spectral and transport properties of the qgp from lattice ......kitpc program “sqgp and extreme...
TRANSCRIPT
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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
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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
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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]
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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↔
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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
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T=11
Quarkonium spectral function – What do we expect!?
!!
+ zero-mode contribution at !=0:
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TÀÀTc T=11
Quarkonium spectral function – What do we expect!?
!!
+ zero-mode contribution at !=0:
+ transport (narrow) peak at small !:
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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 !:
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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 !:
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Free spectral functions – lattice vs. continuum
Free (non-interacting) spectral function [Karsch et al. 03, Aarts et al. 05]
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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
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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
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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
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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
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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
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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
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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]
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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)
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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]
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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
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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]
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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
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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]
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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