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Color Color Superconductivity in Superconductivity in

High Density QCDHigh Density QCD

Roberto CasalbuoniRoberto Casalbuoni

Department of Physics and INFN - FlorenceDepartment of Physics and INFN - Florence

Villasimius, September 21-25, 2004Villasimius, September 21-25, 2004

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IntroductionIntroduction

Motivations for the study of high-density QCD:Motivations for the study of high-density QCD:

● Understanding the interior of CSO’sUnderstanding the interior of CSO’s

● Study of the QCD phase diagram at Study of the QCD phase diagram at T~0 and high T~0 and high

Asymptotic region in Asymptotic region in fairly well fairly well understood: understood: existence of a CS existence of a CS phasephase. Real question: . Real question: does this does this

type of phase persists at relevant type of phase persists at relevant densities ( ~5-6 densities ( ~5-6

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SummarySummary

● Mini review of CFL and 2SC phasesMini review of CFL and 2SC phases

● Pairing of fermions with different Fermi momentaPairing of fermions with different Fermi momenta

● The gapless phases g2SC and gCFLThe gapless phases g2SC and gCFL

● The LOFF phaseThe LOFF phase

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Study of CS back to 1977 (Barrois 1977, Frautschi 1978, Study of CS back to 1977 (Barrois 1977, Frautschi 1978, Bailin and Love 1984) based on Cooper instabilityBailin and Love 1984) based on Cooper instability::

CFL and 2SCCFL and 2SC

At T ~ 0 a degenerate fermion gas is unstableAt T ~ 0 a degenerate fermion gas is unstable

Any weak attractive interaction leads to Any weak attractive interaction leads to Cooper pair formationCooper pair formation

Hard for electrons (Coulomb vs. phonons)Hard for electrons (Coulomb vs. phonons)

Easy in QCD for di-quark formation (attractive Easy in QCD for di-quark formation (attractive channel )channel )3 (3 3 = 3 6)Ä Å

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In QCD, CS easy for large In QCD, CS easy for large due to asymptotic due to asymptotic freedomfreedom

At high At high , m, mss, m, mdd, m, muu ~ 0, 3 colors and 3 flavors ~ 0, 3 colors and 3 flavors

Possible pairings:Possible pairings:

Antisymmetry in color (Antisymmetry in color () for attraction) for attraction

Antisymmetry in spin (a,b) for better use of the Antisymmetry in spin (a,b) for better use of the Fermi surfaceFermi surface

Antisymmetry in flavor (i, j) for Pauli principleAntisymmetry in flavor (i, j) for Pauli principle

α βia jb0 ψ ψ 0

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p

p

s

s

Only possible pairings Only possible pairings

LL and RRLL and RR

Favorite state Favorite state CFLCFL (color-flavor locking) (color-flavor locking) ((Alford, Rajagopal & Wilczek 1999Alford, Rajagopal & Wilczek 1999))

α β α β αβCaL bL aR bR abC0 ψ ψ 0 = - 0 ψ ψ 0 = Δε ε

Symmetry breaking patternSymmetry breaking pattern

c L R c+L+RSU(3) SU(3) SU(3) SU(3)Ä Ä Þ

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What happens going down with What happens going down with ? If ? If << m<< mss we get we get

3 colors and 2 flavors (2SC)3 colors and 2 flavors (2SC)

α β αβ3aL bL ab0 ψ ψ 0 = Δε ε

c L R c L RSU(3) SU(2) SU(2) SU(2) SU(2) SU(2)Ä Ä Þ Ä Ä

But what happens in real world ?

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● Ms not zero

● Neutrality with respect to em and color

● Weak equilibrium

All these effects make Fermi momenta of All these effects make Fermi momenta of different fermions unequal causing problems to different fermions unequal causing problems to

the BCS pairing mechanismthe BCS pairing mechanism

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Consider 2 fermions with mConsider 2 fermions with m1 1 = M, m= M, m22 = 0 at the same = 0 at the same

chemical potential chemical potential . The Fermi momenta are. The Fermi momenta are

221F Mp 2Fp

Effective chemical potential for the massive quarkEffective chemical potential for the massive quark2

2 2eff

MM

2m = m- » m-

m

Mismatch:Mismatch:2M

2dm»

m

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If electrons are present, weak equilibrium makes chemical potentials of quarks of different charges

unequal:

d u ed ue m- m® n =mÞ

In general we have the relation: i i Q( Q )m=m+ m

e Qm=- m

N.B. N.B. e e is not a free parameteris not a free parameter

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Neutrality requires:Neutrality requires:e

VQ 0

¶=- =

¶m

Example 2SC: normal BCS pairing whenExample 2SC: normal BCS pairing when

u d u dn nm =m Þ =

But neutral matter forBut neutral matter for

1/ 3d u d u e d u u

1n 2n 2 0

4» Þ m » m Þ m=m- m » m ¹

Mismatch:Mismatch:d uF F d u e up p

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82 2

- m- m mmd = = = »m ¹

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Also color neutrality requiresAlso color neutrality requires

3 83 8

V VT 0, T 0

¶ ¶= = = =

¶m ¶m

As long as As long as is small no effects on BCS pairing, but is small no effects on BCS pairing, but when increased the BCS pairing is lost and two when increased the BCS pairing is lost and two possibilities arise:possibilities arise:

● The system goes back to the normal phaseThe system goes back to the normal phase

● Other phases can be formedOther phases can be formed

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In a simple model with two fermions at chemical potentials In a simple model with two fermions at chemical potentials the system becomes normal at the the system becomes normal at the Chandrasekhar-Clogston point. Chandrasekhar-Clogston point. Another unstable phase exists.Another unstable phase exists.

BCS1

2

Ddm=

BCSdm=D

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2 2E(p) (p )= dm± - m +D

2 2E(p) 0 p =m± dmÛ - D=

The point The point is special. In the is special. In the presence of a mismatch new features are presence of a mismatch new features are present. The spectrum of quasiparticles ispresent. The spectrum of quasiparticles is

For |For |an an unpairing (blocking) unpairing (blocking) region opens up and region opens up and gapless modesgapless modes are are present.present.

For |For |the gaps the gaps are are and and

2dm Energy cost for pairing

2D Energy gained in pairing

2 2dm> Dbegins to unpair

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g2SCg2SC Same structure of condensates as in 2SC Same structure of condensates as in 2SC ((Huang & Shovkovy, 2003Huang & Shovkovy, 2003))

4x3 fermions:4x3 fermions:

● 2 quarks 2 quarks ungappedungapped q qubub, q, qdb db

● 4 quarks 4 quarks gappedgapped q qurur, q, qugug, q, qdrdr, q, qdg dg

General strategy (NJL model):General strategy (NJL model):

● Write the free energy:Write the free energy:

● Solve: Solve:

NeutralityNeutrality

Gap equationGap equation

3 8 eV( , , , , )mm m m D

e 3 8

V V V0

¶ ¶ ¶= = =

¶m ¶m ¶m

V0

¶=

¶D

α β αβ3aL bL ab0 ψ ψ 0 = Δε ε

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● For For ( (==ee/2/2) 2 gapped quarks become ) 2 gapped quarks become

gapless. The gapless quarks begin to unpair destroying gapless. The gapless quarks begin to unpair destroying the BCS solution. But a new stable phase exists, the the BCS solution. But a new stable phase exists, the gapless 2SC (g2SC) phase. gapless 2SC (g2SC) phase.

● It is the unstable phase which becomes stable in this It is the unstable phase which becomes stable in this case (and CFL, see later) when charge neutrality is case (and CFL, see later) when charge neutrality is required.required.

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g2SCg2SC

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● But evaluation of the gluon masses (5 out of 8 become But evaluation of the gluon masses (5 out of 8 become massive) shows an instability of the g2SC phase. Some of massive) shows an instability of the g2SC phase. Some of the gluon masses are imaginary (the gluon masses are imaginary (Huang and Shovkovy 2004Huang and Shovkovy 2004).).

● Possible solutions are: gluon condensation, or another Possible solutions are: gluon condensation, or another phase takes place as a crystalline phase (see later), or this phase takes place as a crystalline phase (see later), or this phase is unstable against possible mixed phases.phase is unstable against possible mixed phases.

● Potential problem also in gCFL (calculation not yet Potential problem also in gCFL (calculation not yet done).done).

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Generalization to 3 flavors Generalization to 3 flavors ((Alford, Kouvaris & Rajagopal, 2004Alford, Kouvaris & Rajagopal, 2004) )

gCFLgCFL

α β αβ1 αβ2 αβ3aL bL 1 ab1 2 ab2 3 ab30 ψ ψ 0 = Δ ε ε + Δ ε ε + Δ ε ε

3

3

3 2

2

1

1

1

2

g2SC : 0,

g

CFL :

CF

0

L :

D ¹ D =D =

D >D

D =D =D D

>D

=

Different phases are characterized by different values for Different phases are characterized by different values for the gaps. For instance (but many other possibilities exist)the gaps. For instance (but many other possibilities exist)

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Strange quark mass effects:Strange quark mass effects:

● Shift of the chemical potential for the strange Shift of the chemical potential for the strange quarks: quarks: 2

ss s

M

2a am Þ m -m

● Color and electric neutrality in CFL requires Color and electric neutrality in CFL requires 2s

8 3 e

M, 0

2m=- m=m=

m● gs-bd unpairing catalyzes CFL to gCFLgs-bd unpairing catalyzes CFL to gCFL

( )2s

bd gs bd gs 8

1

2

M

2- = m - m =- m=dmm

2s

rd gu e rs bu e

M,

2- -dm =m dm =m-m

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2M

mEnergy cost for pairing

2D Energy gained in pairing

2M2> D

m

begins to unpair

It follows:It follows:

Again, by using NJL model (modelled on one-gluon Again, by using NJL model (modelled on one-gluon exchange):exchange):● Write the free energy:Write the free energy:

● Solve: Solve:

NeutralityNeutrality

Gap equationsGap equations

3 8 e s iV( , , , , M , )mm m m D

e 3 8

V V V0

¶ ¶ ¶= = =

¶m ¶m ¶m

i

V0

¶=

¶D

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● CFL CFL gCFL 2gCFL 2ndnd order order transition at Mtransition at Mss

22// ~ ~

22when the pairing gs-when the pairing gs-bd starts breakingbd starts breaking

● gCFL has gapless gCFL has gapless quasiparticles. Interesting quasiparticles. Interesting transport propertiestransport properties

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● LOFF (LOFF (Larkin, Ovchinnikov, Fulde & Ferrel, 1964Larkin, Ovchinnikov, Fulde & Ferrel, 1964):): ferromagnetic alloy with paramagnetic impurities. ferromagnetic alloy with paramagnetic impurities.

● The impurities produce a constant exchange The impurities produce a constant exchange fieldfield acting upon the electron spins giving rise to acting upon the electron spins giving rise to an an effective difference in the chemical potentials effective difference in the chemical potentials of the opposite spins producing a of the opposite spins producing a mismatchmismatch of the of the Fermi momentaFermi momenta

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According to LOFF, close to first order point (CC point), According to LOFF, close to first order point (CC point), possible condensation withpossible condensation with non zero total momentumnon zero total momentum

qkp1

qkp2

xqi2e)x()x(

m2iq xm m

m

ψ(x)ψ(x) = Δ c e ×år r

More generallyMore generally

q2pp 21

|q|

|q|/q

fixed variationallyfixed variationally

chosen chosen spontaneouslyspontaneously

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Single plane wave:Single plane wave:2 2E(p) E( p q) (p )- m® ± + - m dm» - m +D m

r r rm m

qvF

Also in this case, for Also in this case, for F| | v qm=dm- × >Drr

a unpairing (blocking) region opens up and a unpairing (blocking) region opens up and gapless gapless modes are presentmodes are present

Possibility of a crystalline structure (Possibility of a crystalline structure (Larkin & Larkin &

Ovchinnikov 1964, Bowers & Rajagopal 2002Ovchinnikov 1964, Bowers & Rajagopal 2002))i

i

2iq x

|q |=1.2δμ

ψ(x)ψ(x) = Δ e ×år r

r

The qThe qii’s define the crystal pointing at its vertices.’s define the crystal pointing at its vertices.

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Crystalline Crystalline structures in LOFFstructures in LOFF

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Preferred Preferred structure:structure:

face-centered face-centered cubecube

Analysis via Analysis via GL expansionGL expansion

((Bowers and Bowers and

Rajagopal (2002)Rajagopal (2002)))

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Effective gap equation for the LOFF phaseEffective gap equation for the LOFF phase

((R.C., M. Ciminale, M. Mannarelli, G. Nardulli, M. Ruggieri & R. Gatto, 2004R.C., M. Ciminale, M. Mannarelli, G. Nardulli, M. Ruggieri & R. Gatto, 2004))

See next talk by M. Ruggieri See next talk by M. Ruggieri

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Multiple phase transitions from the CC Multiple phase transitions from the CC point (point (MMss

22//2SC2SCup to the cube up to the cube

case (case (MMss22// ~ 7.5 ~ 7.5 2SC2SCExtrapolating to Extrapolating to

CFL (CFL (2SC2SC ~ 30 MeV) one gets that LOFF ~ 30 MeV) one gets that LOFF

should be favored from about should be favored from about MMss

22// ~120 MeV ~120 MeV up up

MMss22//MeVMeV

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ConclusionsConclusions

● Under realistic conditions (MUnder realistic conditions (Mss not zero, color not zero, color

and electric neutrality) new CS phases might existand electric neutrality) new CS phases might exist

● In these phases gapless modes are present. This In these phases gapless modes are present. This result might be important in relation to the result might be important in relation to the transport properties inside a CSO.transport properties inside a CSO.