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QCD@Work 2003International Workshop onQuantum Chromodynamics
Theory and ExperimentConversano (Bari, Italy)
June 14-18 2003
Inhomogeneous Inhomogeneous color color
superconductivitysuperconductivityRoberto CasalbuoniRoberto Casalbuoni
Department of Physics and INFN – Florence Department of Physics and INFN – Florence & &
CERN TH Division - GenevaCERN TH Division - Geneva
IntroductionIntroduction
Effective theory of CSEffective theory of CS
Gap equation Gap equation
The inhomogeneous phase (LOFF): phase The inhomogeneous phase (LOFF): phase diagram and crystalline structurediagram and crystalline structure
PhononsPhonons
LOFF phase in compact stellar objectsLOFF phase in compact stellar objects
OutlookOutlook
SummarySummary
ab3
bLaL 00
IntroductionIntroduction mmuu, m, mdd,, mms s << << CFL phaseCFL phase
abCC
bRaRbLaL 0000
mmuu, m, mdd << << << m << mss : 2SC phase : 2SC phase
RLcRLc )3(SU)3(SU)3(SU)3(SU
RLcRLc )2(SU)2(SU)2(SU)2(SU)2(SU)3(SU
Possible new inhomogeneous phase of QCDPossible new inhomogeneous phase of QCD
In this situation strange quark decouples. But what In this situation strange quark decouples. But what happens in the intermediate region of happens in the intermediate region of The interesting The interesting
region is forregion is for (see later) (see later) mmss
22//
Effective theory of Effective theory of Color Color
SuperconductivitySuperconductivity
Relevant scales in Relevant scales in CSCS
Fp (gap)(gap)
(cutoff)(cutoff)
Fermi momentum defined byFermi momentum defined by
)p(E F
The cutoff is of order The cutoff is of order D D in in
superconductivity and > superconductivity and > QCD QCD
in QCDin QCD
Fp
Hierarchies of effective Hierarchies of effective lagrangianslagrangians
Microscopic descriptionMicroscopic description LLQCDQCD
Quasi-particles (dressed fermions Quasi-particles (dressed fermions as electrons in metals). Decoupling as electrons in metals). Decoupling
of antiparticles (of antiparticles (Hong 2000Hong 2000))LLHDETHDET
Decoupling of gapped quasi-Decoupling of gapped quasi-particles. Only light modes as particles. Only light modes as
Goldstones, etc. (Goldstones, etc. (R.C. & Gatto; Hong, R.C. & Gatto; Hong,
Rho & Zahed 1999Rho & Zahed 1999))
LLGoldGold
>> p – p>> p – pFF >> >>
p – pp – pFF << <<
ppFF
ppFF
ppFF + +
ppFF + + p – pp – pFF >> >>
Physics near the Fermi Physics near the Fermi surfacesurface
)p( F
Relevant terms in the effective descriptionRelevant terms in the effective description ((see:see: Polchinski, TASI 1992, also Hong 2000; Beane, Bedaque & Polchinski, TASI 1992, also Hong 2000; Beane, Bedaque &
Savage 2000, also R.C., Gatto & Nardulli 2001Savage 2000, also R.C., Gatto & Nardulli 2001))
))p(E(idt)2(
pdS t3
3
R
)p()p()p()p()pppp(dt)2(
pd
2
GS 42314321
34
1k3
k3
M
4-fermi attractive interaction is4-fermi attractive interaction is marginal (relevant at 1-loop)marginal (relevant at 1-loop)
)pp,pp( 4321
SSM M gives risegives rise di-fermion condensation producing a di-fermion condensation producing a
Majorana mass term. Work in the Majorana mass term. Work in the Nambu-GorkovNambu-Gorkov basis: basis:
)p(C
)p(
2
1
Near the Fermi surfaceNear the Fermi surface
)pp(v)pp(p
)p(E)p(E FFF
pp
p
F
FF vp
Fvp
p*
p1
E
ES
p*
p
22p
2 E
E
E
1S
Dispersion relationDispersion relation22
p)p(
Infinite copies of 2-d physicsInfinite copies of 2-d physics
vv11
vv22
At fixed vAt fixed vFF only only energy and energy and
momentum along vmomentum along vFF are relevant are relevant
Gap Gap equationequation
2BCS
224
4
4
|p|p
1
)2(
pdG1
n223
3
),p()T)1n2((
1
)2(
pdGT1
),p(
nn1
)2(
pd
2
G1 du
3
3
1e
1nn
T/),p(du
For TFor T 00
2BCS
23
3
)p(
1
)2(
pd
2
G1
At weak coupling At weak coupling
BCSF
2F
2
2log
v
p
2
G1
)cutoff(
G
2
BCS e2F
2
2F
v
p
density of statesdensity of states
With G fixed by With G fixed by SB at T = 0, requiring SB at T = 0, requiring MMconstconst ~ 400 MeV ~ 400 MeV
and for typical values of and for typical values of ~ 400 – 500 MeV one gets~ 400 – 500 MeV one gets
MeV10010 Evaluation from QCD first principles at asymptotic Evaluation from QCD first principles at asymptotic
((Son 1999Son 1999))
s
2
g2
3
5segb
Notice the behavior exp(-c/g) and not exp(-c/gNotice the behavior exp(-c/g) and not exp(-c/g22) as one ) as one would expect from four-fermi interactionwould expect from four-fermi interaction
For For ~ 400 MeV one finds again ~ 400 MeV one finds again MeV10010
The inhomogeneous The inhomogeneous phase (LOFF)phase (LOFF)
In many different situations the “would be” pairing fermions In many different situations the “would be” pairing fermions belong to Fermi surfaces with different radii:belong to Fermi surfaces with different radii:
• Quarks with different massesQuarks with different masses
• Requiring electrical neutrality and/or weak equilibriumRequiring electrical neutrality and/or weak equilibrium
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
To form a BCS condensate one needs common momenta To form a BCS condensate one needs common momenta of the pair pof the pair pFF
commcomm
4
Mp
2commF
)p()2(
pd2
Fp
03
3 Grand potential at T = 0 Grand potential at T = 0 for a single fermionfor a single fermion
42
1i
commFiFi
commF
2 M))p()(pp(2
Pairing energyPairing energy 22
Pairing possible if Pairing possible if
2M
The problem may be simulated using massless fermions with The problem may be simulated using massless fermions with different chemical potentials (different chemical potentials (Alford, Bowers & Rajagopal 2000Alford, Bowers & Rajagopal 2000))
Analogous problem studied by Analogous problem studied by Larkin & Larkin & Ovchinnikov, Fulde & Ferrel 1964Ovchinnikov, Fulde & Ferrel 1964. Proposal . Proposal
of a new way of pairing. of a new way of pairing. LOFF phaseLOFF phase
LOFF:LOFF: ferromagnetic alloy with paramagnetic ferromagnetic alloy with paramagnetic impurities. impurities.
TheThe impurities produce a constant exchange 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 spinsof the opposite spins. .
Very difficult experimentally but claims of Very difficult experimentally but claims of observations in heavy fermion superconductorsobservations in heavy fermion superconductors ((Gloos & al 1993Gloos & al 1993)) and in quasi-two dimensional layered and in quasi-two dimensional layered organic superconductors (organic superconductors (Nam & al. 1999, Manalo & Klein Nam & al. 1999, Manalo & Klein
20002000))
21 or paramagnetic impurities (or paramagnetic impurities (H) H) give rise to an energy additive termgive rise to an energy additive term
3IH
)2(4
2BCS
2normalBCS
2224
4
4
|p|)ip(
1
)2(
pdG1
Gap equationGap equation
Solution as for BCS Solution as for BCS BCSBCS, up to (for T = , up to (for T =
0) 0) BCS
BCS1 707.0
2
According LOFF, close to first order line, possible According LOFF, close to first order line, possible condensation with condensation with non zero total momentumnon zero total momentum
qkp1
qkp2
xqi2e)x()x(
xqi2
mm
mec)x()x(More generallyMore generally
q2pp 21
|q|
|q|/q
fixed variationallyfixed variationally
chosen chosen spontaneouslyspontaneously
Simple plane wave: Simple plane wave: energy shiftenergy shift
)qk(E)p(E
qvF
Gap equation:Gap equation:),p(
nn1
)2(
pd
2
g1 du
3
3
1e
1n
T/)),p((d,u
du nn
For T For T 00
))()(1(),p(
1
)2(
pd
2
g1
3
3
|| blocking regionblocking region
The blocking region reduces the gap:The blocking region reduces the gap:
BCSLOFF
Possibility of a crystalline structure (Larkin & Possibility of a crystalline structure (Larkin & Ovchinnikov 1964, Bowers & Rajagopal 2002)Ovchinnikov 1964, Bowers & Rajagopal 2002)
xqi2
2.1|q|iq
i
i
e)x()x(
The qThe qii’s define the crystal pointing at its vertices.’s define the crystal pointing at its vertices.
The LOFF phase is studied via a Ginzburg-Landau The LOFF phase is studied via a Ginzburg-Landau expansion of the grand potentialexpansion of the grand potential
see latersee later
642
32
(for regular crystalline structures all the (for regular crystalline structures all the qq are equal) are equal)
The coefficients can be determined microscopically for The coefficients can be determined microscopically for the different structures (the different structures (Bowers and Rajagopal (2002)Bowers and Rajagopal (2002)))
Gap equationGap equation
Propagator expansionPropagator expansion
Insert in the gap equationInsert in the gap equation
We get the equationWe get the equation
053
Which is the same as Which is the same as 0
withwith
3
5
The first coefficient has The first coefficient has universal structure, universal structure,
independent on the crystal. independent on the crystal. From its analysis one draws From its analysis one draws
the following resultsthe following results
22normalLOFF )(44.
)2(4
2BCS
2normalBCS
)(15.1 2LOFF
Small window. Opens up in QCD? Small window. Opens up in QCD? ((Leibovich, Rajagopal & Shuster 2001; Leibovich, Rajagopal & Shuster 2001;
Giannakis, Liu & Ren 2002Giannakis, Liu & Ren 2002))
BCS2
BCS1
754.0
2/
Results of Leibovich, Rajagopal & Shuster (2001)
(MeV) BCS (BCS
LOFF 0.754 0.047
400 1.24 0.53
1000 3.63 2.92
Single plane waveSingle plane wave
Critical line fromCritical line from
0q
,0
Along the critical lineAlong the critical line
)2.1q,0Tat( 2
Preferred Preferred structure:structure:
face-centered face-centered cubecube
General General analysisanalysis
((Bowers and Bowers and
Rajagopal (2002)Rajagopal (2002)))
In the LOFF phase translations and rotations are brokenIn the LOFF phase translations and rotations are broken
phononsphonons
Phonon field through the phase of the condensate (Phonon field through the phase of the condensate (R.C., R.C.,
Gatto, Mannarelli & Nardulli 2002Gatto, Mannarelli & Nardulli 2002):):
)x(ixqi2 ee)x()x(
xq2)x(
Introduce:Introduce: xq2)x()x(f
1
PhononsPhonons
2
22||2
2
2
222
phonon zv
yxv
2
1L
Coupling phonons to fermions (quasi-particles) trough Coupling phonons to fermions (quasi-particles) trough the gap termthe gap term
CeC)x( T)x(iT
It is possible to evaluate the parameters of LIt is possible to evaluate the parameters of Lphononphonon
((R.C., Gatto, Mannarelli & Nardulli 2002R.C., Gatto, Mannarelli & Nardulli 2002))
153.0|q|
12
1v
2
2
694.0
|q|v
2
2||
++
Cubic Cubic structurestructure
i
)i(i
i
iik
;3,2,1i
)x(i
;3,2,1i
x|q|i28
1k
xqi2 eee)x(
i)i( x|q|2)x(
i)i()i( x|q|2)x()x(
f
1
Coupling phonons to fermions (quasi-particles) trough Coupling phonons to fermions (quasi-particles) trough the gap termthe gap term
i
)i(i
;3,2,1i
T)x(iT CeC)x(
Using the symmetry group of the cube one gets:Using the symmetry group of the cube one gets:
3,2,1ji
)j(j
)i(i
2
3,2,1i
)i(i
3,2,1i
2)i(
3,2,1i
2)i(
phonon
c2
b
||2
a
t2
1L
we get for the coefficientswe get for the coefficients
12
1a 0b
1
|q|3
12
1c
2
One can also evaluate the effective lagrangian for the One can also evaluate the effective lagrangian for the gluons in the anisotropic medium. For the cube one findsgluons in the anisotropic medium. For the cube one finds
Isotropic propagationIsotropic propagation
This because the second order invariant for the cube This because the second order invariant for the cube and for the rotation group are the same!and for the rotation group are the same!
Why the interest Why the interest in the LOFF in the LOFF
phase in QCD?phase in QCD?
LOFF phase in CSOLOFF phase in CSO
In neutron stars CS can be studied at T = 0In neutron stars CS can be studied at T = 0
)K10MeV1(
100)MeV(201010T
10
BCS76
BCS
ns
Orders of magnitude from a crude model: 3 free quarksOrders of magnitude from a crude model: 3 free quarks
0M,0MM sdu
For LOFF state from For LOFF state from ppFFBCSBCS 70)MeV(14
n.m.n.m.is the saturation nuclear density ~ .15x10is the saturation nuclear density ~ .15x1015 15 g/cmg/cm33
At the core of the neutron star At the core of the neutron star B B ~ 10~ 101515 g/cm g/cm33
65.m.n
B Choosing Choosing ~ 400 MeV~ 400 MeV
Ms = 200 pF = 25
Ms = 300 pF = 50Right ballpark Right ballpark (14 - 70 MeV) (14 - 70 MeV)
)10Ω/Ω( 6
Glitches: discontinuity in the period of the pulsars.Glitches: discontinuity in the period of the pulsars.
Standard explanations require: metallic crust + Standard explanations require: metallic crust + superfluide inside (neutrons)superfluide inside (neutrons)
LOFF region inside the star might provide a LOFF region inside the star might provide a crystalline structure + superfluid CFL phasecrystalline structure + superfluid CFL phase
New possibilities for strange starsNew possibilities for strange stars
Theoretical problemsTheoretical problems:: Is the cube the optimal Is the cube the optimal structure at T=0? Which is the size of the LOFF structure at T=0? Which is the size of the LOFF window?window?
Phenomenological problemsPhenomenological problems: : Better discussion Better discussion of the glitches (treatment of the vortex lines)of the glitches (treatment of the vortex lines)
New possibilitiesNew possibilities: : Recent achieving of Recent achieving of degenerate degenerate ultracold Fermi gasesultracold Fermi gases opens up new fascinating opens up new fascinating possibilities of reaching the onset of Cooper pairing of possibilities of reaching the onset of Cooper pairing of hyperfine doublets. Possibility of observing the hyperfine doublets. Possibility of observing the LOFF LOFF crystal?crystal?
OutlookOutlook
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