1. Phase Diagram of Dense Neutral Quark Matter with Axial Anomaly and Vector Interaction
Teiji Kunihiro (Kyoto)
In collaboration with
M. Kitazawa, T. Koide, Y. Nemoto
K. Fukushima and Zhao Zhang
New-type of Fermions on the Lattice
2012/02/ /02/ YITP, Kyoto University

2. Contents of talk
Effects of the vector interaction on phase diagram with color superconductivity
Effects of charge and beta-equilibrium constraints
Role of Axial anomaly on phase diagram with CSC
Summary and concluding remarks

3. Fluctuations of chiral order parameter around Tc in Lattice QCD 
the softening of　the  with
increasing T

5. Conjectured QCD phase diagram
Chiral fluctuation effects
2Tc
What are physics contents,
hadrons, partons, or percolated multi-quark states ?
QGP Tc
Critical point
Di-quark
fluctuation effects ?
Hadron
CSC m

6. (Tri-)Critical Point in NJL model
MeV CP
crossover
TCP T
Asakawa,Yazaki,(1989) CP
TCP m

7. Caution!

8. Effects of GV on Chiral Restoration
GV→Large
As GV is increased,
First Order
Cross Over
Chiral restoration is shifted to higher densities.
The phase transition is weakened.
Asakawa,Yazaki ’89 /Klimt,Luts,&Weise ’90 / Buballa,Oertel ’96
What would happen when the CSC joins the game?

9. Importance of Vector-type Interaction for CSC
Instanton-anti-instanton molecule model Shaefer,Shuryak (‘98)
Renormalization-group analysis N.Evans et al. (‘99)
Vector interaction naturally appears in the effective theories. m
density-density correlation
An important observation is that
chiral restoration is suppressed by the vector interaction
or density-density interaction!

10. Contour maps of thermal potential
Intuitive understanding of the effect of vector interaction on chiral restoration
Kitazawa, Koide, Kunihiro & Nemoto (’02)
Contour maps of thermal potential
The possible large
density leading to
CSC is `blamed’
by the vector
interaction.

11. (4) With color superconductivity transition incorporated:
Two critical end point!
M. Kitazawa, T. Koide, Y. Nemoto and T.K., PTP (’02)
(4)
Another end point appears from lower temperature, and hence there can exist two end points in some range of GV !

12. Similarity of the effect of temperature and pairing gap on the chiral condensate.
M. Kitazawa et al.
PTP, 110 (2003), 185:
arXiv:hep-ph/ T Δ

13. Yet another critical point, due to charge neutrality.
Z. Zhang, K. Fukushima, T.K., PRD79, (2009)
G; chiral, H; diquark
“remnant’’
of the 1st-order
Chiral transition
1st order
Large diquark-
pairing
“suviver’’ of the 1st-order
chiral transition
At low(moderate) T,
diquark-pairing is
suppressed(enhanced).
Charge neutrality gives rise to
a mismutch of the Fermi surfaces.

14. Effect of electric chemical potential with neutral CSC
Asymmetric homogenous CSC with charge neutrality
Mismatch cooper paring
nd > nu > ns
Mismatch paring or pair breaking, real case
Standard BSC paring , rare cace
For two flavor asymmetric homogenous CSC
Abnormal thermal behavior of diquark gap
Chromomagnetic instibility, imaginary meissner mass

15. Abnormal thermal behavior of diquark energy gap
Smearing by T induces the pairing! d u p
Melting the condensate
More and more components take part in cooper pairing
Double effects of T :
Competition between these two effects gives rise to abormal thermal behavior of diquark condensate
Enhancing the competition between chiral condensate and diquark condensate for somewhat larger T, leading to a nontrivial impact on chiral phase transition
Shovkovy and Huang, PLB 564, (2003) 205

16. Yet another critical point, due to charge neutrality.
Z. Zhang, K. Fukushima, T.K., PRD79, (2009)
G; chiral, H; diquark
“remnant’’
of the 1st-order
Chiral transition
1st order
Large diquark-
pairing
“suviver’’ of the 1st-order
chiral transition
At low(moderate) T,
diquark-pairing is
suppressed(enhanced).
Charge neutrality gives rise to
a mismutch of the Fermi surfaces.

17. Effects of Charge neutrality constraint on the phase diagram
Z. Zhang, K. Fukushima, T.K., PRD79, (2009)
QCD phase diagram with chiral and CSC transitions with charge neutrality
Pairing with mismatched Fermi surface
Competition between chiral and CSC
Charge neutrality play a role similar to the vector-vector(density-density) interactin and leads to proliferation of critical points.

18. Combined effect of Vector Interaction and Charge Neutrality constraint
Z. Zhang and T. K., Phys.Rev.D80:014015,2009.;
chiral
di-quark
vector
anomaly
for 2+1 flavors
diquark-chiral
density coupl.
Fierts tr.
Kobayashi-Maskawa(’70); ‘t Hooft (’76)

19. Increasing Gv/Gs Model set 2 : M(p=0)= 367.5 MeV , Gd/Gs =0.75
2-flavor case
Z. Zhang and
T. K., PRD80
(2009)
Increasing Gv/Gs
4 critical
points !
4 types of critical point structure
Order of critical-point number : 1, 2, 4, 2,0

20. 2+1 flavor case Z. Zhang and T. K., Phys.Rev.D80:014015,2009.;
Similar to the two-flavor case,
with multiple critical points.

21. the anomaly -induced new CP in the low T region
Incorporating an anomaly term inducing the chiral and diquark mixing
a la Hatsuda-Tachibana-Yamamoto-Baym (2006)
(A) Flavor-symmetric case:
Abuki et al, PRD81 (2010),
125010
the anomaly -induced new CP in the low T region

22. (A’) Role of 2SC in 3-flavor quark matter
H. Basler and M. Buballa,
PRD 82 (2010),094004
with

23. (B) Realistic case with massive strange quark;
<<
H. Basler and M. Buballa,
(2010)
Notice!
Without charge neutrality
nor vector interaction.

24. The role of the anomaly term and G_v under charge-neutrality constraint
Z.Zhang, T.K, Phys. Rev. D83 (2011)
G_V=0:
due to the
Mismatched
Fermi surface
Otherwise,
consistent
with Basler-
Buballa

25. Effect of mismatched Fermi sphere
Z.Zhang, T.K, (2011)
1st
crossover
1st
Owing to the mismatched Fermi sphere inherent in the charge-neutrality
constrained system, the pairing gap is induced by the smearing of Fermi
surface at moderated temperature!

26. Effects of G_v G_V makes the ph.tr. a crossover at intermediate T
with much smaller K’.
A crossover
Region gets to
appear, which
starts from zero T.
Eventually,
the ph. tr becomes
crossover in the
whole T region.
This crossover region is extended to higher temperature region.
Z.Zhang, T.K, Phys. Rev. D83 (2011)

27. G_V varied with K’ /K fixed at 1
1. Effects of mismatched
Fermi sphere by
charge-neutrality
2. Then effect of G_V
comes in to make
ph. tr. at low T
cross over.
Z.Zhang, T.K, Phys. Rev. D83 (2011)

28. Fate of chromomagnetic instability
Z.Zhang, T.K, Phys. Rev. D83 (2011)
Finite G_V
G_v=0

29. The essence of Effects of GV on Chiral Restoration
GV→Large
As GV is increased,
First Order
Cross Over
Chiral restoration is shifted to higher densities.
The phase transition is weakened.
Asakawa,Yazaki ’89 /Klimt,Luts,&Weise ’90 / Buballa,Oertel ’96

30. Philippe de Forcrand and Owe Philipesen (‘08)
responsible for the disappearance of QCD critical point at low density according to recent lattice stimulation ?
Philippe de Forcrand and Owe Philipesen (‘08)
GV → Large
K. Fukushima (‘08)

31. 4. Summary and concluding remarks
QCD phase diagram with vector interaction and axial anomaly
terms under charge neutrality and beta-equilibrium constraints.
There are still a room of other structure of the QCD phase diagram with multiple critical points when the color superconductivity and the vector interaction are incorporated.
G_v is responsible for the appearance of another CP at low T, but not
axial anomaly term in the realistic case.
2. The new anomaly-induced interaction plays the similar role as G_V
under charge- neutrality constraint.
3. The message to be taken in the present MF calculation:
It seems that the QCD matter is very soft along the critical line when
the color superconductivity is incorporated; there can be a good
chance to see large fluctuations of various observables like
chiral-diquark-density mixed fluctuations,
4. Various possibilities at finite rho:

32. G_D dependence without g_V
H .Abuki and T.K. :
Nucl. Phys. A, 768 (2006),118
The phase in the highest temperature is　2SC or g2SC.
　The phase structure involving chiral transition at low density region may be parameter dependent and altered.

33. S.Carignano, D. Nickel and M. Buballa, arXiv:1007.1397
E. Nakano and T.Tatsumi, PRD71 (2005)
Interplay between G_V and Polyakov loop is not incorporated;
see also P. Buescher and T.K., Ginzburg-Levanyuk analysys shows also an
existence of Lifschitz point at finite G_V.
Spatial dependence of Polyakov loop should be considered explicitly.

34. ? Conjectured QCD phase diagram CFL  QGP １st T Precursory hadronic
excitations?
QGP
~150MeV
QCD　CP
１st
Hadron phase ?
CSC
Liq.-Gas
CFL m 
H-dibaryon matter?
A few types of
superfluidity
Meson condensations?

35. Back Ups

36. m Contour of w with GV/GS=0.35 T= 22MeV 15MeV 12MeV 5MeV
M. Kitazawa, et al (’02)
15MeV
12MeV
Very shallow or soft
for creating diquark-chiral
condensation!
5MeV m

37. Effects of the vector interaction on the effective chemical potentials
The vector interaction tends to suppress the mismatch of the Fermi spheres of the
Cooper pairs.
Suppression of the Chromomagnetic instability!

38. Suppression of the Chromomagnetic instability
due to the vector interaction!
Z. Zhang and T. K., Phys.Rev.D80:014015,2009.;
(Partial) resolution of the chromomagnetic instability problem!