Kyushu, 25, Dec., 2008

QCD相図の理論的・実験的理解に向けて北沢正清

Kyushu, 2, Nov., 2007

QCD相図の理論的・実験的理解に向けて～QCD相図上におけるクォーク伝搬関数の振る舞い～北沢正清

Quasi-Particles in the QGP Phase•quarks and gluons•hadronic modes, diquarks, etc….

Why we study excitation modes?
•connection to exp. obs. / bulk thermodynamics•numerical results from lattice simulations•analogy to condensed matter phys.

Quark Propagator in the Phase Diagramhigh-T / dense limit:
Ethermal mass:

TpmT ~ g T 2  m 2 / 3

possible modificationin sQGPgap on
Fermi surface

mchiral symm. broken
 5 S ( , p)  S ( , p) 5  0pFDp

Quarks at Extremely High T•Hard Thermal Loop approx. ( p, , mq<<T )•1-loop (g<<1)(, p) Klimov ‟82, Weldon ‟83
Braaten, Pisarski ‟89

1S ( , p)  0  p  γ  ( , p)

•2 collective excitations
having a “thermal mass” ~ gT
• width ~g2T
•The plasmino mode hasa minimum at finite p.

 / mT•Gauge independent spectrum“plasmino”

gTmT 6p / mT

Decomposition of Quark Propagator 0S (, p)  S (, p)  (p)
 0
 S (, p)  (p)  (p) 
Ep   0 (p    m)
2 Ep
Ep  p 2  m 2

Free quark with mass m  (p) 0  (p) 0
SHTL ( , p)   p     p    (p) 0   (p) 0
Sfree ( , p)   Ep  Ep/m

 / mT

HTL ( high T limit )p / mTp/m

Coupling to Bosonic Modes
Quasi-particle picture of the quark can be modified
by other excitation modes.

— gluons, hadronic modes, …
— chiral soft modesD ( x)  2GS  ( x), (0) ~r(p,)T =1.1Tc

Tspectral func. r in NJL model“pseudogap” region of CSCm

Fermion Spectrum
in Relativistic
Fermion-Boson System

Fermion Spectrum in Yukawa Models
quark + massive scalar boson

1
L  i (i   m f  g )    m m  mb2 2 2Model parameter : g, mf, mb, (T)(1) mf=0, mb>0•light quarks in the QGP phase coupled with massive bosons•quarks and leptons coupled with W and Z at T~mW,Z(2) mf>0, mb=0

MK, Kunihiro, Nemoto, ‟06,‟07
Boyanovsky, „06•massive fermions in gauge theories•electrons at T~me•heavy quarks in QGP
Baym, Blaizot, Svetitsky, ’92

Low T limitmax(m f , mb )  T

High T limitmax(m f , mb )  T

KG-type dispersion

HTL dispersion

 / mT / mfHigh and Low T Limitsp / mfm/T  1
heavy mass limit

w/ fixed Tp / mTm/T  1massless limit

Case2 : mb=0
quark spectral function in 1-loop approx.:

Spectral Function for g =1 , T =1

m f / T  1
thermal mass mT=gT/4
m f / T  1
single peak at mf

m0/T=0.01

r+(,p=0)

0.1
0.3
0.45

/T

0.8

•Two limits are connected
continuously.
•Plasmino peak disappears
as mf /T becomes larger.

•Note: Qualitative result hardly changes with a variation of g, andin Yukawa models with PS, V, AV interactions, and in QED.

Case1 : mf=0
low T / heavy mbquark spc r+(,p) w/ g=1

T/mb=0.4

T/mb=1high T / small mb
T/mb=2.5

T/mb=1.4

Case1 : T dependence for p=0r (p  0,  )g=1
one-loop approx.

Three peak structure is formed at T/mb~1 as an interpolating
behavior between single- and two-pole limits.
Similar result is obtained even with Schwinger-Dyson approach.
Harada, Nemoto, „08

Case1 : T dependence for p=0r (p  0,  )z
g=1 polecomplex
one-loop approx.

Three peak structure is formed at T/mb~1 as an interpolating
behavior between single- and two-pole limits.
Similar result is obtained even with Schwinger-Dyson approach.
Harada, Nemoto, „08

Quark Spectrum in NJL Model
Nambu-Jona-Lasinio model (2-flavor, chiral limit)：
L   i   GS ( )2  ( i 5 τ )2 GS  5.01GeV 2  650MeV

Tc=193.5MeV

Quark self-energy
(k, ) 







Spectral Function of Quark
A(p, p 0 )  r  (p, p 0 )   0  r  (p, p 0 )   0e = 0.05

k [MeV] [MeV]•Three-peak structure emerges.
•The peak around the origin is the sharpest.

T  TceTcSpectral Function of Quark
A(p, p 0 )  r  (p, p 0 )   0  r  (p, p 0 )   0T  TceTc

e = 0.05

 [MeV]k [MeV] [MeV]

Quasi-dispersion relation

Re  S (, p)    | p |  Re  (, p)  0
1

k [MeV]Spectral Function of Quark
A(p, p 0 )  r  (p, p 0 )   0  r  (p, p 0 )   0T  TceTc

e = 0.05

 [MeV]r-(,k)k [MeV]k [MeV]r+(,k)k [MeV]T Dependencem /T 0T  Tc

T =1.1Tc
T  large

T =1.2Tc

T =1.5Tc

As T is raised, the spectral function
and the quasi dispersion relation
approach the free quark spectrum.

Self-energy in Yukawa Model

Decay processesin p plane

Imaginary part for p=0
Im (,0)
T =1
T =1.5
T =2TmB

 / mB
Landau dampingsRe (,0)k-mB

 / mBIm  for Chiral Transition
Origin of the two peaks in Im= Landau damping processes

quarksoft modeannihilate the anti-quark
make a ‘hole’ of the anti-quark
induce the mixing b/w quark & ‘hole’ of anti-quark

Resonant Scatterings of Quarks
quark part:m

Virtual mixing b/w two d.o.f.
induces level repulsionm [MeV]quark

anti-q holequark

anti-quark part:mm

anti-qquark holer-(,k)k [MeV]anti-qr+(,k)k [MeV]Resonant Scatterings of Quarks
quark part:m

Due to the finite mass of soft modes,
the mixing becomes more complicated.

m [MeV]quark

anti-q holequark

anti-quark part:mm

anti-qquark holer-(,k)k [MeV]anti-qr+(,k)k [MeV]Connection between Cases 1 & 2
mf=0

mf/mb=0.2

mf/mb=0.33-peak structure ceases to exist as mf/mb becomes larger.
Mitsutani, et al. „08

Mitsutani, et al. „08

Pole Structure
mf=0mf/mb=0.2“level crossing between two poles!

mf/mb=0.3

Quark Spectrumin Lattice QCD

Simulation Setup•quenched approximation•clover improved Wilson•Landau gauge fixing•2-pole approx. for r+(,p=0)•wall sourcebT
3Tc

7.45

1.5Tc

7.19
6.87
6.64

1.25Tc 6.72
0.9Tc 6.42
0.55Tc 6.13

size
643x16483x16483x12
643x16483x16483x12
643x16483x16483x16483x16# of conf.
51
51
51
44
51
51
40
44
50
56

Case2 : mb=0
quark spectral function in 1-loop approx.:

Spectral Function for g =1 , T =1

m f / T  1
thermal mass mT=gT/4
m f / T  1
single peak at mfm0/T=0.01

r+(,p=0)0.10.30.45

/T

0.8

•Two limits are connected
continuously.
•Plasmino peak disappears
as mf /T becomes larger.

Correlator and Spectral Functione( b / 2 )
C ( )   d b / 2  b / 2 r ( )
e
e


observablein latticedynamical
information

2-pole structure may be a good
assumption for r+().

r ( )  Z1d (  E1 )  Z 2d (  E2 )4-parameter fit E1, E2, Z1, Z2

Z2
E2Z1E1

Correlation Function
643x16, b = 7.459, k = 0.1337, 51confs.

C ( , 0)  C ( )   C ( ) 
 C0 ( ) 0  CS ( )Fitting resultC ( )  z1 e E1  z2 e E2 ( b  )C ( )
 /T
•We neglect 4 points near the source from the fit.
•2-pole ansatz works quite well!! ( c 2/dof.~1 in correlated fit)

Spectral Function
T=3Tc

T = 3Tc 643x16 (b = 7.459)

r ( )  Z1d (  E1 )
 Z 2d (  E2 )

E/T

E2

 = m0Z2 / (Z1+Z2)

E1pole of free quark1 1 1 m0    2  k kc Z2
Z1  Z 2
m0 / T
Z2Z1
E2E1Z1



Z2
E2E1



Spectral Function

Z2 / (Z1+Z2)

E/T

T=3Tc

T = 3Tc 643x16 (b = 7.459)

r ( )  Z1d (  E1 )
 Z 2d (  E2 )E2
E1Z2
Z1  Z 2

 = m0pole of free quark1 1 1 m0    2  k kc m0 / T•Limiting behaviors for m0  0, m0   are as expected.
•Chiral symmetry of quark propagator restores around m0=0.
•Quarks in the chiral limit have a thermal mass!
•E2>E1 : qualitatively different from the 1-loop result.

Temperature Dependence

643x16

Z2 / (Z1+Z2)

E/T

E2
E1minimum of E1

T = 3Tc
T = 1.5Tc
T = 1.25Tc

Z2
Z1  Z 2
m0 / T•mT /T is insensitive to T.
•The slope of E2 and minimum of E1 is much clearer at lower T.
1-loop result might be realized for high T.

Lattice Spacing Dependence
T=3Tc

E/T

E2643x16(b = 7.459)
E1483x12(b = 7.192)
same physical volume
with different a.

m0 / T•No lattice spacing dependence within statistical error.

Spatial Volume Dependence
T=3Tc

E/T

E2643x16(b = 7.459)
E1483x16(b = 7.459)
same lattice spacingwith different aspect ratio.

m0 / T•Excitation spectra have clear volume dependence
even for N /N =4.

Extrapolation of Thermal MassmT /T

Extrapolation of thermal mass to infinite spatial volume limit:

483x16T=1.5Tc
mT /T = 0.800(15)mT = 322(6)MeV643x16

3T=1.25TcmT /T = 0.816(20)mT = 274(8)MeV

3N / N ~ 1/ V

T=3Tc
mT /T = 0.771(18)mT = 625(15)MeV•Small T dependence of mT/T,•while it decreases slightly with increasing T.
•Simulation with much larger volume is desirable.

Quark Dispersion

Karsch, MK, to appear soon.

in quenched approx., 643x16

HTL(1-loop)(plasmino)•Lattice results behave
reasonably as functions of p.
p/TCharm Quarkk from Datta et al. PRD69,094507(2004).
T=1.5Tc
T = 3Tc
T = 1.5Tc

mc threshold 2mc•Charm quark is free-quark like, rather than HTL.
•The J/ peak in MEM seems to exist above 2mc.

Effect of Dynamical Quarks
Quark propagatorin quench approximation:

In full QCD,
 screen gluon field  suppress mT?
 meson loop  will have strong effectif mesonic excitations exist
massless fermion + massive boson
 3 peaks in quark spectrum?
M.K., Kunihiro, Nemoto, „06

Below Tc
C ( ) for 483x16 lattices

T/Tc= 3, 1.5, 1.25,0.9, 0.55.

Quark correlator below Tc•is convex upward.
•inconsistent with positive r ().
•does not approach the chiralsym. one in the chiral limit.

まとめ
有限温度相対論的フェルミオン・ボソン系において、フェルミオンの励起スペクトルは極めて多彩な振る舞いを見せる。
それらの中でも特徴的な3ピーク構造は、QCD相図上でも実現しているかもしれない。
格子QCDを用いた解析により、非閉じ込め相における熱的グルオン場の中を伝搬するクォークは臨界温度付近でも熱質量を持つことが分かった。また、スペクトル関数はベア質量・運動量の関数として自然に振る舞うことが確かめられた。

Some Discussions

Consequences of Nonzero mT
How does the thermal mass affect(1) chiral phase transition(2) mesonic excitation composed of thermally gapped quarks

L   DHTL  GS [( )  ( i 5 ) ]2DHTL   p  mT  HTL (, p) 12

Interaction:+HTL (, p) +Hidaka, MK, PRD75, 011901(R) (2007)

Thermodynamic Potential( ) 24G

T n

d 3k
 (i , k ; m ,  ) trlnSn
T3(2 )

  2G 
S ( , k ; mT ,  ) 1

 0  k  γ   HTL ( , k ; mT )  M/

mT
condensate M

T/Meson Spectral functionD 

mT=0++…

Height of the peaks are
depressed with anincorporation of mT.

mT=0.32mT

Thermal mass is not
responsible for the stabilityof quark composites.

What is the Plasmino?

Weldon ‟89

SHTL ( , p)

 / mT  (p) 0  (p) 0  p     p  

Landaudamping= A quark is scattered by a gluon.

= A quark and a thermallyexcited anti-quark annihilate
and produce a gluon.
The quark turns into
an “anti-quark hole”.