1. Precursory Phenomena in Chiral Transition and Color Superconductivity
国広　悌二（京大基研）
東大大学院特別講義2005年12月5-7日

2. ? ＩＩＩ ＩＩ Ｉ QCD phase diagram QGP cSB CSC CFL  T RHIC QCD CEP
GSI,J-PARC
cSB Ｉ ?
Various phases
CSC
CFL
BCS-BEC? m 
compact stars
H matter?
superconducting
phases
meson condensation?

3. Color Superconductivity; diquark condensation
Dense Quark Matter:
quark (fermion) system q q
with attractive channel in
one-gluon exchange interaction.
[３]c×[３]c＝[３]c＋[６]c
Attractive! p
Cooper instability at sufficiently low T
Fermi surface -p
SU (3)c gauge symmetry is broken!
D~100MeV at moderate density mq~ 400MeV T
confinement
chiral symm.
broken
Color Superconductivity(CSC) m

4. (Dis)Appearance of CFL and Gapless phases in
Charge- and Color-neutral System at T=0
Abuki, Kitazawa
and T.K.;
P.L.B615,102(‘05)
strong coupling
ｇ： gapless
weak
coupling

5. Various CSC phases in plane
H .Abuki and T.K. hep-ph/ :
The phase in the highest temperature is
2SC or g2SC.

6. 2. Precursory Phenomena of Color Superconductivity in Heated Quark Matter
Ref. M. Kitazawa, T. Koide, T. K. and Y. Nemoto
　　　　　Phys. Rev. D70, (2004);
　　　　　Prog. Theor. Phys. 114, 205(2005),
　　　M. Kitazawa, T.K. and Y. Nemoto,
　　　　　hep-ph/ , Phys. Lett.B 631(2005),157

7. ? QCD phase diagram QGP cSB CSC CFL  T QCD CEP m H matter?
　preformed pair fields?
quark spectra modified?
cSB ?
CSC
CFL m 
H matter?
meson condensation?

8. Pairing patterns of CSC
a,b : color
i,j : flavor
for JP=0+ pairing
attractive channel : color anti-symm.
flavor anti-symm.
m<Ms
m>>Ms
Two Flavor Superconductor
(2SC)
Color-Flavor Locked (CFL) r r u d u d s
u d s
u d s

9. The nature of diquark pairs in various coupling
weak coupling
strong coupling!
D ~ MeV
in electric SC
Mean field approx.
works well.
D / EF ~
D / EF ~
Very Short coherence length x.
There exist large fluctuations of pair field even at T>0.
invalidate MFA.
cause precursory phenomena of CSC.
Large pair fluctuations can
relevant to newly born neutron stars
or intermediate states in heavy-ion collisions (GSI, J-Parc)

10. Collective Modes in CSC
Response Function of Pair Field
Linear Response
external field:
expectation value of induced pair field:
Retarded Green function
Fourier transformation
with Matsubara formalism
RPA approx.:
with

11. After analytic continuation to real time,
Equivalent with the gap equation (Thouless criterion)

12. Spectral function of the pair field at T>0
Precursory Mode in CSC
(Kitazawa, Koide, Nemoto and T.K., PRD 65, (2002))
Spectral function of the pair field at T>0
at k=0
ε→０
（Ｔ→ＴＣ）
As T is lowered toward TC,
The peak of r becomes sharp.
(Soft mode)
Pole behavior
The peak survives up to 　　 electric SC：

13. The pair fluctuation as the soft mode;
--- movement of the pole of the precursory mode---
diffusion-mode
like

14. How does the soft mode affect the quark spectra?
---- formation of pseudogap ----

15. T-matrix Approximation　for Quark Propagator　　
Soft mode
Density of States N(w):

16. 1-Particle Spectral Function
= 400 MeV
e=0.01
quasi-particle peak of anti-particle, w = -k-m
quasi-particle peak,
w =w-(k)~ k-m
position of peaks k
kF=400MeV k w
Fermi energy
quasi-particle peaks
at w =w-(k)~ k-m and w =-k-m.
-m =-400MeV w
Quasi-particle peak has a depression around the Fermi energy due to resonant scattering.
: on-shell
: the soft mode

17. Density of states of quarks in heated quark matter
m = 400 MeV
Pseudogap structure appears in N(w).
N(w)/104
The pseudogap survives up to e ~ 0.05
( 5% above TC ). w
Fermi energy

18. Density of States in Superconductor
Quasi-particle energy:
Density of States:
The gap on the Fermi surface becomes smaller
as T is increased, and it closes at Tc.

19. 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.

20. Diquark Coupling Dependence
m= 400 MeV
e=0.01
stronger diquark coupling GC GC
×1.3
×1.5

21. Resonance Scattering of Quarks
GC=4.67GeV-2
Mixing between quarks and holes w k
(M.Kitazawa, Y. Nemoto, T.K.
Phys. Lett.B631 (2005),157)

22. 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
anomalous lepton pair production
eg.1)
collective mode:
(w,k)
(M.Kitazawa and T.K
in progress.)
; cooling of newly born stars
eg.2)

23. 3. Precursory Hadronic Mode and Single Quark Spectrum above Chiral Phase Transition

24. Fermions at finite T free massless quark at T=0 / quark and antiquark
quark at finite T (massless)
several solutions
several solutions

25. Quarks at very high T (T>>>Tc)
1-loop (g<<1) + HTL approx. T
CSC
hadron
QGP m Tc
thermal masses
dispersion relations
plasmino
plasmino

26. Plasmino excitation quark distribution anti-q distribution
particle(q)-like
anti-quark like
Plasmino excitation

27. ? QCD phase diagram and quasi-particles QGP cSB CSC CFL  T QCD CEP m
precursory hadronic
modes?
free quark spectrum? T
QCD CEP
QGP
cSB ?
CSC
CFL m 
H matter?
meson condensation?

28. Interest in the particle picture in QGP
RHIC experiments
robust collective flow
good agreement with rel. hydro models
almost perfect liquid
(quenched) Lattice QCD
charmonium states up to Tc
(Asakawa et al., Datta et al., Matsufuru et al. 2004)
Strongly coupled plasma rather than weakly interacting gas
2Tc
fluctuation effects
QGP
fluctuation effects Tc E
Hadron
CSC m

29. response function in RPA
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) w
response function in RPA
...
spectral function
sharp peak in time-like region k
Hatsuda and T.K,1985
1.2Tc
, they become elementary
modes with small width!
M.Kitazawa,
Y.Nemoto and
T.K. (05)

30. Chiral Transition and the collective modes
para sigma
para pion
c.f. Higgs particle in WSH model
; Higgs field
Higgs particle

31. How does the soft mode affect a single quark spectrum near Tc?
Y. Nemoto, M. Kitazawa ,T. K.
hep-ph/ ;Phys.Lett. B, in press.

32. Model
low-energy effective theory of QCD Fermi type interaction (Nambu-Jona-Lasinio with 2-flavor)
t：SU(2) Pauli matrices
chiral limit
finite : future work
Chiral phase transition takes place at Tc=193.5 MeV(2nd order).
Self-energy of a quark (above Tc)
T->Tc, may be well described with a Yukawa coupling. = + +
+ …
scalar and pseudoscalar parts
: imaginary time real time

33. Formulation (Self-energy)
... = 1-
Quark self-energy:

34. Formulation (Spectral Function)
quark
antiquark

35. Spectral Function of Quark
Quark self-energy
Spectral Function
for free quarks,
quark
antiquark
3 peaks in
also 3 peaks in
|p|
|p|

36. Resonant Scatterings of Quark for CHIRAL Fluctuations= +
+ …
fluctuations of
almost elementary boson at E E
dispersion law

37. 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
lead to quark-”antiquark hole” mixing E
the ‘mass’ of the
elementary modes
w [MeV]
cf: hot QCD (HTL approximation)
r-(w,k)
r+(w,k)
(Klimov, 1981)
quark p
p [MeV]
plasmino

38. Quarks at very high T (T>>>Tc)
1-loop (g<<1) + HTL approx. T
CSC
hadron
QGP m Tc
thermal masses
dispersion relations

39. Quarks at very high T (T>>>Tc)
1-loop (g<<1) + HTL approx. T
CSC
hadron
QGP m Tc
thermal masses
dispersion relations

40. Quarks at very high T (T>>>Tc)
1-loop (g<<1) + HTL approx. T
CSC
hadron
QGP m Tc
thermal masses
spectral function of the space-like region
dispersion relations 10
p/mf
w/mf
-10 10

41. Difference between CSC and CHIRAL
above CSC phase:
One resonant scattering
fluctuations of the order parameter ~ diffusion-like
above chiral transition:
Two resonant scatterings
fluctuations of the order parameter ~ propagating-like

42. Spectral Contour and Dispersion Relationw w
r-(w,k)
r+ (w,k)
r-(w,k)
r+ (w,k)
1.1 Tc
1.05 Tc p p p p
r+ (w,k)
r-(w,k)
r-(w,k)
r+ (w,k)
1.2 Tc
1.4 Tc p p p p

43. For massless gauge field,
Level Repulsions
For massless gauge field,

44. Spectral function of the quarks

45. Finite dependence; asymmetry between and
particle(q)-like
anti-quark like
particle-like states
suppressed

46. 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 w 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 of freedom.
can be reproduced by a Yukawa theory with the boson
being a scalar/pseudoscalar or vector/axial vector one
Future
(Kitazawa, Nemoto and T.K.,in preparation)
finite quark mass effects. (2nd order crossover)
finite coupling with density fluctuation; CEP?

47. ? `QGP’ itself seems surprisingly rich in physics! QGP cSB CSC
Summary of the Talk
QCD phase diagram
2.precursory hadronic
modes? T
strongly modified quark spectra
QCD CEP
1. preformed pair fields?
quark spectra modified?
QGP
`QGP’ itself seems surprisingly
rich in physics!
cSB ?
CSC
Condensed matter physics of strongly
coupled Quark-Gluon systems will
constitute a new field of fundamental physics.
CFL m 
H matter?

48. Back Upps

49. Calculated phase diagram

50. BCS-BEC transition in QM
Y.Nishida and
H. Abuki,
hep-ph/