Download - Color Superconductivity in High Density QCD
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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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IntroductionIntroductionMotivations 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
MM2
m = 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 =mÞIn 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
V Q 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 04
» 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 V 0¶ ¶ ¶= = =¶m ¶m ¶mV 0¶ =¶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 :
CF0
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
M2a am Þ m - m
● Color and electric neutrality in CFL requires Color and electric neutrality in CFL requires 2s
8 3 eM , 02
m=- m=m=m● gs-bd unpairing catalyzes CFL to gCFLgs-bd unpairing catalyzes CFL to gCFL
( ) 2s
bd gs bd gs 812
M2- = m - m =- m=dm m
2s
rd gu e rs bu eM,2- -dm =m dm =m- m
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2Mm
Energy cost for pairing
2D Energy gained in pairing
2M 2> Dmbegins 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 V 0¶ ¶ ¶= = =¶m ¶m ¶m
i
V 0¶ =¶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 ×å r rMore 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 mr r r m 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 ×å r 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 (MMss
22// ~ 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
MMss22// ~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.
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g2SC parameters:g2SC parameters:
S D
2S D S
NJL with chiral (G ) and diquark (G ) couplings:
G 5.0163 GeV , G G , 0.750.6533 GeV, 400 MeV
-= =h h=L = m=
gCFL parameters:gCFL parameters:
0
NJL modelled on one gluon-exchange:25 MeV, 800 MeV, 500 MeVD = L = m=
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0 0 0 -1 +1 -1 +1 0 0ru gd bs rd gu rs bu gs bd
rugdbsrdgursbugsbd
Q%
1D2D3D
3D2D 1D
3- D3- D
2- D2- D
1- D1- D
Gaps Gaps in in
gCFLgCFL
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● gCFL has gCFL has ee not zero, with charge cancelled by not zero, with charge cancelled by unpaired u quarksunpaired u quarks