厳密 中心対称性 保 Nf = 3 QCD
有限温度相転移 解析
Takumi Iritani
Yukawa Institute for Theoretical Physics, Kyoto University

September 4, 2014
TQFT2014 @ RIKENCollaborators:
E. Itou (KEK)
T. Misumi (Keio Univ.)Confinement and Chiral Symmetry Breaking
QCD shows “TWO” important non-perturbative phenomena

Chiral Sym. Breaking

there are no isolated quarks

SU(Nf )L × SU(Nf )R → SU(Nf )V

Temperature

Confinement

deconfinedchiral symmetricconfinedchiral sym. broken

these properties are lostat high temp. and/or density
or some conditions

Density
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Phase Transition at Finite Temperature from Lattice QCD• simultaneous phase transition ??
• confinement and chiral symmetry breaking are related ??
1.0Order Parameters0.8Δ l,s0.6Continuum
N t =16N t =12N t =10N t =80.4

Renormalized
Polyakov Loop

0.2
0.0

100

150

200

250

300

350

T [MeV]
Figure : Wuppertal-Budapest Coll. ’10 (Fig. from Fukushima-Sasaki ’13)
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Real World is Ambiguousperhaps, there are no clear phase transitions and no critical points in QCD

chiral symmetry

confinement
no definite order parameterNuclear Superfluid

mq ̸= 0 ⇒ not exact symmetry

Chemical Potential μ B

Figure : QCD phase diagram (Fukushima-Sasaki ’13)

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Confinement = Center Symmetry ??
Polyakov Loop L = (1/Nc )Treig

∫ 1/T
0

used as “an order parameter”
A4 dτ⟨L⟩ ∼ e−fq /T
T :temperature, fq :single-quark free energy

confined phase
fq → ∞ ⇒ ⟨L⟩ = 0
deconfined phase
fq < ∞ ⇒ ⟨L⟩ ̸= 0

Polyakov Loop: an order parameter of CENTER SYMMETRYby center transformation
L → zL ̸= Lwith z ∈ ZN
center symmetriccenter broken⇒ ⟨L⟩ = 0⇒ ⟨L⟩ ̸= 0But, center symmetry is “explicitly” broken by quarks
4 / 14QCD and Others
QCD: fundamental theory of strong interaction⇒ SU(3) gauge theory with 6 massive quarks in color 3 rep.
but, there are a large class of gauge theories
diﬀerent gauge group, SU(2), ZN , SO(3), G2 , ...
diﬀerent number of flavor, diﬀerent group rep., Higgs fields, ...

⇒ deeper understanding of “our” QCD
For example.

QCD with adjoint fermions : “exact center symmetry”• mismatch of phase transitionsTχ ≃ 8Td — Karsch-Lütgemeier ’99
G2 -gauge theory : “trivial center”• simultaneous 1st order phase transitionsTχ ≃ Td — Danzer-Gattringer-Maas ’09
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Outline of This Workin order to understand non-perturbative nature of QCD,
we focus on “Z3 -center symmetry”

Z3 CENTER SYMMETRY

by introducing some boundary conditions,
we formulate
Nf = 3 QCD with “exact center symmetry”
from lattice QCD simulation, we study▶
▶

center symmetry phase transitioninterplay with chiral properties“center symmetry” would give insights into confinement and chiralsymmetry breaking, and other gauge theories, such asquenched QCD

QCD with adj. quark

Polyakov loop

Polyakov loop

chiral condensate

chiral condensate

temperaturetemperature

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1Nf = 3 QCD with Exact Center Symmetry

2SummaryNon-Abelian Gauge Theory with Exact Center Symmetry
quark breaks center symmetry of Yang-Mills theory
ZN transformation (z ∈ ZN )
ψf (x, τ + 1/T ) = −zψf (x, τ )BOUNDARY CONDITION“flavor” dependent ZN boundary condition asψf (x, τ + 1/T ) = −z f −1 ψf (x, τ )f1ψ̄f (D/ + m) ψf + Tr Gµν Gµν 2and4 x ∑N -tr

S=Z∫s.

this theory possesses center symmetryby relabeling of flavor index
References: PNJL model analysis
Kouno-Sakai-Makiyama-Tokunaga-Sakai-Yahiro, arXiv:1202.5584.
Kouno-Misumi-Kashiwa-Makiyama-Sasaki-Yahiro, Phys. Rev. D88, 016002 (2013).
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Z3-QCD — exact center symmetry with dynamical quarks
Nf = 1 + 1 + 1 flavor QCD with flavor dependent Z3 boundary conditions

etricmmsy-QZ3-Z3s.

anZsu
3 -trdphpo ysicint alCD

Z3-QCD BOUNDARY CONDITION■ twisted boundary z ∈ Z3 corresponds to imaginary chemical potentialµ/T = 2πi/3
at Roberge-Weiss symmetric point, i.e., ZGC (µ/T = 2πi/3) = ZGC (0)• but flavor symmetry is partially broken as
SU(3)L × SU(3)R → [U (1)L ]2 × [U (1)R ]2

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About Lattice QCD Simulation Setup(1 + 1 + 1)-flavor dynamical quark simulation• Iwasaki gauge action and Wilson fermion
DW (x, y) = δx,y − κ4 {∑(1 − γµ )Uµ (x)δx+µ,y + (1 + γµ )Uµ† (x − µ)δx−µ,y}µ=1with flavor dependent Z3 twisted boundary(ψ1 , ψ2 , ψ3 )(x,τ +1/T ) = −(ψ1 , zψ2 , z 2 ψ3 )(x,τ )where z = exp(2πi/3)
simulate at mPS /mV = 0.70
• tuning coupling β and mass κ0.19
0.180.17

κ

lattice size 163 × 4• T = 1/Nt a : a lattice spacing• high β ⇒ high T

0.20

0.16
0.15
0.140.13

compare Nf = 3 QCD
with the same parameters

mPS/mV =0.70mPS/mV =0.75
1.6

1.8

β2.02.2

9 / 14Z3 Symmetric Structure
Polyakov loop shows Z3 -symmetric distributions

Nf = 3 QCD

Z3-QCDβ = 1.7
β = 2.0
β = 2.2

0.4

0.2

Im L

0.2Im Lβ = 1.7
β = 2.0
β = 2.2

0.4

00-0.2-0.2-0.4-0.4-0.4-0.2

0Re L

0.2

0.4

-0.4

-0.2

00.2

0.4

Re L
10 / 140.2

Hysteresis

T > TcT < Tc0.15|L|• T > Tc
⟨L⟩ = 0 is unstable⇒ ⟨L⟩ ̸= 00.10.05• T < Tc⟨L⟩ ̸= 0 is unstable0⇒ ⟨L⟩ = 00200400

HMC trajectory■ Around Tc ,P-loop shows hysteresis⇒ 1st order transition

0.2

β = 1.80, cold start
hot start

|L|

0.15
0.1
0.05
0010002000

3000

HMC trajectory

4000

5000
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Center Symmetry Phase Transition
Polyakov loop

|L |

0.3
0.2

Z3: hot startZ3: cold start
Nf = 3• hot start showssteep transition
around β ∼ 1.8tart

cold s• cold start ⟨L⟩ ̸= 0even around β ∼ 1.6⇒ CHECK lower β0.1
0hot start

1.5 1.6 1.7 1.8 1.9β

• hot start: initial state ⟨L⟩ = 02

2.1 2.2• cold start: initial state ⟨L⟩ ̸= 012 / 14

Chiral Condensate
chiral condensate is dragged by center structure“exact” center symmetric phase ⇒ change becomes steeper
center broken phase⇒ Z3-QCD ≃ Nf = 3

0.8
0.6

"up" quark condensate
Z3: hot startZ3: cold start
Nf = 3• phase transitionis smeared byfinite quark mass

0.4
0.2
01.5 1.6 1.7 1.8 1.9
β22.1 2.2
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Chiral Condensate
chiral condensate is dragged by center structure“exact” center symmetric phase ⇒ change becomes steeper
center broken phase⇒ Z3-QCD ≃ Nf = 30.8
0.60.40.2
0"up" quark condensate

CENTERZ3: hot startZ3: cold start
SY
Nf = 3M
M

ETRI• phase transitionis smeared byfinite quark mass

CCENTER BRO

KEN

1.5 1.6 1.7 1.8 1.9β2

2.1 2.2
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Nf = 3 QCD with Exact Center Symmetry

2SummarySummaryin order to understand the role of “center symmetry” in QCD,
by introducing flavor dependent twisted boundary,we study Nf = 3 QCD with “exact center symmetry”“exact” center symmetry shows 1st order phase transition▶

similar to SU(3) Yang-Mills and QCD with adjoint quark

unlike adj. QCD,
chiral condensate is aﬀected by “center structure” of configuration
mq → 0, it is likely that both chiral & center show 1st order transition
Polyakov loop

Z3-QCDβ = 1.7
β = 2.0
β = 2.2

0.4

0.3

Z3: hot startZ3: cold start
Nf = 3|L |

Im L

0.2
00.2

-0.2

0.1

-0.4

0-0.4

-0.2

0Re L0.20.4tart
cold shot start

1.5 1.6 1.7 1.8 1.9β2

2.1 2.2
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