Quark deconfinement and symmetryHiroaki Kouno

Dept. of Phys., Saga Univ. Collaboration with

K. Kashiwa, Y. Sakai, M. Yahiro 　（ Kyushu. Univ.) and M. Matsuzaki (Fukuoka Univ. of Education)

Confinement of quark

• At zero temperature and zero density, quarks are confined in hadrons (baryons, mesons)

• Color is also confined. Hadrons are white.

If you want to “cut” a meson,…

• you need “Energy” which creates a pair of quark and anti-quark, because of the Einstein’s famous relation E=Mc2.

• You only get two mesons!! Not a isolated quark.

Quark-gluon plasma (QGP)

• However, at finite-temperature and/or finite density, it is expected that hadron will melt and quark-gluon plasma will be formed.

• This phenomenon is regarded as a phase transition.

Lattice QCD (LQCD) simulations

• LQCD is a computer simulation of QCD.

• Since we can not construct continuous space time in computer, we use discrete lattice space-time as an approximation.

• Quark and gluon live in this lattice space time.

Lattice QCD simulation

• At critical temperature, there is a jump in the energy density of the system just like a liquid-gas (vapor) phase transition.

• At finite density, it is difficult to do the Lattice QCD simulation because of the sign problem. However, the phenomenological model calculations predict the quark phase at high density.

Hadron-QGP transition

Chiral symmetry restoration

• There is also chiral phase transition at Tc, where the quark mass becomes small suddenly.

Predicted QCD phase diagram (by Yuji Sakai)

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IntroductionInteraction

Result of NJL model

Result of PNJL model

Summary

Recent development of the dense quark matter

Problems of lattice QCD calculation

Several approaches

?

From NASA

From RHIC

in the finite chemical potential region (T>>μ), lattice QCD calculation is not feasible.

Therefore, the low energy effective theory of QCD is often used in finite chemical potential.

Recent development of the dense QCD study

Problems of lattice QCD calculation

Several approaches

Experiment Inside of compact star

Relativistic Heavy Ion Collider (RHIC)

Large Hadron Collider (LHC)・・・

Many experimental evidences are obtained at RHIC. But there are no absolute evidence.

Equations of state have many ambiguity in quark part.

We do not know the method to calculate the dense QCD at moderate density region exactly !!

IntroductionInteraction

Result of NJL model

Result of PNJL model

Summary

Phase transition and symmetry

• Phases are classified by symmetry and an order parameter φ .

• < φ>=0 ⇒　 symmetry is preserved. Symmetric phase

• < φ>≠0 ⇒　 symmetry is spontaneously broken.

Symmetry broken phase

Discrete mirror symmetry

• Consider the potential which has mirror symmetry with respect to y-axis,

(1) V(x)=x2

(2) V(x)=-2x2+x4

Ground state or vacuum is defined at the minimum of the potential V(x)

Discrete mirror symmetry 　(1) < x>=0 ⇒ 　 symmetry i

s preserved. Symmetric phase

(2) < x>≠0 ⇒ 　 symmetry is spontaneously broken Broken phase

The vacuum solution breaks the symmetry!!

Continuous rotational symmetry

r2=x2+y2

Consider rotational symmetric potential

(1) v(r)=r2

(2) v(r)=-2r2+r4

Symmetry is preserved

(1) < r>=0 ⇒ 　 symmetry is preserved. Symmetric phase

Ground state solution

is (x,y)=(0,0).

Rotational symmetry around (0,0).

Symmetry is spontaneously broken

(2) < r>≠0 ⇒ 　 symmetry is spontaneously broken Broken phase

The vacuum solution breaks the symmetry!!

No rotational symmetry around (x,y)=(1,0).

Nambu-Goldstone bosn

If the symmetry is broken and the vacuum solution is (x,y)=(a,0)

• Square of mass of the particle x is proportional to

• Square of mass of the particle y proportional to

The particle y is a massless particle Nambu-Goldstone boson

02

2

x

V

02

2

y

V

Phase transition

T>Tc

T<Tc <x>≠0

<x>=0

symmetric phase

broken phase

It should be remarked that

• The degeneracy of the ground states induces the discontinuity between symmetric phase and symmetry broken phase.

• If degeneracy disappears, the discontinuity disappears and phase transition disappears.

Phenomenological models

• Since the QCD itself is very complicated and is hard to be solved nonpertubatively,

we use phenomenological model.・　 For chiral phase transition, we use the l

inear sigma model. ・　 For deconfinement transition, we use th

e Polyakov-Nambu-Jona-Lasino (PNJL) model.

Direct interaction of quark

• At low energy, effective direct quark interaction is induced by the gauge interaction at high energy.

Nambu-Jona-Lasinio model

• Consider the direct quark-quark interaction instead of gauge gluon-quark interaction.

　⇒　 Nambu-Jona-Lasinio model 　（ＮＪＬ）

2 20 5 5 2 2( ) ( ) ( ) ( )( )c c

s cq i m q G qq qi q G q i q qi q

L

( )qq ( )qq

Ｍｅｓｏｎ　ｆｉｅｌｄ• If we identify

as σ　 and π meson fields, we obtain the linear sigma model.

qq qq a 5

Linear sigma model

• Rotational invariance in σ–π plane chir⇒al symmetry

),()(),( 22 mesonquark VgMVV

222222 )(4

1)(

2

1),( mesonV

422

4

1

2

1rr

222 r

Spontaneous breaking

• At low temperature and/or low density

is negative. Chiral symmetry is spontaneously broken.

πmeson is a NG boson.

<r>≠0 M≠0⇒Quark becomes heavy.

2

2

1

Restoration of chiral symmetry

• At high temperature,

becomes positive,

chiral symmetry is restored.

<r>=0 　⇒ M=0

quark becomes massless.

2

2

1

Polyakov Loop

• Polyakov Loop is defined by

4

0

exp

c

c

Tr L

Tr P i d A

Polyakov loop and confinement

• The isolated quark free energy F is given by

F 　～　 -log(Φ).

Therefore, if Φ is zero, a quark is confined

since F 　～　 -log(Φ)=-log(0)=∞.

If Φis finite, quarks are deconfined since

F 　～　 -log(Φ)=finite.

Polyakov potential

• Pure LQCD results gives the Polyakov loop potential as

3 3 232 44

( )( , ; )( ) ( )

2 6 4

bb T bU T

T

2 3

0 0 02 0 1 2 3( )

T T Tb T a a a a

T T T

0a 1a 2a 3a 3b 4b

6.75 1.95 2.625 7.44 0.75 7.5

Discrete Z3 symmetry

• Polyakov potential is invariant under discrete Z3 transformation where k is a integer.

2 /3 ,i ke * * 2 /3i ke

Symmetry is preserved

• Z3 symmetry is preserved at low temperature.

• This means F is ∞, since

F 　～　 -log(Φ)

=-log(0)=∞.

Therefore, a quark is confined.

Symmetry is spontaneously broken

• At high temperature, Z3 symmetry is spontaneously broken.

There are three degenerate ground states.

・　 This means F is finite, since

F 　～　 -log(Φ)=finite.

Therefore, quarks are deconfined.

Deconfinement phase transition

T>Tc

T<Tc

<x>≠0

<x>=0

symmetric phase

broken phase

It should be remarked that

• Different from Chiral symmetry, Z3 symmetry is preserved at low temperature and broken at high temperature.

• Since Z3 symmetry is a discrete symmetry, there is no Nambu-Goldstone boson.

• If the effects of quark-anitiquark pair creations are taken into account,

Z3 symmetry is explicitly broken. Therefore Φ is not an exact order parameter any more.

PNJL model

• To include quantum effects of quarks, we use PNJL model, in which the Polyakov loop potential is included as well as the NJL Lagrangian.

PNJL = NJL +Polyakov Loop pot.

+ gauge interaction

PNJL model

●Polyakov potential

パラメータ

C. Ratti, et al. Phys. Rev. D73, 014019 (2006)

O. Kaczmarek, et al., Phys. Lett. B 543 (2002) 41.●Polyakov-loop

PNJL 　＝　 NJL （ chiral symmetry) 　＋　 Polyakov-loop （ confinement ）

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Gauge and direct interactions

Thermodynamic potential

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)()( pEPE )()( pEPE

chemical: potential

Z3 transformation

• The PNJL thermodynamic potential is not invariant under the Z3 transformation

2 /3 ,i ke * * 2 /3i ke

No phase transition

• Since the Z3 symmetry explicitly broken, even at high temperature, the ground state is not degenerate and there no discontinuity between the confined phase and deconfined phase!!

The transition becomes crossover.

Extended Z3 transformation

• However, the PNJL thermodynamic potential is invariant under the extended Z3 transformation with any integer k

2 /3 ,i ke * * 2 /3i ke

3/2 ki

It should be noted that

• Since we change the external variable, chemical potential, the extended Z3 symmetry is not an internal symmetry and the ground state is not degenerate even at high temperature.

• To see the physical meaning of the extended Z3 symmetry, we consider the system with imaginary chemical potential.

(Not a real world!!)

Welcome to Imaginary world!!

• Below we consider imaginary chemical potential.

• Extended Z3

transformation is rewritten by

Tii I

2 /3 ,i ke * * 2 /3i ke

3/2 k

Thermodynamic potential

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extended Z3 trans.

修正版 Polyakov ループ

Thermodynamic potential

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extended Z3 trans.

修正版 Polyakov ループ

Extended Z3 inv.

Roberge-Weiss periodicity

• Since thermodynamic potential

depends on the chemical potential only through the factor ei3θ, it is clear that 　 Ω 　 is invariant under extended Z3 transformation, and there is a Roberge-Weiss periodicity

Extended Z3 symmetry

RW　 even RW odd

periodicity

Same symmetry

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Thermodynamic potentialTRW

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TC

Kratochvila, Forcrand PRD73,114512(2006)

Chiral condensate and quark density

D’Elia, Lombardo PRD67, 014505 (2003)

TRWTC

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D’Elia, Lombardo PRD67, 014505 (2003)

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Chen, Luo PRD72, 034504 (2005)

Wu, Luo, Chen PRD76,034505(2007)

Polyakov LoopTRWTC

RW phase transition

• At high temperature (T>TRW), there is discontinuity at θ=(2k+1)π/3.

It is call Roberge-Weiss (RW) phase transition.

・ The point (θ,T)=(π/3,TRW) is the end point of the RW phase transition.

Phase diagram with imaginary chemical potential

Chiral 転移線

Polyakov 転移線

RW転移線

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Phase diagram with imaginary chemical potential

Chiral trans.

Polyakov trans.

RW trans.

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Comparison with LQCDM. D’Elia et al. PRD76, 114509 (2007)

Phase diagram

Lattice QCD

PNJL model 多項式近似による外挿

CEP

(m=1,2,3,4)

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Remnant of deconfinement phase transition

• The RW phase transition is a remnant of deconfinement phase transition.

• The RW endpoint seems to dominate the deconfinement (crossover) transition at zero and real chemical potential, although it does not exist in the real world.

• It is important to study the properties of RW endpoint.

Summary

• Phase transition can be classified by the symmetry and the order parameter.

• If the order parameter is zero, symmetry is preserved. The symmetry is spontaneously broken, if the order parameter is nonzero.

• There is no internal symmetry and order parameter for the quark deconfinement transition.

Summary

• However, if we were admitted to transform the external variable, the chemical potential, we have the exended Z3 symmetry and the RW phase transition.

• To analyze the deconfinement transition in the real world, it is important to study the properties of the endpoint of RW phase transition. The work is in progress.

Vector-type interaction

PNJL

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カイラル凝縮とクォーク数を見ればベクター相互作用の強さを決めることができる。

Phase diagram with vector-type interaction

Ｇ v=0

Gv の強さに CEP は敏感に反映される。

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Ｇ v=0.25Gs

PNJL

Three ground states in PNJL