gradient ﬂowで探る
格子QCDのエネルギー運動量テンソル
Yusuke Taniguchi
for
WHOT 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)

Our motivation
Energy momentum tensor

Tµ⌫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
pressure5(e+p)/T4

energy
0T00
BT10B
@T20
T30

Poincare symmetry

00100

200

300 400
T (MeV)

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

Our motivation
Energy momentum tensor
energy
0T00
BT10B
@T20
T30

Poincare symmetryTµ⌫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?

Color Superconductivity

Nucleus Neutronstar

Density

hot topics in QGP•Fluctuations and correlations of Tµ⌫
speciﬁc heat, viscosity, …

Quark-Gluon-Plasma

How to calculate Tμν on lattice?
Measure operators on latticeterms in QCD Lagrangian!
aa¯
F(x)F(x)/ (x)µ⌫ ⇢µ⌫ (x) D⇢¯µ⌫ (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 Poincare symmetry)•depends on: lattice action, operator
additive correction for µ⌫ ¯(x) (x)How to calculate Tμν on lattice?
Easier method for renormalization?

Gradient Flow
Flow the gauge ﬁeld@t Aµ (t, x) =SYMAµNarayanan-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 ﬁeldpwithin 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.

itnﬁereaa
Fµ⌫ Fµ⌫ (x, t)

gradient ﬂow

lattice operator

Tr＜1ー＞

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

MS scheme Matching coeﬃcients are calculable perturbatively

Mechanism of gradient ﬂow
gauge ﬂow@t Bµ (t, x) = D⌫ G⌫µ (t, x)

Bµ (t = 0, x) = Aµ (x)Formal solution✓◆ZZ t
Bµ (t, x) = d4 y Kt (x, y)Aµ (y) +dsKt s (x, y)Rµ (s, y)0(x y)2Z4teip(x y)tp2
Kt (x, y) = ee=2(4⇡t)p2

3Rµ (s, y) = bB (s, y) + cB (s, y)fermion ﬂow@t (t, x) = Dµ Dµ (t, x)(t, x) =Z ✓yKt (x, y) (y) +Z t0(t = 0, x) = (x)dsKt s (x, y) 2Bµ (s, y)@µ + Bµ2 (s, y)✓Z

Ks (y, z) (z) + (s, y)z◆◆Mechanism of gradient ﬂow
Graphical representation of gradient ﬂowt

Bμ(t,x)Bμ(t,x)s ◯Bμ(t,x)Bμ(t,x)◯◯◯Aμ(x) AμAν✓◆ZZ t
Bµ (t, x) = d4 y Kt (x, y)Aµ (y) +dsKt s (x, y)Rµ (s, y)0◯◯

no divergenceno loop◯Mechanism of gradient ﬂow
Contribution to a bilinear operatorBµ (t, x)B⌫ (t, x)(x y)2Z¯(t, x) (t, x)K (x, y) = eip(x y) e tp2 = e 4tttpe(4⇡t)2tp2

No UV divergenceAμ¯
t=0Aν4D propagator

4D space-time where QCD in on.

Mechanism of gradient ﬂow
Contribution to a bilinear operatortBµ (t, x)B⌫ (t, x)¯(t, x) (t, x)

No UV divergence at all?
Yes! There is!
eAμ¯Aνtp2Loop is allowedin 4D QCD
div. at t=0 boundary

t=0QCD
Lagrangian

UV div. is renormalized in gauge invariant
dim 4 operator at t=0 boundary

Mechanism of gradient ﬂow
Contribution to a bilinear operatorteBµ (t, x)B⌫ (t, x)¯(t, x) (t, x)

tp2

●sAμ¯
t=0Aνesp2

No UV divergenceMechanism of gradient ﬂow
Contribution to a bilinear operatorteBµ (t, x)B⌫ (t, x)¯(t, x) (t, x)

Careful constructionof ﬂow equation(Suzuki)

tp2

Non trivial!
Aμ¯

s=0 Aν
●UV divergent
div. at t=0 boundary

t=0
UV div. is renormalized in QCD Lagrangian
at t=0 boundary

Mechanism of gradient ﬂow
No div. due to compositenesstWe need to renormalizeAμ¯Aνgauge coupling
quark massquark wave function
gauge wave function

t=0
equivalent to gauge coupling renormalization
aafor gauge invariant operator like Gµ⌫ Gµ⌫
No UV divergence if we adopt

1 a a
L = 2 Fµ⌫ Fµ⌫
4g0

How to calculate Tμν on lattice?
Five steps to calculate Tμν
1. Flow the gauge link and quark ﬁelds
2. Calculate expectation value of ﬂowed operators
3. Multiply the matching coeﬃcients
4. Take a→0 limit
5. Take t→0 limit

H.Suzuki, PTEP 2013, 083B03 (2013)
Makino-Suzuki, PTEP 2014, 063B02 (2014)

two reasons

Solve an operator mixing{Tµ⌫ } (x, t, a) = {Tµ⌫ }WT (x) +t(dim6 operator)

Use of perturbative matching coeﬃcientsCan we extract physics before a→0 ?
Our works and lessons
✓ 2◆
aOt

35

O (t)

T=232 MeV (Nt=12)
linear fit
nonlinear fit
linear+log fit

3025(e+p)/T4WHOT QCD
Collaboration(PRD 2017)(E+P)/T4O t20

21510
5000.51ﬂowed operator on lattice

t/a2

1.5

2

{Tµ⌫ } (x, t, a) = {Tµ⌫ }WT (x) +t(dim6 operator)

thewindow
regiona2+ (dim4 operator) tamed at large t
t+1/log2(t)(dim4) +t2(dim8 operator) tamed at small t
2+O(a T22222, a m , a ⇤QCD )need to take a→0 limit

First topic
1 point function of energy-momentum tensor
Highlights?
Thermodynamical quantity: energy, pressure
01energy T00 T01 T02 T03
BT10 T11 T12 T13 CBC@T20 T21 T22 T23 AT30 T31 T32 T33 pressure✓◆✓◆Entropy density @S@p✏+phT00 + Tii is====@V T@T VTTMaxwellʼs relation

integrable condition of entropy

Comparison with established method4(e+p)/Tt→0 limit by linear extrapolation

4252025

T=174MeV

152025
20
15

1010

105

5

50

0

00.5

1

1.5

2

00.5

35

35

25

20
15
100
00.5

100.5

1.5

235

0

25

0.5

12

1.5

20
15
10T=349MeV
01.5

T=464 MeV (Nt=6)
linear fit
nonlinear fit
linear+log fit

30

15

51t/a22010

T=279MeV

52

T=348 MeV (Nt=8)
linear fit
nonlinear fit
linear+log fit

30(e+p)/T4425

1.5

T=233MeVt/a2

T=279 MeV (Nt=10)
linear fit
nonlinear fit
linear+log fit

30

1(e+p)/T40

T=232 MeV (Nt=12)
linear fit
nonlinear fit
linear+log fit

30

T=199MeV

15t/a2(e+p)/T35

T=199 MeV (Nt=14)
linear fit
nonlinear fit
linear+log fit

30(e+p)/T430(e+p)/T35

T=174 MeV (Nt=16)
linear fit
nonlinear fit
linear+log fit(e+p)/T435

T=465MeV

52000.5

1t/a2t/a2t/a2t/a2t/a2t/a2

1.52

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+p)/T430

gradient ﬂow

25(e-3p)/T412

gradient flow
T-integrationm⇡⇠ 0.6m⇢gradient flow
T-integration

108(e+p)/T4(e-3p)/T420151064205-20-40100200300 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+p)/T435(e-3p)/T440linear fit

353030(e-3p)/T4425(e+p)/Tlinear fit

2015102520151055000100200300 400
T (MeV)

500600-50100200300
T (MeV)

400500600

Second topic
2 point correlation function of ﬂuctuation
Z13Cµ⌫;⇢ (t; x0 ) = 5d x (h Tµ⌫ (t; x0 , ~x) T⇢ (t; 0)i)
T V3ﬂuctuation:Tµ⌫ (t; x) = Tµ⌫ (t; x)hTµ⌫ (t; x)iHighlights?
Conservation law

dC0⌫;⇢ (x0 ) = 0dx0

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

cVC00;00 = 3
TConservation law

ddx0

Z

3d 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;33

C00;00500-50-100

60
40

20C30;30100

lattice artifact is severeh T00 T33 i
h T03 T03 i

0-20

2

4

6time

flow time=0.5
flow time=1.0
flow time=1.5
flow time=2.08100-20

-40
020

-40
0246time

flow time=0.5
flow time=1.0
flow time=1.5
flow time=2.08100246time

flow 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
✏+p
hT00 + Tii i
s====
@V T
@T VTT
integrable 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=0T V3

Nf=2+1 QCD
5hTTi/T020260

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

C00;2240

constant ﬁt middle
3 data points

20

20
0-20
-40

-40
02

46time8

flow time=0.5
flow time=1.0
flow time=1.5
flow time=2.0agrees int→0 limit(e+p)/T4C20;20

40

h T00 T22 i /T 51030
2520
1510
50-5-100

T=232 MeV (Nt=12)

246time810

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

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

Speciﬁc heat
Linear response relation⇣11
1 dhHiH/ThHi = Tr T00 ecV =VZ
V dTZ
13cV = 2d x (h T00 (t; x0 , ~x) T00 (t; 0)i)
T V3
100
C00;00500constant ﬁt middle
3 data points

C00;00h T00 T00 i /T5-50-1000246time

flow time=0.5
flow time=1.0
flow time=1.5
flow time=2.0810cV= 37(17)3T7060504030
20100⌘

T=232 MeV (Nt=12)
linear fit

00.5

11.5
2
linear window
flow time

T=232 MeV (Nt=12)SU(3) Yang-Mills (Quench)
FlowQCD: Kitazawa, Iritani, Asakawa, Hatsuda
a→0 limit done!

Ref[28]: Gavai et al, Phys. Rev.D71(2005) 074013• Ref[16]: Borsanyi et. al., JHEP 1207, 056 (2012)•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

25e/T425

gradient flow
T-integration

1081041564e/T(e-3p)/T420

2p/T410

gradient flow
T-integration

20815

610050-400100

200

300 400
T (MeV)

500

600

425-2

gradient flow
T-integration

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

200

300 400
T (MeV)

500

600

00100

200

300 400
T (MeV)

5006000100200

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;0050-50-10002

46timeflow 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
20
100

T=232 MeV (Nt=12)
linear fit

00.5

1
1.5
flow timecV= 37(17)3T2Future plan
We want to work for viscosity in future.
Shear viscosity

Z 1⇢shear (!)⌘ = lim!!0!

d!K(x0 , !)⇢shear (!) = C12;12 (x0 )01cosh x0 2T!
K(x0 , !) =!
sinh 2T

60

20
0C23;23

C12;12

40

-20
-40

0

2

4

6time

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

8

10

60
50
40
30
20
100-10
-20
02

4

6time

flow time=0.5
flow time=1.0
flow time=1.5
flow time=2.0810Topics dropped from this talk
Chiral condensate
disconnectedchiral susceptibility

0.2(GeV)6

Chiral restoration crossoveru quarks quarkchiral susceptibility

chiral condensate(MeV)3chiral condensate

0.15
0.1
0.05

3.5e-05

u quarks quark

3e-05
2.5e-05
2e-05
1.5e-05
1e-05
5e-06

00100200300 400
T (MeV)

500

600

00

100 200 300 400 500 600
T (MeV)Topics dropped from this talk
Chiral condensate at physical quark mass
disconnectedchiral susceptibility0(GeV)6

Chiral restoration crossoveru quarkchiral susceptibility

subtracted chiral condensate(MeV)3chiral condensate

-0.05
-0.1
-0.15
-0.2

0100200300 400
T (MeV)500600

Very preliminary

0.00025

u quarkheavy mass

0.00020.000150.0001
5e-05
00100

200 300 400
T (MeV)

peak in the previous slide

50060

12hQ i
T =V42hQi
Topological susceptibility
Z
singularity free deﬁnition
14aa2 ⌦⌦↵↵Q=
dx✏
F(x)F(x)
mµ⌫⇢
µ⌫
⇢
200
Q = 2 P P disc64⇡ 2Nf

χT (GeV4)(154(8)MeV)4

1×10

gauge

-3-7.2(0.9)(T/Tc)
ud quark-7.4(1.6)(T/Tc)
DIGA(156(22)MeV)4

1×10-4agrees perfectly!

1×10consistent with
DIGA prediction: ­８-5

one loop PT
0 0.5
same bare mass

1

1.5 2
T/Tc

2.5

3

3.5

aT=1/Nt artifact is severe