1. Chiral symmetry breaking in dense QCD
Naoki Yamamoto (University of Tokyo)
contents
Introduction: QCD critical point at high T
Chiral-super interplay
QCD phase structure from instantons
QCD phase structure at large Nc
Summary & Outlook
(1) T. Hatsuda, M. Tachibana, G. Baym & N.Y., Phys. Rev. Lett. 97 (2006)
(2) N.Y., JHEP 0812 (2008) 060.
駒場原子核理論セミナー
April 15, 2009

2. ? QCD phase diagram T mB Quark-Gluon Plasma Color superconductivity
..But, 2-flavor NJL rather than QCD
Early universe T
Quark-Gluon Plasma ?
RHIC/LHC
Color superconductivity
Hadrons
Neutron star
& quark star mB

3. QCD critical point at high T

4. QCD critical point?
First predicted by 2-flavor NJL model Asakawa-Yazaki, ‘89
Confirmed by other models, e.g., random matrix model Halasz et al. ‘98
Lattice results: still controversial de Forcrand-Philipsen ‘06, ‘08
But models have many ambiguities!
e.g.) NJL-type Lagrangian:
Parameters (to be fitted with pion mass/decay const.): Λ, G, m
→ Calculate phase diagram numerically.
Thermodynamic potential:

5. QCD (tri)critical point (Nf=2)
Potential at lowest order (m=0): T μ
: 1st order
: 2nd order
c.f.) Coefficient in NJL:
N.Y. et al., ‘07

6. No critical point in massless 3-flavor limit
Chiral field:
Pisarski-Wilczek (‘84)
U(1)A anomaly μ T
1st order

7. QCD critical point in 2+1 flavor
0 = mu,d,s
<
0 = mu,d≪ms
< μ T T μ T μ
0<mu,d<ms
As ms increases,
Note) CP in 2-flavor limit is also model-dependent.

8. Some comments
Unknown medium effects on model parameters easily smear out CP!
QCD critical point at high T from 2+1 flavor PNJL model with gD~c0
K. Fukushima, PRD (‘08),
N. Bratovich, T. Hell, S. Rößner + W. Weise (’08)
c.f.)
4-fermi interaction etc. also has medium effects
3-flavor random matrix model with axial anomaly?
Sano-Fujii-Ohtani, (‘09)

9. Location of QCD critical point?
Taken from hep-lat/ , M. Stephanov

10. Chiral-super interplay

11. Chiral vs. Diquark condensates
Chiral condensate
Diquark condensate E p pF
-pF
Y. Nambu (‘60)

12. Chiral-super interplay in models
Phase diagram in 2-flavor NJL model
Berges-Rajagopal, ‘99
Examples of phase diagrams in 2-flavor random matrix model
Vanderheyden-Jackson, ‘00

13. Notes Many ambiguities in NJL:
With vector interaction → coexistence phase appears
Kitazawa et al, ‘02
Possible higher interactions
Kashiwa et al. ‘07
Medium effects on interactions (remember 3-flavor PNJL)
Chen et al. ’09
Favor-dependence, quark masses, ...
However, their topological structures look similar, why?
→ Because all models have QCD symmetries!

14. Ginzburg-Landau approach (Nf=2)
GL potential:
Most general phase diagram Hatsuda-Tachibana-Yamamoto-Baym (‘06) T μ
Precise medium effects on GL coefficients needed

15. Anomaly-induced interplay (Nf=3)
Hatsuda-Tachibana-Yamamoto-Baym (‘06) T μ
: 1st order
: 2nd order
Non-vanishing chiral condensate at high μ due to U(1)A anomaly
The possible 2nd critical point at high μ
Anomaly-induced interplay in NJL Yamamoto-Hatsuda-Baym in progress

16. Realistic QCD phase structure?μ
mu,d,s = 0 (3-flavor limit) ≿ T μ
mu,d = 0, ms=∞ (2-flavor limit) ≿ T μ
0 ≾ mu,d<ms≪∞ (realistic quark masses)
Critical point
Asakawa & Yazaki, 89
Hatsuda, Tachibana, Yamamoto & Baym 06
2nd critical point

17. QCD phase structure from instantons

18. Instantons and chiral symmetry breaking
Why instanton? : mechanism for chiral symm. breaking/restoration
“instanton liquid” (metal)
“instanton molecule” (insulator)
T=0
T>Tc
Schäfer-Shuryak, Rev. Mod. Phys. (‘97)
Origin of NJL model:
nonlocal NJL model
See, e.g., Hell-Rößner-Cristoforetti-Weise, arXiv:
Then, χSB in dense QCD from instantons?

19. Low-energy dynamics in dense QCD
Dense QCD : U(1)A is asymptotically restored.
Low-energy effective Lagrangian of η’
Manuel-Tytgat, PL(‘00)
Son-Stephanov-Zhitnitsky, PRL(‘01)
Schäfer, PRD(‘02)
convergent!

20. Coulomb gas representation
: topological charge
: 4-dim Coulomb potential
Instanton density, topological susceptibility
Witten-Veneziano relation：

21. Renormalization group analysis
Fluctuations：
RG scale：
Change of potential after RG：
RG trans.：
kinetic vs. potential
D＝2： potential irrelevant → vortex molecule phase
potential relevant → vortex plasma phase
D≧3： potential relevant → plasma phase

22. Phase transition induced by instantons
D-dim sine-Gordon model：
　　　　　System 　　　　　　parameter α Topological excitations Order of trans.
2D O(2) spin system vortex 2nd
3D compact QED magnetic monopole crossover
4D dense QCD instanton crossover
Note: weak coupling QCD:
Unpaired instanton plasma in dense QCD
→Coexistence phase: 　　　
Actually,

23. Phase diagram of “instantons” (Nf=3)mB
QGP
CFL
χSB
“instanton molecule”
“instanton liquid”
“instanton
gas“
Chiral phase transition at high μ: instanton-induced crossover.
4-dim. generalization of Kosterlitz-Thouless transition.
N. Yamamoto, JHEP 0812:060 (2008)

24. QCD phase structure at large Nc

25. QCD phase diagram at large Nc
Gluodynamics (~Nc2) dominates independent of μB (~Nc).
McLerran-Pisarski, NPA (‘07)
see also, Horigome-Tanii, JHEP (‘07)

26. CSC at large Nc? ★ Diquarks are suppressed at large Nc! qq scattering
Double-line notation
qq scattering
Deryagin-Grigoriev-Rubakov (‘92)
Shuster-Son (‘00)
Ohnishi-Oka-Yasui (‘07)
★ Diquarks are suppressed at large Nc!

27. Color Superconductivity
Conjectured Phase Diagram for Nc = 3
RHIC
LHC
SPS
FAIR
AGS
Confined
N ~0(1)
Not Chiral
Baryons
N ~ NcNf
Chiral
Debye Screened
Baryons Number
N ~ Nc 2
Color Superconductivity
Liquid Gas Transition
Critical Point
Quark Gluon Plasma
Quarkyonic Matter
Confined Matter T
From McLerran at QM2009
Not correct for 3-flavor limit: deconfinement earlier than χSR.
Note that large Nc leads to
No color superconductivity
Weak axial anomaly indep. of μ
A dynamical question: subtleness of quark masses. (flavor-dep.)
A puzzle: how χSB occurs after χSR?

28. Summary & Outlook QCD phase structure
Consensus is highly model-dependent.
The QCD critical point at high T?
Possible 2nd critical point at high μ.
2. Instanton plasma from low μ to high μ
Instantons play crucial roles everywhere.
Non-vanishing chiral condensate even at high μ.
Future problems
Quarkyonic vs. CSC?
QCD phase structure from QCD itself?
AdS/CFT application?

29. Finite-volume QCD at high μ
N. Yamamoto, T. Kanazawa, arXiv:
microscopic regime:
Exact analytical results;
Partition function (zero topological sector): a novel correspondence!
Spectral sum rules: Dirac spectra at high μ are governed by the CSC gap Δ.
Lee-Yang zeros: conventional random matrix model fails to reproduce CSC.
Application to dense 2-color QCD is also possible.
T. Kanazawa, T. Wettig, N. Yamamoto, to appear soon.
at μ=0.
at high μ.

30. Hadron-quark continuity
Continuity between hadronic matter and quark matter (Color superconductivity)
Hadrons (3-flavor)
SU(3)L×SU(3)R
→ SU(3) L+R
Chiral condensate
NG bosons (π etc)
Vector mesons (ρ etc)
Baryons
Color superconductivity
SU(3)L×SU(3)R×SU(3)C×U(1)B
→ SU(3)L+R+C
Diquark condensate
NG bosons
Gluons
Quarks
Phases
Symmetry
breaking
Order parameter
Elementary
excitations
Conjectured by Schäfer & Wilczek, PRL 1999

31. Back up slides

32. Order of the thermal transition
Z(3) GL theory
O(4) GL theory
SUL(3)xSUR(3) GL theory

33. Color Superconductivity
QCD at high density
asymptotic freedom
Attractive channel
[3]C×[3]C=[3]C+[6]C
Fermi surface q 3
Cooper instability
dL,R :diquark E p pF
-pF
3-flavor case
u,d,s
r,g,b u d s
Color-Flavor Locking (CFL) phase
Alford-Rajagopal-Wilczek (‘99)

34. Color superconductivity phase transition
Diquark field:
Iida-Baym (‘00) μ T
2nd order