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

Introduction to color superconductivityIntroduction to color superconductivity

Effective theory of CSEffective theory of CS

Gap equation Gap equation

The anisotropic phase (LOFF): phase diagram The anisotropic phase (LOFF): phase diagram and crystalline structureand crystalline structure

PhononsPhonons

LOFF phase in compact stellar objectsLOFF phase in compact stellar objects

OutlookOutlook

SummarySummary

LiteratureLiterature

Reviews of color superconductivityReviews of color superconductivity::

T. Schaefer, hep-ph/0304281T. Schaefer, hep-ph/0304281

K. Rajagopal and F. Wilczek, hep-ph/0011333K. Rajagopal and F. Wilczek, hep-ph/0011333

G. Nardulli, hep-ph/0202037G. Nardulli, hep-ph/0202037

Original LOFF papers:Original LOFF papers:

A.J. Larkin and Y. N. Ovchinnikov, Zh. Exsp. Teor. Fiz. A.J. Larkin and Y. N. Ovchinnikov, Zh. Exsp. Teor. Fiz. 47 (1964) 113647 (1964) 1136

P. Fulde and R.A. Ferrel, Phys. Rev. 135 (1964) A550P. Fulde and R.A. Ferrel, Phys. Rev. 135 (1964) A550

Review of the LOFF phase:Review of the LOFF phase:

R. Casalbuoni and G. Nardulli, hep-ph/0305069R. Casalbuoni and G. Nardulli, hep-ph/0305069

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 instability:Bailin and Love 1984) based on Cooper instability:

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 )6333(

IntroductionIntroduction

CS can be treated perturbatively for large CS can be treated perturbatively for large due to asymptotic freedomdue to asymptotic freedom

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

00 jbia

p

p

s

s Only possible pairings Only possible pairings

LL and RRLL and RR

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

abCC

bRaRbLaL 0000

Symmetry breaking patternSymmetry breaking pattern

RLcRLc )3(SU)3(SU)3(SU)3(SU

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)

ab3

bLaL 00

RLcRLc )2(SU)2(SU)2(SU)2(SU)2(SU)3(SU

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)

mmss22//

Possible new anisotropic phase of QCDPossible new anisotropic phase of QCD

Effective theory of Effective theory of Color Color

SuperconductivitySuperconductivity

Relevant scales in Relevant scales in CSCS

Fp ((gapgap))

(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 (Hong 2000)of antiparticles (Hong 2000)LLHDETHDET

Decoupling of gapped quasi-Decoupling of gapped quasi-particles. Only light modes as particles. Only light modes as

Goldstones, etc. (R.C. & Gatto; Goldstones, etc. (R.C. & Gatto; Hong, Rho & Zahed 1999)Hong, Rho & Zahed 1999)

LLGoldGold

p – pp – pFF >> >>

p – pp – pFF << <<

ppFF

ppFF

ppFF + +

ppFF + +

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

Marginal term in the effective descriptionMarginal term in the effective description )pp,pp( 4321

and attractive interactionand attractive interaction

The marginal term becomes relevant at 1 – loopThe marginal term becomes relevant at 1 – loop

BCS instability solved by condensation and BCS instability solved by condensation and formation of Cooper pairsformation of Cooper pairs

resT**

3

3

M S)p(C)p()p(C)p(dt)2(

pd

2

1S

SSresres is neglected in the mean field approximation is neglected in the mean field approximation

G

2)x(C)x(

G

2)x(C)x(xdS

**T3

res

The first term in SThe first term in SM M behaves as a Majorana mass term behaves as a Majorana mass term

and it is convenient to work in theand it is convenient to 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

2

p2 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 relevantare 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 couplingAt 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

MeV100Evaluationd from QCD first principles at asymptotic Evaluationd 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 MeV100

The anisotropic The anisotropic phase (LOFF)phase (LOFF)

In many different situations pairing may happen between In many different situations pairing may happen between fermions belonging to Fermi surfaces with different radius, fermions belonging to Fermi surfaces with different radius, for instance:for instance:

• Quarks with different massesQuarks with different masses

• Requiring electric neutralityRequiring electric neutrality

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 ifPairing possible if

2M

The problem may be simulated using massless fermions The problem may be simulated using massless fermions with different chemical potentials (Alford, Bowers & with different chemical potentials (Alford, Bowers &

Rajagopal 2000)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

ppF2 F2 = =

ppF1 F1 = = – M– M22/2/2

ppFFcc = = – M – M22/4/44

M2

4M2

EE11(p(pFFcc)) = =

4M2

4M2

EEF1F1== EEF2F2 = =

EE22(p(pFFcc) = ) = MM22/4/4

2

4cFiFi

cF 16

M))p()(pp(

LOFF:LOFF: ferromagnetic alloy with paramagnetic ferromagnetic alloy with paramagnetic impurities. 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 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

First order transitionFirst order transition, , since forsince for 11,,

For For , , usual BCSusual BCS second order transitionsecond order transition at T= 0.5669 at T= 0.5669 BCSBCS

Existence of aExistence of a tricritical point tricritical point in the plane (in the plane (T)T)

According LOFF possible condensation with According LOFF possible condensation with non zero total momentum of the pairnon zero total momentum of the pair

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.

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 asWhich 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

2/BCS1 BCS2 754.0

Small window. Opens Small window. Opens up in QCD? (Leibovich, up in QCD? (Leibovich,

Rajagopal & Shuster Rajagopal & Shuster 2001; Giannakis, Liu & 2001; Giannakis, Liu &

Ren 2002)Ren 2002)

Results of Leibovich, Rajagopal & Shuster (2001)

(MeV) BCS (BCS

(LOFF) 0.754 0.047

400 1.24 0.53

1000 3.63 2.92

Corrections for non weak couplingCorrections for non weak coupling

NormalNormal

LOFFLOFF

BCSBCS

weak couplingweak coupling strong couplingstrong coupling

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

Tricritical point

General study by Combescot and Mora (2002). General study by Combescot and Mora (2002). Favored structureFavored structure 2 antipodal vectors2 antipodal vectors

At T = 0 the antipodal At T = 0 the antipodal vector leads to a second order vector leads to a second order phase transition. Another phase transition. Another tricritical point ? (Matsuo et tricritical point ? (Matsuo et al. 1998)al. 1998)

Change of crystalline structure from tricritical Change of crystalline structure from tricritical

to zero temperature?to zero temperature?

Two-dimensional case (Two-dimensional case (Shimahara 1998Shimahara 1998))

2

c

caNa T

TTb

2

1

0qqq

)]rqcos(

)rqcos()rq[cos(2)r(

]eee[)r(

)]qycos()qx[cos(2)r(

)rqcos(2)r(

e)r(

321

3

21hexa

rqirqirqitra

sqa

FFLOa

rqiFFa

321

Analysis close to the critical lineAnalysis close to the critical line

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 (R.C., Phonon field through the phase of the condensate (R.C., Gatto, Mannarelli & Nardulli 2002):Gatto, Mannarelli & Nardulli 2002):

)x(ixqi2 ee)x()x(

xq2)x(

introducingintroducing 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 2002)(R.C., Gatto, Mannarelli & Nardulli 2002)

153.0|q|

12

1v

2

2

694.0

|q|v

2

2||

++

Cubic structureCubic structure

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

0)x(

4)x(

4)x(

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(

(i)(i)(x) transforms under the group O(x) transforms under the group Ohh of the cube. of the cube.

Its e.v. ~ xIts e.v. ~ xi i breaks O(3)xObreaks O(3)xOhh ~ ~ OOhhdiagdiag. Therefore we get. Therefore we get

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 evaluate the effective lagrangian for the gluons in One can evaluate the effective lagrangian for the gluons in tha anisotropic medium. For the cube one findstha 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 fromFor LOFF state from ppFFBCSBCS 70)MeV(14

s,d,ui

iqeqees,d,ui

ii NNQNNN

0Qe

2

s,d,ui

iF2B )p(

3

1

3

1

Weak equilibrium:Weak equilibrium:

2s

2s

sFes

ddFed

uuFeu

Mp,3

1

p,3

1

p,3

2

Electric neutralityElectric neutrality::

n.m.n.m.is the saturation nuclear density ~ .15x10is the saturation nuclear density ~ .15x1015 15 g/cmg/cm

At the core of the neutron star At the core of the neutron star B B ~ 10~ 101515 g/cm g/cm

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 explanation: metallic crust + neutron Standard explanation: metallic crust + neutron superfluide insidesuperfluide inside

LOFF region inside the star providing the crystalline LOFF region inside the star providing the crystalline structure + superfluid CFL phasestructure + superfluid CFL phase

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 ofRecent 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. hyperfine doublets. However reaching equal populations However reaching equal populations is a big technical problemis a big technical problem ((Combescot 2001Combescot 2001). ). LOFF phase?LOFF phase?

OutlookOutlook