1. What do we know about the ground state of the color superconducting phase of QCD?
Roberto Casalbuoni
Dept. of Physics, University of Florence
INFN & GGI - Florence
Origin of Mass and Strong Coupling Gauge Theories
Nagoya November 21-24, 2006
R. Casalbuoni: What do we…

2. R. Casalbuoni: What do we…
Outline of the talk
Introduction
Pairing fermions with different Fermi momenta
The gapless phase gCFL
The LOFF phase
Results about the LOFF phase with 3 flavors
Astrophysical consequences (if enough time)
Conclusions and outlook
Nagoya November 21-24, 2006
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3. R. Casalbuoni: What do we…
Introduction
Motivations for the study of high-density QCD
Understanding the interior of CSO’s
Study of the QCD phase diagram at T~ 0 and moderate m
Asymptotic region in m fairly well understood: existence of a CS phase. Real question: does this type of phase persist at relevant densities (~5-6 r0)?
Nagoya November 21-24, 2006
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4. Pairing fermions with different Fermi momenta
Ms not zero
Neutrality with respect to em and color
Weak equilibrium
no free energy cost in neutral singlet, (Amore et al. 2003)
All these effects make Fermi momenta of different fermions unequal causing problems to the BCS pairing mechanism
Nagoya November 21-24, 2006
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5. Effective chemical potential for the massive quark
Consider 2 fermions with m1 = M, m2 = 0 at the same chemical potential m. The Fermi momenta are
Effective chemical potential for the massive quark
M2/2 effective chemical potential
Mismatch:
Nagoya November 21-24, 2006
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6. R. Casalbuoni: What do we…
If electrons are present, weak equilibrium makes chemical potentials of quarks of different charges unequal:
From this: with
N.B. me is not a free parameter: neutrality requires:
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7. Neutrality and β equilibrium
Non interacting quarks
If the strange quark is massless this equation has solution
Nu = Nd = Ns , Ne = 0; quark matter electrically neutral with no electrons
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8. R. Casalbuoni: What do we…
By taking into account Ms
Fermi surfaces for neutral and color singlet unpaired quark matter at the  equilibrium and Ms not zero.
In the normal phase 3 = 8 = 0.
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9. R. Casalbuoni: What do we…
As long as dm is small no effects on BCS pairing, but when increased the BCS pairing is lost and two possibilities arise:
The system goes back to the normal phase
Other phases can be formed
Notice that there are also color neutrality conditions
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10. R. Casalbuoni: What do we…
In a simple model without supplementary conditions, with two fermions at chemical potentials m1 and m2, the system becomes normal at the Chandrasekhar - Clogston point. Another unstable phase exists.
Nagoya November 21-24, 2006
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11. R. Casalbuoni: What do we…
The point |dm| = D is special. In the presence of a mismatch new features are present. The spectrum of quasiparticles is
For |dm| < D, the gaps are D - dm and D + dm
For |dm| = D, an unpairing (blocking) region opens up and gapless modes are present (relevant in astrophysical applications)
Energy cost for pairing
begins to unpair
Energy gained in pairing
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12. (Alford, Kouvaris & Rajagopal, 2005)
gCFL
The case of 3 flavors
(Alford, Kouvaris & Rajagopal, 2005)
Different phases are characterized by different values for the gaps. For instance (but many other possibilities exist)
Nagoya November 21-24, 2006
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13. R. Casalbuoni: What do we…-1 +1 ru gd bs rd gu rs bu gs bd
Gaps in gCFL
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14. R. Casalbuoni: What do we…
Strange quark mass effects:
Shift of the chemical potential for the strange quarks:
Color and electric neutrality in CFL requires
The transition CFL to gCFL starts with the unpairing of the pair ds with (close to the transition)
Nagoya November 21-24, 2006
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15. R. Casalbuoni: What do we…
It follows:
Energy cost for pairing
begins to unpair
Energy gained in pairing
Calculations within a NJL model (modelled on one-gluon exchange):
Write the free energy:
Solve:
Neutrality
Gap equations
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16. (Alford, Kouvaris & Rajagopal, 2005)
CFL # gCFL 2nd order transition at Ms2/m ~ 2D, when the pairing ds starts breaking
(Alford, Kouvaris & Rajagopal, 2005)
(0 = 25 MeV,  = 500 MeV)
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17. R. Casalbuoni: What do we…
gCFL has gapless quasiparticles, and there are gluon imaginary masses also in this phase (RC et al. 2004, Fukushima 2005).
Instability present also in g2SC (Huang & Shovkovy 2004; Alford & Wang 2005)
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18. Solving the chromomagnetic instability
Gluon condensation. Assuming artificially <A 3> or <A 8> not zero (of order 10 MeV) this can be done (RC et al. 2004) . In g2SC the chromomagnetic instability can be cured by a chromo-magnetic condensate (Gorbar, Hashimoto, Miransky, 2005 & 2006; Kiriyama, Rischke, Shovkovy, 2006) . Rotational symmetry is broken and this makes a connection with the inhomogeneous LOFF phase (see later). At the moment no extension to the three flavor case.
Nagoya November 21-24, 2006
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19. R. Casalbuoni: What do we…
CFL-K0 phase. When the stress is not too large (high density) the CFL pattern might be modified by a flavor rotation of the condensate equivalent to a condensate of K0 mesons (Bedaque, Schafer 2002). This occurs for ms > m1/3 2/3. Also in this phase gapless modes are present and the gluonic instability arises (Kryjevski, Schafer 2005, Kryjevski, Yamada 2005). With a space dependent condensate a current can be generated which resolves the instability. Again some relations with the LOFF phase. No extension to the three flavor case.
Nagoya November 21-24, 2006
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Single flavor pairing. If the stress is too big single flavor pairing could occur but the gap is generally too small. It could be important at low  before the nuclear phase (see for instance Alford 2006)
Secondary pairing. The gapless modes could pair forming a secondary gap, but the gap is far too small (Huang, Shovkovy, 2003; Hong 2005; Alford, Wang, 2005)
Mixed phases of nuclear and quark matter (Alford, Rajagopal, Reddy, Wilczek, 2001) as well as mixed phases between different CS phases, have been found either unstable or energetically disfavored (Neumann, Buballa, Oertel, 2002; Alford, Kouvaris, Rajagopal, 2004).
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21. This makes the LOFF phase very interesting
Chromomagnetic instability of g2SC makes the crystalline phase (LOFF) with two flavors energetically favored (Giannakis & Ren 2004), also there are no chromomagnetic instability although it has gapless modes (Giannakis & Ren 2005), however see talk by Hashimoto.
This makes the LOFF phase very interesting
Nagoya November 21-24, 2006
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22. R. Casalbuoni: What do we…
LOFF phase
LOFF (Larkin, Ovchinnikov, Fulde & Ferrel, 1964): ferromagnetic alloy with paramagnetic impurities.
The impurities produce a constant exchange field acting upon the electron spin giving rise to an effective difference in the chemical potentials of the opposite spin producing a mismatch of the Fermi momenta
Studied also in the QCD context (Alford, Bowers & Rajagopal, 2000)
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23. R. Casalbuoni: What do we…
According to LOFF, close to first order point (CC point), possible condensation with non zero total momentum
More generally
fixed variationally
chosen spontaneously
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24. The qi’s define the crystal pointing at its vertices.
Single plane wave:
Also in this case, for
an unpairing (blocking) region opens up and gapless modes are present
More general possibilities include a crystalline structure (Larkin & Ovchinnikov 1964, Bowers & Rajagopal 2002)
The qi’s define the crystal pointing at its vertices.
Nagoya November 21-24, 2006
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25. R. Casalbuoni: What do we…
Small window. Opens up in QCD? (Leibovich, Rajagopal & Shuster 2001; Giannakis, Liu & Ren 2002)
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26. Preliminary results about LOFF with three flavors
Recent study of LOFF with 3 flavors within the following simplifying hypothesis (RC, Gatto, Ippolito, Nardulli & Ruggieri, 2005)
Study within the Landau-Ginzburg approximation.
Only electrical neutrality imposed (chemical potentials 3 and 8 taken equal to zero).
Ms treated as in gCFL. Pairing similar to gCFL with inhomogeneity in terms of simple plane waves, as for the simplest LOFF phase.
Nagoya November 21-24, 2006
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27. R. Casalbuoni: What do we…
A further simplifications is to assume only the following geometrical configurations for the vectors qI, I=1,2,3 (a more general angular dependence will be considered in future work) 1 2 3 4
The free energy, in the GL expansion, has the form
with coefficients I, I and IJ calculable from an effective NJL four-fermi interaction simulating one-gluon exchange
Nagoya November 21-24, 2006
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Nagoya November 21-24, 2006
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We require:
At the lowest order in I
since I depends only on qI and i we get the same result as in the usual LOFF case:
In the GL approximation we expect to be pretty close to the normal phase, therefore we will assume 3 = 8 = 0. At the same order we expect 2 = 3 (equal mismatch) and 1 = 0 (ds mismatch is twice the ud and us).
Nagoya November 21-24, 2006
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30. R. Casalbuoni: What do we…
Once assumed  1 = 0, only two configurations for q2 and q3, parallel or antiparallel. The antiparallel is disfavored due to the lack of configurations space for the up fermions.
Nagoya November 21-24, 2006
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(we have assumed the same parameters as in Alford et al. in gCFL, 0 = 25 MeV,  = 500 MeV)
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32. Comparison with other phases
LOFF phase takes over gCFL at about 128 MeV and goes over to the normal phase at about 150 MeV
(RC, Gatto, Ippolito, Nardulli, Ruggieri, 2005)
Confirmed by an exact solution of the gap equation (Mannarelli, Rajagopal, Sharma, 2006)
Nagoya November 21-24, 2006
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No chromo-magnetic instability in the LOFF phase with three flavors (Ciminale, Gatto, Nardulli, Ruggieri, 2006)
Transverse masses
Longitudinal masses
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Extension to a crystalline structure (Rajagopal, Sharma 2006), always within the simplifying assumption 1 = 0 and 2 = 3
The sum over the index a goes up to 8 qia. Assuming also 2 = 3 the favored structures (always in the GL approximation up to 6) among 11 structures analyzed are
2Cube45z
CubeX
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36. Specific heat and neutrino emissivity
Cv ~ T for electrons and gapless or free quarks;
Cv ~ exp(-/T) 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])
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37. 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:
V = volumes of matter, nm (nuclear), qm (quark)
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38. Cooling of neutron star
Cooling faster for a core made up of unpaired or LOFF-paired quarks (Angiani, Nardulli & Ruggieri, 2006)
nm:
R ~ 12 KM
qm:
R1~5 Km, R2~10 Km
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39. R. Casalbuoni: What do we…
Conclusions
The problem of the QCD phases at moderate densities and low temperature is still open.
Various phases are competing, many of them having gapless modes. However, when such modes are present a chromomagnetic instability arises.
Also the LOFF phase is gapless but the gluon instability does not seem to appear.
Recent studies of the LOFF phase with three flavors seem to suggest that this should be the favored phase after CFL, although this study is very much simplified and more careful investigations should be performed.
Nagoya November 21-24, 2006
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