d x (cid:104) ( θ † θ + φ φ † )( τ , x )( θ † θ + φ φ † )( , ) (cid:105) = (cid:104) G θ ( β , ; 0 ) (cid:105) . (2.18)Then, the thermal averaged Sommerfeld factors become S = P P , S = N c P − P ( N c − ) P (2.19)with P = N c Re (cid:104) G θαα ; ii ( β , ; 0 , ) (cid:105) , P = N c (cid:104) G θασ ; i j ( β , ; 0 , ) G θ † σα ; ji ( β , ; 0 , ) (cid:105) , (2.20) P = N c (cid:104) G θαα ; i j ( β , ; 0 , ) G θ † σσ ; ji ( β , ; 0 , ) (cid:105) , (2.21)for the color singlet channel and the color octet channel respectively. The Sommerfeld factor ( S )usually means the enhancement due to the attractive long range interaction between slowly moving,5 eavy Flavors in QGP Seyong Kim annihilating heavy particles (heavy quark and heavy anti-quark for QCD) compared to the Bornmatrix element for the pair annihilation ( | M resummed | = S | M tree | ) [17]. Then, Γ = Im f ( S ) M χ f S N c , Γ = Im f ( S ) M χ f ( N c − ) S N c . (2.22)or if Boltzmann equation is considered [7], Γ chem (cid:39) πα s M (cid:18) MT π (cid:19) / e − M / T (cid:20) ¯ S + (cid:18) + N f (cid:19) ¯ S (cid:21) (2.23)with the assumption that the octet Sommerfeld factors are spin-independent.Using FASTSUM configurations [18] (24 × N τ , a s / a τ = . a -fixed ( a s = . ( ) fm), N f = + m π (cid:39) m K (cid:39)
500 MeV), [9] calculated S and S (Fig. 3). A naive per-turbative estimate for the singlet Sommerfeld factor, S , (the right figure, the red line) ∼ O ( ) .The lattice value ranges ∼ O ( − ) . Only when a bound state contribution is included inthe model spectral function of heavy quark/heavy anti-quark system, the analytic estimate becomesclose to the lattice value (the right figure, the black line). For the repulsive octet channel, S , boththe lattice value and the analytic estimate stays ∼ O ( ) . The huge increase in the singlet Sommer-feld factor illustrates the importance of non-perturbative effect near T c . For a phenomenologically T / T c S , M = 4.7 GeV _ S , M = 2.4 GeV _ S , M = 4.7 GeV _ S , M = 2.4 GeV _ T / T c S _ S _ S _ M = 4.5 GeV
Figure 3:
The thermal averaged singlet channel Sommerfeld factor ( S ) and octet channel factor ( S ) [9](left). Note that the vertical axis is in the logarithmic scale. The right shows a perturbative calculation ofthese quantities using a spectral function with a bound state (black) and using a spectral function without abound state (red) for the singlet channel interesting case, one can estimate the chemical equilibriation rate for a charm quark ( M ∼ . T ∼
400 MeV, with S ∼ , S ∼ .
8, which is Γ − ∼
150 fm/c. This suggeststhat within the QGP phase lifetime ∼
10 fm/c of the current relativistic heavy ion collision, charmquark does not chemical equilibrate, as expected.
3. Quarkonium
Despite long and intense efforts by the lattice community, a quantitative understanding onwhether bound states of heavy quark and heavy anti-quark pair can exist in thermal medium based6 eavy Flavors in QGP
Seyong Kim on first principle calculation is still lacking. An early (qualitative) picture of quarkonium “melting”based on a screened potential [19] is too simplistic since there is an imaginary part of potential inthermal medium [20]. The recent directions taken to study “quarkonium in medium” are roughlycategorized as, • (1) calculate the potential between heavy quarks from the spectral function of Wilson loop/Wilsonline correlator using T (cid:54) = T (cid:54) = • calculate a fully relativistic heavy quark propagator using T (cid:54) = • calculate heavy quark propagator using NRQCD under the background of T (cid:54) = ∼ M scale is in-tegrated out) allows us to focus on the binding and melting features of states in the spectralfunction.Here we concentrate on the results from lattice NRQCD study of quarkonium in non-zero tem-perature, i.e., studies of T (cid:54) = G ( τ / a τ ) , τ / a τ = , · · · ( N τ − ) , a τ is thetemporal lattice spacing) by calculating a heavy quark propagator using lattice NRQCD and ob-tain the spectral functions of the NRQCD quarkonium correlators using Bayesian method. Thereare many practical advantages of NRQCD formalism over full QCD for a lattice study of quarko-nium: quarkonium correlators from a heavy quark propagator which is calculated with NRQCDis highly accurate (typically the statistical error is ∼ O ( − ) ) compared to the quarkonium cor-relators calculated with relativistic QCD. NRQCD heavy quark propagator can be calculated fastsince NRQCD Lagrangian is a first order in time and is numerically an initial value problem whichallows a larger τ than a relativistic propagator. However, the continuum limit of the correlators cannot be taken because the spatial lattice spacing must satisfy Ma s ∼ O ( ) .Given G ( τ ) , G ( τ ) = (cid:90) ∞ d ω π K ( ω , τ ) ρ ( ω ) ≤ τ < T (3.1)where the kernel K ( τ , ω ) becomes ∼ e − ωτ for NRQCD and ( e − ωτ + e − ω ( / T − τ ) ) / ( − e − ω T ) = cosh ( ω ( τ − / T )) / sinh ( ω / T ) for full QCD, non-zero temperature behavior is studied by thetemperature dependence of the spectral function, ρ ( ω ) . The integral transform, Eq. 3.1, showsthe crux of studying the in-medium property of quarkonium through the spectral function of theEuclidean correlator. Since we do not know the analytic structure of the spectral function, thisinverse integral transform problem is ill-posed. To make the numerical problem worse, the numberof temporal lattice sites for G ( τ ) at T (cid:54) = eavy Flavors in QGP Seyong Kim spectral structure is expected to be more complicated at T (cid:54) = τ / a τ of G ( τ / a τ ) can extend its range to 1 / Ta τ −
1. (3)the inverse integraltransform becomes the inverse Laplace transform. For a given lattice G ( τ / a τ ) with finite errorbars, there still exist numerous ρ ( ω ) ’s which satisfy Eq. 3.1. To this problem, Bayes theorem forthe conditional probability P [ X | Y ] = P [ Y | X ] P [ X ] / P [ Y ] , (3.2)is applied where P [ X | Y ] is the probability of observing the event X given that the event Y is true.For a given Data D and a prior knowledge ( H ), the probability for the spectral function ( ρ ) is P [ ρ | D , H ] ∝ P [ D | ρ , H ] P [ ρ | H ] , P [ D | ρ , H ] = e − L , L = ∑ i ( D i − D ρ i ) / σ i , P [ ρ | H ] = e − S , S = S [ ρ ( ω ) , m ( ω )] , (3.3)where P [ ρ | H ] = e − S , S = S [ ρ ( ω ) , m ( ω )] . (3.4)Here S is a prior and m ( ω ) is a default model. Currently, two different priors, Shannon-Jaynesentropy for S S SJ = α (cid:90) d ω (cid:16) ρ − m − ρ log ( ρ m ) (cid:17) (3.5)which is called “Maximum Entropy Method [38] and new prior [39] S BR = α (cid:90) d ω (cid:16) − ρ m + log ( ρ m ) (cid:17) (3.6)are under studies. Since only with infinite number of data points in G ( τ ) and zero statistical errorsall methods should agree [38], many different priors must be tested with finite number of datapoints and non-zero statistical errors. ρ ( ω ) / m b ω (GeV) 9 10 11 12 13 14 15 Υ T/T c = 0 . .
84 0 . .
95 0 . . . .
27 1 . .
52 1 . .
90 00.10.20.30.400.10.20.30.4 9 10 11 12 13 14 15 ρ ( ω ) / m b ω (GeV) 9 10 11 12 13 14 15 χ b T/T c = 0 . .
84 0 . .
95 0 . . . .
27 1 . .
52 1 . . Figure 4:
Spectral functions for the ϒ channel (left) and those for the χ b channel (right) at successivetemperatures from FASTSUM collaboration using NRQCD heavy quark propagator and MEM for the re-construction of spectral function [18]. Using anisotropic lattices with a fixed lattice scale and temperature change by N τ , FASTSUMcollaboration calculated T (cid:54) = eavy Flavors in QGP Seyong Kim functions of S-wave and P-wave quarkonium correlators using MEM. Using the 1st generationconfigurations (12 × N τ , a s / a τ = . , N f = m π / m ρ ≈ . , T c ≈ T (cid:54) = Ma s ≥ a τ , the zero point energy shift associated with NRQCD formalism needs tobe fixed only once. ρ Υ - B R ( ω ) ω [GeV] S-waveT ≈ C C C C C C C C C C C C C C ρ χ b1 - B R ( ω ) ω [GeV] P-waveT ≈ C C C C C C C C C C C C C C Figure 5:
Spectral functions for the ϒ channel (left) and those for the χ b channel at successive temperaturefrom [34] using NRQCD heavy quark propagator and new Bayesian prior [39] for the reconstruction ofspectral function. From the spectral function of ϒ channel, it is found that ϒ ( S ) survives upto T ∼ . T c while ϒ ( S ) and ϒ ( S ) are sequentially suppressed in agreement with [2]. χ b melts immediately above T c [33]. A detailed study of systematic errors is performed: default-model dependence in theMEM prior, energy window dependence in the spectral function reconstruction, dependence onthe number of configurations and dependence on the Euclidean time window and found that thereconstructed spectral functions are stable. However, with non-zero statistical errors in the dataof quarkonium correlators and finite number of the temporal lattice sites, choice of the prior S isnot unique and reconstructed spectral functions may depend on the choice. Using the same setupon the 2nd generation configurations (24 × N τ , a s / a τ = . , N f = + ϒ ( S ) upto ∼ . T c and the melting of χ b just above T c . Fig. 4 shows the spectral functions for the ϒ channel and that for the χ b channel.For a quantitative understanding of the quarkonium melting, FASTSUM’s results need to bechecked with a setup which has different systematics. Authors of [34] calculated NRQCD heavyquark propagator using a isotropic lattice configurations from HotQCD (48 × N f = + m π ∼
160 MeV and T c =
154 MeV) [40] where the temperature is changed by changing the latticespacing a with fixed N τ =
12. The pro is that the temperature can be changed continuously and adetailed temperature scan is possible. The con is that the zero energy shift for NRQCD needs to befixed by accompanying T = eavy Flavors in QGP Seyong Kim functions by using two different priors, MEM and a new Bayesian prior [39]. [34] found thatqualitatively similar behavior for the ϒ channel found by FASTSUM: the survival of ϒ ( S ) upto thehighest temperature they studied ( ∼ . T c ) from two different priors. The left figure of Fig. 5 showsthe spectral function of the ϒ channel obtained with new Bayesian prior. However, qualitativelydifferent behaviors for the χ b channel is found: the spectral function from MEM shows meltingbut the spectral function from new Bayesian prior shows survival of χ b upto ∼ . T c (the rightfigure in Fig. 5).Reconstruction of spectral function with new Bayesian prior is generally stronger in identi-fying peaks [41] but is susceptible to spurious “ringing” (i.e., reconstructs peaks for the spectralfunction from lattice free quarkonium correlators although an analytic result for the spectral func-tion does not show such peaks [34]). Bayesian reconstructions of spectral functions for the correlators with finite data points andfinite statistical errors are expected to show prior dependence. Between MEM prior and newBayesian prior, which one is closer to the “Bayesian continuum limit” (infinite data points andzero statistical errors) can only be tested by going toward the Bayesian continuum limit. ω [GeV]0510152025 ρ ( ω ) ( a r b it r a r y no r m a li s a ti on ) "Gen3" Nt=32Gen2 Nt=16 Gen3 (Run31-untuned versus Gen2)
Upsilon ω [GeV]010203040 ρ ( ω ) ( a r b it r a r y no r m a li s a ti on ) "Gen3" t=2-30, ω min a τ =-0.08"Gen3" t=2,4,6...28, ω min a τ =-0.08Gen2 t=2-14, ω min a τ =-0.12 Gen3 (run31-untuned) versus Gen2
P-wave χ b1 Figure 6:
Preliminary spectral functions for the ϒ channel (left) and those for the χ b channel (right) fromthe 3rd Generation configurations (see the text for parameters). For the further study, FASTSUM collaboration focus on testing a direction of the Bayesiancontinuum limit by halving the lattice spacing (i.e., increasing the temporal lattice data pointsby twice) while keeping other conditions the same. In Fig. 6, preliminary results of the spectralfunctions from the “3rd generation configurations (32 × N τ , a s / a τ = . , N f = +
1) show nosignificant differences between the results from the 2nd generation configuration and those fromthe preliminary 3rd generation configurations. On the other hand, authors of [34] concentrate onincreasing statistics ( ∼ T = ( . T c ) , ( . T c ) , ( . T c ) us-ing new HotQCD gauge configurations[42]. Fig. 7 show preliminary results of the reconstructedspectral functions of bottomonium correlators at higher statistics and higher temperature. Interest-ingly, at higher temperature, χ b finally shows melting behavior.10 eavy Flavors in QGP Seyong Kim T = = = = = = = = = = = = = = = = = �� �� �� �� �� ω [ ��� ] �� - � ������������ ρ ( ω ) S Channel @ T >
0, n = T = = = = = = = = = = = = = = = = = �� �� �� �� �� ω [ ��� ] ��������������������� ρ ( ω ) P Channel @ T >
0, n = Figure 7:
Preliminary spectral functions for the ϒ channel (left) and those for the χ b channel (right) fromS.K. A. Rothkopf, P. Petreczky with four times higher statistics correlators and higher temperature (see thetext for parameters).
4. Summary
Some of recent progresses in the lattice studies of heavy flavours in thermal environment arediscussed after looking over the opportunities offered by studying the properties of QGP throughheavy quark flavours and the challenges faced in such lattice studies. Focuses in this talk are on thetransport phenomena of heavy quarks in thermal medium and the spectral properties of quarkoniain thermal medium.Various effective field theories for heavy quark and quarkonia on a T = ∼ α s T / M ), this alternative formulation avoids the numerical difficulty of accessing a narrow peakin the study of kinetic equilibriation and also new observable defined in the formulation givesless complicated spectral functions. It gives us a robust determination of the momentum diffusioncoefficient for heavy quark at least in the quenched approximation. κ / T = . − . O ( ) fm/c as the kinetic equilibriation time scale which is similar to the lightparton kinetic equilibriation time scale. Although it is difficult to generalize the same setup to thecase of fully dynamical lattices (because the new observable is gluonic), quenched non-perturbativeestimate of the momentum diffusion constant is a significant step toward understanding fast kineticequilibriation of charm quark in relativistic heavy ion collisions.In the case of heavy quark chemical equilibriation, “tail of a spectral function as a transportcoefficient” allows a non-perturbative definition of the thermal averaged Sommerfeld factor (andrelated Γ chem ) as a ratio of two-point correlators. This factor is calculable using lattice NRQCDmethod (and the chemical equilibriation rate, given as the heavy quark number susceptibility timesthe Sommerfeld factor). Using dynamical N f = + m π (cid:39) m K (cid:39)
500 MeV), [9]11 eavy Flavors in QGP
Seyong Kim found much larger Sommerfeld enhancement ( ∼ O ( ) ) than estimated by a perturbative method.Phenomenologically, at T ∼
400 MeV, for a charm quark mass M ∼ . Γ − ∼
150 fm/c,which suggests that within the lifetime ∼
10 fm/c of QGP phase in the current relativistic heavy ioncollision experiment, the chemical equilibriation is not achieved. Since the chemical equilibriationrate changes rapid with temperature, however, the planned Future Circular Collider [43] heavy ionprogram may yield charm quark chemical equilibriation.Over the years, lattice community studied thermal behavior of quarkonium by various meth-ods. Recently, using lattice NRQCD to calculate a heavy quark propagator and using Bayesianmethod to obtain spectral functions of temporal quarkonium correlators constructed from thisNRQCD heavy quark propagator overcame various difficulties identified in studying quarkoniumon a lattice. Up until this year’s lattice conference, two groups which have used this method indifferent lattice setup reported that the survival of ϒ ( S ) upto ∼ . T c ( or1 . T c ) and sequential sup-pression of higher ϒ states as temperature increases. But the results on χ b states differ: FASTSUMreports the melting of P-wave immediately above T c but the result of [34] hints at the survival of χ b upto ∼ . T c . In these studies, quarkonium correlators themselves can be calculated to high ac-curacy. However, since Bayesian methods can give a unique result only in the Bayesian continuumlimit, further studies are needed. FASTSUM group focuses on increasing the data points of tem-poral quarkonium correlators. Preliminary result shows that doubling the number of the temporallattice sites does not change significantly reconstructed spectral functions of ϒ and χ b . Authors of[34] focus on increasing temperature and increasing statistics. Their preliminary result show thatat higher temperature, χ b melts and higher statistics reduces jack-knife errors in the reconstructedspectral functions. More studies by both groups are forthcoming. Also, further investigations onthe Bayesian reconstruction method for the spectral functions are needed. Acknowledgements
I would like to acknowledge that many discussions with G. Aarts, C. Allton, N. Brambilla,Y. Burnier, A. Francis, O. Kaczmarek, M. Laine, M.P. Lombardo, P. Petreczky and A. Vairo werehelpful to my understanding on “heavy flavors in non-zero temperature”. This work is supportedby the National Research Foundation of Korea under grant No. 2015R1A2A2A01005916 fundedby the Korean government (MEST).
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