Euclidean correlation functions for transport coefficients under gradient flow

Hai-Tao Shu1

Lattice 2019Wuhan, China, 16-22 June, 2019

1

in collaboration with

L. Altenkort1, O. Kaczmarek1, 2, L. Mazur1

1Bielefeld University, 2 Central China Normal University

Outline

2

• Motivation

• Transport coefficients from euclidean correlation functions

• Application of gradient flow on the lattice as noise reduction technique

• Correlation functions on the lattice

• Summary & Outlook

Motivation

3

0 1 2 3 4pT [GeV]

0

5

10

15

20

25

v 2 (p

erce

nt)

STAR non-flow corrected (est.)STAR event-plane

Glauber

η/s=10-4

η/s=0.08

η/s=0.16

elliptic flow

[Luzum and Romatschke 08]

Relativistic viscous hydrodynamics – p.20

[M. Luzum and P. Romatschke, Phys. Rev. C 78, 034915 (2008) ]

• Transport coefficients as crucial inputs for hydro/transport models to describe the experimental data

[PHENIX, PRL98(2007)172301]

✴ Heavy quark (momentum) diffusion coefficient ✴ Shear viscosity & bulk viscosity

4

Analytic continuation

Transport coefficients from the lattice

spectral reconstruction: see Anna-Lena Kruse & Hiroshi Ohno’s talk

⇣(T ) =⇡

18

3X

i,j=1

lim!!0

limp!0

⇢ii,jj(!,p, T )

!

⌘(T ) = ⇡ lim!!0

limp!0

⇢12,12(!,p, T )

2!Shear viscosity:

Bulk viscosity:

OX(x) = T12(x)

OX(x) = Tii(x)

HQ momentum diffusion coefficient:

OX(x) = Ei(x) = lim!!0

2T⇢ii(!)

!

G

X

(⌧, ~p) =

Zd

3~x e

�i~p·~xh ˆOX

(⌧, ~x)

ˆOX

(0,

~

0)i =Z

d!

2⇡

cosh(!(⌧T � 1/2)/T )

sinh(!/2T )

⇢

X

(!, ~p, T )

For correlators of topological charge, see Lukas Mazur’s talk

5

Color-electric correlatorsLarge-mass limit drives current-current correlator to be a “color-electric” correlator (in pure gauge) along a Polyakov loop

GJJ(⌧) ! GEE(⌧) = �1

3

3X

ii=1

hRe Tr[U(�, ⌧)gEi(⌧,~0)U(⌧, 0)gEi(0,~0)]ihRe Tr[U(�, 0)]i

⌧

�

• Lattice calculations of color-electric correlators:

๏ available on the lattice๏ gives access to “mass shift” of quarkonia

(vacuum subtraction required)

[A.Francis, et al., Lattice 2011, PRD92(2015)116003]

✤Multi-level ✤Link-integration

0.0 0.1 0.2 0.3 0.4 0.5τ T

-3

-2

-1

0

1

2

3

4

5

6

7

Gla

t / G

no

rm

standard

multi + link

T = 2.25 Tc, β = 7.457, 64

3x 24

⌧T

2.25Tc, � = 7.457, 643 ⇥ 24

[Luscher & Weisz, JHEP1102(2011)051][Narayanan & Neuberger, JHEP0603(2006)064]

[Luscher & Weisz, JHEP09 (2001)010][Forcrand &Roiesnel, PLB151(1985)77]

However, only works for gluonic fieldsNeed Gradient Flow for dynamic quarks, but start from gauge field first …

�m =3

2a20�, � = �

Z �

0d⌧ GEE(⌧)

[S. Caron-Huot et al., JHEP 0904 (2009) 053]

[A M. Eller, J. Ghiglieri, G. Moore, PRD.99.094042]

see A M. Eller’s poster

Gradient flow as a “diffusion” equation along extra dimension “t”

with initial condition:@tB(x, t) = D⌫G⌫µ(x, t) B⌫(x, t)|t=0 = A⌫(x)

Small flow time expansion

t ! 0O(x, t)X

k

ck(t)ORk (x)

Example: construction of T𝜇𝜐

E(t, x) =1

4G

a⇢�(x, t)G

a⇢�(x, t)

Uµ⌫(x, t) = G

aµ⇢(x, t)G

a⌫⇢(x, t)� �µ⌫E(t, x)

T

Rµ⌫(x) = lim

t!0

⇣Uµ⌫(t, x)

↵U (t)+

�µ⌫

4↵E(t)E(t, x)subt

⌘

Applications: running coupling / topo. charge / scale setting

defining operators / noise reduction / ...

{

Gradient flow

2p8t

4D b

ound

ary

t

x0 t

OX(⌧, ~x, t)

7

Lattice set-up

currently correlators are measured at 1.5Tc on 3 different quenched lattices (and beta)

� a[fm](a�1[GeV]) N� N⌧ T/Tc #confs. #meas.

6.8736 0.026 (7.496) 6416 1.50 10000 1000064 0.00 10000 -

7.0350 0.022 (9.119) 80 20 1.50 10000 10000

7.1920 0.018 (11.19) 96

16 2.25 10000 -24 1.50 10000 300028 1.29 10000 -32 1.13 10000 -48 0.75 10000 -

7.5440 0.012 (17.01) 144 36 1.50 - -7.7930 0.009 (22.78) 192 48 1.50 - -

Table 2: Lattice setup used to perform the continuum extrapolation. For the finest lattice, weused five quark sources for the two kappa values closest to bottomonium and charmonium

2

8

GEE under flow (643x16)

✤ Gradient flow reduces the error✤ How much can we flow ?

0.0 0.1 0.2 0.3 0.4 0.5

⌧T0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

Z(t=0)pert

GEE(⌧T, t)

G freeEE (⌧T )

pure SU(3) | T ⇡ 1.5TC | 643⇥16 | #conf = 10000 | imp. dist.

p8tT

0.0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045

0.05

0.055

0.06

0.065

0.07

0.075

Gfree(⌧) ⌘ ⇡2T 4hcos

2(⇡⌧T )

sin

4(⇡⌧T )

+

1

3 sin

2(⇡⌧T )

i

[S. Caron-Huot & M. Laine & G.D. Moore]

Renormalized to NLO:[C. Christensen & M. Laine]

Tree-level improved: [PoS LATTICE 2011,202 ]

9

Flow time limit for GEE (643x16)

LO perturbative limit for flow time:

[A M. Eller and G. Moore, PRD97 (2018) 11, 114507] 0.000 0.025 0.050 0.075 0.100 0.125 0.150 0.175 0.200

p⌧F ⌘

p8tT

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

Z(t=0)pert

GEE(⌧T, t)

G freeEE (⌧T )

pure SU(3) | T ⇡ 1.5TC | 643⇥16 | #conf = 10000 | imp. dist.

⌧T

0.06883

0.1085

0.1617

0.2247

0.2915

0.3572

0.4179

0.4527

0.0 0.1 0.2 0.3 0.4 0.5

x0/�

�0.2

0.0

0.2

0.4

0.6

0.8

1.0

1.2

hEa i(x

0 ,⌧ F

)Eb j(

0,⌧ F

) i �hE

a i(x

0 ,0)

Eb j(

0,0)i �

Scaled E-Field E-Field Correlator

⌧F = 0.001

⌧F = 0.01

⌧F = 0.05

⌧F = 0.1

⌧F = 0.2

⌧F = 0

GEE(t > 0)

GEE(t ⇡ 0)

⌧T

✤ Good signal within flow limit✤ Future steps: continuum limit & t → 0 limit

⌧F < 0.1136(⌧T )2

10

Gradient Flow v.s. Multi-Level (1)

✤ Comparable errors in both methods✤ Almost continuum limit in ML

p8tT = 0.075

(all 𝜏𝑇>0.25 within flow limit)

���

�

���

�

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11

Gradient Flow v.s. Multi-Level (2)

p8tT = 0.075

(all 𝜏𝑇>0.25 within flow limit)

cont. data from PRD92,116003]

✤ Lattice effects in GF can be seen✤ Data points under flow move in “correct” way✤ Continuum-extrapolation at fixed flow time is anticipated

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12

GTT under flow (643x16)

✤ Flow effects can also be seen in energy-momentum tensor correlators✤ Need more statistics in shear channel✤ Understand the behavior of correlators within & beyond the flow time limit

with one-loop Suzuki coefficients[Hiroshi Suzuki, PTEP 2013 (2013) 083B03]

bulkshear

1e-09

1e-08

1e-07

1e-06

1e-05

1e-04

1e-03

1e-02

1e-01

0 0.05 0.1 0.15 0.2 0.25

G(⌧, t)

p8tT

⌧/a

quenched 64

3 ⇥ 16,� = 6.873, 1.5TC , p = 0, #conf=12000, shear

0

1

2

3

4

5

6

7

8

1e-07

1e-06

1e-05

1e-04

1e-03

1e-02

1e-01

1e+00

0 0.1 0.2 0.3 0.4 0.5

G(⌧, t)

p8tT

⌧/a

quenched 64

3 ⇥ 16,� = 6.873, 1.5TC , p = 0, #conf=12000, bulk

0

1

2

3

4

5

6

7

8

13

Summary & Outlook

‣ GEE and GTT are measured at 1.5Tc on 3 different lattices under GF ‣ Good signal for GEE under GF, comparable to those from ML algorithm ‣ Need more statistics for GTT ‣ Confident in GF for dynamic quarks

✴ Move on to finer & larger lattices and different temperatures ✴ Perform continuum & t → 0 extrapolation for the correlators ✴ Extract spectral functions from correlators and estimate 𝜂, 𝜁, 𝜅 (and 𝛾)

accordingly (consider perturbative constraints) ✴ Extend to full QCD using large and fine 2+1-flavor HISQ lattices

Thanks!

14

15

Backup: perturbative flow time limit

[S. Eller and G. Moore, PRD97 (2018) 11, 114507]

0.0 0.1 0.2 0.3 0.4 0.5

x0/�

�0.2

0.0

0.2

0.4

0.6

0.8

1.0

1.2

R d3 x(h

Tij(x

,⌧F)T

ij(0

,⌧F)i

��

1/3h

Tll(x

,⌧F)T

kk(0

,⌧F)i

�)

R d3x(h

Tij(x

,0.0

0000

1)T

ij(0

,0.0

0000

1)i �

�1/

3hT

ll(x

,0.0

0000

1)T

kk(0

,0.0

0000

1)i �)

Scaled Shear Channel Two-Point Function of Flowed EMT

⌧F = 0.005

⌧F = 0.015

⌧F = 0.03

⌧F = 0.05

⌧F = 0.1

⌧F = 0.2

⌧F = 0.000001

Gshear(t > 0)

Gshear(t ⇡ 0)

⌧T