Naoki Yamamoto (University of Tokyo)

Chiral symmetry breaking in dense QCD

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) 122001.(2) N.Y., JHEP 0812 (2008) 060.

駒場原子核理論セミナー

April 15, 2009

QCD phase diagram

T

mB

Quark-Gluon Plasma

Color superconductivity

Hadrons Neutron star & quark star

?

Early universe

RHIC/LHC

..But, 2-flavor NJL rather than QCD

QCD critical point at high T

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:

QCD (tri)critical point (Nf=2)

T

μ

: 1st order: 2nd order

c.f.) Coefficient in NJL: N.Y. et al., ‘07

Potential at lowest order (m=0):

No critical point in massless 3-flavor limit

1st order

Pisarski-Wilczek (‘84)

μ

T

Chiral field:

U(1)A anomaly

QCD critical point in 2+1 flavor

μ

T T

μ

0 = mu,d,s 0 = mu,d m≪ s

T

μ

0<mu,d<ms

<

As ms increases,

<

Note) CP in 2-flavor limit is also model-dependent.

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)

4-fermi interaction etc. also has medium effects 3-flavor random matrix model with axial anomaly?

Sano-Fujii-Ohtani, (‘09)

c.f.)

Location of QCD critical point?

Taken from hep-lat/0701002, M. Stephanov

Chiral-super interplay

Chiral vs. Diquark condensates

E

p

pF

-pF

Diquark condensate Chiral condensate

Y. Nambu (‘60)

Chiral-super interplay in models

Phase diagram in 2-flavor NJL modelBerges-Rajagopal, ‘99

Examples of phase diagrams in 2-flavor random matrix model Vanderheyden-Jackson, ‘00

Notes Many ambiguities in NJL:

With vector interaction → coexistence phase appearsKitazawa et al, ‘02

Possible higher interactionsKashiwa 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!

Ginzburg-Landau approach (Nf=2) GL potential:

T

μ

Most general phase diagram Hatsuda-Tachibana-Yamamoto-Baym (‘06)

Precise medium effects on GL coefficients needed

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

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

QCD phase structurefrom instantons

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:

Phase diagram of “instantons” (Nf=3)

T

mB

QGP

CFLχSB“instanton liquid”

“instanton molecule”

“instanton gas“

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

N. Yamamoto, JHEP 0812:060 (2008)

QCD phase structureat large Nc

QCD phase diagram at large Nc

McLerran-Pisarski, NPA (‘07)

see also, Horigome-Tanii, JHEP (‘07)

Gluodynamics (~Nc2) dominates independent of μB (~Nc).

CSC 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)

Conjectured Phase Diagram for Nc = 3

RHIC

LHC

SPS

FAIR

AGS

Confined

N ~0(1)

Not Chiral

Confined

Baryons

N ~ NcNf

Chiral

Debye Screened

Baryons Number

N ~ Nc 2

Chiral

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?

1. 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.

3. Future problems• Quarkyonic vs. CSC?• QCD phase structure from QCD itself?• AdS/CFT application?

Summary & Outlook

Finite-volume QCD at high μ

microscopic regime:

Exact analytical results;I. Partition function (zero topological sector): a novel correspondence!

II. Spectral sum rules: Dirac spectra at high μ are governed by the CSC gap Δ.

III. 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.

N. Yamamoto, T. Kanazawa, arXiv:0902.4533.

at μ=0.

at high μ.

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 superconductivitySU(3)L×SU(3)R×SU(3)C×U(1)B

→ SU(3)L+R+C

Diquark condensate

NG bosons

Gluons

Quarks

PhasesSymmetry breaking

Order parameter

Elementaryexcitations

Hadron-quark continuity

Continuity between hadronic matter and quark matter (Color

superconductivity)

Conjectured by Schäfer & Wilczek, PRL 1999

Back up slides

Order of the thermal transition Z(3) GL theoryO(4) GL theory

SUL(3)xSUR(3) GL theory

Color Superconductivity

QCD at high density asymptotic freedom Attractive channel [3]C×[3]C=[3]C+[6]C

Fermi surface

Cooper instability

E

p

pF

-pF

3-flavor case

ud s Color-Flavor Locking

(CFL) phase

r,g,b

q q

3

dL,R :diquark

u,d,s

Alford-Rajagopal-Wilczek (‘99)

2nd order

Color superconductivity phase transition

Iida-Baym (‘00)

μ

T

Diquark field: