spectral and transport properties of the qgp from lattice ......kitpc program “sqgp and extreme...

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Spectral and transport properties of the QGP from Lattice QCD calculations Olaf Kaczmarek University of Bielefeld KITPC program “sQGP and Extreme QCD“ Beijing 20.05.2015

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  • 1

    Geistes-, Natur-, Sozial- und Technikwissenschaften – gemeinsam unter einem Dach

    Spectral and transport properties of the QGP from Lattice QCD calculations

    Olaf Kaczmarek

    University of Bielefeld

    KITPC program “sQGP and Extreme QCD“ Beijing

    20.05.2015

  • Lattice calculations of hadronic correlation functions ... and how we try to

    extract transport properties and spectral properties from them

    1) Vector meson correlation functions for light quarks

    continuum extrapolation

    comparison to perturbation theory

    àà Electrical conductivity

    àà Thermal dilepton rates and thermal photon rates 2) Color electric field correlation function

    Heavy quark momentum diffusion coefficient ·· 3) Vector meson correlation functions for heavy quarks

    Heavy quark diffusion coefficients

    Charmonium and Bottomonium dissociation patterns

    with A.Francis, M. Laine, T.Neuhaus, H.Ohno

    with H-T.Ding, H.Ohno et al.

    with H-T.Ding, F.Meyer, et al.

    with J.Ghiglieri, M.Laine, F.Meyer

  • Transport Coefficients are important

    ingredients into hydro/transport models

    for the evolution of the system.

    Usually determined by matching to

    experiment (see right plot)

    Need to be determined from QCD

    using first principle lattice calculations!

    here heavy flavour:

    Heavy Quark Diffusion Constant D

    Heavy Quark Momentum Diffusion ·

    or for light quarks:

    Light quark flavour diffusion

    Electrical conductivity

    Motivation - Transport Coefficients

    [H.T.Ding, OK et al., PRD86(2012)014509]

    [A.Francis, OK et al., PRD83(2011)034504]

    [PHENIX Collaboration, Adare et al.,PRC84(2011)044905 & PRL98(2007)172301]

    [OK, arXiv:1409.3724]

  • Heavy Ion Collision QGP Expansion+Cooling Hadronization

    Charmonium+Bottmonium is produced (mainly) in the early stage of the collision

    Depending on the Dissociation Temperature

    - remain as bound states in the whole evolution

    - release their constituents in the plasma

    Motivation - Quarkonium in Heavy Ion Collisions

    J/ψψ

    J/ψψ

    cc

    J/ψψ J/ψψ

    DD

    D

    ? ?

    [S.Bass]

    ]2) [GeV/c-µ+µMass(7 8 9 10 11 12 13 14

    )2Ev

    ents

    / ( 0

    .1 G

    eV/c

    0

    10

    20

    30

    40

    50 = 2.76 TeVsCMS pp

    |y| < 2.4 > 4 GeV/cµ

    Tp

    -1 = 230 nbintL

    datatotal fitbackground

    ]2) [GeV/c-µ+µMass(

    7 8 9 10 11 12 13 14

    )2

    Eve

    nts

    / (

    0.1

    Ge

    V/c

    0

    100

    200

    300

    400

    500

    600

    700

    800

    = 2.76 TeVNNsCMS PbPb

    Cent. 0-100%, |y| < 2.4

    > 4 GeV/cµ

    Tp

    -1bµ = 150 intL

    data

    total fit

    background

    Sequential suppression

    for bottomonium

    observed at CMS

  • Heavy Ion Collision QGP Expansion+Cooling Hadronization

    Charmonium+Bottmonium is produced (mainly) in the early stage of the collision

    Depending on the Dissociation Temperature

    - remain as bound states in the whole evolution

    - release their constituents in the plasma

    Motivation - Quarkonium in Heavy Ion Collisions

    [Kaczmarek, Zantow, 2005]

    Charmonium yields at SPS/RHIC First estimates on

    Dissociation Temperatures from detailed knowledge of Heavy Quark Free Energies and Potential Models

    J/ψψ

    J/ψψ

    cc

    J/ψψ J/ψψ

    DD

    D

    ? ?

    [S.Bass]

  • Heavy Ion Collision QGP Expansion+Cooling Hadronization

    Heavy quark potential complex valued at finite temperature

    Motivation - Quarkonium in Heavy Ion Collisions

    [Y.Burnier, OK, A.Rothkopf, PRL114(2015)082001]

    J/ψψ

    J/ψψ

    cc

    J/ψψ J/ψψ

    DD

    D

    ? ?

    [S.Bass]

    VðrÞ ¼ limt→∞

    i∂tWðt; rÞWðt; rÞ

    : VðrÞ ¼ limt→∞

    Zdωωe−iωtρðω; rÞ=

    Zdωe−iωtρðω; rÞ↔

  • Lattice observables: only correlation functions calculable on lattice but Transport coefficient determined by slope of spectral function at !=0 (Kubo formula)

    Transport coefficients from Lattice QCD – Flavour Diffusion

    related to a conserved current

    (0,0) (¿,x)

    Transport coefficients usually calculated using correlation function of conserved currents

  • T=11

    Quarkonium spectral function – What do we expect!?

    !!

    + zero-mode contribution at !=0:

  • TÀÀTc T=11

    Quarkonium spectral function – What do we expect!?

    !!

    + zero-mode contribution at !=0:

    + transport (narrow) peak at small !:

  • T>Tc TÀÀTc T=11

    Quarkonium spectral function – What do we expect!?

    !!

    + zero-mode contribution at !=0:

    + transport (narrow) peak at small !:

  • TTc TÀÀTc

    Quarkonium spectral function – What do we expect!?

    !!

    + zero-mode contribution at !=0:

    + transport (narrow) peak at small !:

  • T=0 T

  • T=0 TTc TÀÀTc T=11

    Quarkonium spectral function – What do we expect!?

    !!

    + zero-mode contribution at !=0:

    + transport (narrow) peak at small !:

  • 2mq

    fl(Ê)w

    Ê

    T > Tc

    T >> Tc

    T = Tc

    Vector spectral function – hard to separate different scales Different contributions and scales enter in the spectral function

    - continuum at large frequencies

    - possible bound states at intermediate frequencies

    - transport contributions at small frequencies

    - in addition cut-off effects on the lattice

    notoriously difficult to extract from correlation functions

    transport (narrow) peak at small !:

  • Free spectral functions – lattice vs. continuum

    Free (non-interacting) spectral function [Karsch et al. 03, Aarts et al. 05]

  • Free (non-interacting) spectral function [Karsch et al. 03, Aarts et al. 05]

    Lattice cut-off effects: we will perform the continuum extrapolation to get rid of the cut-off effects

    0

    0.02

    0.04

    0.06

    0.08

    0.1

    0.12

    0.14

    0.16

    0.18

    0 20 40 60 80 100 120 140 160 180 200

    ω/T

    ρV(ω)/ω2 Cont.

    1283x241283x321283x48

    1/a=6.5 GeV

    1/a=13 GeV

    1/a=19 GeV

    Free spectral functions – lattice vs. continuum

  • Free (non-interacting) spectral function [Karsch et al. 03, Aarts et al. 05]

    [Petreczky+Teaney 06 Aarts et al. 05]

    zero mode contribution at ! ' 0 [Umeda 07]

    with interactions:

    Free spectral functions – lattice vs. continuum

  • What are the contribution of the different scales to the correlator?

    illustrative example:

    - continuum contribution dominates at small ¿T

    - bound state contribution over a wide range, only one state in this example

    - only small contribution from the transport peak at large ¿T

    Vector spectral function – hard to separate different scales

    0

    0.2

    0.4

    0.6

    0.8

    1

    0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5τT

    Gpart(τT)/G(τT)

    transportboundstate

    massive continuum

    0

    5

    10

    15

    20

    25

    30

    35

    40

    0 5 10 15 20 25 30

    σ/ωT transportboundstate

    massive continuum

  • Heavy Quark Effective Theory (HQET) in the large quark mass limit

    for a single quark in medium

    leads to a (pure gluonic) “color-electric correlator” [J.Casalderrey-Solana, D.Teaney, PRD74(2006)085012, S.Caron-Huot,M.Laine,G.D. Moore,JHEP04(2009)053]

    Heavy quark (momentum) diffusion: ,

    Heavy Quark Momentum Diffusion Constant – Single Quark in the Medium

    ¿

    ¾

  • Heavy Quark Momentum Diffusion Constant – Perturbation Theory

    0

    0.1

    0.2

    0.3

    0.4

    0.5

    0.6

    0.5 1 1.5 2 2.5

    0.050.01 0.40.30.20.1

    �=g4 T

    3

    gs

    � s

    Next-‐to-‐leading order (eq. (2.5) )L eading order (eq. (2.4) )

    T runcated leading order (eq. (2.5) with C= 0)

    can be related to the thermalization rate:

    NLO in perturbation theory: [Caron-Huot, G.Moore, JHEP 0802 (2008) 081]

    very poor convergence

    à Lattice QCD study required in the relevant temperature region

  • NLO spectral function in perturbation theory: [Caron-Huot, M.Laine, G.Moore, JHEP 0904 (2009) 053]

    in contrast to a narrow transport peak, from this a smooth limit

    is expected

    Qualitatively similar behaviour also found in AdS/CFT [S.Gubser, Nucl.Phys.B790 (2008)175]

    Heavy Quark Momentum Diffusion Constant – Perturbation Theory

    0 1 2ω / mD

    -1

    0

    1

    2

    3

    4

    [κ(ω

    )-κ] / [g

    2 CF T m

    D2 / 6π]

    Landau cuts

    full result - 2 (ω/mD)2

    full result

  • Heavy Quark Momentum Diffusion Constant – Lattice algorithms [A.Francis,OK,M.Laine,J.Langelage, arXiv:1109.3941 and arXiv:1311.3759]

    due to the gluonic nature of the operator, signal is extremely noisy

    à multilevel combined with link-integration techniques to improve the signal [Lüscher,Weisz JHEP 0109 (2001)010 [Parisi,Petronzio,Rapuano PLB 128 (1983) 418, and H.B.Meyer PRD (2007) 101701] and de Forcrand PLB 151 (1985) 77]

  • Heavy Quark Momentum Diffusion Constant – Tree-Level Improvement [A.Francis,OK,M.Laine,J.Langelage, arXiv:1109.3941 and arXiv:1311.3759]

    normalized by the LO-perturbative correlation function:

    and renormalized using NLO renormalization constants Z(g2)

  • lattice cut-off effects visible at small separations (left figure)

    à  tree-level improvement (right figure) to reduce discretization effects

    leads to an effective reduction of cut-off effect for all ¿T

    Heavy Quark Momentum Diffusion Constant – Tree-Level Improvement [A.Francis,OK,M.Laine,J.Langelage, arXiv:1109.3941 and arXiv:1311.3759]

  • Heavy Quark Momentum Diffusion Constant – Lattice results

    Quenched Lattice QCD on large and fine isotropic lattices at T' 1.4 Tc - standard Wilson gauge action

    - algorithmic improvements to enhance signal/noise ratio

    - fixed aspect ration Ns/Nt = 4, i.e. fixed physical volume (2fm)3

    - perform the continuum limit, a! 0 $ Nt ! 1

    - determine · in the continuum using an Ansatz for the spectral fct. ½(!)

  • Heavy Quark Momentum Diffusion Constant – Lattice results

    finest lattices still quite noisy at large ¿T

    but only

    small cut-off effects at intermediate ¿¿T

    cut-off effects become visible at small ¿T

    need to extrapolate to the continuum

    perturbative behavior in the limit ¿¿T !! 0

    allows to perform continuum extrapolation, a! 0 $ Nt ! 1 , at fixed T=1/a Nt

  • Heavy Quark Momentum Diffusion Constant – Continuum extrapolation

    finest lattices still quite noisy at large ¿T

    but only

    small cut-off effects at intermediate ¿¿T

    cut-off effects become visible at small ¿T

    need to extrapolate to the continuum

    perturbative behavior in the limit ¿¿T !! 0

    well behaved continuum extrapolation for 0.05 ·· ¿¿ T ·· 0.5 finest lattice already close to the continuum coarser lattices at larger ¿T close to the continuum how to extract the spectral function from the correlator?

  • Model spectral function: transport contribution + NNLO [Y.Burnier et al. JHEP 1008 (2010) 094)]

    some contribution at intermediate distance/frequency seems to be missing

    Heavy Quark Momentum Diffusion Constant – Model Spectral Function

    continuum extrapolation

    NNLO (vacuum) perturbation theory

    ··/T3 = 3.0 ··/T3 = 2.0 ··/T3 = 1.0

    ··/T3 = 4.0

  • Model spectral function: transport contribution + NNLO + correction

    used to fit the continuum extrapolated data àà first continuum estimate of ·· :

    Heavy Quark Momentum Diffusion Constant – Model Spectral Function

    result of the fit to ½½mmooddeell (!!)

    with three parameters: ·,A,B

    NNLO (vacuum) perturbation theory

  • Model spectral function: transport contribution + NNLO + correction

    used to fit the continuum extrapolated data àà first continuum estimate of ·· :

    result of the fit to ½½mmooddeell (!!)

    Heavy Quark Momentum Diffusion Constant – Model Spectral Function

    NNLO (vacuum) perturbation theory

    A ½NNLO(!) + B w3

    small but relevant contribution at ¿T > 0.2 !

  • Conclusions and Outlook

    àà Continuum extrapolation for the color electric correlation function

    extracted from quenched Lattice QCD

    - using noise reduction techniques to improve signal

    - and an Ansatz for the spectral function

    àà first continuum estimate for the Heavy Quark Momentum Diffusion Coefficient ··

    - still based on a simple Ansatz for the spectral function

    àà More detailed analysis of the systematic uncertainties needed

    - different Ansätze for the spectral function

    - using contributions from thermal perturbation theory

    - other techniques to extract the spectral function

    Other Transport coefficients from Effective Field Theories?

  • Bottomonium - Lattice NRQCD [G.Aarts et al., JHEP11(2011)103]

    In non-relativistic QCD the Lagrangian is expanded in terms of v=|p|/M

    with

    and

    which is correct up to order O(v^4) [G.T.Bodwin,E.Braaten,G.P.Lepage, PRD 51 (1995) 1125]

  • NRQCD is more sensitive to the

    bound state region

    Kernel is T-independent

    à contributions at ! < 2M absent

    à no small-! contribution

    à no information on transport properties

    requires anisotropic lattices with asMÀ1 no continuum limit in NRQCD

    only small energy region accessible

    Bottomonium - Lattice NRQCD [G.Aarts et al., JHEP1407(2014)097]

    0

    2

    4

    6

    8

    10

    12

    0

    1

    2

    3

    4

    5

    9 10 11 12 13 14 15

    ρ(ω

    )/m

    2 b

    9 10 11 12 13 14 15ω (GeV)

    9 10 11 12 13 14 15

    T/Tc = 0.76

    0.84

    0.840.95

    0.951.09

    1.091.27

    1.271.52

    1.521.90

    0

    0.1

    0.2

    0.3

    0.4

    0

    0.1

    0.2

    0.3

    0.4

    9 10 11 12 13 14 15

    ρ(ω

    )/m

    4 b

    9 10 11 12 13 14 15ω (GeV)

    9 10 11 12 13 14 15

    χb1T/Tc = 0.76

    0.84

    0.840.95

    0.951.09

    1.091.27

    1.271.52

    1.521.90

  • 0

    5

    10

    15

    20

    25

    30

    35

    40

    45

    0 1 2 3 4 5 6 7

    ρ(ω

    )(G

    eV2)

    ω − 2mb (GeV)0 1 2 3 4 5 6 7 8

    0

    200

    400

    600

    800

    1000

    1200

    1400

    1600

    ρ(ω

    )(G

    eV4)

    S wave P waveFirst gen.

    Second gen.First gen. cont.

    Second gen. cont.

    cut-off effects and energy resolution determined by spatial lattice spacing

    no continuum limit in NRQCD, asMÀ1 continuum limit straight forward, but expensive only small energy region accessible transport properties accessible

    Lattice cut-off effects – free spectral functions [G.Aarts et al., JHEP1407(2014)097]

    0

    0.2

    0.4

    0.6

    0.8

    1

    1.2

    0 20 40 60 80 100 120 140 160 180

    ρvc(ω)/ω2

    ω/T

    Nτ=24Nτ=32Nτ=48continuum

    [H.T.Ding, OK et al., arXiv:1204.4945] gauge configurations from nf=2+1 dynamical Wilson fermion action

    as ' 0.16 fm as ' 0.13 fm 1/at ' 7.35 GeV 1/at ' 5.63 GeV

    gauge configurations from quenched action

    a ' 0.01 fm 1/a ' 19 GeV

    [see also F.Karsch et al., PRD68 (2003) 014504]

    0GeV 16GeV 40GeV 73GeV

    anisotropic NRQCD isotropic Wilson

  • Correlations of conserved charges – open charm sector [A.Bazavov, H.T.Ding, P.Hegde, OK et al., PLB737 (2014) 210]

    2+1 flavor HISQ with almost physical quark masses

    323£8 and 243£ 6 lattices with ml = ms/20 and physical ms and quenched charm quarks

    generalized susceptibilities of conserved charges

    are sensitive to the underlying degrees of freedom

    àà indications that open charm hadrons start to dissolve already close to the chiral crossover

    χ BQ SCklmn =∂(k+l+m+n)[P (µ̂B , µ̂Q , µ̂S , µ̂C )/T 4]

    ∂µ̂kB∂µ̂lQ µ̂

    mS ∂µ̂

    nC

    ∣∣∣∣µ⃗=0

    charmed baryon sector open charm meson sector

  • Lattice observables:

    Quarkonium Spectral Functions – HQ Diffusion and Dissociation

    related to a conserved current in the vector channel

    (0,0) (¿,x)

    In the following: Meson Correlation Functions

  • Lattice observables: only correlation functions calculable on lattice but Transport coefficient determined by slope of spectral function at !=0 (Kubo formula)

    Quarkonium Spectral Functions – HQ Diffusion and Dissociation

    In the following: Meson Correlation Functions

    2mq

    fl(Ê)w

    Ê

    T > Tc

    T >> Tc

    T = Tc

  • [H.T.Ding, OK et al., PRD86(2012)014509] from Maximum Entropy Method analysis on a fine but finite lattice: statistical error band from Jackknife analysis no clear signal for bound states at and above 1.46 Tc study of the continuum limit and quark mass dependence required!

    Charmonium Spectral function

  • 0

    0.5

    1

    1.5

    2

    2.5

    3

    3.5

    4

    1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3

    2πTD

    T/Tc 0

    0.5

    1

    1.5

    2

    2.5

    3

    3.5

    4

    1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3

    2πTD

    T/Tc

    Charmonium Spectral function – Transport Peak

    Perturbative estimate (®s~0.2, g~1.6): Strong coupling limit:

    LO: 2¼TD ' 71.2 2¼TD = 1 NLO: 2¼TD ' 8.4

    [Moore&Teaney, PRD71(2005)064904, [Kovtun, Son & Starinets,JHEP 0310(2004)064] Caron-Huot&Moore, PRL100(2008)052301]

    [H.T.Ding, OK et al., PRD86(2012)014509]

  • 0

    2

    4

    6

    8

    10

    12

    14

    16

    1 1.5 2 2.5 3T/Tc

    2πTD

    0

    2

    4

    6

    8

    10

    12

    14

    16

    1 1.5 2 2.5 3T/Tc

    2πTD

    αs≈0.2NLO pQCD

    (Ding et al.)

    (Banerjee et al.)

    (Kaczmarek et al.)

    charm quark

    static quark

    static quark

    AdS/CFT

    Charmonium Spectral function – Transport Peak

    Perturbative estimate (®s~0.2, g~1.6): Strong coupling limit:

    LO: 2¼TD ' 71.2 2¼TD = 1 NLO: 2¼TD ' 8.4

    [Moore&Teaney, PRD71(2005)064904, [Kovtun, Son & Starinets,JHEP 0310(2004)064] Caron-Huot&Moore, PRL100(2008)052301]

  • Charmonium and Bottomonium correlators [H.T.Ding, H.Ohno, OK, M.Laine, T.Neuhaus, work in progress]

    •  standard plaquette gauge & O(a)-improved Wilson quarks •  quenched gauge field configurations •  on fine and large isotropic lattices •  T = 0.7-1.4Tc •  2 different lattice spacing

    analysis of cut-off effects continuum limit (in the future)

    •  both charm & bottom tuned close to their physical masses

    Experimental values: mJ/Ψ = 3.096.916(11) GeV, mΥ = 9.46030(26) GeV

  • Correlation functions along the spatial direction are related to the meson spectral function at non-zero spatial momentum exponential decay defines screening mass Mscr : bound state contribution high-T limit (non-interacting free limit)

    Spatial correlation function and screening masses

    indications for medium modifications/dissociation

    [“Signatures of charmonium modification in spatial correlation functions“, F.Karsch, E.Laermann, S.Mukherjee, P.Petreczky, (2012) arXiv:1203.3770]

  • Spatial Correlation Function and Screening Masses

  • exponential decay defines screening mass Mscr : bound state contribution high-T limit (non-interacting free limit)

    Spatial Correlation Function and Screening Masses

    indications for medium modifications/dissociation

  • Spatial correlation function and screening masses

    0.5

    1

    1.5

    2

    2.5

    3

    3.5

    100 150 200 250 300 350 400 450 500

    M [GeV]

    T [MeV]

    ss-

    1++

    0++

    1−−

    0−+

    2

    2.5

    3

    3.5

    100 150 200 250 300 350 400 450 500

    M [GeV]

    T [MeV]

    sc-

    1+

    0+

    1−

    0−

    2.8

    3

    3.2

    3.4

    3.6

    3.8

    4

    4.2

    100 150 200 250 300 350 400 450 500

    M [GeV]

    T [MeV]

    cc-

    1++

    0++

    1−−

    0−+

    q p

    − ~ϕðxÞ Γ JPC ss̄ sc̄ cc̄

    MS− γ4γ5 0−þ ηss̄ Ds ηc

    1MSþ 1 0

    þþ D�s0 χc0

    MPS− γ5 0−þ ηss̄ Ds ηc

    ð−1ÞxþyþzMPSþ γ4 0

    þ− – –MAV− γiγ4 1

    −− ϕ D�s J=ψð−1Þx, ð−1Þy

    MAVþ γiγ5 1þþ f1ð1420Þ Ds1 χc1

    MV− γi 1−− ϕ D�s J=ψ

    ð−1Þxþz, ð−1ÞyþzMVþ γjγk 1

    þ− hc

    [A.Bazavov, F.Karsch, Y.Maezawa et al., PRD91 (2015) 054503]

    2+1 flavor HISQ with almost physical quark masses

    483£12 lattices with ml = ms/20 and physical ms

    ‘’ss and sc possibly dissolve close to crossover temperature’’

    ‘’cc in line with the sequential melting of charmonia states’’

  • Reconstructed correlation function

    r

    S.  Da'a  et  al.,  PRD  69  (2004)  094507

       equals  to  unity  at  all  τ  

    if  the  spectral  funcCon  doesn’t  vary  with  temperature  

    H.-‐T.  Ding  et  al.,  PRD  86  (2012)  014509

    can be calculated directly from correlation function for suitable ratios of N’¿ / N¿ without knowledge of spectral function:

  • r

    Charmonium and Bottomonium correlators

    bottom

    charm

    charm

    bottom

  • Charmonium and Bottomonium correlators – S-wave channels

    r

    different behavior in pseudo-scalar (left) and vector (right) channel

    strong modification at large ¿¿ in vector channel

    stronger for charm compared to bottom

    related to transport contribution

  • Charmonium and Bottomonium correlators – P-wave channels

    r

    comparable behavior in scalar (left) and axial-vector (right) channel

    strong modification at large ¿¿ in both channels

    stronger for bottom compared to charm

    related to transport/constant contribution

  • Mid-point subtracted correlators – P-wave channels

    r mid-point subtracted correlator

    small !! region gives (almost) constant contribution to correlators

    effectively removed by mid-point subtraction

    pseudo-scalar (left) and vector (right) very comparable

    need to understand cut-off effects and quark-mass effects

  • Outlook

    work is still in progress

    continuum extrapolation for the quarkonium correlators still needed

    detailed analysis of the systematic uncertainties

    Extract spectral properties (on continuum extrapolated correlators) by

    - comparing to perturbation theory

    - Fits using Ansätze for the spectral function

    - Bayesian techniques to extract the spectral function

    Final goal:

    understand the temperature and quark mass dependence of

    heavy quark diffusion coefficient

    dissociation temperatures for different states