The axial anomaly and the phases of dense QCD Gordon Baym University of Illinois
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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
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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
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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
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Phase diagram of equilibrated quark gluon plasma
Karsch & Laermann, 2003
Critical pointAsakawa-Yazaki 1989.
1st order
crossover
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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
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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 »»
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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
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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
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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
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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
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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
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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
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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
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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.
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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)
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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)
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Schematic phase structure of dense QCD with two light u,d quarks and a medium heavy s quark
without anomaly
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Schematic phase structure of dense QCD with two light u,d quarks and a medium heavy s quark
with anomaly
New critical point
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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)
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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
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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|
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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
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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
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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
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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.
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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)
»»
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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
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Pion mass splittingPion mass splitting
unstable
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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
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THE ENDTHE END
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Toy ModelToy Model
Lagrangian:Lagrangian:
Two complex scalar fields:
Diagonalize to find mass relations:
light “pion”
heavy “pion”
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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 .