Naoki Yamamoto (University of Tokyo)

高密度 QCD におけるカイラル対称性

contents

• Introduction: color superconductivity• The role of U(1)A anomaly and chiral symmetry

breaking• Partition function zeros and chiral symmetry

breaking• Summary & Outlook(1) T. Hatsuda, M. Tachibana, N.Y. and G. Baym, Phys. Rev. Lett. 97 (2006) 122001.(2) N.Y., JHEP 0812 (2008) 060.(3) N.Y. and T. Kanazawa, Phys. Rev. Lett. 103 (2009) 032001.

KEK 理論センター研究会「原子核・ハドロン物理」　2009.8.11.

QCD phase diagram

T

mB

Quark-Gluon Plasma

Hadrons

RHIC/LHC

CFL

Color superconductivity

quark matter

Neutron star

Color Superconductivity

QCD at high density → asymptotic free Fermi surface

Attractive channel → Cooper instability

[3]C×[3]C=[6]C+[3]C

E

p

μ

q q

3

“diquark condensate”

“Fermi sea”

“Dirac sea”

Color-Flavor Locking (CFL)

ud s

r,g,bu,d,s

Pairing channel • s-wave pairing, spin singlet → Dirac antisymmetric• Attractive channel → color antisymmetric• Pauli principle → flavor antisymmetric• U(1)A anomaly → Lorentz scalar

3-flavor limit: Color-Flavor Locking (CFL) Alford-Rajagopal-Wilczek (NPB1999)

Gauge-invariant order parameter

e.g.)

Symmetry breaking pattern:

CFL is positive parity

... due to the presence of U(1)A anomaly.

Consider the Kobayashi-Maskawa-’t Hooft (KMT) vertex with quark mass:

VKMT is minimized when

and the positive parity state is energetically favored.

Alford-Rajagopal-Wilczek (NPB1999)

Kobayashi-Maskawa (PTP1970);‘t Hooft (PRD1976)

G G

T. Schafer (PRD2002)

Chiral symmetry breaking in CFL

The chiral condensate:

Exactly calculated thanks to the screening of instantons at high μ:

[Point]

1. Chiral symmetry is broken not only by the diquark condensate but also the chiral condensate in CFL.

2. Nonzero chiral condensate in CFL is model-independent.

3. Chiral-super interplay of the type is inevitable.

Alford-Rajagopal-Wilczek (NPB1999)

T. Schafer (PRD2002); NY (JHEP2008)

Possible phase structure I

Anomaly-induced critical point at high μ. Hatsuda-Tachibana-NY-Baym (PRL2006) A realization of quark-hadron continuity. Schafer-Wilczek (PRL1999) Critical point(s) of other origins. Kitazawa-Koide-Kunihiro-Nemoto

(PTP2002); Zhang-Fukushima-Kunihiro (PRD2009);

Zhang-Kunihiro, arXiv:0904.1062.

T

mB

Quark-Gluon Plasma

HadronsColor

superconductivity

Possible phase structure III

Is there this possibility? [see also Hidaka-san’s talk]

T

mB

Quark-Gluon Plasma

Hadrons CFLquark matter

Phase diagram of “instantons” (Nf=3)

T

mB

“instanton liquid”

“instanton molecule”

“instanton gas“

Chiral phase transition at high μ: instanton-induced crossover. 4-dim. generalization of Kosterlitz-Thouless transition.

NY (JHEP2008)

Another viewpoint: Lee-Yang zeros

The partition function zeros in the complex plane at V<∞ reflects the information of the chiral condensate at V=∞:

Nonzero chiral condensate at V=∞ requires a cut through m=0.

Halasz-Jackson-Verbaarschot (PRD97)

[Lee-Yang zeros at μ=0] Leutwyler-Smilga (PRD92)

Predictions of Random Matrix Theory (RMT)

Halasz-Jackson-Verbaarschot (PRD97); Halasz, et al. (PRD98) RMT predictions:

1. Chiral symmetry restores at μ=μc.

2. The cut will move away from origin as μ increases.

→ Is it consistent with the chiral symmetry breaking at high μ?

[Random Matrix Theory → Ohtani-san’s talk]

Finite-volume QCD at high density

QCD in a large but finite torus:

ε-regime:

Elementary excitations in CFL;• 9 quarks: mass gap~Δ due to the color superconductivity. • 8 gluons: mass gap~Δ due to the Higgs mechanism.• 8+1(+1) Nambu-Goldstone (NG) modes: nearly (or exactly) massless.

In ε-regime,• Non-NG modes negligible since . • Kinetic terms of NG modes negligible.

NY-Kanazawa (PRL2009)

Partition functions in ε-regime

Chiral Lagrangian at high μ (flavor-symmetric): Son-Stephanov (PRD2000)

Exact partition function at high μ:

a novel correspondence between hadronic phase and CFL phase

related to quark-hadron continuity!

Dirac spectrum...

at μ=0.

at high μ.

NY-Kanazawa (PRL2009)

Exact Lee-Yang zeros at high density

Asymptotic partition function and Lee-Yang zeros at μ=∞:

Chiral condensate vanishes at μ=∞. However, many Lee-Yang zeros exist near origin even at high μ

and the chiral condensate can be nonzero for μ<∞.

NY-Kanazawa (PRL2009)

1. Phases in dense QCD• The U(1)A anomaly (or instanton) plays crucial role.• Non-vanishing chiral condensate even at high μ.• Chiral-super interplay is inevitable.• Possible critical point(s) in dense QCD.

2. Partition function zeros in dense QCD• Exact X-shaped cut in the complex mass plane at μ=∞.• Chiral condensate can be nonzero for μ<∞.

3. Future problems• Phases at lower or intermediate densities?• Anomaly-induced interplay in NJL. Baym-Hatsuda-NY, in progress.

• Confinement-deconfinement transition?• Microscopic understanding based on QCD?

Summary & Outlook

Back up slides

Chiral vs. Diquark condensates

E

p

pF

-pF

Diquark condensate Chiral condensate

Y. Nambu (‘60)

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-flavor locking

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

Elementaryexcitations

quark-hadron continuity

Continuity between hadronic matter and quark matter (color-flavor locking)

Conjectured by Schäfer & Wilczek, PRL 1999

Instantons and chiral symmetry breaking

Why instanton? : mechanism for chiral symm. breaking/restoration

T=0 T>Tc

“instanton liquid” (metal) “instanton molecule” (insulator)

Schäfer-Shuryak, Rev. Mod. Phys. (‘97)

See, e.g., Hell-Rößner-Cristoforetti-Weise, arXiv: 0810.1099

nonlocal NJL model

Origin of NJL model:

Then, χSB in dense QCD from instantons?

Dense QCD : U(1)A is asymptotically restored.

Low-energy dynamics in dense QCD

convergent!

Low-energy effective Lagrangian of η’

Manuel-Tytgat, PL(‘00)Son-Stephanov-Zhitnitsky, PRL(‘01)Schäfer, PRD(‘02)

Coulomb gas representation

: topological charge

: 4-dim Coulomb potential

Instanton density, topological susceptibility

Witten-Veneziano relation ：

Renormalization group analysis

Fluctuations ：

Change of potential after RG ：

RG trans. ：

RG scale ：

kinetic vs. potential

D ＝ 2 ： potential irrelevant → vortex molecule phase potential relevant → vortex plasma phase

D 3≧ ： potential relevant → plasma phase

Phase transition induced by instantons

Unpaired instanton plasma in dense QCD

→Coexistence phase: 　　　

Actually,

　　　　　 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

D-dim sine-Gordon model ：

Note: weak coupling QCD:

Color superconductivity at large Nc

qq scattering

qq scattering

Double-line notation

★ Diquarks are suppressed at large Nc!

Deryagin-Grigoriev-Rubakov (‘92)

Shuster-Son (‘00)

Ohnishi-Oka-Yasui (‘07)

0 ≾ mu,d<ms ∞ (realistic quark masses)≪

Realistic QCD phase structure?

2nd critical point

Critical pointAsakawa & Yazaki, 89

mu,d,s = 0 (3-flavor limit) mu,d = 0, ms=∞ (2-flavor limit)≿ ≿T

μ

T

μT

μ

Hatsuda, Tachibana, Yamamoto & Baym 06

Possible phase structure II

T

mB

Quark-Gluon Plasma

HadronsColor

superconductivity

Of course, 1st order chiral phase transition at T=0 is still possible.