Investigation of QCD phase structurefrom imaginary chemical potential

Kouji Kashiwa

RIKEN BNL Research Center

2014/02/06 BNL

Introduction : Quark and gluon

Theme 　：　 Phase structure of Quantum Chromodynamics at finite T and m.

Are there any different states? If those exist, where can they appear?

Those are confined inside Hadrons.

From experiments at Relativistic Heavy ion Collider (RHIC) and Large Hadron Collider (LHC),Some data are obtained which can not be understood from Hadronic state only.

Question :

Quarks and gluons can not be observed directly.

What states? When and where those can be seen?

Introduction : Phase diagram

Phase diagram： quark-gluon system

Recent conceptual drawing

Several phases were predicted so far…

There is no quantitative discussion at finite m.

K. Fukushima, T. Hatsuda, Rept. Prog. Phys. 74 (2011) 014001.

Introduction : Phase diagram

LHCRHIC

GSI

JPARC

Early

uni

vers

e

Compact star

ρ0

AGS

SPS

KEK-PS

It is quite important for experiments and observation.Phase diagram： quark-gluon system

Phase transition

Deconfinement phase transition

Chiral phase transition Chiral symmetry ：

Polyakov-loop:

0

0

Free energy for one quark excitation

0 　/~ F Te 1 0F

F

For example, L. D. McLerran and B. Svetitsky, Phys. Rev. D 24 (1981) 450.

F, , T Vm

qq

1 Tr c

LN

Symmetry under transformations of left- and right-handed components of quark independently.

Order parameter : Chiral condensate

Origin of the mass of proton, neutron, pion and so on.

Z3 symmetry (center of SU(3) ) ：

It exists in pure gauge.（ twist at temporal boundary ）

（ Zero quark mass）

Phase transition considered in this talk.

Phase transition

In this talk, we assume there is the order-parameters for the deconfinement transition.

There is the different clarification for confinement/deconfinement

Topological orderIt was proposed in solid state physics (fractional quantum hall state)

Order parameter Spontaneous symmetry breaking

Example: chiral condensate Chiral symmetry breaking

There is no order parameter.

Difference between those sates are characterized by the non-trivial degeneracy of the vacuum.

We need the non-trivial topology.

Masatoshi Sato, PRD 77 (2008) 0450013.

Problem?

Lattice QCD simulation : first principle calculation of QCD

Sig problem

( ) U det[ ] expq GZ Sm D M

1 U det[ ] exp GSZ O D O M

probability

( ) exp( )ii

Z Probability

11

exp( )( ) 1( )

PZ

Partition function

Statistical dynamics

4 0( )q qD mmmm m M

Dirac operator :

Probability can becomes complex (also minus)

Several approaches to circumvent the sign problem:Taylor expansion

ReweightingThese can not reach very high m.

Analytic continuation

Canonical approach

Problem?

Ambiguity in effective models

M. Stephanov, Prog. Theor. Phys. Suppl. 153 (2004) 139.

M. Kitazawa, T. Koide, T. Kunihiro and Y. Nemoto, Prog. Theor. Phys. 108 (2002) 929.

NJL+CSC+Gv case

Ginzburg-Landau approach

T. Hatsuda, M. Tachibana, N. Yamamoto and G. Baym, Phys. Rev. lett, 97 (2006) 122001.

Multi critical endpoint ?

Sign problem free systems

No sign problem ：

Can we use these system?

We construct the effective model by combining the LQCD data

Imaginary chemical potentialIso-spin chemical potential (Baryon chemical potential = 0)

Two color QCD

Our approach:at imaginary chemical potential

Because those reasons, we can not obtain reliable QCD phase diagram.

Z3 symmetryZ3 symmetry

Quark contribution(explicit center symmetry breaking)

Pure gauge : Contour Plot Im

Re

DeconfinedConfined

Three degenerate minima are came from Z3 symmetry

Quark contribution breaks Z3 symmetry explicitly.

Two of them become metastable.

What happen at finite m?

Imaginary chemical potential

QCD has characteristic properties at finite imaginary m!( It is similar to AB phase, but different )

Roberge Weiss (RW) phase transition line

RW endpoint

2p/3

Phase diagram： Imaginary chemical potential

It is completely different from that at real m.

Non-trivial periodicity

Roberge-Weiss (RW) periodicity

First-order transition along T-axis

RW transition

A. Roberge and N. Weiss, Nucl. Phys. B275 (1986) 734.

Imaginary chemical potential

/Canonical Grand Canonical( , ) IiB TIZ T B d e Z

Tmm

/Grand Canonical Canonical( , ) ( , )RB T

RB

Z T e Z T Bmm

Fugacity expansion:

Fourier representation:

2p/3

A. Roberge and N. Weiss, Nucl. Phys. B275 (1986) 734.

Even function has cusp.Odd function has gap.

This coexistence:K.K., M. Yahiro, H. Kouno, M. Matsuzaki, Y. Sakai,J. Phys. G 36 (2009) 105001.

Imaginary chemical potential

Imaginary chemical potential

/Canonical Grand Canonical( , ) IiB TIZ T B d e Z

Tmm

/Grand Canonical Canonical( , ) ( , )RB T

RB

Z T e Z T Bmm

Fugacity expansion:

Fourier representation:

Imaginary chemical potentialThis relation means that the imaginary chemical potential has almost all information of the real chemical potential region.

Actually, there are some method to use above relation in lattice QCD simulations.

Analytic continuation method

Canonical approach

Standard methods : Analytic continuation

Analytic continuationFig ：　 P. de Forcrand, S. Kratochvila, Nucl. Phys. B (Proc. Suppl.) 153 (2006) 62.

0 1

0 1

[ , ]( )M

i M ii N

i N i

a a aP N M

b b bm m

mm m

Data are collected at imaginary m.

Data are fitted by analytic functions.

Example：

Based on Lattice QCD simulation only：

Standard methods : Canonical approach

Canonical approachFig ：　 P. de Forcrand, S. Kratochvila, Nucl. Phys. B (Proc. Suppl.) 153 (2006) 62.

Check Maxwell contraction

If there is first-order transition,S sharp structure is there （ in finite size system）

We should investigate (T,r) where S sharp structure is vanished.

/Canonical Grand Canonical( , ) IiB TIZ T B d e Z

Tmm

Problem?

Convergence radius 　 (Analytic continuation)

Order of phase transition 　 (Analytic continuation)

Finite size system 　（ Canonical approach ）

Color superconductivity 　（ Canonical approach, Analytic continuation ）

We combine effective model and Lattice results.Dynamics of phase transition are included.

Parameters can be determined at finite imaginary m.Imaginary m has information of real m region.

Recent model development

Fermion part

What model should we use?

Nambu—Jona-Lasinio (NJL) model

2 20 5( ) ( )sq i m q G qq qi qm

m L

If the gluonic contribution is not correctly introduced,the RW periodicity should be vanished.

By using some approximations and ansatz, we can derive the NJL model from QCD.

01( )4

q i m q F F gq A qm m mm m m L

4expaQCD QCDZ DqDq DA i d xm L

40exp ( ) [ ]DqDq i d x q i m q iW jm

m

4 41[ ] ln exp ,4

a aa aiW j DA d x F F ig d x A jm

m

mm m

2

aaj q qm m

1 2 1 1 2

1

1 1 1

1

(1) 41 1 1

2,(2) 4 4

1 2 , 1 2 1 2

...( ) 4 41 ... 1 1

[ ] [0] ( ) ( )

( , ) ( ) ( ) ...2

( ... ) ( )... ( ) ...!

n n n

n

a

a a a a

na a aan

n n n

iW j iW g W x j x d x

g W x x j x j x d x d x

g W x x j x j x d x d xn

mm

m mm m

mmm m

...

W(n) is the connected n-point function of gauge boson without quark loops.2

(2)0( ) ( ) ( , ) ( )

2a bab

GCMgq i m q j x W x y j ym m

m m L2 28

0 50

( )2 2

i i

NJL sa

q i m q G q q qi qmm

L

NJL model(This model only has 2p periodicity)

For example:

Quark color current :

Recent model development

Fermion part

What model should we use?

Quark-meson model can be also used. (Basically it is almost equivalent with NJL model)

2 20 5( ) ( )sq i m q G qq qi qm

m L

2 20 5( ) ( ) ( , )sq i D m q G qq qi q Um

m L

3

332 ( ) ln 1 ( )

(2 )E E E

P M f cd pU U N N E p T e e e

V

p

3ln 1 ( )E E ET e e e 2 2M s vU G G

Polyakov-loop extended Nambu—Jonal-Lasinio (PNJL) model

Mean field approximation

Thermodynamic potential

If the gluonic contribution is not correctly introduced,the RW periodicity should be vanished.

Gluonic contribution

Nambu—Jona-Lasinio (NJL) model

NJL model(This model only has 2p periodicity)

Gluon partRecent model development

Polyakov-loop potential

Meisinger-Miller- Ogilvie model

Matrix model for deconfinement

Effective potential from (Landau gauge) gluon and ghost propagator

Strong coupling expansion

Mass like parameter is introduced. Up to the second order term of high T expansion is included.

Extension of MMO model.

Gluon and ghost propagators in Landau gauge are used.

To reproduce LQCD data in the pure gauge limit.

P. N. Meisinger, T. R. Miller, M. C. Ogilvie, PRD 65 (2002) 034009.

A. Dumitru, Y. Guo, Y. Hidaka, C. P. K. Altes, R. D. Pisarski, PRD 83 (2011) 034022.

K. Fukushima, Phys. Lett. B 591 (2004) 277.

+

K. Fukushima, K.K. , Phys. Lett. B 723 (2013) 360.

RW periodicity can be reproduced by using following models.

(RW periodicity is the remnant of the Z3 symmetry)

U

U

U

U

Results : Model ambiguities

2

vG q qmVector-type interaction

It relates with ω0 mode.

Fermion part : Phase diagram

If the vector-type interaction is sufficiently large,CEP should be vanished.

For example,

K. K., H. Kouno, M. Matsuzaki and M. Yahiro, Phys. Lett. B662 (2008) 26. 　　

This behavior also appears in the NJL model.

Results : Vector interaction 2

vG q qmVector-type interaction

It relates with ω0 mode.Vector-type interaction

Set C0

0.4 0.8 1.0

Y. Sakai, K. K, H. Kouno, M. Matsuzaki and M. Yahiro, Phys. Rev. D 79 (2009) 096001. 　　　　　　

P. de Forcrand and O. Philipsen, Nucl. Phys. B 642 (2002) 290.L. K. Wu, X. Q. Luo and H. S. Chen, Phys. Rev. D 76 (2007) 034505.

Lattice data:

Results : Columbia plot

Gluonic contribution

Zero chemical potential RW endpointColombia plot

Order of phase transition

Gluonic part also has strong ambiguity even in perturbative regime of quark contribution

Ambiguity appears even at large quark mass region.

Larger ambiguity may be seen on the RW endpoint.

Matrix

There is the possibility thatRegion can be first order region.

There is no phase boundary until 1 GeV in the case of Polyakov-Log.

K.K., V. V. Skokov, R. D. Pisarski, Phys. Rev. D85 (2012) 114029.K.K., R. D. Pisarski, Phys. Rev. D87 (2013) 096009.

Related topic : Hosotani mechanism

Imaginary chemical potential may be important for other topics.For the physics beyond the standard model

nf = 2pT (n + 1/2) + mI

Matsubara frequency

n f = 2pT (n + f)

Matsubara frequency with arbitral boundary condition

n f = 2pT (n + 1/) – pT + 2pTf

Angle represents the arbitral boundary condition

Imaginary m

Boundary condition for temporal direction

Fermion boundary condition is important for Hosotani mechanism.If the extra-dimension is not simply connected with the system, the gauge symmetry breaking vacuum expectation value can affect the system.

Higgs can be understood as the fluctuation of extra-dimensional gauge boson component.

Hosotani mechanism：For example: Y. Hosotani, Phys. Lett. B 126 (1983) 309; Ann. Phys. 190 (1989) 233.

0

β, 1/Lφ

Com

pact

ed d

irecti

on

Related topic : Hosotani mechanism

q1 = q2 ≠ q3 : SU(2) ×U(1) ,

Gauge symmetry breaking is happen

Eigen value

For example, q1 ≠ q2 ≠ q3 : U(1) ×U(1)

Wilson loop in compacted directionNth phases (qi) for SU(N)

Temporal direction is taken as compact dimension in following.

Perturbative one-loop effective potential (free gas limit)

Start from QCD Lagrangian density

Decompose A4 to “expectation value + fluctuation”

Drop the interactions from the action

Calculate the ln det (2n+p2)

Inverse of perturbative propagator.

Divergences can be subtracted in 5D as same as 4D.

Free gas calculation

Perturbative one-loop effective potential (free gas limit)

=

=

~After summing up each integrations:

4D 5D

+c

D. Gross, R. Pisarski, L. Yaffe, Rev. Mod. Phys 53 (1981) 43.N. Weiss, Phys.Rev.D 24 (1981) 475.

Phase structure

aPBC adjoint

SU(3) SU(2)×U(1) U(1)×U(1)

Large m Medium m Small m

PBC adjoint

Actual forms:

Gauge boson

Adjoint fermion

Boundary angleFermion : 1/2Boson : 0, p

Phase :

Number of flavor : Na

Fermion mass : mf, ma

Arbitral dimensional form can be obtained similar form.

Phase diagram

Phase Structure

D : Deconfined phase

S : Split (skewed) phaseR : Re-confined phase

C : Confined phase

In previous studies for Hosotani mechanism, fermion mass effects were almost neglected.

We use the perturbative one-loop potential.

SU(3)

U(1)×U(1)

SU(2)×U(1)

K.K., T. Misumi, JHEP 05 (2013) 042.

Lattice gauge results

Phase Structure Scatter plot of Polyakov-loop

Lattice setup: 2 flavor, 3 color and adjoint staggered fermion

Lattice data : G. Cossu, M. D’Elia, JHEP 07(2009), 048.

Comparison

Phase StructureWe can understand it from Hosotani mechanism! SU(3)

U(1)×U(1)

SU(2)×U(1)

K.K., T. Misumi, JHEP 05 (2013) 042.

Problem from confinement and U(1) ×U(1) phases

Phase structure

H. Nishimura, M. Ogilvie, Phys. Rev. D 81 (2010) 014018.K.K., T. Misumi, JHEP 05 (2013) 042.

In their calculation, the confined and U(1)×U(1) phases are same...

Phase structure

C

Unknown

In their calculation, the confined and U(1)×U(1) phases are same...

Problem from confinement and U(1) ×U(1) phases

H. Nishimura, M. Ogilvie, Phys. Rev. D 81 (2010) 014018.K.K., T. Misumi, JHEP 05 (2013) 042.

Chiral properties K.K., T. Misumi, JHEP 05 (2013) 042.

With adjoint fermion With adjoint and fundamental fermion

To describe the chiral symmetry breaking and restoration, we use the Nambu—Jona-Lasinio type model.

2+1+1 dimensional system

QCD-like theory at finite temperature and one compactified spatial dimension is interesting.

Standard local NJL model with moment cutoff can not be used.

We use the nonlocal NJL model.

K.K., T. Misumi, in preparation.

This system may be useful to understood the system under the strong external magnetic field.

The summation came from the Landau quantization appears as same as the Kaluza-Klein summation.

Non-trivial chiral properties are obtained and almost all effective model can not explain it…

2+1+1 dimensional system

K. Farakov and P. Pasipoularides, Nucl. Phys. B 705 (2005) 92.M. Sakamoto and K. Takenaga, Phys. Rev. D76 (2007) 085016.

Our results (these are still 3+1 dimensional system)

Perturbative one-loop effective potential for massive particle

Integral representation

Poisson formula

K.K., T. Misumi, in preparation.

Distribution function

5-dimensional SU(3) lattice gauge theory E. Itou, K.K., T. Nakamoto, in preparation.

Investigation of the 5-dimensional system is important.

However, phase structures of the 5-dimensional SU(3) lattice (pure) gauge theory is not well understood yet.

We should know the critical where bulk first order transition vanished.

Small extra-dimensional system Multi-(4-dimensional) layered system4-dimensional layer

large a5

5-dimensional SU(3) lattice gauge theory E. Itou, K.K., T. Nakamoto, in preparation.

Summary

We study the QCD phase diagram from the imaginary chemical potential.

Imaginary chemical potential has almost all information of real chemical potential.

There is no sign problem and thus lattice QCD simulation is possible.

We determined the vector-type interaction at the imaginary chemical potential and draw the phase diagram.

To obtain more accurate diagram, we need more accurate data,We show the usefulness of the imaginary chemical potential to study it.

Imaginary chemical potential can be converted to the boundary condition. It may be useful to understand the Hosotani mechanism.

Physics beyond the standard model