Teiji Kunihiro (Dep. of Physics, Kyoto)

RBRC Workshop at Brookhaven National Laboratory“Understanding QGP Through Spectral Functions and Euclidean Correlators”,

at BNL, April 23-25, 2008

Quark Quasi-Particle Pictureat Finite Temperature and Density

in Effective Models

In collaboration withM. Kitazawa, T. Koide, K. Mitsutani and Y. Nemoto

YITP

Thoretical (half-conjectured) Phase Diagram of QCDThoreticalThoretical (half(half--conjectured) Phase Diagram of QCDconjectured) Phase Diagram of QCD

RHIC

TT

μμ

CEPTc

2Tc

chiral sym. broken (antiquark-quark condensate)

confinement

chiral sym. restored deconfinement

quark-quark condensate

GSI,J-PARC

compact stars

diquark fluctuations

chiral fluctuations

Phase transitions large fluctuations owing to strong coupling around the critical points. There may exist also other low-lying (hadronic) excitations in the QGP phase.

density fluctuations

Here we explore how they affectquark quasi-particle picture!

ContentsContentsContents

1.Introduction2. diquark fluctuations and pseudogap in quark

specral function in hot and dense quark matter --- a lesson from condensed matter physics ---

3.Soft modes of chiral transition and anomalous quasi-quark spectrum

4.Summary and concluding remarks

2. Diquark fluctuations and pseudogap in quark spectral function in hot and dense quark matter

2. 2. DiquarkDiquark fluctuations and fluctuations and pseudogappseudogap in in quark spectral function in hot and dense quark spectral function in hot and dense quark matterquark matter

The mechanism of the pseudogap in High-TcSC is still controversial, but see,Y. Yanase et al, Phys. Rep. 387 (2003),1, where the essential role of pair fluc. is shown.

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

Phase diagram of cupratesto be high-Tc

superconductor

Renner et al.(‘96)

Lesson from condensed matter physics on strong correlations

(hole doping)

A typical non-Fermi liq. behavior!

Fermi energy

( )ρ ω

μ = 400 MeV

Possible pseudogap formation in heated quark matter

Fermi energy

N(ω)/104

ω

3

3( ) ( , )(2 )dN ω ρ ωπ

= ∫k k

M. Kitazawa, T. Koide, T. K. and Y. NemotoPhys. Rev. D70, 956003(2004);Prog. Theor. Phys. 114, 205(2005),

T

μ

Pseudogap ?

pseudogap !

c

c

T TT

ε−

=

CSC

Pseudogap isformed above Tcof CSC in heatedquark matter!

How?

for μ = 400 MeVT

= 1.05Tc

1 1( ) Im1

( )DSe βωω

πω−= −

−kk

Dynamical Structure Factorof diquark fluctuations

sharp peak at the origin(= diffusive over-damping mode)

( , )nD ω =k + + ⋅⋅⋅

Mechanism of the pseudogap

formation1. Development of precursory diquark fluctuations above Tc

2.Coupling of quarks with the diquark fluctuations

+ + +⋅⋅⋅

Cherenkov-like emission

of diquark mode around Fermi energy

( , )kωΣ = μquark self-energy

Depression of the quark spectralFunction around the Fermi energy

= 400 MeV, =0.01ε

GC

=4.67GeV-2

ω

kkF

Janko, Maly, Levin, PRB56,R11407 (1995)

Mixing between particles and holesowing to the Landau damp. by thecollective diquark mode.

level repulsion of energy level

Level repulsion or Pseudo gap due to resonant scatteringLevel repulsion or Pseudo gap due to resonant scatteringLevel repulsion or Pseudo gap due to resonant scattering

M. Kitazawa, T.K. and Y. Nemoto, Phys. Lett.B 631(2005),157

particle hole particle

particleholeparticlehole

particle #-1

particle #+1

3. Soft modes of chiral transition and anomalous quasi-quark spectrum

3. Soft modes of 3. Soft modes of chiralchiral transition and transition and anomalous quasianomalous quasi--quark spectrumquark spectrum

Chiral Transition and the collective modesChiralChiral Transition and the collective modesTransition and the collective modes

c.f. The sigma as the Higgs particle in QCD

; Higgs fieldHiggs particle

0

para sigmapara pion

The low mass sigma in vacuum is now established:pi-pi scattering; Colangero, Gasser, Leutwyler(’06) and many others Full lattice QCD ; SCALAR collaboration (’03)

q-qbar, tetra quark, glue balls, or their mixed st’s?

-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0.0

M (GeV)- Γ

/2 =

Im M

(G

eV

) Basdevant et al. (1972) Achasov, Shestakov (1994) Zou, Bugg (1994) Janssen et al. (1995) Tornqvist (1996) Kaminski (1997) Ishida (1997) Locher, et al. (1998) Oller, Oset (1999) Oller, Oset (1999) Kaminski (1999) Hannah (1999) Anisovich, Nikonov (2000)

0.2 0.4 0.6 0.8 1.0 1.2 1.4

Digression:The poles of the S matrix in the complex mass plane fthe sigma meson channel:

complied in Z. Xiao and H.Z. Zheng (2001)

Softening ?

See also, I. Caprini, G. Colangero and H. Leutwyler, PRL(2006);H. Leutwyler, hep-ph/0608218 ; M_sigma=441 – i 272 MeV

0.0

0.1

0.2

0.3

0.4

0.5

0.6

5.24 5.26 5.28 5.3 5.32 5.34

m = 0.02L = 8

πa0 [δ]f0 [σ]

σπ

the softening of the σ

withincreasing T

T

μ

TFluctuations of chiral

order parameter

around Tc

in Lattice QCD

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

T. Hatsuda and T.K. (1985)

T

μ

T

(NJL model cal.)

[ ]2 ( ), (0)( ) SGD x xσ ψψ ψψ= −

= + +

( ,1( , ) m )Ip D pσσρ ωπ

ω= −

PNJL model

S. Rößner, T. Hell, C. Ratti,W. Weise.arXiv:0712.3152

[hep-ph]

K. Fukushima (2004)C. Ratti, M. Thaler, W. Weise (2006)

Para-pion

and para-sigma modes are still seen in PNJL model

P=Polyakov-loop coupled

W. Weise, talkat NFQCD2008 atYITP, Kyoto.

Quark Self-energy at finite T),(),(),(),( 0000

00 ppppppppp SV Σ+⋅Σ−Σ=Σ γγ

=0Quark Spectral function

particle antiparticle

0 0 0 0( , ) ~ ( ) ( )p p p p p pρ δ δ− + +c.f: for a free quark

0 0( , ) ( , )ρ ρ+ −= −p p p pat zero density

Quark Spectral FunctionQuark Spectral FunctionQuark Spectral Function

),(),(),(),( 00000

0 pppppppp SV ρργρρ +⋅−= pγ

=0

(chiral limit)

0 0 0 0 0( , ) ( , ) ( , )p p p p p pρ ρ γ ρ γ− −+ +Λ Λ= +

00 0

0

1( , )( ( ,, ) )

G p pG p p p pΣ

=−

0

0 0

00

0

1( , ) Im ( , )

( , ) ( , )

p p G p p

p p p pγ

ρπ

ρ ρ γ+ + −−

= −

= Λ Λ+

Digression: Quarks at very high T (T>>Tc) ---- physical origin of plasmino and thermal mass ---

Digression:Digression: Quarks at very high T (T>>Quarks at very high T (T>>TcTc)) -------- physical origin of physical origin of plasminoplasmino and thermal massand thermal mass ------

1-loop (g<<1) + HTL approx. T),,( <<qmp ω

=Σ ),( pω

T

CSChadron

QGP

μTc

ω

p

Tm

Tm−

quasi-q

anti- plasmino

plasmino

quasi-anti-q

p

plasmino

quasi-anti-q

quarkanti-quark

( , )pρ ω+( , )pρ ω−

quark anti-qth hole quark

Thermally excitedanti-quarks

=

Klimov, Weldon(’82),Pisarski(’89), A.Schaefer, Thoma (’99)

We incorporate the fluctuation mode into a single particle Green

function of a quark through a self-energy.

+ +Σ Σ Σ

+ ⋅⋅⋅=−

=)(),(

1),(0 ,pωΣpG

pGnn

n ωω

+ + + ⋅⋅⋅

≡, n mi iω ω+ +k q

, miωq=

Σ=Σ ),( pnω

),(),()2( 03

3

qGqpDqdT mm

mn ωωωπ∑∫ ++=

Non self-consistent T-approximation (1-loop of the fluctuation mode)

Quarks coupled to Quarks coupled to chiralchiral soft modessoft modes near near TcTc

N.B. This is a complicated multiple integral owing to the compositeness of the para-sigma and para-pion modes.

k

[MeV]

k

[MeV]

ω [MeV]

ρ+

(ω,k)ρ-

(ω,k)

k

[MeV]

ε = 0.05

c

c

T TT

ε −=Spectral Function of Quark Spectral Function of Quark Spectral Function of Quark

0 0 0 0 0( , ) ( , ) ( , )p p p p p pρ ρ γ ρ γ− −+ +Λ Λ= + Kitazawa, Nemoto and T.K.,Phys. Lett. B633, 269 (2006),

[ ] 1Re ( , ) | | Re ( , ) 0S ω ω ω−+ += − − Σ =p p p

Quasi-dispersion relation for eye-guide;

•Three-peak structure emerges.•The peak around the origin isthe sharpest.

00 0 0( , ) ( , ) ( , )±Σ = Σ Σ∓ Vp p p p p p

Fluctuations of the chiral codensateFluctuations of the Fluctuations of the chiralchiral codensatecodensatesharp peak in time-like region

σ , π-mode

Spectrum of the fluctuations

p

ω

2 2 ( )ω ≈ ± +s sp m T

propagating mode

T = 1.1Tc μ=0

c.f.: diffusion-like mode in diquark fluctuations

sm

sm−

CSC

T

σm

πm( )sm T

Tc

m

p

2~ pω

para

quark + massive boson.( , )niωΣ =p

Near the critical point , the soft-modes may be representedby an elementary boson. Yukawa model!

Quark Spectrum in Yukawa modelsQuark Spectrum in Yukawa modelsQuark Spectrum in Yukawa models

0

0.1

0.2

0.3

0.4

p0

-0.8 -0.4

0 0.4

0.8

p

2

4

6

8

10

ρ+ (p,p0)

0

0.1

0.2

0.3

0.4

p0

-0.8 -0.4

0 0.4

0.8

p

T=1.2mB

ω

The 3-peak structure emerges irrespective of the type of the boson at .BT m∼

Kitazawa, Nemoto and T.K.,Prog. Theor. Phys.117, 103(2007),

g=1 , T/m=1.5

Massie scalar/pseudoscalar boson Massie vector/axial vector boson

Remark: Bosonic excitations in QGP may includeσ, π, ρ, J/ψ, … /glue balls… How about the case of Vector manifestaion.(B-R,H-Y-S)

(at one-loop)

quark anti-q hole quark

anti-q quark hole anti-q

quark part:

anti-quark part:

The level crossing is shifted by the mass of the fluctuation modes.

Mechanism of the 3Mechanism of the 3--peak peak fromationfromation

p

[MeV]

ω [MeV]

p

[MeV]

ρ-

(

,p) ρ+

(

,p)

level repulsion

level repulsion

: the HTL result only with the normal quark and plasmino.0bm →

0bm →

ω ωbω bω

bω bω

DiscussionsDiscussionsDiscussions

The complex quasi-quark pole --- ‘Gauge independence’The complex quasiThe complex quasi--quark pole quark pole ------ ‘‘Gauge independenceGauge independence’’

The three residues comparable at T ~ mb which support 3-peak structure

There are three poles corresponding to the three peaks in the spectral function; the pole distribution is symmetric with respect to the imaginary axis because mf = μ

= 0

The sum of the three residues approximately satisfy the sum rule

Pole position

T-dependence of the residuesT↑

Mitsutani, Kitazawa, Nemoto, T.K.Phys. Rev. D77, 045034 (2008)

Finite quark mass effects Finite quark mass effects Finite quark mass effects

•There still exist three poles.•The pole at T=0 (red) moves toward the origin as T is raised.•The pole in the ω

< 0-region has a

larger imaginary part than that in the positive-ω

region for the same T.

•The residue at the pole in the negative ω

region is suppressed at T ~ mb ,

corresponding to the suppression of the peak in the negative-energy region.•The sum of the residues approximately satisfy the sum rule also in this case.

mf / mb = 0.1

Mitsutani, Kitazawa, Nemoto, T.K.Phys. Rev. D77, 045034 (2008)

Re Z

mf / mb = 0.3mf / mb = 0.2

The pole at T=0 moves toward the origin as T is raised. This behavior is qualitatively the same as in the case of lower masses.

The pole at T=0 moves toward the large-ω

region as T is raised.

This behavior is qualitatively different from that in the smaller mass cases..

Structure change of the pole behaviorStructure change of the pole behaviorStructure change of the pole behavior

critfm

The physics contents of the three poles change at a critical mass . We find / 0.21crit

f bm m ≈

Beyond one-loopBeyond oneBeyond one--looploopSchwinger-Dyson approach for lin. sigma model; Harada-Nemoto(’08)

The three peak structure

in the quark spectral function is still therefor small momenta,although the central peak gets tohave a width

owing to

multiple scattering.

Possible confirmation in Lattice QCDPossible confirmation in Lattice QCDPossible confirmation in Lattice QCD

Unquenched lattice simulation with hopefully chiral fermionaction on a large lattice isnecessary for accommodate the possible chiral fluctuationswith energy comparable to 200cT ∼ MeV.

Harada, Nemoto,0803.3257(hep-ph)

Future problems:Full self-consistent calculationConfirmatin in the lattice QCDexperimental observables ; eg. Lepton-pair production (PHENIX?)

transport coefficientsSoft mode (density-fluctuations) at the CEP and quark spectrum

4. Summary and concluding remarks4. Summary and concluding remarks4. Summary and concluding remarks

In the fermion-boson system with mF

<<mB

, the fermion spectral function has a 3-peak structure at 1-loop approximation at T ~ mB

.

The physical origin of the 3-peak structure is the Landau damping ofquarks and anti-quarks owing to the thermally excited massive boson,which induces a mixing between quarks and anti-quark hole,

If the chiral transition is close to a second order, quarks mayhave a 3-peak structure in the QGP

phase near Tc

.

If a QCD phase transition is of a second order or close to that, thereshould exist specific soft modes, which may be easily thermally excited.

The boson may be vector-type or glueballs.