1. Color Superconductivity
Giuseppe Nardulli
Physics Department and INFN Bari
Varenna, June 29, 2006

2. Collaboration Bari - Firenze- Geneva
R. Casalbuoni, M. Ciminale, R. Gatto, N. Ippolito, M. Mannarelli, G. Nardulli, M. Ruggieri
Review papers
F. Wilczek, K. Rajagopal, hep-ph/
G. N. Riv. N. Cim.(2002) hep-ph/
R. Casalbuoni and G.N. Rev. Mod. Phys, (2004) hep-ph/

3. Nuclear matter and QCD
At small nuclear density and low temperature the relevant degrees of freedom of nuclear matter are the hadrons: nucleons, hyperons, mesons.
At high density and/or high temperature the relevant dof are quarks and gluons. The corresponding theory is Quantum ChromoDynamics
QCD is a gauge theory based on the local gauge group SU(3), with quarks belonging to the fundamental and gluons to the adjoint representation
QCD enjoys the property of Asymptotic Freedom

5. Accelerators (RHIC,LHC, GSI) and compact stars

6. Continuous emission of e.m. radiation and slowing down
Structure of a Pulsar
Continuous emission of e.m. radiation and slowing down

7. Phenomenology of pulsar: glitches
Sudden increases of pulsar rotational frequency
Vela Psr

8. Further phenomenology
Effects on EOS
Cooling by neutrino emission
Neutrinos from supernovae
Shear viscosity and other transport properties

9. Nuclear matter at T=0, large μ
Quarks have S=1/2. They are in three colors (r,g,b) and, if light, in three flavors (u,d,s).
When two quarks scatter (3 x 3 ) they are either in a sextet (color symmetric) or in antitriplet (color antisymmetric) state
Gluon exchange in the antitriplet is attractive

10. Free fermion gas and BCS mechanism
Per
f(E) E

11. Free energy
Adding a particle at the Fermi surface
Extracting a particle, i.e. forming a hole
Free energy
unchanged

12. SC more efficient in QCD
For an arbitrary attractive interaction it is energetically convenient to form particle pairs (Cooper pairs)
In metals SC realized only in special conditions, when the attractive interactions between electrons, due to phonon exchange beats the repulsive e.m. forces
QCD:attractive channels with
SC more efficient in QCD

13. QCD phases at T=0, large μ
As a consequence of the attractive gluonic interaction in the color antisymmetric channel and the Cooper’s theorem:
Formation of Cooper pairs of quarks (diquarks) and color superconductivity
The condensate is colored, not white as in the quark-antiquark channel, relevant at zero density

14. Cooper pairs
a) Normal preferred state: Ppair=0

15. b) Preferred total spin Spair=0
Two particle w.f.
ψ(r) = χ Σk gk eikr r = r1 – r2
The spin wf χ with S=0 is antisymmetric in spin therefore the orbital part has behavior
cos kr (symmetric in r), while S=1 has behavior sin kr; coupling disfavored at small r

16. Color Superconductivity at extreme densities
QCD with 3 flavors: attractive interaction in color antisymmetric (antitriplet) channel
For S=0 state Pauli principle implies antisymmetry in flavor
Asymptotically (very large densities):Color Flavor Locking (CFL)
< 0 | ψi α ψj β |0> = Δ εijγ ε αβγ
Valid for μ >> mq

17. CFL (color flavor locking) phase
Symmetries of the
CFL (color flavor locking) phase
< 0 | ψ ψ |0>  0
broken spontaneously. Also U(1) global symmetry
(baryonic number) broken. 8 gluons acquire mass
(Higgs-Anderson). 8+1 Goldstone bosons similar
to π and K of chiral symmetry breaking

18. CS at intermediate densities
Analysis complicated because the three light quarks are different: strange quark has mass comparable with μ for μ of the order of 500 MeV, while up, down are massless
These are the densities expected in the core of neutron stars where CS most likely takes place
In stars quark matter must be neutral and in beta equilibriun due to
d -> u  e and u e -> d 

19. Neutrality and β equilibrium
Non interacting quarks
μd,s = μu + μe
If the strange quark is massless this equation has solution
Nu = Nd = Ns , Ne = 0; quark matter electrically neutral with no electrons

20. Strange quark mass

21. Choosing the true vacuum:
Mass effects, neutrality, beta equilibrium:
The result is that there are different gaps for quarks of different flavor.
Gapless phases: uniform (gCFL) or not uniform (LOFF). Three gaps
< 0 | ψi α ψj β |0> = γΔγ εijγ ε αβγ
Choosing the true vacuum:
The true ground state has the minimum value of the free energy (grand potential)

22. Free energy Gapless CFL Gap Parameters
M.Alford, P.Jotwani, C. Kouvaris, J. Kundu, K.Rajagopal
Gapless CFL

23. The importance of being gapless
Gapless modes: relevant for astrophysical effects
(phenomenology of compact stars)
µ= 500 MeV
Δ= 25 MeV
M2s/2µ= 80 MeV

24. Meissner masses in the gCFL phase
Solid line a=1,2;
Gluons 3, 8:
same behavior;
4,5,6,7: no instability
R. Casalbuoni,
M. Mannarelli,
G. N., M. Ruggieri, R. Gatto
Imaginary masses: instability
Symptom that it is not the true vacuum

25. LOFF= Larkin-Ovchinnikov&
Loff Phase
If δμ0 it can be energetically
favourable to have
states where the Cooper pair
has total momentum 2q 0
LOFF= Larkin-Ovchinnikov&
Fulde-Ferrel (1964);
R. Casalbuoni, G.N.
Rev. Mod. Phys.(2004)
Δ ( r) = < 0 | ψ( r)ψ( r)|0> = Δ exp[i(p1 +p2 )r]

26. Two flavors
Condensate BCS
(independent of δμ)
Condensate
LOFF
Ω(BCS) - Ω (normal)
Ω(LOFF)-Ω (normal)

27. Symmetry breaking for LOFF phase with 3 flavors
T(3) x O(3) Oh
Cube group

28. LOFF phase: three flavors
Needed for realistic calculations
R.Gatto,R.Casalbuoni,N.Ippolito,GN, M.Ruggieri : Ginzburg Landau expansion of the gap equation and free energy and M.Mannarelli, K. Rajagopal, R. Sharma: one plane wave
K. Rajagopal, R. Sharma: Cristallography
< 0 | ψi α ψj β |0> = γΔγ εijγ ε αβγ q exp iqr

29. Ginzburg-Landau expansion of the gap equation Δ (small)

30. Gap parameters: one plane wave
K( r) = K e2i qk r
For each inhomogeneous pairing a Fulde-Ferrell ansatz; qk
represents the momentum of the Cooper pair.
This is the simplest ansatz, other structures should be examined
Three independent functions
1( r), 2( r), 3( r),
describing respectively d-s, u-s and u-d pairing.

31. Expected: μe =m2s/4μ Δ1 = 0; Δ2 = Δ3
Near the transition point du,us couplings, equal strength, no ds pairing
Expected:
μe =m2s/4μ
Δ1 = 0; Δ2 = Δ3

32. q2 q3 parallel favored by phase space

33. =n+ 1/4I=13 (2I I2+ II4 + JIIJI2J2) +O(6)
Ginzburg-Landau expansion
=n+ 1/4I=13 (2I I2+ II4 + JIIJI2J2) +O(6)
n = -3/122(u4+d4+s4) -1/ 122 e4
-k  = 0 minimum in k
-e = 0 electrical neutrality
Approximation: 3'8' 0 (confirmed by M.Ciminale et al., in preparation: Valid near the transition point, since this is the result for the normal phase)
The numerical results in this regime confirm 1=0; 2  3
R. Casalbuoni, R.Gatto,,
N.Ippolito,GN,
M.Ruggieri

34. Comparison among different CSC phases
Ginzburg Landau Approximation
Calculation at
leading order in μ
R.Gatto,
R.Casalbuoni,
N.Ippolito,
G.Nardulli,
M.Ruggieri
M.Mannarelli,
K. Rajagopal,
R. Sharma

35. Free energy expansion for more plane waves
Ω= αΔ2+βΔ4+γΔ6 +δΔ8…
The favored structures for both 2 and 3 flavors have cubic symmetry (Bowers, Rajagopal, Sharma).

37. Chromomagnetic instability
In gCFL : imaginary gluon Meissner masses
Studied in LOFF model with 2 flavors (Giannakis&Ren, Fukushima, Gorbar, Hashimoto&Miransky).
LOFF favored in comparison to gapless uniform conductive states. At least in the GL region (small gap) no chromomagnetic instability
For 3 flavors: recent results (Ciminale,Gatto,GN,Ruggieri)

38. Feynman Diagrams at O(Δ4)

39. Gluon Meissner masses:no instability
Transverse mass
Longitudinal mass
M.Ciminale,
R.Gatto,GN,
M.Ruggieri

40. Compact stars and CS Model of neutron star with quark core
Laboratory for the study ogf color superconductivity
Model of neutron star with quark core

41. Specific heat and neutrino emissivity
Cv  T for electrons and gapless or free quarks;
Cv exponentially suppressed for gapped quarks
E=neutrino emissivity
E T6 for gapless or free quarks;
E T8 for nuclear matter (URCA [beta decay] processes forbidden, modified URCA allowed [ n+X->p+X+e+ν]

42. Cooling of a compact star
L dt = heath lost in time dt
-V Cv dT =change in the energy of star
Taking into account quark and nuclear matter for neutrino emission as well as photon emission, the rate of change in T:
dT = -- Vnm Enm + Vqm Eqm + Lγ
dt Vnm Cnm + Vqm Cqm
V=volumes of nm (nuclear) or qm (quark)

43. Cooling of neutron star (preliminary)

44. Conclusions
Color superconductivity is a theoretical prediction based on the theory of hadronic interactions, QCD
Best laboratory to study it: future dedicated experiments with heavy ions or the core of compact stars
Compact star lab: remote, but experimental signatures can be found and should allow to find the correct QCD vacuum at high, but not very high nuclear density