QCD@Work 2003 International Workshop on Quantum Chromodynamics Theory and Experiment Conversano (Bari, Italy) June 14-18 2003 Inhomogeneous color superconductivity.

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