Some Topics on Chiral Transition and Color Superconductivity
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Transcript of 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
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
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
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
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
Quarks at very high T (T>>>Tc) 1-loop (g<<1) + HTL approx. T),,( qmp
),( 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
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 EEEpE
1hE
2hE
2hE
dispersion relations
T
CSChadron
QGP
Tc
Quarks at very high T (T>>>Tc) 1-loop (g<<1) + HTL approx. T),,( qmp
),( p
thermal masses 222
6
1Tgm f
0)],(Re[ pD
2,1, hhp EEE 0)],(Re[ pD
2,1, hhp EEE2hE
2hE
dispersion relations
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
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.
Futurefinite quark mass effects. (2nd order crossover)
finite coupling with density fluctuation; CEP?
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.
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
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