2020/6/15 @ TCHoU

グラジエントフローによる
QGP熱力学
Thermodynamics of QGP
with the gradient flow
— Application of the SFtX method to lattice QCD at finite temperatures —WHOT-QCD Collaboration:
K. Kanaya, Y. Taniguchi, A. Baba, A. Suzuki (Univ. Tsukuba)
S. Ejiri, S. Itagaki, R. Iwami, M. Shirogane, N. Wakabayashi (Niigata Univ.)
M. Kitazawa, A. Kiyohara (Osaka Univ.)
T. Umeda (Hiroshima Univ.)
H. Suzuki (Kyushu Univ.)
Tomonaga Centerfor the History of the Universe

Gradient Flow
Narayanan-Neuberger (2006), Lüscher (2010-)

Gradient Flow
(example) Yang-Mills theory in the continuum

Bµ(t,x)

Original theory: gauge field Aµ(x) in D=4 dim. space-time,
SYM [Aµ ] =<latexit sha1_base64="Xb+SJBMLuKD6YQK7tJfnfCwTSI4=">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</latexit>
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1
2g02

Z

1
dD x tr[Fµ⌫ Fµ⌫ ] = 2
2g0

Z

t

a
a
dD x Fµ⌫Fµ⌫Introduce a fictitious "time" t, and evolve ("flow") the field Aµ by
with.

This is a kind of diffusion equation.
Its perturbative solution reads

t=0

Aµ(x)

Bµ ~ smeared Aµ over a physical range of √(8t).
("8" = 2 x D with D=4 dim.)def.

Quantum expectation values = path-integration over the original fields Aµ
def.

Flowed operators are free from UV divergences and short-distance singularities.
Lüscher-Weisz (2011)

SFtX method based on GF
H. Suzuki, PTEP 2013, 083B03 (2013) [E: 2015, 079201]

Making use of the finiteness of the GF, H. Suzuki developed a general method to correctly calculate
any renormalized observables non-perturbatively on the lattice.

Small Flow-time eXpansion (SFtX) method
physical obs. we wantcontinuum
t=0
a→0

latticet=0, a>0

GF
t→0

continuum
t>0

t → 0 by linear window

GF

finite,physically well-defineda→0latticet>0, a>0

evaluate their values
non-perturbatively

construct corresponding operators on the lattice

Because we can construct a lattice operator directly from the continuum operator,
this method is applicable also to observables whose base symmetry is broken on the lattice(Poincaré inv., chiral sym., etc.)➯ energy-momentum tensor➯ QCD with Wilson-type quarks, to cope with the problems due to chiral violation.
When we can identify a proper window, we may exchange the order of two extrapolations.

[1]NF = 2+1 QCD
with slightly heavy u,dand ≈physical s quarks
Taniguchi-Ejiri-Iwami-KK-Kitazawa-Suzuki-Umeda-Wakabayashi, Phys.Rev. D 96, 014509 (2017)
Taniguchi-KK-Suzuki-Umeda, Phys.Rev. D 95, 054502 (2017)

test of SFtX with dynamical quarks
Taniguchi-Ejiri-Iwami-KK-Kitazawa-Suzuki-Umeda-Wakabayashi, Phys.Rev. D 96, 014509 (2017)
As the 1st step with dynamical quarks:➤

Heavy ud quarks (m𝝅/m𝝆 ≈0.63) with ≈physical s quark (m𝜼ss/m𝝓 ≈0.74).

Fine lattice (a≈0.07fm with improved action) using the fixed-scale approach.
➤ Compare with EoS by the conventional T-integration method.
➤

WHOT-QCD Collab., Phys.Rev. D 85, 094508 (2012)20

Nf =2+1 QCD, RG-improved Iwasaki gauge + NP O(a)-improved Wilson quarks
CP-PACS+JLQCD's T = 0 config. (ß = 2.05, 283x56, a ≈ 0.07fm, m𝝅/m𝝆 ≈0.63):
the lightest and the finest among the 3ß x 5mud x 2ms data points available.
T > 0 by fixed-scale approach, WHOT-QCD config.(323xNt, Nt = 4, 6, 8, 10, 12, 14, 16)gauge measurements at every config.
quark measurements every 10 config's, using a noisy estimator method.
continuum
physicalpoint extrapolation
14 12 10 =>9 to do8765(β=1.90)mπ/mρ~0.6

Nt= 16 14 12

8

10

6

(β=2.05)
100

T[MeV]
200

300

Tpc ≈190 MeV

400

500

4

3p/T
4
ε/T
4
(ε−3p)/T
15

10

5

T[MeV]
0
100

200

300

400

500

600

700

energy-momentum tensor
In continuum, EMT is defined as the generator of Poincaré transformation.
for YM theory

source of the gravity
conserved Noether current associated with the Poincaré inv.
a fundamental observable of the theory to extract
EoS (energy, pressute), momentum, shear stress, ...
fluctuation/correlation functions => specific heat, viscosity, ...

0energy
T00 T01
BT10 T11
B@T20 T21
T30 T31

momentum

shear stress

T02
T12
T22
T32

1

T03
T13 C
C
T23 A
T33

pressure

On the lattice, the Poincaré invariance
explicitly
broken.
EMT on isthelattice (Caracciolo
et al. (1989–))
We have to
EMTlattice(Caracciolo
et al.
(1989–))
Underonthe thehypercubic
symmetry,
the operator
reproducing
the correct
EMTandof QCDfor a →coefficients
0 is given byfine-tune the renormalization
mixingof many operators
Under the hypercubic symmetry, the operator reproducing the correct7
to make the current conserved and to getEMT
the correct
density!
of QCD values
for a →of0 en.
is givenby etc. in the continuum limit.

Zi Oiµν (x)|lattice − VEV,
7
i=1!
{Tµν }R (x) =Zi Oiµν (x)|lattice − VEV,

{Tµν }R (x) =Caracciolo et al., NP B309, 612 (1988); Ann.Phys. 197,
119 (1990)
where
i=1
!
!
a
a
a
a
O1µν
(x) ≡
Fµρ (x)Fνρ (x),O2µν (x) ≡ δµνFρσ(x)Fρσ
(x),
where
ρ !
ρ,σ!
a
a
a
a
"
#
O1µν (x) ≡Fµρ←(x)Fνρ (x),O2µν (x) ≡ δµν ←Fρσ(x)Fρσ(x),→→←→O3µν (x) ≡ ψ̄(x)ρ γµ D ν + γν D µ ψ(x), O4µν (x) ≡ δµν ψ̄(x)/ ψ(x),
ρ,σD
" ←
←
→
→
←
→ #
O5µν
(x)(x)≡ δ≡
m0 ψ̄(x)ψ(x),µνψ̄(x)O3µνγµ D ν + γν D µ ψ(x), O4µν (x) ≡ δµν ψ̄(x) D/ ψ(x),and,Lorentzones:O5µν(x) ≡non-covariantδµν m0 ψ̄(x)ψ(x),!
aaO6µνLorentz(x) ≡ δµνFµρ(x)Fµρ
(x),
non-covariant
ones:
allowed by the lattice rotation symmetry => and,ρ!

←→O7µν (x) ≡ δµν ψ̄(x)γµ D µ ψ(x)←→

GF with dynamical quarks
Lüscher, JHEP1304.123(2013)

For the finiteness, the flow action can be different from the original action as far as the gauge-covariance is preserved.
To include quarks (matter fields), Lüscher proposed a simple method, in which the gauge flow is the same as the puregauge case.

gauge flow

the same as the pure YM caseoriginal gauge field at t = 0

quark flow

original quark field at t = 0

only gauge fields involved1) quark flow preserves the gauge and chiral symmetries.
𝜒f has the same gauge and chiral transformation properties as 𝜓f.
2) quark flow is independent of spinor and flavor indices.
3) quark fields need renormalization <= can be handled numerically a la Makino-Suzuki

full QCD EMT by SFtX
Makino-Suzuki, PTEP 2014, 063B02 [E: 2015. 079202]

Measure flowed operators at t ≠ 0:and combine them as
etc.

Physical EMT extracted by t→0 extrapolation.
ci : matching coefficientsto make t→0 smoother by removing known small-t mixings & t-dep. in the continuum
to match the renormalization schemes when the observable is scheme-dependent
perturbation theory applicable to calculate ci at t ≈ 0 in AF theories

In this study, we mainly use 1-loop ci by Makino-Suzuki. We revisit the issue with 2-loop ci later.

EMT with dynamical quarks
T ≈ 199 MeV
Nt = 14

30

20
15

25
15

20
15

25
20
15

10

10

5

5

5

5

0

0

0

0.5

1

1.5

2

0

0.5

0

0.5

1
2

t/a

1.5

2

0

0.5

t/a2

1.5

2

Tpc

25
20
15
10
5
0
-5
-10
-15
-20
0

0.5

1
2

t/a

1

1.5

0

2

0

t/a2

(e-3p)/T4

25
20
15
10
5
0
-5
-10
-15
-20

1

1.5

2

25
20
15
10
5
0
-5
-10
-15
-20
0

0.5

1
2

t/a

0.5

1

1.5

2

1.5

2

2

t/a

(e-3p)/T40(e-3p)/T4

4

30

10

2

(e-3p)/T25

T ≈ 464 MeV
Nt = 6

35

10

t/a

e-3p

30

20

40

T ≈ 279 MeV
Nt = 1035435(e+p)/TT ≈ 174 MeV
Nt = 163040(e+p)/T44035(e+p)/T425(e+p)/T

e+p

40

4

Nf=2+1 EMT with heavy u,d

1.5

2

25
20
15
10
5
0
-5
-10
-15
-20
0

0.5

1
2

t/a

t1/2

a2/t-like behavior at t ≈ 0 visible.
Linear behavior visible below t1/2. (Nt=6 may be marginal.)
a2/t term looks negligible in the "linear windows" => Linear fit using the windows.
At T ≈ 697 MeV (Nt=4), no linear windows found.
Smaller errors for e+p <= no T=0 subtraction required

EMT with dynamical quarks
Taniguchi-Ejiri-Iwami-KK-Kitazawa-Suzuki-Umeda-Wakabayashi, Phys.Rev. D 96, 014509 (2017)

Nf=2+1 EoS with heavy u,d30

12gradient flow
T-integration

25

10
8
(e-3p)/T420(e+p)/T4

gradient flow
T-integration

15
10

6
4
2
0

5

-2

0

-4
0

100

200

300
T (MeV)

400

500

600

0

100

200

300 400
T (MeV)

500

600

EoS by SFtX agrees with conventional method at T≤300 MeV (Nt ≥10).
Suggest a≈0.07fm close to the cont. limit.
Disagreement at T≥350 MeV due to O((aT)2 =1/Nt2) lattice artifact at Nt <~ 8.
[Note that this lattice artifact is independent of a.]chiral condensate / susceptibility
Taniguchi-Ejiri-Iwami-KK-Kitazawa-Suzuki-Umeda-Wakabayashi, Phys.Rev. D 96, 014509 (2017)

Nf=2+1 chiral cond. / disconnected susceptibility(GeV)3(GeV)6

MS scheme at µ=2 GeVCrossover suggested around Tpc≈190 MeV, consistent with previous study.
Peak higher with decreasing mq, as expected.
=> Physically expected results even with Wilson-type quarks!
SFtX powerful to extract physical properties.

topological susceptibility
Taniguchi-KK-Suzuki-Umeda, Phys.Rev. D 95, 054502 (2017)(GeV)4gluonic(T/Tpc)-7.2(0.9)
fermionic-7.3(1.7)(T/Tpc)DIGA

1×10-3χt.56(14)x10-3.60(34)x10-3

1×10-4

1×10-5
0

0.5

1

1.5

2

2.5

T/Tpc

Two definitions agree well => SFtX enables us reliable predictions.
Power low consistent with a prediction of Dilute Instanton Gas model.

[1A]
Issue of renormalization-scalein NF = 2+1 QCD
with slightly heavy u,dTaniguchi-Ejiri-KK-Kitazawa-Suzuki-Umeda, arXiv: 2005.00251 (2020)

Also, the parameter g denotes the running gauge coupling in the MS scheme at therenormalization scale µ. The coeﬃcients of the beta function are given by
11
4
C A − TF ,
3
3
!
"
34 2
20
β1 = CA − 4CF + CA TF .
3
3

renormalization scale µβ0 =(1.30)(1.31)

Then the coeﬃcients are explicitly given by,1• matchingcoefficients of the SFtX methodč (t)
1

1

!

g2

#

7

3

c1(t) = g2 1 + (4π)2 −β0 L(µ, t) − 3 CA + 2 TF

GF

$

%
!
"
g4
14482
16546
1187
2
+
−β1 L(µ, t) + CA−−
ln 2 +
ln 3
4
(4π)
405
135
10
#
! "
$
59
1
10873 73 2 2773
302
+ C A TF
Li2
+
+ π −
ln 2 +
ln 3
9
4
810
54
135
45
#
! "
$&"
256
1
2587 7 2 106
161+ C F TF −
Li2
+
− π −
ln 2 −
ln 3
,
9
4
108
9
9
18

ci
µ

MS
2GeV

(1.32) with
etc.

Harlander-Kluth-Lange, EPJC 78:944 (2018)č2 (t)!
#$1 are calculatedg2
25
ci at small
t
in
terms
of
the MS-bar running coupling g(µ) and mass m(µ).
= 2 −1 +
β
L(µ,
t)
+
C
−
3T
0
A
F
4g
(4π)2
6
The MS-bar
renorm.
scaleµ
isfree to choose, as%!
" far as the perturbative expansions are OK.
4g56713 1187
16546
2
+ should
β1 L(µ,bet) +indep.
CA−µ.
ln 3 +
ln 2
Final results
of
4
(4π)
1620
10
135
#

! "
$
59
1
6071 73 2 2287
361
+ CA TF − Li2−− π +ln 2 −
ln 3
9
4
405
54
135
90
#
! "
$&"
d
220
1
1757 10 2 164
247+ C F TF
Li2
−
+ π −
ln 2 +
ln 3
,
9
4
54
9
9
9

A conventional choice is μ(t) = μ (t) ≡HKL suggested
1

1

8t

, a natural scale of flowed operators.
(1.33)

which makes L(µ,t) = 0 and suppresses NNLO in a similar level as µd.

Li2 (z) denotes the dilogarithm function.

Practically µ0(t) ≈ 1.5 µd(t)4=> µ0 more perturbativeextends the perturbative region towards larger t[ A larger µ(t) is even more perturbative, but a huge L(µ,t) breaks the perturbative expansion. ]EoS with heavy u,dTaniguchi-Ejiri-KK-Kitazawa-Suzuki-Umeda, arXiv: 2005.00251 (2020)

entropy density (e+p)/T4

Results with the µ0-scaleµ0 and µd results consistent with each otherEoS with heavy u,dTaniguchi-Ejiri-KK-Kitazawa-Suzuki-Umeda, arXiv: 2005.00251 (2020)

trace anomaly (e-3p)/T4

Results with the µ0-scaleµ0 and µd results consistent with each otherchiral condensate with heavy u,dTaniguchi-Ejiri-KK-Kitazawa-Suzuki-Umeda, arXiv: 2005.00251 (2020)

ud- and s-chiral cond. (VEV-subtracted)

Results with the µ0-scaleµ0 and µd results consistent with each otherµ0 improves linear behavior at large t => more reliable linear extrapolations

chiral susceptibility with heavy u,dTaniguchi-Ejiri-KK-Kitazawa-Suzuki-Umeda, arXiv: 2005.00251 (2020)

ud- and s-chiral suscept. (disconnected)

Results with the µ0-scaleµ0 and µd results consistent with each otherµ0 improves linear behavior at large t=> µ0 extend the reliability/applicability of the SFtX method => helps the phys. pt. study[2]NF = 2+1 QCD
with physical u,d,s quarks
KK-Baba-SuzukiA-Ejiri-Kitazawa-SuzukiH-Taniguchi-Umeda, PoS Lattice2019, 088 (2020)

(2+1)-flavor phys.pt. QCD

WHOT-QCD, EPJ Conf. 175, 07023 (2018)+ New data at T≈122–146 MeV (prelim.)

RG-improved Iwasaki gauge + NP O(a)-improved Wilson quarks
T=0 configs. of PACS-CS (ß=1.9, 323x64, a≈0.09fm) [Phys.Rev.D79, 034503 (2009)] 80 configs.
All quarks fine-tuned to the phys.pt. by reweighting [Phys.Rev.D81, 074503 (2010)] using mπ, mK, m𝝮 inputs.
T>0 by fixed-scale approach, (323xNt, Nt = 4, 5, ... , 18): T≈122 – 549 MeV.

Odd Nt too, to have a finer T-resolution.
TABLE Ks=0.13663377
II. Parameters for the numerical
Generated directly at the phys.pt. w/o reweighting [ß=1.9, Kud=0.13779625,]. simulation: Temperature in MeV, T /190 MeV, the temporal lattice size Nt , t1/2 defined by Eq. (??), and the numbused in gauge and fermion measurements. Spatial box size is 323 .
T [MeV] T /Tpc Nt

physical point

14 12

10 9

8

7

6

0

5

mπ/mρ~0.6(β=2.05)100

16 14 128

10

Tpc ≈190 MeV
200

300

400

Expect Tpcphys < 190 MeV.

64

gauge confs. fermion confs.

32

80

122

18 10.125

129

17 9.03125

6

137

16

146

T[MeV](β=1.90)

on-going

0

t1/2

500

Where is Tpc for physical mq?
Lattice is slightly coarser than the heavy QCD case (a≈0.07fm).
Expect a-indep. lattice artifacts of O((aT)2=1/Nt2) at Nt ≤ 8 (T≥274 MeV)

8

o

308

g
n

oi

ng

80

308

239

239

15 7.03125

143

143

157

14 6.125

650

65

169

13 5.28125

550

55

183

12

610

61

199

11 3.78125

890

89

219

10 3.125

690

69

244

9 2.53125

780

78

274

8

2

680

68

313

7 1.53125

220

22

366

6

1.125

280

280

439

5 0.78125

130

130

548

4

70

70

4.5

0.5

renormalization scale µ
Lattice at a≈0.09fm is slightly coarser than the heavy QCD case (a≈0.07fm).
=> Perturbative behavior worse --- µ0 may help.

30

2

T=219 MeV (Nt=10) (chi /dof=0.01)
1 loop µ0
1 loop µd

2520(e+p)/T4

ud quark chiral susceptibilityg(µ(t)) becomes large at t/a2≈1.5 with µd(t), but remains small up to ≈3 with µ0(t).

15
10
5
0
-5
0

0.5

1

1.5
2

t/a

2

2.5x10-6

2

T=137 MeV (Nt=16) (chi /dof=0.1)-62x10 1 loop µ0
1 loop µd
1.5x10-6
1x10-6
5x10-7
0
0

0.5

1
t/a2

µ0 and µd results consistent with each otherµ0 improves linear behavior at large t=> µ0 extend the reliability/applicability of the SFtX method

1.5

2EoS at the physical point
entropy density
30

30

2

T=122 MeV (Nt=18) (chi /dof=0.01)
1 loop µ0
1 loop µd

25

15
10

20
15
10

5

5

0

0

-5

-5
0

0.5

1

1.5

T=137 MeV (Nt=16) (chi2/dof=0.01)
1 loop µ0
1 loop µd

25(e+p)/T4(e+p)/T

4

20

1-loopµ0-scale
Preliminary(e+p)/T4

2

O((aT)2) lattice artifacts at Nt ≤ 8
0

0.5

t/a2

15
10

35

30

T=219 MeV (Nt=10) (chi2/dof=0.01)
1 loop µ0
1 loop µd

2520(e+p)/T4(e+p)/T

4

20

2

t/a2

T=169 MeV (Nt=13) (chi2/dof=0.01)
1 loop µ0
1 loop µd

25

1.5

25

15
10

20
15

5

5

10

0

0

5

-5

-5
0

0.5

1
t/a2

1.5

2

0

0.5

1
t/a2

1.5

chi2/dof=0.01
1 loop µ030(e+p)/T4

30

1

2

0
0

100

200

300

400

500

600

T (MeV)µ0 and µd results consistent with each other
Results with the µ0-scaleµ0 improves linear behavior at large t=> µ0 extend the reliability/applicability of the SFtX method

chi2/dof=0.01
1 loop µ030(e+p)/T425
20

11(2010)077

35
Figure 11. The energy density normalized by T 4 as a function of the temperature on Nt = 6, 8
and 10 lattices. The Stefan-Boltzmann limit ϵS B = 3pS B is indicated by an arrow.

15
10
5
0
0

100

200

300

400

500

Borsany et al., JHEP 1011, 077 (2010), with KS(stout).

600

T (MeV)

Figure 12. The entropy density normalized by T 3 as a function of the temperature on Nt = 6, 8
and 10 lattices. The Stefan-Boltzmann limit sS B = 4pS B /T is indicated by an arrow.

30

chi2/dof=0.01
1 loop µ0 – 18 –25
(e-3p)/T4

20
15
10
5
the
the

trace
anomalynormalized
nf
=
2
+
1
parameters
from

by
T
table

4

together
2.

with

the

-5
0

100

200

300

400

500

600

JHEP1 (2010)07

0
Figure15.
Continuum
estimateforusingparametrizationofequation(3.1)

T (MeV)
Figure16.
The
normalized
trace
anomalyfor
two
diﬀerent
values
of
the
light
quark
ys
ys
Nt
=
8
lattices:thephysical
mud
=
mph
and
the
three
degenerate
flavor
mud
=
mph
s
ud

–

ud chiral susceptibility

0.0001421–

chi2/dof=0.1
u quark 1 loop µ00.00012

massescase.

onTpcphys < 157 MeV (T≈122-146MeV critical ??)(cf.) Result with 2+1 staggered quarks
156.5 ± 1.5 MeV

0.0001

Bazavov et al. PLB795, 15 (2019), HISQ

8x10-56x10-54x10-52x10-5
0
0

100

200

300

T (MeV)

400

500

600

Need more statistics / more data points at low T's. (on-going)
A definite conclusion possible only after continuum extrapolation.

summarysummary: SFtX method in 2+1 flavor QCD
1. 2+1 flavor QCD with slightly heavy u,d and ≈physical s quarks➤

fine a≈0.07fm lattice with improved Wilson quarks, 323xNt (Nt=4-16): T≈174-697MeV
EoS agrees well with conventional integral method at T≤300 MeV (Nt ≥10), while O((aT)2 =1/Nt2)
lattice artifacts suggested at Nt≤8.
Chiral suscept. show clear peak at Tpc≈190 MeV expected from Polyakov loop etc.
Topological suscepts. by gluonic and fermionic definitions agree well.
µ0-scale extends the reliability/applicability of the SFtX method.
1- and 2-loop matching coefficients lead to consistent results, while EoM gets O((aT)2 =1/Nt2) lattice
artifacts at Nt≤10.

=> SFtX powerful in evaluating physical observables.
A definite conclusion possible only after continuum extrapolation, though our results suggest thata≈0.07fm is fine enough.

➤less fine a≈0.09fm lattice, 323xNt (Nt=4-18): T≈122-549MeV
The µ0-scale helps much.
Tpcphys < 157 MeV (T≈122-146MeV critical ??)Need more statistics / more data points at low T's. => on-going.
Data at larger t/a2 may help. => on-going.
Need continuum extrapolation too. => being started.

ud chiral susceptibility

2. 2+1 flavor QCD with physical u,d,s quarks

0.00014chi2/dof=0.1
u quark 1 loop µ00.000120.00018x10-5
6x10

-5

4x10

-5

2x10

-5

0
0

100

200

300

400

500

600

T (MeV)

preliminaryprospects / to doother observables
EMT correlation functionstransport coefficients of QGP: shear/bulk
viscosity, etc.
test: thermodynamic relations vs. linear
response relations

chiral observables

Shear viscosity

preliminary

matrix elements: BK, etc.

topological observables at the physical point

continuum extrapolation
Slightly heavy ud + ≈phys. s on a less fine lattice (a≈0.097fm), 243xNt (Nt=8-12): T≈170-254MeV
Look similar to the fine lattice caseLinear windows narrower than the fine lattice case. => µ0 will help
a-dep. looks small up to this aneed more statistics + a finer point
PACS10 configurations (T=0) at the physical point
generation of finite temperature configurations => started!