1. Ginzburg-Landau approach to QCD phase transitions
Motoi Tachibana (Saga University, Japan)
contents
Introduction to QCD and its phase structure
Chiral-super interplay in high density QCD
(Comments on UcA/QCD correspondence )
Summary
(1) Tetsuo Hatsuda, Naoki Yamamoto, Gordon Baym and M. T., Phys. Rev. Lett. 97 (2006)
N. Yamamoto, T. Hatsuda, G. Baym and M. T., Phys. Rev. D 76 (2007)
T. Hatsuda, N. Yamamoto and M. T., Phys. Rev. D 78 (2008)
EMMI, Feb. 20, 2009

2. In this workshop, we have heard
QCD, QCD phase diagram
Phase transitions, phase coexistence
Chiral symmetry breaking
Axial anomaly
Color superconductivity
Critical point(s)
Bose-Einstein condensation (BEC)
My talk is also related to these.

3. Constituents

4. Quark Flavors Tc ~ 200 MeV μc ~ 400 MeV ms~100MeV Quark flavors
Kobayashi & Maskawa (1973)
Light quarks
mu~3MeV
md~5MeV
ms~100MeV
Heavy quarks
mc~1.3 GeV
mb~4.3 GeV
mt~171 GeV
Tc ~ 200 MeV
μc　 ~ 400 MeV 　
ms~100MeV
same order
As far as we are interested in physics around several
hundred MeV, we can take heavy quark mass to ∞.

5. Dynamics

6. Quark Colors and Quantum Chromo Dynamics
SUc(3) YM theory as a model of strong interaction
Nambu (’66)
Precursors of asymptotic freedom
　Vanyashin & Terenteev (’65), Khriplovich (’69), ’t Hooft (’72)
1973 Discovery of asymptotic freedom
Gross & Wilczek, Politzer (’73)

7. Finally, I discovered QUARK! which is more like
liquid than I expected.
It says this is QUARK,
but I cannot see its color.
Maybe this should be
called “QUARKs” ?
“QUARK” in Germany

8. Features

9. QCD running coupling “Bergkatze” Color Confinement Asymptotic freedom
scale （ＧｅV)
Running coupling
(αs = ｇ２/４π） αs
Color
Confinement　
Asymptotic
freedom　
(int)/(kin) << 1
“Bergkatze”

10. Dynamical Breaking of Chiral Symmetry
Chiral basis :
QCD Lagrangian :
classical QCD symmetry (massless quarks)
Quantum QCD vacuum (massless quarks)
Chiral condensate :
spontaneous mass generation
Axial anomaly :
quantum violation of U(1)A

11. Phases of QCD
H2O 　
4He

12. T Schematic phase diagram in QCD QGP (quark-gluon plasma)
CSC (color superconductivity)
QGP (quark-gluon plasma)
cSB
(chiral symmetry breaking)

13. Color superconductivity at high baryon density
CFL
dSC
uSC
2SC
flavor color
“Majorana mass”
major differences from the standard BCS superconductor
1. Relativistic fermi system
color-magnetic int. dominant
High Tc : Tc/eF ~ 0.1
Compact pair : r~ 1-10 fm
Son, PRD59 (’99),
Schafer & Wilczek, PRD60 (’99)
Pisarski & Rischke, PRD61 (’00)
2. Color-flavor entanglement
Various phases (c.f. Ice, 3He)
2SC, uSC, dSC, CFL etc

14. Origin of each “phase” and Extreme QCD (EQCD)
strong residual int.
pre-formed pairs
Hatsuda & Kunihiro,
　Phys. Rev. Lett. 55 (‘85)
DeTar, Phys.Rev. D32 (‘85)
Asymptotic freedom
+ Debye screening
 　deconfinement
Collins & Perry,
Phys.Rev.Lett. 34 (‘75)
cSB
CSC
QGP mB T
Asakawa & Yazaki,
　Nucl. Phys. A504 (’89)
Critical point
quark-anti-quark pairing
Chiral instability
Nambu & Jona-Lasinio,
　Phys.Rev. 122 (‘61)
quark-quark pairing
 Cooper instability
Bailin & Love,
Phys.Rep.107 (‘84)

15. QCD and high temperature superconductivity (HTS)
cSB
CSC
QGP
1. Competition between different orders
2. Strong coupling 　
Common features in QCD, HTS, and ultracold atoms
Sigrist and Ueda, (‘91)
Babaev, Int. J. Mod. Phys. A16 (‘01)
Kitazawa, Nemoto, Kunihiro, PTP (‘02)
Abuki, Itakura & Hatsuda, PRD (’02) 　　
Chen, Stajic, Tan & Levin, Phys. Rep. (’05)
Baym, Hatsuda, Tachibana & Yamamoto (’06)

17. Chiral-super interplay in high density QCD　　
cSB
CSC
QGP ?

18. How to study phase structure ?
Ginzburg-Landau-Wilson (GLW) approach : model independent, analytic
1. Topological structure of the phase diagram
2. Order of the phase transition
3. Critical properties
Lattice QCD : exact, limited to μ~0, numerically heavy
Models (NJL model, strong coupling QCD etc) : qualitative, semi-analytic
σ (x) : Order parameter field
Same symmetry with underlying theory
K = {T, m, μ, … } : External parameters
Recipe
Ginzburg-Landau = Saddle point approximation
Wilson = Fluctuations by renormalization group method
・ Valid for continuous or weak 1st order transitions
・ Choice of σ (x) is an “art”
・ Results should be eventually checked by e.g. lattice QCD
Caution

19. GL analysis for chiral-super interplay in QCD (Nf=3)
QCD Symmetry:
Chiral field:
diquark field:
GL potential
Pisarski & Wilczek,
PRD 29 (’84)
・Iida & Baym, PRD 63 (’01)
・Iida, Matsuura, Tachibana
　& Hatsuda, PRD 71 (’05)
Yamamoto, Tachibana, Baym
& Hatsuda, PRL 97 (’06)

20. Complete classification of the GL potential (m=0)
Axial anomaly L L R L R R
= anomaly-induced terms
Yamamoto, Tachibana, Baym & Hatsuda,
PRL 97 (’06), PRD 76 (’07)

21. Chiral-CFL interplay in three massless flavors
Chiral-CFL interplay in Nf=3
Color-flavor
Locking (CFL)
Natural
parameter
relations
: anomaly
γ term acts as an external field for σ,
washing out the 1st order transition for
large γd as in magnetic system with
an external field. 2
Appearance of New Critical Point

22. Possible phase diagram in QCD
cSB
CSC
QGP
“Anomaly driven critical point
in high density QCD”
Yamamoto, Tachibana, Baym & Hatsuda,
PRL 97 (’06)

23. Comments Finding precise location of new critical point requires
phenomenological models, and lattice QCD simulation.
(GL approach just can tell us the topological structures)
To make schematic phase diagram more realistic should include
*realistic quark masses
*for neutron stars, charge neutrality and beta equilibrium
*Interplay with confinement (characterized by Polyakov loop) [e.g., R. Pisarski, PRD62 (2000); K. Fukushima, PLB591 (2004); C.Ratti, M. Thaler, W. Weise PRD73 (2006); C.Ratti, S. Rössner and W. Weise, PRD (2007) hep-ph/ ].
* thermal gluon fluctuations
* possible spatial inhomogeneities (LOFF states)

24. Two massless flavors tetracritical pt. bicritical point
Assume
2-flavor CSC
phase (2SC)
then
(No cubic terms; cf. three flavors)
tetracritical pt.
bicritical point

25. Instanton-induced crossover in dense QCD
N. Yamamoto, JHEP0812:060(2008)
Phase Diagram of “Instantons” T mB
QGP
CFL
χSB
“instanton molecule”
“instanton liquid”
“instanton
plasma“

26. Comment on UcA/QCD correspondence
cSB
CSC
QGP

27. GL free energy for multicomponent ultracold fermion system
with superfluidity and magnetism (Demler et al., 2007)
magnetization
pairing gap

28. BEC-BCS crossover in quark matter ?
Abuki, Itakura & Hatsuda, PRD65 (’02)
BEC-like
BCS-like
μ(MeV)
ξc/dq
Ladder QCD at finite m ξc
: coherenth
length dq
: interquark
distance
40K : JILA group, PRL 92 (2004)
6Li : Innsbruck group, PRL 92 (2004)
MIT group, PRL 92 (2004)
40K
Cond. of Fermionic-Atom Pairs
N0/N = 10% % % ξc dq
tightly bound
loosely bound

29. Bose-Fermi mixture in Ultracold Atoms and Dense QCD
-- superfluid of composite-fermions --
K. Maeda, Master thesis (’09)
Maeda, Baym & Hatsuda, (’09)
b,f norml gas
N normal gas
b-BEC
N-BCS
weak
strong T
Cold Atom
40K
87Rb
dense QCD

30. Summary and Future 1. QCD phase structure and Extreme QCD (EQCD)
・ Three major phases in QCD: ChiSB,　QGP and CSC
・ Axial anomaly (Kobayashi-Maskawa-’tHooft) plays crucial roles
2. Chiral-super interplay in dense QCD
・ Possible new critical point driven by axial-anomaly
・ Hadron-quark continuity
・ Connection to similar system in UcA
・ Instanton-induced crossover
3. Comment on UcA/QCD correspondence
・ BEC-BCS crossover in UcF <-> CSC in QCD
・ composite fermion superfluid in UcBF <-> neutron superfluid in QCD
・　UcA as a tabletop lab. for dense QCD
Exp.
Supercom.
Theory
UcA

31. phase diagram (without d-σ coupling)
: 1st order
: 2nd order μ T

32. phase diagram (with d-σ coupling)μ
A new critical point driven by the axial anomaly

33. m condensates Continuity in the ground state Low m High m p(8) & H
NGs
Vectors
Fermions
excitation
V (9)
gluons (8)
baryons (8)
Quarks (9)
Continuity in the excited state??
Schafer and Wilczek, PRL 82 (1999)
Generalized Gell-Mann-Oakes-Renner relation :
Yamamoto, Tachibana,
Baym & Hatsuda., PR D76 (’07) ○
Continuity of vector mesons
Octet vector meson <-> octet gluon 　
Tachibana,Yamamoto & Hatsuda,
PRD78 (2008)
Explicit realization of spectral continuity

34. Backup slides

35. Chiral Transition at Finite T　　
cSB
CSC
QGP ?

36. X ~ GLW analysis of hot QCD Symmetry: Chiral field:
Pisarski & Wilczek (’84)
Axial
anomaly X
Symmetry: ~
Chiral field:
Chiral transformation:
SU(Nf)LxSU(Nf)RxU(1)A
SU(Nf)LxSU(Nf)R
quark mass term

37. Some examples of GL potential
・ 2nd order phase transition
Z(2) Ising model
Nf=2 QCD
・ 1st order phase transition
Z(3) Potts model
Nf=3 QCD
・ Tri-critical behavior
Meta-magnet
Nf=2+1 QCD

38. Order of the thermal QCD transition (μ=0)
Pisarski and Wilczek, PRD29 (’84)
Svetitsky & Yaffe, NPB210 (’82)
1st
cross over
2nd m s
small
large
u,d
Nf=0
Nf=1
Nf=2
Nf=3

39. Chiral phase transition (Nf=3)μ T
Chiral field: ?
(Note: 2nd order for Nf=2)
1st order

40. Color superconductivity
phase transition μ T ?
Diquark field:
2nd order

41. Natural parameter relations:μ T
Chiral-super interplay
Hatsuda-Tachibana-
Yamamoto-Baym (‘06) ?
Natural parameter relations:

42. phase diagram (with d-σ coupling)μ
A new critical point driven by the axial anomaly

43. Realistic phase diagram in Nf=2+1 ?
mu,d,s = 0 (3-flavor limit) T μ
mu,d = 0, ms=∞ (2-flavor limit) T μ T μ
0 ≾ mu,d<ms≪∞ (realistic quark masses)
High T critical point
Low T critical point

44. 高密度星に関係する様々なカラー超伝導相 （電荷中性条件とβ平衡条件が重要）
高密度星に関係する様々なカラー超伝導相 （電荷中性条件とβ平衡条件が重要）
nd > nu > ns u d s
CFL
dSC
uSC
2SC
cSB
CFL
2SC
uSC
dSC
FFLO
2SC : Bailin and Love, Phys. Rep. (’84)
CFL : Alford, Rajagopal and Wilczek, NPB (’99)
dSC : Iida, Matsuura, Tachibana and Hatsuda, PRL (’04)
uSC : Ruster, Werth, Buballa, Shovkovy and Rischke, PRD (’05)
FFLO, gapless phase, CSL, K-cond. etc

45. Symmetry realization in hot/dense QCD（for mu,d,s=0 case）
QGP :
Collins & Perry, PRL 34 (1975)
cSB
CSC
QGP
χSB :
Nambu, PRL 4 (1960)
CSC :
Alford, Rajagopal &
Wilczek, NP B537 (1999)

46. Spectral continuity at finite μ
cSB
CSC
QGP

47. QCD sum rules in the superconducting medium
Vector current：
Current correlation function：
Operator Product Expansion (OPE) up to O(1/Q6) :
4-quark condensate
Diquark condensate
Chiral condensate

48. Mass formula from Finite Energy Sum Rules
At low density:
At intermediate
density:
At high density:

49. Spectral continuity of vector mesons
T.H., Tachibana and Yamamoto,
PRD78 (2008)
Nonet vector mesons
(heavy)
Octet
vector mesons
(light)
Octet gluons in CFL: mg=1.362Δ
Gusynin & Shovkovy, NPA700 (2002)
Malekzadeh & Rischke, PRD73 (2006)

50. QCDの相構造
バリオン密度 温度

51. magnetically controllable ・density 1014 - 1015 cm-3
K. Maeda, Master thesis (2009)
10 12
10 9
10 7
T[K]
10 2 10 1
10 -3
10 -7
Quark-gluon plasma
Superfluid neutron matter
Center of sun
Boiling water
Freezing water
Liquid nitrogen
Superfluid, superconductor
Superfluid of 3He
Ultracold atoms
Ultracold Atoms (UcA)
　　・T ~ 10-7　K
・hyperfine states
magnetically controllable
　　・density cm-3
( cf. Air ~ 1019cm-3 )

52. Y. Nambu, Nobel Lecture (Dec.8, 2008), page 24/25

53. Strangely looking? CSC (color QGP (quark-gluon plasma) cSB
superconductivity)
QGP
(quark-gluon plasma)
cSB
(chiral symmetry breaking)