1. 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
Title

2. Color superconductivity

3. Color superconductivity

4. Phase diagram of equilibrated quark gluon plasma
Critical point
Asakawa-Yazaki 1989.
1st order
crossover
Karsch & Laermann, 2003

5. New critical point in phase diagram:
induced by chiral condensate – diquark pairing coupling
via axial anomaly
Hatsuda, Tachibana, Yamamoto & GB, PRL 97, (2006)
Yamamoto, Hatsuda, Tachibana & GB, PRD76, (2007)
Hadronic
Normal QGP
Color SC
(as ms increases)

6. Order parameters In hadronic (NG) phase: = color singlet chiral field
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 => Coupling via ‘t Hooft 6-quark interaction dR    
» 3
dLy
» dLy dR

7. 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

8. Chiral free energy (from anomaly) m02» 3
Pisarski & Wilczek, 1984
m02» 3
(from anomaly)
a0 becoming negative => 2nd order transition to broken chiral symmetry

9. Quark BCS pairing (diquark) free energy (Iida & GB 2001)
Transition to color superconductivity when 0 becomes negative
d fully invariant under:
G = SU(3)L×SU(3)R×U(1)B×U(1)A×SU(3)C

10. G = SU(3)L×SU(3)R×U(1)B×U(1)A×SU(3)C
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)L×SU(3)R×U(1)B×U(1)A×SU(3)C

11. Three massless flavors
Simplest assumption:
Color-flavor
locking (CFL)
Alford, Rajagopal
& Wilczek (1998)
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)
¿ 1
If b < 0, need 6 f-term to stabilize system.

12. Warm-up problem: first ignore -d couplings : ==0, b>0
=>
1st order chiral transition
2nd order pairing transition
Hadronic (NG)
 ≠ 0, d=0
Normal (NOR)
 = d=0 T
NG= Nambu-Goldstone
NOR
2nd order NG
CSC
COE
Coexistence
(COE)
 ≠ 0, d ≠ 0
Color sup (CSC)
 = 0, d ≠ 0 μ
Schematic
phase diagram
1st order

13. Major modification of phase diagram via chiral-diquark interplay!
Full G-L free energy with chiral-diquark coupling ( > 0,  ≥ 0)
Locate phase boundaries and order of transitions by comparing free energies:
 > 0,  = 0
b > 0, f = 0
no -d coupling
( =  = 0)
A= new critical point
Major modification of phase diagram via chiral-diquark interplay!
Non-zero  <<  produces no qualitative changes
b < 0 with f > 0 => qualitatively similar results

14. 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.

15. Two massless flavors Assume 2-flavor CSC phase (2SC) then
(Nf= 2 GL parameters / Nf=3 parameters)
No cubic terms;
cf. three flavors:
tetracritical pt.
bicritical point

16. 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
CSC
Hadronic NG
QGP
No anomaly-induced critical
point for Nf=2 in SU(3)C or SU(2)C T 
COE
(NG-like)
(CSC-like)
Hadronic NG
QGP
“Hadron”-quark continuity at low T (Schäfer-Wilczek 1999)

17. Schematic phase structure of dense QCD with two light u,d quarks and a medium heavy s quark without anomaly

18. Schematic phase structure of dense QCD with two light u,d quarks and a medium heavy s quark with anomaly
New critical point

19. 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/ ]. Delineate nature of NG-like coexistence phase.
* thermal gluon fluctuations
* possible spatial inhomogeneities (FFLO states)

20. 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 » Tc /Tf » e-1/kfa

21. Phase diagram of cold fermions vs. interaction strength
BCS
BEC of
di-fermion
molecules
Temperature Tc
Free fermions +di-fermion molecules
Free fermions
-1/kf a
a>0
a<0
Tc/EF» 0.23
Tc» EFe-p/2kF|a|
(magnetic field B)
Unitary regime (Feshbach resonance) -- crossover
No phase transition through crossover

22. Deconfinement transition vs. BEC-BCS crossoverTc
free fermions
molecules
BCS
Abuki, Itakura & Hatsuda, PRD65, 2002
BCS paired
quark matter
BCS-BEC
crossover
Hadrons
Hadronic
Normal
Color SC
BCS
Possible structure of crossover (Fukushima 2004 )
In SU(2)C :
Hadrons <=> 2 fermion molecules. Paired deconfined phase <=> BCS paired fermions B

23. Quark 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.
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 (M>1.8M¯) are observed (e.g., Vela X-1, Cyg X-2) => no quark matter cores

24. More realistically, expect gradual onset of quark degrees of freedom in dense matter
Hadronic
Normal
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 rn3) fm-3
Quarks can still be bound even if deconfined.
Calculation of equation of state remains a challenge for theorists

25. 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
Gell-Mann-Oakes-Renner (GOR) relation
Alford, Rajagopal, & Wilczek, 1999
Hadrons
Color
superconductivity ? mB
Mass spectrum and form of pions at intermediate density?

26. Ginzburg-Landau effective Lagrangian
Pion  at low density
Generalized pion  at high density
Under SU(3)R,L
and
Axial anomaly couples  to  and to quark masses, mq
: to O(M)

27. Generalized pion mass spectrum
Mass eigenstates:
= 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

28. Pion mass splitting
unstable

29. 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
=> 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 =>
pion as generalized meson; generalized GOR relation
Vector mesons, nucleons and
other heavy excitations
(Hatsuda, Tachibana, & Yamamoto,
in preparation)
Vector meson continuity

30. THE END

31. Toy Model Two complex scalar fields: Lagrangian: light “pion”
heavy “pion”
Diagonalize to find mass relations:

32. 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 .