Correlation functions for heavy quarks in the QGP from lattice ...theor.jinr.ru/~diastp/dm15/talks/Kaczmarek_L2.pdfHeavy Quark Momentum Diffusion Constant – Lattice algorithms [A.Francis,OK,M.Laine,J.Langelage,

1

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

Correlation functions for heavy quarks in the QGP from lattice QCD calculations

Olaf Kaczmarek

University of Bielefeld

Helmholtz International Summer School “Dense Matter“ Dubna

04.07.2015

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


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


[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 quark potential complex valued at finite temperature


[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

2mq

fl(Ê)w

Ê

T > Tc

T >> Tc

T = Œ

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

(narrow) transport peak at small !:

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 behavior 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 ·· :


result of the fit to ½½mmooddeell (!!)

with three parameters: ·,A,B


Model spectral function: transport contribution + NNLO + correction

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




A ½NNLO(!) + B w3

small but relevant contribution at ¿T > 0.2 !

!¿T: linear behavior motivated at small frequencies !ÀT: vacuum perturbative results and leading order thermal correction: using a renormalization scale for leading order becomes here we used 4-loop running of the coupling model the spectral function using these asymptotics with two free parameters

Heavy Quark Momentum Diffusion Constant – IR and UV asymptotics

Model spectral function: transport contribution + UV-asymtotics


already closer to the data

Model spectral function: transport contribution + UV-asymtotics

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



A ½UV(!)

small but relevant contribution at ¿T > 0.2 !

model corrections to ½IR by a power series in !

analysis of the systematic uncertainties àà continuum estimate of ·· :

Heavy Quark Momentum Diffusion Constant – systematic uncertainties

Conclusions and Outlook – Heavy Quark Momentum Diffusion

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 detailed analysis of the systematic uncertainties

- 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

χb1

T/Tc = 0.760.84

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 80

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=48

continuum

[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

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

2mq

fl(Ê)w

Ê

T > Tc

T >> Tc

T = Œ

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

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

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]

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

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 – S-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

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 [H.T.Ding, H.Ohno, OK, M.Laine, T.Neuhaus, work in progress]

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

[H.T.Ding, H.Ohno, OK, M.Laine, T.Neuhaus, work in progress]

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

1MS

þ 1 0þþ D�s0 χc0

MPS− γ5 0−þ ηss̄ Ds ηc

ð−1Þxþyþz

MPSþ γ4 0þ− – –

MAV− γiγ4 1−− ϕ D�

s J=ψð−1Þx, ð−1Þy

MAVþ γiγ5 1þþ f1ð1420Þ Ds1 χc1

MV− γi 1−− ϕ D�

s J=ψð−1Þxþz, ð−1Þyþz

MVþ γjγk 1þ− 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’’

Correlations of conserved charges – open charm sector Taylor expansion of pressure in terms of chemical potentials related to conserved charges

generalized susceptibilities of conserved charges

are sensitive to the underlying degrees of freedom

charm contributions to pressure in a hadron gas:

partial pressure of open-charm mesons and charmed baryons depends on hadron spectra

ratios independent of the detailed spectrum and sensitive to special sectors:

=1 when DoF are hadronic

=3 when DoF are quarks

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

∂µ̂kB∂µ̂

lQ µ̂m

S ∂µ̂nC

∣∣∣∣µ⃗=0

relative contribution of C=2 and C=3 baryons negligible

always charmed baryon sector


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



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

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


∂µ̂kB∂µ̂

lQ µ̂m

S ∂µ̂nC

∣∣∣∣µ⃗=0

charmed baryon sector

Correlations of conserved charges – open charm sector Taylor expansion of pressure in terms of chemical potentials related to conserved charges



charm contributions to pressure in a hadron gas:

partial pressure of open-charm mesons and charmed baryons depends on hadron spectra

ratios independent of the detailed spectrum and sensitive to special sectors:


∂µ̂kB∂µ̂

lQ µ̂m

S ∂µ̂nC

∣∣∣∣µ⃗=0

partial pressure of open-charm mesons: open charm meson sector

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


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



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


∂µ̂kB∂µ̂

lQ µ̂m

S ∂µ̂nC

∣∣∣∣µ⃗=0

open charm meson sector

partial pressure of open-charm mesons:

Documents

Correlation functions for heavy quarks in the QGP from lattice ...theor.jinr.ru/~diastp/dm15/talks/Kaczmarek_L2.pdfHeavy Quark Momentum Diffusion Constant – Lattice algorithms [A.Francis,OK,M.Laine,J.Langelage,