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The three flavor LOFF phase of QCD
N. D. IppolitoUniversity and INFN, Bari, Italy
HISS : Dense Matter in HIC and Astrophysics, Dubna, 2006HISS : Dense Matter in HIC and Astrophysics, Dubna, 2006
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Very high densities ( >> mquark) and low temperature ( T 0 )
CFL superconductive phase CFL superconductive phase (Color Flavor Locking; Alford, Rajagopal and Wilczek 1999)(Color Flavor Locking; Alford, Rajagopal and Wilczek 1999)
( Nf = 3 )
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Note the presence of just oneone gap parameter for all the
pairs.
Form of the CFL condensate
3
1IijI
αβIβj
αi εεΔ0ψψ0 ~
(Neglecting the condensation in the symmetric 6 channel)
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Going down with the density, we cannot still neglect the strange quark
mass.
The condition >> ms is not more fulfilled !
• ms 0
• Color and electrical neutrality must be imposed• Equilibrium under weak interactions
Different gaps for pairs of
different flavors
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Gapless CFL phase(Alford, Kouvaris, Rajagopal 2004)(Alford, Kouvaris, Rajagopal 2004)
Pairing ansatz
3
1IijI
αβII
βj
αi εεΔ0ψψ0 ~
1 ~ ds 2 ~ us 3 ~ ud
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Results of gCFL phase
Gap parameters
Free energy
( Alford, Kouvaris, Rajagopal : hep-ph/0406137 )
BUT…
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Imaginary Meissner masses
Gluon 8
Gluons 1,2
Gluon 3
( Casalbuoni, Gatto, Mannarelli, Nardulli, Ruggieri : hep-ph/0410401 )
Signal of instability of Signal of instability of gCFL phasegCFL phase
Problem not yet solved. Probably indicates that gCFL is not the true
vacuum
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LOFF phase
An inhomogeneousinhomogeneous side of Superconductivity
Larkin, Ovchinnikov 1964; Fulde, Ferrell 1964 ;
Alford, Bowers, Rajagopal 2001;
Casalbuoni, Nardulli 2004
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In presence of a difference of chemical potentials :
Two flavor Superconductivity(not necessarily CSC)
BCSBCS . 70702
1
BCS survives until
up
down
For > 1 it’s difficult to form pairs with zero total momentum
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LOFF :In a window 1 < < 2 0.754 BCS it can be
energetically favourable to form pairs with non zero total momentum
Ptot = p1+ p2 = 2q 0
Simplest ansatz for the condensate (one plane wave)
~ ei2q•r(r)
In general, more plane waves:
rqiP
mm
me)r(
2
1
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LOFF phase in QCD with LOFF phase in QCD with three flavors three flavors
Casalbuoni, Gatto, NDI, Nardulli, Ruggieri. PLB 2005Casalbuoni, Gatto, NDI, Nardulli, Ruggieri. PLB 2005
Pairing ansatz
ijII
IIji )r(C
3
15
rqiII
Ie)r( 2
with
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Requirements and Requirements and approximationsapproximations
-equilibrated quark matter • Non zero strange quark mass 3= 8=0• HDET(High Density Effective
Theory) approximation• Mean field approximation• Ginzburg-Landau approximation for
the free energy and the gap• Imposition of electrical neutrality
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-equilibrium: d= u+ e ; s= u+ e;Strange quark mass treated at the leading order in 1/: s s-ms
2/2 ; 3= 8=0 ; (recently justified by Casalbuoni,
Ciminale, Gatto, Nardulli, Ruggieri; June 2006)
The chemical potential term in the Lagrangean has the form
αβijie
αβij δ)δQμ(μμ
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Explicitely we have
2μ
2s
m
eμ
3
1μ
sμ
eμ
3
1μ
dμ
eμ
3
2μ
uμ
strange
downup
where
)Mdiag(0,0,δM sαβαβ
ij ijαβ
aaijαβαβ
ij δTigAδδD ;
So the starting point is the free
Lagrangean
L= βj0αβij
αβij
αβijiα )ψγμM(iDψ
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High Density Effective Theory
Large component
Small residual momentum
In four dimensions
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In this way we can consider just the degrees of freedom near the Fermi surface, i.e. the residual component of quark momenta, and integrate only on a small region near it.
Within HDET, the free Lagrangean reads
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To this free Lagrangean we add a NJL coupling treated in the mean field approximation
where
is the pairing ansatz.
with
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(This change is performed by matrices that are combinations of Gell-Mann matrices)
and introduce the Nambu-Gor’kov field.
So the complete Lagrangean reads
Let’s change the basis for the spinor fields
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Ginzburg-Landau expansion
Gap Equation
0Δ
Ω
I
321 ,,I,
Electrical neutrality0μ
Ω
e
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The norm of qI is fixed minimizing the Free Energy.
At the first order in
0q
Ω
IqI 1.2 I
As to the directions of the qI , one should look for the energetically favored orientations
CrystallographCrystallographyy
In our work we consider just four
structures, with the qI parallel or
antiparallel to the same axis
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Results
The favorite structure has
1=0, 2 = 3 and q2,q3 parallel
The result of imposing electrical neutrality is just
4
2s
e
M
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Free energy diagram
Loff phase with three flavors DOES NOT suffer of chromomagnetic instabilities!
(Ciminale, Nardulli, Ruggieri, Gatto hep-ph/0602180)
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Very good result, but recently other good news!!
( Rajagopal, Sharma hep-ph/0605316 )
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Free energy
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Conclusions
•Three flavor LOFF phase is chromomagnetically stable
•It has lower free energies than the normal phase and the homogeneous phases in a wide window of Ms
2/
It is a serious candidate for being It is a serious candidate for being the true vacuum at intermediate the true vacuum at intermediate
densitiesdensities