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

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 )

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)

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

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

Results of gCFL phase

Gap parameters

Free energy

( Alford, Kouvaris, Rajagopal : hep-ph/0406137 )

BUT…

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

LOFF phase

An inhomogeneousinhomogeneous side of Superconductivity

Larkin, Ovchinnikov 1964; Fulde, Ferrell 1964 ;

Alford, Bowers, Rajagopal 2001;

Casalbuoni, Nardulli 2004

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

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

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

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

-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

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

High Density Effective Theory

Large component

Small residual momentum

In four dimensions

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

To this free Lagrangean we add a NJL coupling treated in the mean field approximation

where

is the pairing ansatz.

with

(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

Ginzburg-Landau expansion

Gap Equation

0Δ

Ω

I

321 ,,I,

Electrical neutrality0μ

Ω

e

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

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

Free energy diagram

Loff phase with three flavors DOES NOT suffer of chromomagnetic instabilities!

(Ciminale, Nardulli, Ruggieri, Gatto hep-ph/0602180)

Very good result, but recently other good news!!

( Rajagopal, Sharma hep-ph/0605316 )

Free energy

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