16/8/22 @ 熱場2016

Thermodynamics in (2+1)-flavor QCD
by the gradient flow method
Kazuyuki Kanaya

Center for Integrated Research in Fundamental Science and Engineering (CiRfSE)
Univ. Tsukuba

in collaboration withS.Ejiri, R.Iwami, M.Kitazawa, H.Suzuki, Y.Taniguchi, T.Umeda, N.Wakabayashi(WHOT-QCD Collaboration)

QCD Thermodynamics with Gradient Flow
Gradient flow

Lüscher(2009–), Narayanan-Neuberger(2006)

Imaginary evolution of the system into a fictitious "time" t preserving gauge sym. etc.:(ex) pure gauge theory

original gauge field

We may view the flowed field Bµ as a smeared Aµ over aphysical range of √(8t).
It was shown that operators of flowed fields have no UV divergences norshort-dist. singularities at t > 0.
Lüscher-Weisz(2011)
GF provides us with a new physical (i.e. non-perturbative) renormalization
scheme, which is directly calculable on the lattice in the a → 0 limit.
This opened many possibilities to drastically simplify lattice evaluation ofphysical observables.

Our project: Application of GF to thermodyn. of (2+1)-flavor QCD
EMT / EOS
H. Suzuki (2013), Makino-Suzuki (2014)
Chiral condensate
Hieda-Suzuki (2016)
Topological charge / susceptibility, ...

Iwasaki gauge + NP clover at a ≈ 0.07fm, mPS/mV ≈ 0.63

Main Results

physical point 14 12 10 9876T/Tpc ≈ 0.92 -- 2.44
with the fixed-scale
approach.
(β=1.90)mπ/mρ~0.6

Energy-momentum tensor and EOS
1230

gradient flow
T-integration10

Nt=16 14 126(β=2.05)

T[MeV]

100

200

gradient flow
T-integration

258105300

400500Tpc

206(e+p)/T4(e-3p)/T484201510-25-400100

200300 400
T (MeV)

500

600

0100200300 400
T (MeV)

500600Good agreement with theconventional method at T ≤300 MeV (Nt≥10), suggestingsmall-Nt artifacts at Nt < 10.

03e-05u quarks quark-0.1

chiral susceptibility

subtracted chiral condensate

Chiral condensate / disconnected susceptibility

VEV-subtracted

-0.2
-0.3
-0.4
-0.5
-0.6

u quarks quark

2.5e-05

Disconnectedchiralsusceptibility

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

0100200300 400
T (MeV)

50060000100 200 300 400 500 600
T (MeV)

Clear signal of Tpc at≈190 MeV, with Wilsontype quarks! Peak higher
with lighter quark.

Iwasaki gauge + NP clover at a ≈ 0.07fm, mPS/mV ≈ 0.63

Main Results

physical point 14 12 10 9876T/Tpc ≈ 0.92 -- 2.44
with the fixed-scale
approach.
(β=1.90)mπ/mρ~0.6

Topological susceptibility
We test two alternative definitions:

Nt=16 14 128106(β=2.05)
100

5T[MeV]
200300400500Lüscher(’10), Consonni-Engel-Giusti(’15) Tpc

Guisti-Rossi-Testa(’04)
using chiral W-T identities

GeV4gauge(T/Tc)-7.2(9)ud quark(T/Tc)-7.4(1.6)

χT

1×10-3Gauge definition and fermion definition
agree well with each other at T ≤ 1.5,even with Wilson-type quarks!

1×10-4

1×10-5

0

0.5

1

1.5 2
T/Tc

2.5

3

3.5

Flow for t < t1/2 insufficient for gauge definition, giving upper bounds.
Small-Nt error severe for Nt≤8 (from our study of EMT).
At T/Tpc > 2, Q is freezed to 0. Need a different method, e.g. Frison etal, arXiv:1606.07175.

Power-low behavior of gauge and fermion
definitions consistent with the diluteinstant gas approximation (DIGA) which
predicts the exponent to be –8.

Thermodynamics in (2+1)-flavor QCD
by the gradient flow method
WHOT-QCD Collaboration: Kazuyuki Kanaya*, Shinji Ejiri1, Ryo Iwami1, Masakiyo Kitazawa2,Hiroshi Suzuki3, Yusuke Taniguchi*, Takashi Umeda4, Naoki Wakabayashi1*: U. Tsukuba, 1: Niigata U., 2: Osaka U., 3: Kyushu U. 4: Hiroshima U.
a fine lattice with approximately physical squark and heavy u and d quarks (mPS/mV ≈0.63) in the range T ≈ 174-697 MeV.

As our first study with dynamical quarks,
we reuse configurations generated in ourprevious study of EOS in (2+1)-flavor QCD on

We found that GF with Suzuki’s strategy
works quite well for all of these observables.

4. Energy-momentum tensor and EOS
GF enables us to determine EMT andEOS without additional inputs such as
the beta-function.

We may view the flowed field Bµ as asmeared Aµ over a physical range √(8t).
It was shown that operators in terms of Bµ have no UVdivergences nor short-dist. singularities at t > 0. Lüscher-Weisz(’11)

Clear linear windows found for t → 0at Nt ≥ 6.
T ≈ 199 MeV

42020-430 01525

Nt = 1425100 flow
200 300 400
gradient
T-integration T (MeV)

500

600

500

600

2000.5

11.5(e+p)/T4022t/a

Good agreement with the conventional
method at T ≤ 300 MeV (Nt≥10),
suggesting small-Nt artifacts at Nt < 10.

1510
5001005. Chiral condensate

200300 400
T (MeV)GF works well also for the chiral cond., which has been difficult
with Wilson-type quarks due to explicit chiral violation.
0subtracted chiral condensate

This avoids difficulties in LQCD to define observable related toa symmetry which is violated on the lattice.
We apply the strategy to 2+1 flavor QCD at finite temperature:* Energy-momentum tensor and EOS
Makino-Suzuki, PTEP 2014, 063B02 (2014)* Chiral condensate and its susceptibility* Topological susceptibility

3e-05

u quarks quark-0.1

VEV-subtracted

-0.2
-0.3
-0.4
-0.5
-0.6

u quarks quark

2.5e-05

Disconnectedchiralsusceptibility

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

0100200300 400
T (MeV)

50060000100 200 300 400 500 600
T (MeV)

Clear signal of Tpc at ≈190 MeV. Peak higher with lighter quark.

Hieda-Suzuki, arXiv:1606.04193 (2016)

6. Topological susceptibility

which are otherwise difficult to calculate on the lattice due toexplicit violation of relevant symmetry (continuous coordinatetransformation, chiral symmetry, etc.).

We test two alternative definitions:
Lüscher(’10), Consonni-Engel-Giusti(’15)

3. Simulation parameters

Guisti-Rossi-Testa(’04)
using chiral W-T identities
GeV4Gauge definition and fermion
definition agree well with eachother at T ≤ 1.5, even withWilson-type quarks!

gauge(T/Tc)-7.2(9)ud quark(T/Tc)-7.4(1.6)

1×10-3χT

Nf=2+1 QCD, Iwasaki gauge + NP-clover // fine lattice, physical s &heavy ud
CP-PACS+JLQCD's T = 0 config. (ß = 2.05, 283x56, a ≈ 0.07fm, mPS/mV ≈ 0.63) available on ILDG/JLDG
WHOT-QCD's T > 0 fixed-scale config.set (323xNt, Nt = 4, 6, 8, 10, 12,14, 16)
WHOT-QCD, Phys.Rev.D85, 094508 (’12)
gauge measurements at every config..
EOS by T-integration method
quark measurements every 5(10) config's
continuum extrapolation => next step study

1×10-4

20

43p/T
4
ε/T
4(ε−3p)/T

15

1×10

10

T[MeV]
200

300

400

500

600

-5

00.5

1

1.5 2
T/Tc

2.5

3

3.5

Flow for t < t1/2 insufficient for gauge definition, giving upper bounds.
Small-Nt error severe for Nt≤8 (from our study of EMT).
At T/Tpc > 2, Q is freezed to 0. Need a different method, e.g. Frison et al,arXiv:1606.07175.

5700

To avoid oversmearing. wrapping aroundthe lattice,√(8t/a2) ≤ min(Ns/2, Nt/2)
Tpc ≈ 190 MeV assumed

6-2

30

5

The strategy of H. Suzuki to compute observables with GF:
1) Define observable in a continuum regularization.
2) Relate it with lattice operator thru finite quantities at t > 0.
3) Remove unwanted contributions at t>0 by taking t → 0.

0100

810Suzuki (’12–)

gradient flow
T-integration

1035=> GF defines a physical (non-pert.) renormalization
scheme, which can be calculated directly on the lattice.
This opened many possibilities to drastically simplify
lattice evaluation of renormalized observables.

2. QCD thermodynamics with GF

12(e-3p)/T4Lüscher(’09–), Narayanan-Neuberger(’06)

Imaginary evolution of the system into a fictitious "time" t.

chiral susceptibility

1. Gradient flow

the relevant symmetry on the lattice. To
calculate them, we adopt the strategy of H.
Suzuki using the gradient (GF) flow method.

(e+p)/T4

We study the energy momentum tensor(EMT), chiral condensate, and topologicalsusceptibility in (2+1)-flavor QCD withimproved Wilson quarks. These quantities
have been not straightforward to be studied
on the lattice due to the explicit violation ofi.e., t/a2 ≤ t1/2 = [min(Ns/2, Nt/2)]2 /√8Power-low behavior of gauge
and fermion definitions
consistent with the diluteinstant gas approximation(DIGA) which predicts theexponent to be –8.

7. Discussions
GF opens us a big possibility to accelerate the study of QCD thermodynamics.
Though a definite conclusion possible only after cont. extrapolation, our niceresults at Nt ≥10 suggests that our lattices are sufficiently fine, while small-Nt
artifact visible at Nt < 10.
Further studies called.

please visit our poster
for details

SUMMARY➤

We apply gradient flow ideas to investigate thermodynamics of (2+1)-flavor
QCD. As the first test, we choose heavy ud quarks with physical s quark, on afine lattice (a ≈ 0.07fm, mPS/mV ≈ 0.63), and adopt the fixed-scale approach.

➤

EOS agrees with conventional T-integration method at T≤300 MeV (Nt ≥10).

➤Chiral condensate and its disconnected susceptibility also calculated. Even withthe explicit chiral violation of Wilson-type quarks, we obtain reasonable results,➤

Topological susceptibility by gauge definition and by fermion definition beautifully
agree with each other, and reproduce the T-dep. of DIGA.

➤But, note that our mπ ~ 400 MeV and finite a. A definite conclusion possible onlyafter cont. extrapolation.

➤Our good results at Nt ≥10, however, suggests that our lattices are sufficiently
fine, while small-Nt artifacts visible at Nt < 10.

➤

Further study needed to complete the cont. extrapolation, and at lighter mud.