The axial anomaly and the phases ofThe axial anomaly and the phases of dense QCDdense QCD

Gordon BaymGordon BaymUniversity of IllinoisUniversity of Illinois

In collaboration with In collaboration with Tetsuo Hatsuda, Motoi Tachibana, & Naoki YamamotoTetsuo Hatsuda, Motoi Tachibana, & Naoki Yamamoto

Quark Matter 2008Quark Matter 2008

JaipurJaipur

6 February 20086 February 2008

Quark-gluon plasma

Hadronic matter2SC

CFL

1 GeV

150 MeV

0

Tem

pera

ture

Baryon chemical potential

Neutron stars

?

Ultrarelativistic heavy-ion collisions

Nuclear liquid-gas

Color superconductivity

Quark-gluon plasma

Hadronic matter2SC

CFL

1 GeV

150 MeV

0

Tem

pera

ture

Baryon chemical potential

Neutron stars

?

Ultrarelativistic heavy-ion collisions

Nuclear liquid-gas

Color superconductivity

Phase diagram of equilibrated quark gluon plasma

Karsch & Laermann, 2003

Critical pointAsakawa-Yazaki 1989.

1st order

crossover

Hatsuda, Tachibana, Yamamoto & GB, PRL 97, 122001 (2006)Yamamoto, Hatsuda, Tachibana & GB, PRD76, 074001 (2007)

New critical point in phase diagram: induced by chiral condensate – diquark pairing coupling

via axial anomaly

Hadronic

Normal QGP

Color SC

(as ms increases)

q q 0

q q 0

qq 0

Order parametersOrder parameters

In hadronic (NG) phase: In hadronic (NG) phase:

= color singlet chiral field= color singlet chiral field

In color superconducting phase :In color superconducting phase :

» » 33 » » ddLLyy d dRR

U(1)U(1)AA axial anomaly => Coupling via ‘t Hooft axial anomaly => Coupling via ‘t Hooft 6-quark6-quark interaction interaction

a,b,c = colori,j,k = flavorC: charge conjugation

ddRR

ddLLyy

det i, j q RjqL

i »»

Ginzburg-Landau approach

In neighborhood of transitions, d (pair field) and In neighborhood of transitions, d (pair field) and (chiral field) are (chiral field) are small. Expand free energy small. Expand free energy (cf. with free energy for d = (cf. with free energy for d = = 0) in = 0) in powers of d and powers of d and

d int .

= chiral + pairing + = chiral + pairing + chiral-pairing interactionschiral-pairing interactions

Chiral free energyChiral free energyPisarski & Wilczek, 1984

(from anomaly)

aa00 becoming negative => 2 becoming negative => 2ndnd order transition to broken chiral symmetry order transition to broken chiral symmetry

m0

2» 3

Quark BCS pairing (diquark) free energy (Iida & GB 2001)(Iida & GB 2001)

Transition to color superconductivity when Transition to color superconductivity when 00 becomes negative becomes negative

d fully invariant under:

G G = SU(3)= SU(3)LL×SU(3)×SU(3)RR×U(1)×U(1)BB×U(1)×U(1)AA×SU(3)×SU(3)CC

dL d L e2i(B A )VLdLVCT

dR d R e2i(B A )VR dRVCT

dLdR e 4 iAVL dLdR

VR

Chiral-diquark coupling:

int . 1tr dR dL dLdR

1tr dLdL

dR dR 2tr dLdL

dR dR tr

3 dettr dLdR 1 h.c.

Leading term ("triple boson" coupling) » 1 arises from axial anomaly.Pairing fields generate mass for chiral field.

terms invariant under:

(to fourth order in the fields)

G G = SU(3)= SU(3)LL×SU(3)×SU(3)RR×U(1)×U(1)BB×U(1)×U(1)AA×SU(3)×SU(3)CC

tr over flavor

Three massless flavors

Simplest assumption:

dL dR d

dd

Color-flavor locking (CFL)

then

c and terms arise from the anomaly. ‘t Hooft interaction => has same sign as c (>0) and similar magnitude

From microscopic computations (weak-coupling QCD, NJL)

~ Tcr.

2

ln Tcr.

Alford, Rajagopal& Wilczek (1998)

mu md ms 0

det i, j q RjqL

i

If b < 0, need If b < 0, need 66 f-term to stabilize system. f-term to stabilize system.

¿ 1

Warm-up problem: first ignore -d couplings : ==0, b>0

3F (,d) a2 2

c3 3

b4 4

2

d2 4

d4

1st order chiral transition 2nd order pairing transition

Hadronic (NG) ≠ 0, d=0

a

Normal (NOR)= d=0

Coexistence (COE)≠ 0, d ≠ 0

Color sup (CSC)= 0, d ≠ 0

0

T

μ

NOR

CSCCOE

NG

d

Schematicphase diagram

=>=>

22ndnd order order

11stst order order

NG= Nambu-Goldstone

Full G-L free energy with chiral-diquark coupling (Full G-L free energy with chiral-diquark coupling (> 0, > 0, ≥ 0) ≥ 0)

Locate phase boundaries and order of transitions by comparing free energies:

(NOR ) 0,0 ,(CSC ) 0,d ,(NG ) ,0 ,(COE ) ,d

no no -d coupling -d coupling ((= = = 0) = 0)

> 0, > 0, = 0= 0

b > 0, f = 0b > 0, f = 0

Major modification of phase diagram via chiral-diquark interplay!

A= new critical pointA= new critical point

Non-zero Non-zero << << produces no qualitative changes produces no qualitative changes

b < 0 with f > 0 => qualitatively similar resultsb < 0 with f > 0 => qualitatively similar results

Critical point arises because d2, in -d2 term, acts as external field for , washing out 1st order transition for large d2 -- as in magnetic system in external field.

With axial anomaly, NG-like and CSC-like coexistence phases have same symmetry, allowing crossover.

NG and COE phases realize U(1)B differently and boundary is sharp.

Two massless flavors

Assume 2-flavor CSCphase (2SC)

then

mu md 0,ms

0

dL dR 0

0d

2F (,d) a2 2

b4 4

f6 6

2

d2 4

d4

d2 2

No cubic terms;No cubic terms;

cf. three flavors:cf. three flavors:

tetracritical pt.tetracritical pt. bicritical pointbicritical point

(Nf= 2 GL parameters / Nf=3 parameters)

Phase structure in T vs.

Mapping the phase diagram from the (a, α) plane to the (T, μ) plane requires dynamical picture to calculate G-L parameters.

T

COE

CSCHadronic NG

QGP

mu,d 0,ms

N f 2

No anomaly-induced critical point for Nf=2 in SU(3)C or SU(2)C

T

COE(NG-like)

COE(CSC-like)

Hadronic NG

QGP

mu,d ,s 0

N f 3

“Hadron”-quark continuity at low T (Schäfer-Wilczek 1999)

Schematic phase structure of dense QCD with two light u,d quarks and a medium heavy s quark

without anomaly

Schematic phase structure of dense QCD with two light u,d quarks and a medium heavy s quark

with anomaly

New critical point

Finding precise location of new critical point requiresphenomenological models, and lattice QCD simulation. Too cold to be accessible experimentally.

To make schematic phase diagram more realistic should include

* realistic quark masses

* for neutron stars, charge neutrality and beta equilibrium

* interplay with confinement (characterize 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/0609281 ]. Delineate nature of NG-like coexistence phase.

* thermal gluon fluctuations

* possible spatial inhomogeneities (FFLO states)

Hadron-quark matter deconfinement transition vs.Hadron-quark matter deconfinement transition vs.BEC-BCS crossover in cold atomic fermion systemsBEC-BCS crossover in cold atomic fermion systems

In trapped atoms continuously transform from molecules to Cooper pairs: D.M. Eagles (1969) ; A.J. Leggett, J. Phys. (Paris) C7, 19 (1980); P. Nozières and S. Schmitt-Rink, J. Low Temp Phys. 59, 195 (1985)

Tc/Tf » 0.2 Tc /Tf » e-1/kfa

Pairs shrink

6Li

Phase diagram of cold fermionsvs. interaction strength

(magnetic field B)

Unitary regime (Feshbach resonance) -- crossoverNo phase transition through crossover

BCS

BEC of di-fermionmolecules

Temperature

Tc

Free fermions +di-fermion molecules

Free fermions

-1/kf a0

a>0a<0

Tc/EF» 0.23Tc» EFe-/2kF|a|

B

In SU(2)C :

Hadrons <=> 2 fermion molecules. Paired deconfined phase <=> BCS

paired fermions

Deconfinement transition vs. BEC-BCS crossover

Possible structure of crossover (Fukushima 2004 )

Abuki, Itakura & Hatsuda, PRD65, 2002

BCS paired quark matter

BCS-BEC crossoverHadrons

Hadronic

Normal

Color SC

BCS

Tc

molecules BCS

free fermions

Quark matter cores in neutron starsQuark matter cores in neutron stars

Canonical picture: compare calculations of eqs. of state of hadronic matter and quark matter. Crossing of thermodynamic potentials => first order phase transition.

Typically conclude transition at »10nm -- not reached in neutron stars if high mass neutron stars (M>1.8M¯) are observed (e.g., Vela X-1, Cyg X-2) => no quark matter cores

ex. nuclear matter using 2 & 3 body interactions, vs. pert. expansion or bag models. Akmal, Pandharipande, Ravenhall 1998

More realistically, expect gradual onset of quark degrees of freedom in dense matter

HadronicNormal

Color SC

New critical point suggests transition to quark matter is a crossover at low T

Consistent with percolation picture, that as nucleons begin to overlap, quarks percolate [GB, Physica (1979)] :

nperc » 0.34 (3/4 rn

3) fm-3

Quarks can still be bound even if deconfined.

Calculation of equation of state remains a challenge for theorists

T

Color Color superconductivitysuperconductivity

HadronsHadrons

Quark-Gluon PlasmaQuark-Gluon Plasma

?Mass spectrum and form of pions at intermediate densityMass spectrum and form of pions at intermediate density?

Continuity of pionic excitations with increasing densityContinuity of pionic excitations with increasing density

Gell-Mann-Oakes-Renner (GOR) Gell-Mann-Oakes-Renner (GOR) relationrelation Alford, Rajagopal, & Wilczek, 1999Alford, Rajagopal, & Wilczek, 1999

Low pseudoscalar octet (,K,) goes continuously to high diquark pseudoscalar. Octet hadron-quark continuity in excited states as well.

Ginzburg-Landau effective LagrangianGinzburg-Landau effective Lagrangian

Pion at low density

Generalized pion at high density

Under SU(3)R,L andand

»»

AAxial anomaly couples to and to quark masses, mq

　　 : to O(M)

»»

Generalized pionGeneralized pion mass spectrummass spectrum

Generalized Gell-Mann-Oakes-Renner relation Generalized Gell-Mann-Oakes-Renner relation

Hadron-quark continuity also in excited states

Axial anomaly plays crucial role in pion mass spectrum

Axial Axial anomalyanomaly （（ breakinbreakingg U(1)U(1)AA ））

Mass eigenstates:Mass eigenstates:

= mixed state of & with mixing = mixed state of & with mixing angleangle . .

at very high density

Pion mass splittingPion mass splitting

unstable

ConclusionConclusion Phase structure of dense quark matter

Collective modes in intermediate density

Intriguing interplay of chiral and diquark condensatesU(1)A axial anomaly in 3 flavor massless quark matter => new low temperature critical point in phase structure of QCD at finite

Concrete realization of quark-hadron continuityEffective field theory at moderate density => pion as generalized meson; generalized GOR relation

Vector mesons, nucleons and other heavy excitations(Hatsuda, Tachibana, & Yamamoto,in preparation)

Vector meson continuity

THE ENDTHE END

Toy ModelToy Model

Lagrangian:Lagrangian:

Two complex scalar fields:

Diagonalize to find mass relations:

light “pion”

heavy “pion”

dLdR e 4 iAVL dLdR

VR

'e2iAVLVR

Continuous crossover from NG to CSC phases allowed by symmetryContinuous crossover from NG to CSC phases allowed by symmetry

In CFLIn CFL phase: dLdRy breaks chiral symmetry but

preserves Z4 discrete subgroup of U(1)A.

For = 0, different symmetry breaking in two phases. term has Z6 symmetry, with Z2 as subgroup. With axial anomaly, NG and CSC-like coexistence phases have same symmetry, and can be continuously connected.

NG and COE phases realize U(1)B differently and boundary is not smoothed out. In COE phase: breaks chiral symmetry, preserving only Z2 .