Upload
others
View
6
Download
0
Embed Size (px)
Citation preview
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)
���
�
���
�
���� ��� ���� ��� ���� ���
��
���������������
������� �������� ��� ���������������� �������� ��� ��������������� �������� ��� ���������������� �������� ��� ��������������� �������� ��� ��������
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
���
�
���
�
���� ��� ���� ��� ���� ���
��
���������������
������� �������� ��� ���������������� �������� ��� ���������������� �������� ��� ��������������� �������� ��� ��������������� �������� ��� ��������
��� ���������
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