10th Symposium

学際大規模情報基盤共同利用・共同研究拠点 萌芽型共同研究 採択課題

EX18803 (九州大学情報基盤研究開発センター推薦課題)

鈴木博 (九州大学 理学研究院)

グラディエント・フローによる量子色力学の状態方程式

1. Gradient Flow (GF) (Narayanan-Neuberger (’06), Lüscher (’09~)) is
the evolu1on of the gauge ﬁeld Aμ along a ﬁc11ous 1me t≥0:(d/dt)B=DνGνμ, Bμ|t=0=Aμ.
This may be regarded as the smearing of Aμ over the region |x|=√(8t).
Any local product of the ﬂowed ﬁelds is a renormalized ﬁnite
operator (Lüscher-Weisz (’11)).

03.5e-05

u quarks quark-0.04-0.06-0.08-0.1

u quarks quark

3e-05
chiral susceptibility-0.02

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

0100

200300 400
T (MeV)

500

0600

0100 200 300 400 500 600
T (MeV)

Crossover around Tpc≈190MeV.
Peak higher with decreasing mq as expected.
Physically expected results even with the Wilson-type quarks.
l Topological suscep1bility

1×10

χt

To understand the state of maUer at very high temperature/density
such as in the early universe, the inside of the neutron star, and theheavy ion collision etc., it is crucial to know the Equa1on of State(EoS) in the Quantum Chromo Dynamics (QCD), the fundamental
theory of the strong interac1on. The ul1mate goal of our project is tocompute the EoS by employing the laZce QCD on the basis of theWilson-type quark ac1on. For this aim, we are now employing theenergy-momentum tensor (EMT) deﬁned through the gradient ﬂow.
In this poster, we summarize achievements we made so far to set up
the star1ng point in this JHPCN exploratory project.

l Chiral condensate/disconnected suscep1bility:
subtracted chiral condensate

WHOT-QCD Collabora1on: Hiroshi Suzuki*, Shinji Ejiri1, Ryo Iwami1,Kazuyuki Kanaya2, Masakiyo Kitazawa3, Takanori Shimojyo*, Yusuke
Taniguchi2, Takashi Umeda4 (*Kyushu U., 1Niigata U., 2U. Tsukuba,
3Osaka U., 4Hiroshima U.)

gluonic(T/Tpc)-7.2(0.9)
fermionic(T/Tpc)-7.3(1.7)
DIGA-3

1×10-4

2. The above property of GF can be used to construct a universal
expression for EMT: H.S., PTEP 2013, 083B03 (’13).
(1) EMT with the dimensional regulariza1on sa1sﬁes Ward-Takahashi
rela1ons associated with the transla1onal invariance.
(2) We can construct an operator of ﬂowed ﬁelds that coincides withthe above EMT for t->0. This expression is universal.
(3) This provides a method to deﬁne EMT on the laZce. This method
was shown to work quite well in the quenched QCD: FlowQCD,
PRD90, 011501 (’14), PRD94, 114512 (’16).
(4) We apply this method to Nf=2+1 QCD. Formula1on: Makino-H.S.,
PTEP 2014, 063B02 (’14). This enable us to determine EoS
without addi1onal input as the beta func1on.
(5) Similar idea can be applied to scalar/current operators: HiedaH.S., Mod. Phys. LeU. A31, 1650214 (’16) , and the topologicalcharge, etc.

Gluonic and fermionic deﬁni1ons of χt agree even with the Wilsontype quarks.
Power-low behavior Is consistent with the dilute instanton gas
approxima1on (DIGA) which predicts the exponent -8.

3. QCD with heavy ud quarks: WHOT-QCD, PRD96, 014509 (’17);
D95, 054502 (’17) .

l EoS (preliminary):

gradient flow
T-integration

108(e-3p)/T(e+p)/T44201510420-20-40100200300 400
T (MeV)

500

600

22.5

Nf=2+1 QCD, Iwasaki gauge ac1on+NP improved clover ac1on. a≈0.09fm,
physical point
PACS-CS T=0 conﬁgura1ons (β=1.90, 323×64). WHOT-QCD T>0 ﬁxed-scaleconﬁgura1ons (323×Nt, Nt=4, 5, …, 14).

3540linear fit

30linear fit

3530(e-3p)/T4252015252015
1050050 100 150 200 250 300 350 400
T (MeV)-50100200300 400
T (MeV)

500

600

Similar to the heavy quark case.
The method seems to be working. e+p≈staggered, bute-3p≈3×staggered with large errors.

651.5

4. QCD at physical point

0121T/Tpc

5gradient flow
T-integration

250.510l EoS:
300(e+p)/T4

Nf=2+1 QCD, Iwasaki gauge ac1on+NP improved clover ac1on. Fine laZce(a≈0.07fm), but ud quarks are heavy (mPS/mV≈0.63) , physical s quark mass
CP-PACS+JLQCD T=0 conﬁgura1ons (β=2.05, 283×56). WHOT-QCD T>0
ﬁxed-scale conﬁgura1ons (323×Nt, Nt=4 ,6, …, 16 ) WHOT-QCD, PRD85,094508 (’12).

1×10-50100

200300 400
T (MeV)

500

600

EoS by GF well agrees with the conven1onal method for T≤300MeV(Nt≥10).
Suggests a≈0.07fm is rather close to the con1nuum.
Disagreement at T≥350MeV may be aUributed to O((aT)2=1/Nt2)
ar1facts at Nt≤85. Within the scope of this JHPCN exploratory study, we are planning,p Heavy ud quark with a coarse laZce a≈0.1fm. Conﬁgura1on
genera1on and the thermodynamic measurement (in progress).
p Physical point, a≈0.09fm with Nt=16 and 15. Conﬁgura1on
genera1on and the thermodynamic measurement.
p Heavy ud quark with a ﬁner laZce a≈0.05fm. Thermaliza1on.
p We want to obtain solid results which can be con1nued to a fullscale JHPCN project.

学際大規模情報基盤共同利用・共同研究拠点 第10回シンポジウム
2018年 7月12日, 13日