The axial anomaly and the phases of dense QCD

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The axial anomaly and the phases of dense
QCD Gordon Baym University of Illinois In
collaboration with Tetsuo Hatsuda, Motoi
Tachibana, Naoki Yamamoto Quark Matter
2008 Jaipur 6 February 2008
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Color superconductivity
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Color superconductivity
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Phase diagram of equilibrated quark gluon plasma
Critical point Asakawa-Yazaki 1989.
1st order
crossover
Karsch Laermann, 2003
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New critical point in phase diagram induced
by chiral condensate diquark pairing
coupling via axial anomaly
Hatsuda, Tachibana, Yamamoto GB, PRL 97, 122001
(2006) Yamamoto, Hatsuda, Tachibana GB, PRD76,
074001 (2007)
(as ms increases)
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Order parameters
a,b,c color i,j,k flavor C charge conjugation
In hadronic (NG) phase
color singlet chiral field In color
superconducting phase
U(1)A axial anomaly gt Coupling via t Hooft
6-quark interaction



dR
?
?
?
?3
dLy dR?
?
dLy
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Ginzburg-Landau approach
In neighborhood of transitions, d (pair field)
and ? (chiral field) are small. Expand free
energy ? (cf. with free energy for d ? 0) in
powers of d and ?

chiral pairing chiral-pairing interactions
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Chiral free energy
Pisarski Wilczek, 1984
(from anomaly)
a0 becoming negative gt 2nd order transition to
broken chiral symmetry
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Quark BCS pairing (diquark) free energy (Iida
GB 2001)
Transition to color superconductivity when ?0
becomes negative
?d fully invariant under G
SU(3)LSU(3)RU(1)BU(1)ASU(3)C
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Chiral-diquark coupling
(to fourth order in the fields)
tr over flavor
Leading term ("triple boson" coupling) ?1
arises from axial anomaly. Pairing fields
generate mass for chiral field. ? terms
invariant under
G SU(3)LSU(3)RU(1)BU(1)ASU(3)C
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Three massless flavors
Simplest assumption
Color-flavor locking (CFL)
Alford, Rajagopal Wilczek (1998)
then
c and ? terms arise from the anomaly. t
Hooft interaction gt
? has same sign as c (gt0) and similar magnitude

From microscopic computations
(weak-coupling
QCD, NJL)
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If b lt 0, need ?6 f-term to stabilize system.
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Warm-up problem first ignore ?-d couplings
??0, bgt0
gt
1st order chiral transition
2nd order pairing transition
Hadronic (NG) ? ? 0, d0
Normal (NOR) ? d0
T
NG Nambu-Goldstone
NOR
NG
2nd order
CSC
COE
Coexistence (COE) ? ? 0, d ? 0
Color sup (CSC) ? 0, d ? 0
µ
Schematic phase diagram
1st order
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Full G-L free energy with chiral-diquark coupling
(? gt 0, ? 0)
Locate phase boundaries and order of transitions
by comparing free energies
? gt 0, ? 0 b gt 0, f 0
no ?-d coupling (? ? 0)
A new critical point
Major modification of phase diagram via
chiral-diquark interplay!
Non-zero ? ltlt ? produces no qualitative changes
b lt 0 with f gt 0 gt 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 CSC phase (2SC)
then
(Nf 2 GL parameters / Nf3 parameters)
No cubic terms cf. three flavors
tetracritical pt.
bicritical point
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Phase structure in T vs. ?
Mapping the phase diagram from the (a, a) plane
to the (T, µ) plane requires dynamical picture to
calculate G-L parameters.
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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
requires phenomenological 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. BEC-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)
Pairs shrink
6Li
Tc/Tf 0.2 Tc /Tf
e-1/kfa
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Phase diagram of cold fermionsvs. interaction
strength
(magnetic field B)
Unitary regime (Feshbach resonance) --
crossover No phase transition through crossover
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Deconfinement transition vs. BEC-BCS crossover
Tc
free fermions
molecules
BCS
BCS

Possible structure of crossover (Fukushima 2004 )
In SU(2)C Hadrons ltgt 2 fermion molecules.
Paired deconfined phase ltgt BCS paired fermions
?B
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Quark matter cores in neutron stars
Canonical picture compare calculations of eqs.
of state of hadronic matter and quark matter.
Crossing of thermodynamic potentials gt first
order phase transition.
ex. nuclear matter using 2 3 body interactions,
vs. pert. expansion or bag models. Akmal,
Pandharipande, Ravenhall 1998

Typically conclude transition at ?10?nm --
not reached in neutron stars if high mass
neutron stars (Mgt1.8M) are observed (e.g., Vela
X-1, Cyg X-2) gt no quark matter cores
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More realistically, expect gradual onset of
quark degrees of freedom in dense matter
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?
rn3) fm-3 Quarks can still be bound even if
deconfined.
Calculation of equation of state remains a
challenge for theorists
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Continuity of pionic excitations with increasing
density
Low ? pseudoscalar octet (?,K,?) goes
continuously to high ? diquark pseudoscalar.
Octet hadron-quark continuity in excited states
as well.
T
Quark-Gluon Plasma
Hadrons
Color superconductivity
?
mB
Mass spectrum and form of pions at intermediate
density?
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Ginzburg-Landau effective Lagrangian

Generalized pion ? at high density
Pion ? at low density
Under SU(3)R,L
and
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Generalized pion mass spectrum
mixed state of
with mixing angle .
Generalized Gell-Mann-Oakes-Renner relation
Axial anomaly(breaking U(1)A)
at very high density
Hadron-quark continuity also in excited
states Axial anomaly plays crucial role in pion
mass spectrum
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Pion mass splitting
unstable
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Conclusion
Phase structure of dense quark matter
Intriguing interplay of chiral and diquark
condensates U(1)A axial anomaly in 3 flavor
massless quark matter gt new low temperature
critical point in phase structure of QCD
at finite ?
Collective modes in intermediate density
Concrete realization of quark-hadron
continuity Effective field theory at moderate
density gt 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 END
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Toy Model
Two complex scalar fields
Lagrangian
light pion heavy pion
Diagonalize to find mass relations
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Continuous crossover from NG to CSC phases
allowed by symmetry
In 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 .