The instanton andthe phases of QCD
Naoki Yamamoto (University of Tokyo)

contents••••Introduction
QCD phase structure from QCD symmetries(1)QCD phase structure from instantons(2)Summary & Outlook(1) T. Hatsuda, M. Tachibana, G. Baym & N.Y., Phys. Rev. Lett. 97 (2006) 122001.
(2) N.Y., JHEP 0812 (2008) 060.

九大若手研究会「量子色力学の相構造研究の現状と展望」
Dec. 25. 2008

QCD phase diagram
Early universe

T
Quark-Gluon Plasma
RHIC/LHC

Hadrons?
Neutron star& quark starColor
superconductivitymBQCD phase structurefrom QCD symmetriesGinzburg-Landau (GL) approach
Symmetry of the system & order parameter
1. Topological structure of the phase diagram
2. Order of the phase transition
3. Critical behaviorse.g.) Ising modelZ(2) symmetry : m ⇔－mOrder parameter: m (magnetization)

2nd order phase transition

unbroken phase (T>Tc)

broken phase (T<Tc)

Chiral vs. Diquark condensates
Axial anomaly

QCD symmetry :Chiral field:

Diquark field:
u,d,spF
Ep-pF

r,g,bGinzburg-Landau potential
Axial anomaly(instanton-induced int.)
Pisarski-Wilczek („84)
Iida-Baym („00)

Hatsuda-TachibanaYamamoto-Baym („06)

Chiral-super interplay (Nf=3)

1st order
Natural parameter
relations:

2nd orderQCD phase diagram (Nf=3)

T: 1st order: 2nd orderμ
2nd critical point driven by the axial anomaly (instanton)
See also, Kitazawa, Koide, Kunihiro & Nemoto (’02)

Realistic QCD phase structure?
Tmu,d,s ≿= 0 (3-flavor limit)TTmu,d =≿ 0, ms=∞ (2-flavor limit)

μμ0 ≾ mu,d<ms≪∞ (realistic quark masses)

Critical point
Asakawa & Yazaki, 89

Hatsuda, Tachibana,Yamamoto & Baym 06

2nd critical pointμQCD phase structurefrom instantons

Instantons and chiral symmetry breaking
 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 Nambu-Jona-Lasinio (NJL) model
nonlocal NJL model
See, e.g., Hell-Rößner-Cristoforetti-Weise,arXiv: 0810.1099

 Then, χSB in dense QCD from instantons?

Low-energy dynamics in dense QCD
 Dense QCD : U(1)A is asymptotically restored.

NG boson field:
 Low-energy effective Lagrangian of η’c.f.
Manuel-Tytgat, PL(„00)
Son-Stephanov-Zhitnitsky, PRL(„01)
Schäfer, PRD(„02)

convergent!

Coulomb gas representation

: topological charge: 4-dim Coulomb potential

 Instanton density, topological susceptibility

 Witten-Veneziano relation：
N.Y., JHEP („08)

Renormalization group analysis
RG scale：Fluctuations：Change of potential after RG：
 RG trans.：

kinetic vs. potential

 D＝2：

potential irrelevant → vortex molecule phasepotential relevant → vortex plasma phase D≧3： potential relevant → plasma phasePhase transition induced by instantons
D-dim sine-Gordon model：
parameter αSystem

Topological excitations

Order of trans.

2D O(2) spin systemvortex

2nd

3D compact QED

magnetic monopole

crossover

4D dense QCD

instantoncrossoverNote: weak coupling QCD:
 Unpaired instanton plasma in dense QCD
Coexistence phase:
Actually,N.Y., JHEP („08)

“Phase diagram” of instantons (Nf=3)
T“instanton molecule”

QGP“instanton liquid”

χ SBCFL“instantonplasma“mBSome comments
 Effects of instantons may be also important in hot QCD.
QCD critical point at high Tfrom PNJL model with gD
K. Fukushima, PRD („08),
N. Bratovich, T. Hell, S. Rößner + W.
Weise (‟08)

QCD critical points candisappear due to the medium
effects on instantons!

Large Nc?
qq scattering

Double-line notation

qq scattering★ Diquarks are suppressed at large Nc!

Deryagin-Grigoriev-Rubakov („92)
Shuster-Son („00)
Ohnishi-Oka-Yasui („07)

See, also, McLerran-Pisarski („07)

Summary & Outlook
1.

Chiral-super interplay driven by the instanton••2.

2nd critical point at low T and high μ
Continuity of χSB phase and CSC phaseInstanton plasma from low μ to high μ•3.

The instantons play crucial roles everywhere!

Future problems•••

Real location of the 2nd critical point?
How to detect it in lab. exp. or observation of NS?
AdS/CFT application to phases of dense QCD?

Back up slides

QCD phase diagram at large Nc
Gluodynamics (~Nc2) dominates independent of μB (~Nc).

McLerran-Pisarski, NPA („07)
see also, Horigome-Tanii, JHEP („07)

Hadron-quark continuity
Continuity between hadronic matter andquark matter (Color superconductivity)
Hadrons (3-flavor)

Color superconductivity

Symmetrybreaking

SU(3)L×SU(3)R

SU(3)L×SU(3)R×SU(3)C×U(1)B

Order
parameter

Chiral condensate

Diquark condensate

NG bosons (π etc)

NG bosons

Vector mesons (ρ etc)

GluonsBaryons

QuarksPhasesElementary
excitations→ SU(3) L+R→ SU(3)L+R+C

Conjectured by Schäfer & Wilczek, PRL 1999

Prediction for the location of the critical point

Taken from hep-lat/0701002, M. StephanovOrder of the thermal transition
O(4) GL theory

SUL(3)xSUR(3) GL theory

Z(3) GL theory

Ginzburg-Landau theory of tricritical/critical point
Taken from “Quark-Gluon Plasma”
Yagi, Hatsuda and Miake(Cambridge Univ. Press, 2005)