… 1

● At asymptotic densities and T = 0, the ground state of QCD is the CFL phase (highly symmetric diquark condensate)

● Understanding the interior of CSO’s

● Study of the QCD phase diagram at T~ 0 and moderate density (phenomenological handle?)

Real question: does this type of phase persist at relevant densities

(~5-6 0)?

What do we know about the ground state of the color superconducting

phase of QCD?

… 2

Pairing fermions with different Fermi

momenta● Ms not zero

● Neutrality with respect to em and color

● Weak equilibrium

All these effects make Fermi momenta of different fermions unequal causing

problems to the BCS pairing mechanism

no free energy cost in neutral singlet, (Amore et al. 2003)

!

… 3

Weak equilibrium makes chemical potentials of quarks of different charges unequal:

From this: and

N.B. e is not a free parameter: neutrality requires:

¹ i ¹ Qi¹ Q¹ e ¡ ¹ Q

Q ¡ @V@¹ e

d ! ueº ) ¹ d ¡ ¹ u ¹ e

… 4

Neutrality and β equilibrium

If the strange quark is massless this equation has solutionNu = Nd = Ns , Ne = 0; quark matter electrically neutral with no electrons

Non interacting quarks

¹ d;s ¹ u ¹ e

… 5

● Fermi surfaces for neutral and color singlet unpaired quark matter at the equilibrium and Ms not zero.

● In the normal phase 3 = 8 = 0.

By taking into account Ms

¹ e ¼pdF ¡ puF ¼puF ¡ psF ¼M s = ¹

pdF ¡ psF ¼ ¹ e

… 6

Notice that there are also color neutrality conditions

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

● The system goes back to the normal phase● Other phases can be formed

@V@¹

T ; @V@¹

T

… 7

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

For For , an unpairing , an unpairing (blocking) region opens up and(blocking) region opens up and gapless modesgapless modes are presentare present (relevant in astrophysical (relevant in astrophysical applications)applications)

For For , the gaps are, the gaps are andand

2dm Energy cost for pairing

2D Energy gained in pairing

begins to unpair

E p j±¹ §qp ¡ ¹ j

E

p

gapless m odes

blocking region

E p , p ¹ §q

±¹ ¡

±¹ >

… 8

The case of 3 flavors gCFLgCFL

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 the gaps. For instance (but many other possibilities exist)

h jîaLïbL j i ²®̄ ²ab ²®̄ ²ab ²®̄ ²ab

(Alford, Kouvaris & Rajagopal, 2005)

… 9

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 in

gCFL

2

1

3

: us pairing: ds pairin

: ud pairing

g

… 10

Strange quark mass effects:

● Shift of the chemical potential for the strange quarks:

● Color and electric neutrality in CFL requires

●The transition CFL to gCFL starts with the unpairing of the pair ds with (close to the transition)

¹ ®s ) ¹ ®s ¡ M s ¹

¹ ¡ M s ¹ ; ¹ ¹ e

±¹ ds M s ¹

… 11

Energy cost for pairing

Energy gained in pairing

begins to unpair

It follows:

Calculations within a NJL model (modelled on one-gluon exchange):

● Write the free energy:

● Solve:

Neutrality

Gap equations

M s¹

M s¹ >

@V@¹ e

@V@¹

@V@¹

@V@ i

V ¹ ; ¹ ; ¹ ; ¹ e; i

… 12

● CFLgCFL 2nd order transition at Ms

2, when the pairing ds starts breaking

0 25 50 75 100 125 150M S

2/ [MeV]0

5

10

15

20

25

30

Gap P

aram

eters

[MeV

]

3

2

1

0 25 50 75 100 125 150M 2 / [MeV]

-50

-40

-30

-20

-10

0

10

Ener

gy D

iffere

nce

[10

6 MeV

4 ]

gCFL

CFL

unpaired

2SC

g2SC

s

(Alford, Kouvaris & Rajagopal, 2005)

(0 = 25 MeV, = 500 MeV)

… 13

0 20 40 60 80 100 120

-4-3-2-101

m (M )m (0)

M

M

s

M s2

2

2

1,2

3

8

● gCFL has gapless quasiparticles, and there are gluon imaginary masses (RC et al. 2004, Fukushima 2005).

0 20 40 60 80 100 120-0.25

00.250.5

0.751

1.25m (M )m (0)

M

M

s

M s2

2

2

4,5

6,7

● Instability present also in g2SC (Huang & Shovkovy 2004; Alford & Wang 2005)

… 14

● Gluon condensation. Assuming artificially <A 3> or <A 8> not zero (of order 10 MeV) this can be done (RC et

al. 2004) . In g2SC the chromomagnetic instability can be cured by a chromo-magnetic condensate (Gorbar, Hashimoto, Miransky, 2005 & 2006; Kiriyama, Rischke, Shovkovy, 2006). Rotational symmetry is broken and this makes a connection with the inhomogeneous LOFF phase (see later). At the moment no extension to the three flavor case.

How to solve the chromomagnetic instability

… 15

● CFL-K0 phase. When the stress is not too large (high density) the CFL pattern might be modified by a flavor rotation of the condensate equivalent to a condensate of K0 mesons (Bedaque, Schafer 2002). This occurs for ms > m1/3 2/3. Also in this phase gapless modes are present and the gluonic instability arises (Kryjevski, Schafer 2005, Kryjevski, Yamada 2005). With a space dependent condensate a current can be generated which resolves the instability. Again some relations with the LOFF phase. No extension to the three flavor case.

… 16

● Single flavor pairing. If the stress is too big single flavor pairing could occur but the gap is generally too small. It could be important at low before the nuclear phase (see for instance Alford 2006)

● Secondary pairing. The gapless modes could pair forming a secondary gap, but the gap is far too small (Huang, Shovkovy, 2003; Hong 2005; Alford, Wang, 2005)

● Mixed phases of nuclear and quark matter (Alford, Rajagopal, Reddy, Wilczek, 2001) as well as mixed phases between different CS phases, have been found either unstable or energetically disfavored (Neumann, Buballa, Oertel, 2002; Alford, Kouvaris, Rajagopal, 2004).

… 17

● Chromomagnetic instability of g2SC makes the crystalline phase (LOFF) with two flavors energetically favored (Giannakis & Ren 2004), also there are no chromomagnetic instability although it has gapless modes (Giannakis & Ren 2005), however see talk by Hashimoto.

… 18

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

● The impurities produce a constant exchange field acting upon the electron spin giving rise to an effective difference in the chemical potentials of the electrons producing a mismatch of the Fermi momenta● Studied also in the QCD context (Alford, Bowers &

Rajagopal, 2000)

LOFF phase

… 19

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

More generally

fixed variationally

chosen spontaneously

~p ~k ~q; ~p ¡ ~k ~q hÃxÃx i e i~q¢~x

hÃxÃx i X

m me i~qm¢~x

~p ~p ~q

j~qj

~q=j~qj

rings

… 20

Single plane wave:

Also in this case, for an unpairing (blocking) region opens up and gapless modes are present

More general possibilities include a crystalline structure ((Larkin & Ovchinnikov 1964, Bowers & Rajagopal 2002))

The qi’s define the crystal pointing at its vertices.

¹ ±¹ ¡ ~vF ¢~q >

hà xÃx i X

~qie i~qi ¢~x

E ~k ¡ ¹ ) E § ~k ~q ¡ ¹ ¨ ±¹ ¼q j~kj ¡ ¹ ¨ ¹

¹ ±¹ ¡ ~vF ¢~q

… 21

1e1n T/)),p((d,u

ForFor T T 00

))()(1(),p(

1)2(pd

2g1 3

3

||blocking blocking regionregion

The blocking region reduces the gap:The blocking region reduces the gap:

BCSLOFF

… 22

642

32(for regular crystalline structures all the

q are equal)The coefficients can be determined microscopically for the different structures (Bowers and Rajagopal (2002)Bowers and Rajagopal (2002) ))

The LOFF phase has been studied via a Ginzburg-Landau expansion of the grand potential

… 23

Gap equationGap equation

Propagator expansion

Insert in the gap equation

General strategy

… 24

We get the equation

053

Which is the same as

0

with

3

5

The first coefficient has universal structure, independent on the

crystal. From its analysis one draws the

following results

… 25

22normalLOFF )(44.

)2(4

2BCS

2normalBCS

)(15.1 2LOFF

2/BCS1 BCS2 754.0

Small window. Opens up in QCD?

(Leibovich, Rajagopal & Shuster 2001; Giannakis, Liu

& Ren 2002)

… 26

Single plane wave

Critical line from

0q

,0

Along the critical line

)2.1q,0Tat( 2

… 27

Preferred structure:

face-centered

cube

Bowers and Bowers and Rajagopal Rajagopal

(2002)(2002)

… 28

0)x(

4)x(

4)x(

… 29

In the LOFF phase translations and rotations are broken

phononsPhonon field through the phase of the

condensate (R.C., Gatto, Mannarelli & Nardulli 2002R.C., Gatto, Mannarelli & Nardulli 2002):)x(ixqi2 ee)x()x(

xq2)x(

introducing

xq2)x()x(f1

PhononPhononss

… 30

2

22||2

2

2

222

phonon zv

yxv

21L

Coupling phonons to fermions (quasi-particles) trough the gap term

CeC)x( T)x(iT

It is possible to evaluate the parameters of LLphononphonon (R.C., Gatto, Mannarelli & R.C., Gatto, Mannarelli &

Nardulli 2002Nardulli 2002)

153.0|q|

121v

22

694.0

|q|v

22||

++

… 31

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(

f1

… 32

Coupling phonons to fermions (quasi-particles) trough the gap term

i

)i(i

;3,2,1i

T)x(iT CeC)x(

(i)(i)(x) (x) transforms under the group O Ohh of the cube. . Its e.v. ~ xi breaks O(3)xO O(3)xOhh ~ ~ OOhh

diagdiag

2(i)(i) 2

phononi 1,2,3 i 1,2,3

2(i) (i) ( j)i i j

i 1,2,3 i j 1,2,3

1 aL | |2 t 2

b c2

… 33

we get for the coefficients

121a 0b

1

|q|3

121c

2

One can evaluate the effective lagrangian for the gluons in tha

anisotropic medium. For the cube one findsIsotropic propagationIsotropic propagation

This because the second order invariant for the cube and for the rotation group

are the same!

… 34

Preliminary results about LOFF with three flavors

Recent study of LOFF with 3 flavors within the following simplifying hypothesis (RC, Gatto, Ippolito, Nardulli & Ruggieri, 2005)

● Study within the Landau-Ginzburg approximation.

● Only electrical neutrality imposed (chemical potentials 3 and 8

taken equal to zero).

● Ms treated as in gCFL. Pairing similar to gCFL with inhomogeneity in terms of simple plane waves, as for the simplest LOFF phase.

hîaLïbL i

X

I I ~x ²®̄ I ²abI ; I ~x I e i~qI ¢~x

… 35

A further simplifications is to assume only the following geometrical configurations for the vectors qI, I=1,2,3 (a more general angular dependence will be considered in future work)

The free energy, in the GL expansion, has the form

with coefficients and IJ calculable from an effective NJL four-fermi interaction simulating one-gluon exchange

1 2 3 4

¡ normal X

I

0@®I I

¯I I

X

I 6 J

¯ I J

I J

1A O

normal ¡

¼ ¹ u ¹ d ¹ s ¡

¼¹ e

… 36

¯ I qI ; ±¹ I ¹ ¼

qI ¡ ±¹ I

®I qI ;±¹ I ¡ ¹ ¼

à ¡ ±¹ I

qI

¯̄¯̄¯qI ±¹ IqI ¡ ±¹ I

¯̄¯̄¯ ¡

¯̄¯̄¯ qI ¡ ±¹ I

¯̄¯̄¯

!

¯ ¡ ¹ ¼

Z dn¼

q1 ¢n ¹ s ¡ ¹ d q2 ¢n ¹ s ¡ ¹ u

! ; ¹ s $ ¹ d

! ; ¹ s $ ¹ u

´ BCS; ¹ u ¹ ¡

¹ e; ¹ d ¹

¹ e; ¹ s ¹

¹ e¡ M s ¹

… 37

We require:@

@ I

@@qI

@@¹ e

At the lowest order in I

@@qI

) @®I@qI

since I depends only on qI and i

we get the same result as in the simplest LOFF case:

j~qI j :±¹ I

In the GL approximation we expect to be pretty close to the normal phase, therefore we will assume 3 = 8 = 0. At the same order we expect 2 = 3 (equal mismatch) and 1 = 0 (ds mismatch is twice the ud and us).

… 38

Once assumed 1 = 0, only two configurations for q2 and q3, parallel or antiparallel. The antiparallel is disfavored due to the lack of configurations space for the up fermions.

… 39

2

1

3

: us pairing: ds pairin

: ud pairing

g

(we have assumed the same parameters as in

Alford et al. in gCFL, 0 = 25 MeV, = 500 MeV)

;

… 40

LOFF phase takes over gCFL at about 128 MeV and goes over to the normal phase at about 150 MeV

(RC, Gatto, Ippolito, Nardulli, Ruggieri, 2005)

Confirmed by an exact solution of the gap equation (Mannarelli, Rajagopal, Sharma, 2006)

Comparison with other phases

… 41

Longitudinal masses

Transverse masses

No chromo-magnetic instability in the LOFF phase with three flavors (Ciminale, Gatto, Nardulli, Ruggieri, 2006)

M M M M

M M

… 42

Extension to a crystalline structure (Rajagopal, Sharma 2006), always within the simplifying assumption 1 = 0 and 2 = 3

hudi ¼ Xa i~qa

¢~r; husi ¼ Xa i~qa

¢~r

The sum over the index a goes up to 8 qia. Assuming also 2 = 3

the favored structures (always in the GL approximation up to 6) among 11 structures analyzed are

CubeX 2Cube45z

… 43

… 44

Conclusions

● Various phases are competing, many of them having gapless modes. However, when such modes are present a chromomagnetic instability arises.

● Also the LOFF phase is gapless but the gluon instability does not seem to appear.

● Recent studies of the LOFF phase with three flavors seem to suggest that this should be the favored phase after CFL, although this study is very much simplified and more careful investigations should be performed.

● The problem of the QCD phases at moderate densities and low temperature is still open.