some topics on chiral transition and color superconductivity

Some Topics on Chiral Transition and Color Superconductivity

Teiji Kunihiro (YITP)

HIMNov. 4-5, 2005APCTP, Pohang

CSC

T

SB

QGPQCD CEP

meson condensation?

?

0

CFL

QCD phase diagram

H matter?

cT

superconductingphases

BCS-BEC?

Various phases

RHIC

GSI,J-PARC

compact stars

ＩＩＩ

ＩＩＩ

Color Superconductivity; Color Superconductivity; diquark condensationColor Superconductivity; Color Superconductivity; diquark condensation

chiral symm. broken

Color Superconductivity(CSC)

with attractive channel in one-gluon exchange interaction.

quark (fermion) systemDense Quark Matter:

Cooper instability at sufficiently low T

SU (3)c gauge symmetry is broken!

[３ ]c×[３ ]c＝ [３ ]c＋ [６ ]c

~100MeV at moderate density q~ 400MeV

T

confinement

Attractive!p

-pFermi surface

q q

Abuki, Kitazawa and T.K.;P.L.B615,102(‘05)

(Dis)Appearance of CFL and Gapless phases in Charge- and Color-neutral System at T=0

strong coupling

weakcoupling

ｇ： gapless

Various CSC phases in plane T

The phase in the highest temperature is2SC or g2SC.

H .Abuki and T.K. hep-ph/0509172:

Abuki, Kitazawa and T.K.;P.L.B615,102(‘05)

2. Precursory Phenomena of Color Superconductivity in Heated Quark Matter

Ref. M. Kitazawa, T. Koide, T. K. and Y. Nemoto　　　　　 Phys. Rev. D70, 956003(2004);　　　　　 Prog. Theor. Phys. 114, 205(2005),　　　 M. Kitazawa, T.K. and Y. Nemoto, 　　　　　 hep-ph/0505070 , Phys. Lett.B, in press;　　　　　 hep-ph/0501167

CSC

T

SB

QGPQCD CEP

meson condensation?

?

0

CFL

QCD phase diagram

H matter?

　 preformed pair fields? quark spectra modified?

T

0

The nature of diquark pairs in various couplingThe nature of diquark pairs in various couplingThe nature of diquark pairs in various couplingThe nature of diquark pairs in various coupling

strong coupling! weak coupling

~ 50-100MeV/ EF ~ 0.1-0.3 / EF ~ 0.0001

in electric SC

Large pair fluctuations can

Very Short coherence length .

Mean field approx.works well.

invalidate MFA.cause precursory phenomena of CSC.

There exist large fluctuations of pair field even at T>0.

relevant to newly born neutron stars or intermediate states in heavy-ion collisions (GSI, J-Parc)

Collective Modes in CSCResponse Function of Pair Field

5 2 2† h.c.ex

CexH d i x

expectation value of induced pair field:

external field:

0

5 2 2( ) ( ) ( ), ( , )tC

exex tx i x i ds H s O t x

5 2 22 ( )( ) ( '( ,( )) ' )'indC

C x

Rexe

G x i x Dd x x xtx d x

Linear Response

5 2 2 5 2 22 ( ) ( ), (0) (0) (, ) )( CR CCG x i xD t i t x

Retarded Green function

Fourier transformationwith Matsubara formalism

( , )nQ k

RPA approx.:

( , )

1 ( , )C n

C n

G Q

G Q

k

kwith

After analytic continuation to real time,

Equivalent with the gap equation (Thouless criterion)

Precursory Mode in CSC

C

C

T T

T

The peak survives up to 　　 electric SC ：

ε→ ０（Ｔ→ＴＣ）

The peak of becomes sharp.

As T is lowered toward TC,

(Soft mode)

(Kitazawa, Koide, Nemoto and T.K., PRD 65, 091504(2002))

at k=0

Pole behavior

Spectral function of

the pair field at T>0

0.2 0.005

The pair fluctuation as the soft mode; --- movement of the pole of the precursory mode---

diffusion-modelike

How does the soft mode affect the quark spectra?

---- formation of pseudogap ----

0

1( , )

( , ) ( , )nn n

GG i i

k

kk

( , )n k 3

03

q( , ) ( , )

(2 ) n m mm

dT G

k q q

, n mi i k q

, miq

10 0( , ) ( )n nG i i

k k

T-matrix Approximation 　 for Quark Propagator 　　

Soft mode

30

3( ) ( , )

(2 )

dN

kk 0 01

( , ) Tr Im ( , )4

RG k k

3 0N d x Density of States N():

quasi-particle peak of anti-particle, = k

quasi-particle peak,=k)~ k

Fermi energy

Quasi-particle peak has a depression around the Fermi energy due to resonant scatteringresonant scattering.

= 400 MeV=0.01

=400MeV 0

kF=400MeVk

position of peaks

quasi-particle peaksat =k)~ k and =k.

k

1-Particle Spectral Function1-Particle Spectral Function1-Particle Spectral Function1-Particle Spectral Function

,00

,k k

: the soft modethe soft mode

: on-shell

= 400 MeV

The pseudogap survives up to ~ 0.05 ( 5% above TC ).

Density of states of quarks in heated quark matter

Pseudogap structure appears in N().

Fermi energy

N(

)/10

4

k

2 2sgn( ) ( )k k

2 2( )

d k

dk k

Quasi-particle energy:

( )N

2

Density of States:

The gap on the Fermi surface becomes smalleras T is increased, and it closes at Tc.

Density of States in SuperconductorDensity of States in SuperconductorDensity of States in SuperconductorDensity of States in Superconductor

d

dkkN 2)(

The origin of the pseudogap in HTSC is still controversial.

:Anomalous depression of the density of state near the Fermi surface in the normal phase.

Pseudogap

Conceptual phase diagram of HTSC cuprates

Renner et al.(‘96)

stronger diquark coupling GC

Diquark Coupling DependenceDiquark Coupling DependenceDiquark Coupling DependenceDiquark Coupling Dependence

GC ×1.3 ×1.5

= 400 MeV=0.01

Resonance Scattering of QuarksResonance Scattering of QuarksResonance Scattering of QuarksResonance Scattering of Quarks

GC=4.67GeV-2Mixing between quarks and holes

k

(M.Kitazawa, Y. Nemoto, T.K.hep-ph/0505070; Phys. Lett.B , in press)

There may exist a wide T region where the precursory soft mode of CSC has a large strength.

Summary of this section

effects of the soft mode on . H-I coll. & proto neutron stars

(,k)collective mode:

The soft mode induces the pseudogap, Typical Non-Fermi liquid behavior

Future problems:

eg.1)anomalous lepton pair production

eg.2) ; cooling of newly born stars(M.Kitazawa and T.Kin progress.)

resonant scattering

3. Precursory Hadronic Mode and

Single Quark Spectrum above

Chiral Phase Transition

Quarks at very high T (T>>>Tc) 1-loop (g<<1) + HTL approx. T),,( qmp

),( p

pE

thermal masses 222

6

1Tgm f

0)],(Re[ pD

2,1, hhp EEE

1hE0)],(Re[ pD

2,1, hhp EEE pE

1hEdispersion relations plasmino

plasmino

T

CSChadron

QGP

Tc

E E

)(En )(En

particle(q)-likeanti-quark like

Plasmino excitation

quark distribution anti-q distribution

CSC

T

SB

QGPQCD CEP

meson condensation?

?

precursory hadronic modes?free quark spectrum?

0

CFL

QCD phase diagram and quasi-particles

H matter?

fluctuation effects

Interest in the particle picture in QGPInterest in the particle picture in QGP

TT

ETc

2Tc

fluctuationeffects

RHIC experimentsrobust collective flow

• good agreement with rel. hydro models• almost perfect liquid

(quenched) Lattice QCD

charmonium states up to 1.6-2.0 Tc(Asakawa et al., Datta et al., Matsufuru et al. 2004)

Strongly coupled plasma rather than weakly interacting gas

The spectral function of the degenerate hadronic ``para-pion” and the ``para-sigma” at T>Tc for the chiral transition: Tc=164 MeV

T. Hatsuda and T.K. (1985)

response function in RPA

( , )D k ...

(1( ) Im )DA kk

spectral function

0,1.1 CTT

sharp peak in time-like region

k

Hatsuda and T.K,1985

1.2Tc

M.Kitazawa,Y.Nemoto andT.K. (05)

, they become elementary modes with small width!

cT T

Chiral Transition and the collective modes

0

c.f. Higgs particle in WSH model

; Higgs fieldHiggs particle

para sigmapara pion

How does the soft mode affect

a single quark spectrum near Tc?

Y. Nemoto, M. Kitazawa ,T. K.

hep-ph/0510167

Model low-energy effective theory of QCD 4-Fermi type interaction (Nambu-Jona-Lasinio with 2-flavor)

])()[( 25

2 qiqqqGqiqL S ： SU(2) Pauli matrices

MeV631,GeV105.5 -26 SG

0 du mm chiral limit finite : future workdu mm ,

Chiral phase transition takes place at Tc=193.5 MeV(2nd order).

Self-energy of a quark (above Tc)),(),(

)2(

dT),( 03

3

qGqpDq

p mmnm

n

),( qpD mn

),(0 qG m

),( pD n

= + + + …

iinR

npp |),(),( : imaginary time real time

scalar and pseudoscalar parts

T->Tc, may be well described with a Yukawa coupling.

|p|

( , )ni k

Spectral Function of QuarkSpectral Function of Quark

Quark self-energy

Spectral Function

0 0 0 0 0( , ) ( , ) ( , )A p p p p p p

0,05.1 CTT

0Re||0 pp

0p

quark

3 peaks in also 3 peaks in

|p| 0p

01ˆ( ) (1 )

2p p

for free quarks,)p()p,( 00 pp

antiquark

qq

Resonant Scatterings of Quark for Resonant Scatterings of Quark for CHIRALCHIRAL Fluctuations Fluctuations

:),( 0pp = + + …

E

0pE

0p

0,08.1 CTT

dispersion law0Re||0 pp

0,05.1 CTT

Im

Re

0p 0pfluctuations of

0p

0p

almost elementary boson at cT T

p [MeV]p

[MeV]

+(,k)-(,k)

Resonant Scatterings of Quark for Resonant Scatterings of Quark for CHIRALCHIRAL Fluctuations Fluctuations

E

0p

E0p

“quark hole”: annihilation mode of a thermally excited quark

“antiquark hole”: annihilation mode of a thermally excited antiquark(Weldon, 1989)

lead to quark-”antiquark hole” mixing

0

0

pp

pp

cf: hot QCD (HTL approximation)

(Klimov, 1981)

0,05.1 CTT

quark

plasmino

the ‘mass’ of the elementary modes


),( p

thermal masses 222

6

1Tgm f

0)],(Re[ pD

2,1, hhp EEE 0)],(Re[ pD

2,1, hhp EEE

pE

1hEpE

1hE

2hE

2hE

dispersion relations

T

CSChadron

QGP

Tc


),( p

pE

thermal masses 222

6

1Tgm f

0)],(Re[ pD

2,1, hhp EEE

1hE0)],(Re[ pD

2,1, hhp EEEpE

1hE

2hE

2hE


T

CSChadron

QGP

Tc


),( p

thermal masses 222

6

1Tgm f

0)],(Re[ pD

2,1, hhp EEE 0)],(Re[ pD

2,1, hhp EEE2hE

2hE


spectral function of the space-like region

p/mf mf

T

CSChadron

QGP

Tc

0

10 -10

0

10

above CSC phase: One resonant scattering

above chiral transition: Two resonant scatterings

Difference between Difference between CSCCSC and and CHIRALCHIRAL

fluctuations of the order parameter ~ diffusion-like

)0~( ~)( 2 ppp

fluctuations of the order parameter ~ propagating-like

)0~( 0)( ~)( 0 pp

Level Repulsions

00

For massless gauge field,

( , ) ~ ( ) ( )A p p p

2 2 2 20 0( , ) ~ ( ) ( )A p p p

0

0

pp

p

0

0

pp

p

Spectral function of the quarks

0,0 T

0,0 T

q q

E E

E E

)(En

)(En

)(En

)(En

Finite dependence; asymmetry between and q q

particle(q)-likeanti-quark like

particle-like statessuppressed

Summary of the second part

We have investigated how the fluctuations of affect the quark spectrum in symmetry-restored phase near Tc.

Near (above) Tc, the quark spectrum at long-frequency and low momentum is strongly modified by the fluctuation of the chiral condensate, .

The many-peak structure of the spectral function can be understood in terms of two resonant scatterings at small and p of a quark and an antiquark off the fluctuation mode.

This feature near Tc is model-independent if the fluctuation of is dominant over the other degrees offreedom.

qq

Futurefinite quark mass effects. (2nd order crossover)

finite coupling with density fluctuation; CEP?

qq

qq

can be reproduced by a Yukawa theory with the bosonbeing a scalar/pseudoscalar or vector/axial vector one

(Kitazawa, Nemoto and T.K.,in preparation)

CSC

T

SB

QGPQCD CEP

?

2.precursory hadronic modes?

0

CFL

QCD phase diagram

H matter?

1. preformed pair fields? quark spectra modified?

strongly modified quark spectra

`QGP’ itself seems surprisingly rich in physics!

Summary of the Talk

Condensed matter physics of strongly coupled Quark-Gluon systems will constitute a new field of fundamental physics.

Back Upps

Pairing patterns of CSCPairing patterns of CSCPairing patterns of CSCPairing patterns of CSC

u d su d s

Two Flavor Superconductor(2SC)

u d u d

s

<Ms >>Ms

Color-Flavor Locked (CFL)

5ij i jCi for JP=0+ pairing

ij ijk k d

a,b : colori,j : flavor

attractive channel : color anti-symm.

flavor anti-symm.

00

d 1

2

3

d

(3) (2)c cSU SU (3) (3) (3)

(3)c L R

c L R

SU SU SU

SU

BCS-BEC transition in QM

0.001

0.01

0.1

1

T/E

F

1.21.11.00.90.8

G/G0

Tmf

TBEC

TBEC_RL

Y.Nishida and H. Abuki,hep-ph/0504083

Calculated phase diagram

0qq

0qq

Fermions at finite T

0),( pD 0,0 hp EE

||

ˆ

2

1

||

ˆ

2

11),( 00

0 p

p

p

p

ppS

　 free massless quark at T=0

|| p

quark and antiquark quark at finite T (massless)

),(

ˆ

2

1

),(

ˆ

2

1

),(),(

1),( 00

0 pD

p

pD

p

pCpApS

/

CAD 0),( pD 0),( pD

several solutionsseveral solutions

0),( pD 0,0 ph EE

Formulation (Self-energy)Formulation (Self-energy)3

0 30 0

( ) ( ) ( ) ( )( , ) ( ( ))

(2 )f C p q p p q p

q p pq p p p q p p q p

N N f E f E f E f Ed pp p E E p q p

E E p E E i p E E i

0 0

1 ( ) ( ) 1 ( ) ( )( ( )) p q p p q p

q p pp q p p q p

f E f E f E f EE E p q p

p E E i p E E i

... =1-

0( , )p p

Quark self-energy:

( , )D k

400 0

40 0

Im ( , )1ˆ( ) coth tanh

2 (2 ) 2 2

qD p q q qd qq

q p p i T T

400 0

40 0

Im ( , )1ˆ( ) coth tanh

2 (2 ) 2 2

qD p q q qd qq

q p p i T T

p

p q

Formulation (Spectral Function)Formulation (Spectral Function)

Spectral function:

00 0 0 0 ˆ( , ) ( , ) ( , )VA p p p p p p p

00 0[ ( , ) ( ) ( , ) ( )]p p p p p p

01ˆ( ) (1 )

2p p

quark antiquark

00 0

1 1( , ) Im

( , )p p

p p p p i

1.4 Tc1.2 Tc

1.1 Tc1.05 Tc

Spectral Contour and Dispersion RelationSpectral Contour and Dispersion Relation

p p

p p

p

p

p

p

+ (,k)

+ (,k)

+ (,k)

+ (,k)

-(,k)

-(,k)

-(,k)

-(,k)

some topics on chiral transition and color superconductivity

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