Energy-MomentumTensor Correlation
function in Nf=2+1 QCD
at ﬁnite temperature
Yusuke Taniguchi
forWHOT QCD collaboration

S.Ejiri, R.Iwami, K.Kanaya, M.Kitazawa, M.Shirogane,A.Suzuki, H.Suzuki, Y.T, T.Umeda, N.Wakabayashi
PRD 95, 054502 (2017), PRD 96, 014509 (2017)

Energy-Momentum Tensor
Relativistic extension of energy density, momentum
density, energy ﬂow and stress
RHS of Einsteinʼs equation Rµ⌫

Perfect ﬂuid assumption

18⇡GRgµ⌫ = 4 Tµ⌫2cSource of gravityµ ⌫Tµ⌫ = (⇢ + p)u u + g

Simpler in quantum ﬁeld theory
EM tensor = conserved current
Noetherʼs theorem
translation invarianceµ⌫pEnergy-Momentum Tensor
Use for a study of Quark-Gluon Plasma
Very high temperature state of nuclear
Observed in heavy ion collision experiment
Lattice QCD

Energy-Momentum Tensor
Use for a study of Quark-Gluon Plasma
momentum
T01
T11
T21
T31

T02
T12
T22
T32

stress•If we have Tµ⌫

entropy density
130

gradient flow
T-integration

25
T03
20
T13 CC
15
T23 A
10T33
pressure 5(e+p)/T4

energy
0T00
BT10B
@T20
T30

00100

200300 400
T (MeV)

500600direct measurement of thermodynamic quantity
Successful for 1 point function
Todayʼs 1st topic

Energy-Momentum Tensor
Use for a study of Quark-Gluon Plasmaenergy
0T00
BT10B
@T20
T30

momentum
T01
T11
T21
T31

T02
T12
T22
T32

stress•If we have Tµ⌫

From Nuclear Matter to Quark Matter

1Temperature

T03
T13 CC
T23 A
T33
pressure

Early Evolutionof the Universe?

Todayʼs 2nd topic?

Nuclear Matter
Nucleus?

Color Superconductivity
NeutronstarDensityhot topics in QGP•Fluctuations and correlations of Tµ⌫
speciﬁc heat, viscosity, …

Quark-Gluon-Plasma

How to calculate Tμν on lattice?
Diﬃculty: No translation invariance of lattice
Measure expectation value of operators on latticeterms in QCD Lagrangian!
aa¯/ (x)µ⌫ F⇢ (x)F⇢ (x)µ⌫ (x) Dµ⌫¯(x) (x)terms in QCD Lagrangian when trace is taken
⇣ !
⌘!
aa
¯(x) µ D ⌫ + ⌫ D µFµ⇢ (x)F⌫⇢ (x)(x)

Renormalization

Well established for E and P
problems

Karsch coeﬃcientsnon universal (No translation symmetry)•depends on: lattice action, operator

How to calculate Tμν on lattice?
Easier method for renormalization?

Gradient Flow
Flow the gauge ﬁeld
@t Aµ (t, x) =SYM
AµNarayanan-Neuberger(2006)
Lüscher(2009–)

Aµ (t = 0, x) = Aµ (x)2t: ﬂow time, dim=[length ]A kind of diﬀusion equation @t Aµ (t, x) = D⌫ G⌫µZ
4Solution Aµ (t, x) = d yKt (x y)Aµ (y) + interactions
x2 /4teheat kernel Kt (x) =(4⇡t)D/2smear ﬁeldp
within 8t

How to calculate Tμν on lattice?
Easier method for renormalization?

Gradient Flow as a renormalization scheme
Narayanan-Neuberger(2006), Lüscher(2010), Lüscher-Weisz(2011)

Gauge operators with ﬂowed ﬁeld Aµ (t, x)does not have UV divergence
1operators are renormalized scale: µ = p8t
visit t<<1/ΛQCD2 region non-perturbatively
NP renormalized operator
universaln.

itnﬁereaaFµ⌫ Fµ⌫ (x, t)

gradient ﬂowlattice operator

Tr＜1ー＞

H. Suzuki (2016) take a→0 limit safely

MS scheme Matching coeﬃcients are calculable perturbatively

First topic
1 point function of energy-momentum tensor
Highlights?
Thermodynamical quantity: energy, pressure
01energy T00 T01 T02 T03
BT10 T11 T12 T13 C
BC@T20 T21 T22 T23 A
T30 T31 T32 T33 pressure✓◆✓◆
Entropy density @S@p✏+phT00 + Tii is====@V T@T VT
T
Maxwellʼs relation

integrable condition of entropy

Comparison with established method

SU(3) Yang-Mills (Quench)
FlowQCD: Kitazawa, Iritani, Asakawa, Hatsuda, Suzuki
a→0 limit done!

Ref[1]: Boyd et. al., Nucl. Phys. B 469, 419 (1996)• Ref[4]: Borsanyi et. al., JHEP 1207, 056 (2012)•Nf=2+1 QCD
WHOT QCD collaboration
a〜0.07 [fm], heavy ud quark
e/T4

10

gradient flow
T-integration

20

8

15

6p/T4e/T

425

gradient ﬂow

10m⇡⇠ 0.6m⇢p/T4

gradient flow
T-integration

42

5000100

200300 400
T (MeV)

500600

WHOT-QCD, Phys. Rev. D 85, 094508 (2012)
integration method

0100200300 400
T (MeV)

500600aT=1/Nt artifact is severeNf=2+1 QCD
WHOT QCD collaboration
a〜0.090 [fm], physical quark mass
e/T430p/T410nonlinear fit

206e/T

p/T4842515

nonlinear fit

421005-200

100

200

300
T (MeV)

400

500

600

0100

200

300
T (MeV)

400

500

600

Second topic
2 point correlation function of ﬂuctuationZ
1
3Cµ⌫;⇢ (t; x0 ) = 5
d x (h Tµ⌫ (t; x0 , ~x) T⇢ (t; 0)i)
T V3
ﬂuctuation:Tµ⌫ (t; x) = Tµ⌫ (t; x)hTµ⌫ (t; x)iHighlights?
Conservation lawdC0⌫;⇢ (x0 ) = 0dx0

Linear response relations✏+pC0i;0i = C00;ii =T4

cVC00;00 = 3T

Conservation lawddx0Z3d x (h T0⌫ (t; x0 , ~x) T⇢ (t; 0)i) = 0
V3PµSU(3) Yang-Mills (Quench)
FlowQCD: Kitazawa, Iritani, Asakawa, Hatsuda
h T00 T00 i

correlation function is very ﬂat in the middle

Nf=2+1 QCD
a〜0.07 [fm], heavy ud quark WHOT QCD collaboration
T=232 MeV, Nt=12h T00 T00 i

40C00;33C00;00

500-50-10060
40

20C30;30100

lattice artifact is severeh T00 T33 i
h T03 T03 i

0-20

246timeflow time=0.5
flow time=1.0
flow time=1.5
flow time=2.08100-20-40
020-400246timeflow time=0.5
flow time=1.0
flow time=1.5
flow time=2.0

8

100

2

46timeflow time=0.5
flow time=1.0
flow time=1.5
flow time=2.0

correlation function is very ﬂat in the middle

810

Entropy density

compareThermodynamical relation
Maxwellʼs relation✓◆✓◆@S@p✏+phT00 + Tii is====@V T@T VT
Tintegrable condition of entropy
Linear response relation✓◆⇣⌘@p@hTii i
1H/Ts==hTii i = Tr Tii e@T V@T ZZ
113s = 2 h H Tii i = 2d x (h T00 (t; x0 , ~x) Tii (t; 0)i)
TT VZ3@hT01 i13✏+p==d x (h T0i (t; x0 , ~x) T0i (t; 0)i)@v1 ~v=0
T V3

Nf=2+1 QCD
5hT
Ti
/T02
02
60

T=232 MeV (Nt=12)0-20

C00;2240

constant ﬁt middle
3 data points20200-20-40-400

246time8

flow time=0.5
flow time=1.0
flow time=1.5
flow time=2.0

agrees in
t→0 limit(e+p)/T4C20;20

40

h T00 T22 i /T 5

1030
2520
15
10
50-5-100

T=232 MeV (Nt=12)

246time8

10

flow time=0.5
flow time=1.0
flow time=1.5
flow time=2.0

results from
1pt function
00.5

11.5
2
flow time linear windowSummary
Flow method works well for EM tensor!
as powerful as the derivative method.
More suitable for Wilson fermion.
Good agreement with T integration method
12gradient flow
T-integration

25

e/T425

gradient flow
T-integration

1081041564e/T(e-3p)/T4202p/T410

gradient flow
T-integration

20815

610050-400100200300 400
T (MeV)

500

600

425-2

gradient flow
T-integration

430(e-3p)/T4p/T(e+p)/T40100

200

300 400
T (MeV)500

60000100200300 400
T (MeV)

500

600

0100200

Lattice artifact is severe for Nt=4, 6, 8300 400
T (MeV)

50060

Summary
Gradient ﬂow works well for EMT correlation function
We have good results:
Conservation law
Linear response relation0(e+p)/T4

C00;0050-50-1000246time

flow time=0.5
flow time=1.0
flow time=1.5
flow time=2.0

8

1030
25
20
15
1050-5-10

T=232 MeV (Nt=12)

C00;0010000.5

1
1.5
flow time

270
60
50
40
30
20100

T=232 MeV (Nt=12)
linear fit

00.5

1
1.5
flow timecV= 37(17)3T2