Transcript The sigma meson and chiral transition in nuclear medium
Some Topics on Chiral Transition and Color Superconductivity Teiji Kunihiro (YITP)
HIM Nov. 4-5, 2005 APCTP, Pohang
QCD phase diagram
T T c
ＩＩＩ
RHIC
QGP QCD CEP
GSI,
ＩＩ c SB CSC Ｉ 0 superconducting phases meson condensation? Various phases
CFL
m
Color Superconductivity;
diquark condensation
Dense Quark Matter: quark (fermion) system with attractive channel in one-gluon exchange interaction.
q q [ ３ ] c × [ ３ ] c ＝
[
３
] c
＋ [ ６ ] c Cooper instability at sufficiently low
T
p
SU
(3)
c
gauge symmetry is broken!
D ~100MeV at moderate density m
q
~ 400MeV Attractive!
Fermi surface -p
T
confinement chiral symm. broken Color Superconductivity(CSC) m
(Dis)Appearance of CFL and Gapless phases in Charge- and Color-neutral System at T=0
strong coupling
Abuki, Kitazawa and T.K.; P.L.B615,102( ‘ 05) ｇ：
gapless weak coupling
m H .Abuki and T.K. hep-ph/0509172: Abuki, Kitazawa and T.K.; P.L.B615,102( ‘ 05) The phase in the highest temperature is 2SC or g2SC.
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
QCD phase diagram
T
QCD CEP QGP
preformed pair fields?
quark spectra modified?
c SB
?
CSC 0 H matter?
meson condensation?
CFL
m
The nature of diquark pairs in various coupling
T
0
strong coupling!
weak coupling
m D
~ 50-100MeV
D
/ E F
~ 0.1-0.3
in electric SC D
/ E F
~ 0.0001
Mean field approx.
works well.
Very Short coherence length x .
There exist large fluctuations of pair field even at T>0 .
Large pair fluctuations can invalidate MFA.
cause precursory phenomena of CSC.
relevant to
newly born neutron stars or intermediate states in heavy-ion collisions (GSI, J-Parc)
Collective Modes in CSC
Response Function of Pair Field
Linear Response external field:
H ex
d
x
D †
ex
5 2 2 h.c.
expectation value of induced pair field: 5 2 2
C ex
t t
0
ex
D
ind D R
(
x
2
G C
2
G C
5 2 2
C
(
x
) 5 2 2
C e x
d
x
D R
i
5 2 2
C
(0) ' ) D
ex
x
) (
t
) Retarded Green function Fourier transformation with Matsubara formalism RPA approx.: 1
G Q C
G Q C
n
)
n
) with
Q
k
n
)
After analytic continuation to real time
, Equivalent with the gap equation (Thouless criterion)
Precursory Mode in CSC
(Kitazawa, Koide, Nemoto and T.K., PRD 65, 091504(2002)) Spectral function of the pair field at T>0 ε→ ０ （Ｔ → Ｔ Ｃ ） at
k
=0
T
T C T C
As
T
is lowered toward
T C
, The peak of becomes sharp.
The peak survives up to 0.2
(Soft mode) electric SC ： Pole behavior 0.005
The pair fluctuation as the soft mode;
--- movement of the pole of the precursory mode-- diffusion-mode like
How does the soft mode affect the quark spectra?
---- formation of pseudogap ----
T-matrix Approximation for Quark Propagator
1
G
n
)
G
0
i
n
)
k
i
n
)
G
0
i
n
) (
i
n
0 1
n
)
q
,
i
m
,
i
n
i
m
T
m d
3 q 3 (
k q
,
n
m
)
G
0
m
) Soft mode Density of States N ( ):
N
d
3
k
3 0 0
N
3
d x
1 4 Tr 0 Im
G R
1-Particle Spectral Function
quasi-particle peak of anti-particle,
=
k
m m = 400 MeV =0.01
quasi-particle peak,
=
k)~ k
m position of peaks
k k F
=400 MeV Fermi energy quasi-particle peaks at =
k
)~
k
m and =
k
m .
Quasi-particle peak has a depression around the Fermi energy due to
resonant scattering
.
k
m = 400 MeV 0
k
,
k
m : on-shell
0
, 0 : the soft mode
Density of states of quarks in heated quark matter
m = 400 MeV
Pseudogap structure appears in N(
).
The pseudogap survives up to ~ 0.05 ( 5% above T C ).
Fermi energy
Density of States in Superconductor
Quasi-particle energy: sgn(
k
m ) (
k
m ) 2 D 2 D D m Density of States:
N
( )
d
dk
k
2
k dk d
m (
k
m ) 2 D 2
N
2 D m The gap on the Fermi surface becomes smaller as
T
is increased, and it closes at
T c
.
k
Pseudogap
:Anomalous depression of the density of state near the Fermi surface in the normal phase. Conceptual phase diagram of HTSC cuprates Renner et al.(‘96) The origin of the pseudogap in HTSC is
still controversial
.
Diquark Coupling Dependence
stronger diquark coupling
G C G C
× 1.3
m = 400 MeV =0.01
× 1.5
Resonance Scattering of Quarks
G C
=4.67GeV
-2 Mixing between quarks and holes
k
(M.Kitazawa, Y. Nemoto, T.K.
hep-ph/0505070; Phys. Lett.B , in press)
Summary of this section
There may exist a wide T region where the precursory soft mode of CSC has a large strength.
The soft mode induces the pseudogap , Typical Non-Fermi liquid behavior resonant scattering
Future problems :
effects of the soft mode on .
H-I coll. & proto neutron stars
eg.1) collective mode: ( ,
k
) anomalous lepton pair production eg.2) ; cooling of newly born stars (M.Kitazawa and T.K
in progress.)
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. (
p
, ,
m q
T) ( ,
p
)
thermal masses
m f
2 1 6
g
2
T
2
T Tc hadron QGP CSC
E p
dispersion relations
Re[
D
( ,
E p p
)] , 0
E h
1 , Re[
D
( ,
p
)] 0
E p
,
E h
1 ,
E h
2
E h
2
E p E h
1
E h
1
plasmino plasmino
m
quark distribution anti-q distribution
n
(
E
) anti-quark like
E n
(
E
) particle(q)-like
E
Plasmino excitation
QCD phase diagram and quasi-particles
T
precursory hadronic modes?
free quark spectrum?
QCD CEP QGP c SB
?
CSC 0 H matter?
meson condensation?
CFL
m
Interest in the particle picture in QGP
2Tc T
RHIC experiments robust 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 fluctuation effects Tc fluctuation effects E
m
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
...
spectral function
A
(
k
) 1 Im
D
(
k
) Hatsuda and T.K,1985
T
T c
, they become elementary modes with small width!
1.2Tc
sharp peak in time-like region M.Kitazawa, Y.Nemoto and T.K. (05)
k T
1 .
1
T C
, m 0
Chiral Transition and the collective modes
para sigma para pion 0 c.f. Higgs particle in WSH model ; Higgs field Higgs particle
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)
L
q i
q
G S
[(
q q
) 2 (
q i
5
q
) 2 ] ：
SU(2) Pauli matrices
m u
G S m d
5 .
5 10 6 GeV -2 , 0 chiral limit 631 MeV finite : future work
u d
Chiral phase transition takes place at Tc=193.5 MeV(2 nd order).
Self-energy of a quark (above Tc)
(
n
,
D
(
n
,
p
)
p
) T
m
d 3
q
( 2 ) 3
D
(
n
=
m
,
p
q
)
G
0 (
m
,
q
) T->Tc, may be well described with a Yukawa coupling.
+ +
D
(
n
m
,
p G
0 (
m
,
q
)
q
)
scalar and pseudoscalar parts
+ …
R
( ,
p
) (
n
,
p
) |
i
n
i
: imaginary time real time
Spectral Function of Quark Quark self-energy
i
n
)
Spectral Function
A
p
p
0 ) quark
p
0 ) 0
p
0 antiquark ) 0 1 2 (1 ) for free quarks, ( p,
p
0 ) (
p
0 p ) 3 peaks in also 3 peaks in
p
0
p
0 |
p
| Re 0
|p| |p| p
0
T
1 .
05
T C
, m 0
Resonant Scatterings of Quark for CHIRAL Fluctuations
(
p
,
p
0 ) :
p
0 Im
p
0 = + + … fluctuations of
almost elementary boson at
T
T c E p
0
E p
0
p
0 Re
p
0 dispersion law |
p
| Re 0
p
0
T
1 .
08
T C
, m 0
T
1 .
05
T C
, m 0
Resonant Scatterings of Quark for CHIRAL Fluctuations
“quark hole”: annihilation mode of a thermally excited quark “antiquark hole”: annihilation mode of a thermally excited antiquark (Weldon, 1989)
E
the ‘mass’ of the elementary modes
p
0
p
0
E
(
,k)
lead to quark ”antiquark hole” mixing [MeV]
+ (
,k)
cf: hot QCD (HTL approximation) (Klimov, 1981) quark
p
0 0
p p p
[MeV]
p p T
1 .
05
T C
, m 0 plasmino
Quarks at very high T (T>>>Tc)
1-loop (g<<1) + HTL approx. (
p
, ,
m q
T) ( ,
p
)
thermal masses
m f
2 1 6
g
2
T
2
T Tc hadron QGP CSC
E p E h
1
dispersion relations
Re[
D
( ,
E p p
)] , 0
E h
1 , Re[
D
( ,
p
)] 0
E p
,
E h
1 ,
E h
2
E h
2
E h
2
E p
E h
2
E h
1 m
Quarks at very high T (T>>>Tc)
1-loop (g<<1) + HTL approx. (
p
, ,
m q
T) ( ,
p
)
T QGP Tc hadron CSC thermal masses
m f
2 1 6
g
2
T
2
E p E h
1
dispersion relations
Re[
D
( ,
E p p
)] , 0
E h
1 , Re[
D
( ,
p
)] 0
E p
,
E h
1 ,
E h
2
E h
2
E p E h
2
E h
2
E h
1 m
Quarks at very high T (T>>>Tc)
1-loop (g<<1) + HTL approx. (
p
, ,
m q
T)
T
( ,
p
)
QGP Tc CSC thermal masses
m f
2 1 6
g
2
T
2
hadron spectral function of the space-like region dispersion relations
Re[
D
( ,
E p p
)] , 0
E h
1 , Re[
D
( ,
p
)] 0
E p
,
E h
1 ,
E h
2
E h
2 0
E h
2
E h
2 10 0 /
mf p/mf
10 -10 m
Difference between CSC and CHIRAL
above CSC phase: One resonant scattering fluctuations of the order parameter ~ diffusion-like (
p
) ~
p
2 (
p
~ 0 ) above chiral transition: Two resonant scatterings fluctuations of the order parameter ~ propagating-like (
p
) ~ 0 ( 0) (
p
~ 0 )
Level Repulsions
For massless gauge field,
A
p
p
)
p
)
p p A
p
p
2 0 2 )
p
2 0 2 )
p
0 0
p
0 0 0 0
p p
Spectral function of the quarks
m
E q E T
0 , m 0 anti-quark like
n
(
E
)
q T
0 , m m 0
E n
(
E
) particle(q)-like
q
particle-like states suppressed
E n
(
E
)
n
(
E
)
Summary of the second part
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
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 freedom.
can be reproduced by a Yukawa theory with the boson Future being a scalar/pseudoscalar or vector/axial vector one
(Kitazawa, Nemoto and T.K.,in preparation) finite quark mass effects. (2 nd m order crossover)
Summary of the Talk
T
2.precursory hadronic QCD phase diagram modes?
strongly modified quark spectra
c 0 QCD CEP
`QGP’ itself seems surprisingly rich in physics!
?
CSC
1. preformed pair fields?
quark spectra modified?
CFL
coupled Quark-Gluon systems will H matter?
m
Back Upps
Pairing patterns of CSC
for J P =0 + pairing D
ij
i
Ci
5
j
attractive channel : color anti-symm.
flavor anti-symm.
m <
M s
Two Flavor Superconductor (2SC)
u d a,b : color i,j : flavor D
ij ijk
d
k
m >>
M s
Color-Flavor Locked (CFL)
u d u d s
d
SU
(3)
c
SU
(2)
c
0 0 D s u d s
d
D 1
SU
(3)
c
SU
(3)
SU
(3)
L
SU
(3)
R
D 2 D 3
BCS-BEC transition in QM
1
TBEC_RL
Y.Nishida and H. Abuki, hep-ph/0504083 0.1
Tmf TBEC
0.01
0.001
0.8
0.9
1.0
G
/
G
0 1.1
1.2
Calculated phase diagram
q q
0
q q
0
Fermions at finite T
free massless quark at T=0
S
0 ( ,
p
) 1 /
p
1 2 0 |
p
| |
p
| 1 2 0 |
p
| quark and antiquark
quark at finite T (massless)
S
( ,
D D
p
) ( , ( ,
D
( ,
p p
)
p
) )
A
( , 0 0 0
p
) 0 1
C
( ,
p
)
E p
1 2
several solutions several solutions
0
D
( ,
p
) 1 2 0
D
( ,
D
p
)
A
C
0 ,
E h
0
D
( ,
p
) 0
E h
0 ,
E p
0
p
Formulation (Self-energy)
( , 0 ) 3
d p
3
E C E p
(
E
(
E E p
p
))
p
0
p
)
E p
E E p
p
)) 1
p
0
E p p
)
E
)
i
p
0
E p
)
p E
i
) 1
p
0
E p p
)
E
)
i
i
) ... = 1
D
Quark self-energy: ( , 0 ) 1 1 4
d q
4 Im ( , 0
q
0
p
0
p
i
) ( 0 ) coth
q
0 2
T
tanh
q
2
T
4
d q
4 Im ( , 0
q
0
p
0
p
i
) ( 0 ) coth
q
0 2
T
tanh
q
2
T
Formulation (Spectral Function)
Spectral function: ( , 0 ) 0 ( , 0 ) 0
V
( , 0 ) [ ( , 0 ) ( , 0 ) quark antiquark ( , 0 ) 1 Im
p
0
p
1 ( , 0 )
i
0 1 2 (1 )
p
Spectral Contour and Dispersion Relation
(
,k)
+ (
,k)
(
,k)
+ (
,k)
1.05 Tc 1.1 Tc p p p p
(
,k)
1.2 Tc
+ (
,k)
p p
(
,k)
1.4 Tc
+ (
,k)
p