Color Superconductivity in High Density QCD

Roberto Casalbuoni Dept. of Physics, University of Florence INFN - Florence Budapest, August 1-3, 2005 JHW Budapest, August 1-3, 2005 R. Casalbuoni:Superconductivity in high density QCD

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Outline of the talk



Introduction

    

Pairing fermions with different Fermi momenta The gapless phases g2SC and gCFL The LOFF phase Preliminary results about the LOFF phase with 3 flavors Conclusions and outlook JHW Budapest, August 1-3, 2005 R. Casalbuoni:Superconductivity in high density QCD

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Introduction

Motivations for the study of high-density QCD



Understanding the interior of CSO’s



Study of the QCD phase diagram at T~ 0 and moderate

m

Asymptotic region in

m

fairly well understood : existence of a CS phase

.

Real question : does this type of phase persist at relevant densities (~5-6

r

0 )?

JHW Budapest, August 1-3, 2005 R. Casalbuoni:Superconductivity in high density QCD

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Pairing fermions with different Fermi momenta

●

M s not zero

●

Neutrality with respect to em and color no free energy cost in

!

( Amore et al. 2003 )

●

Weak equilibrium

All these effects make Fermi momenta of different fermions unequal causing problems to the BCS pairing mechanism

JHW Budapest, August 1-3, 2005 R. Casalbuoni:Superconductivity in high density QCD

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Consider 2 fermions with m 1 = M, m 2 chemical potential = 0 at the same

m

. The Fermi momenta are

p F   q ¹  ¡ M  p F   ¹

Effective chemical potential for the massive quark

¹   q ¹  ¡ M  ¼ ¹ ¡ M  ¹  Mismatch: ±¹ ¼ M   ¹

M 2 /2

m

effective chemical potential JHW Budapest, August 1-3, 2005 R. Casalbuoni:Superconductivity in high density QCD

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

If electrons are present, weak equilibrium makes chemical potentials of quarks of different charges unequal:

 d !

ue º  ) ¹  ¹ ¹ e  ¹ d ¡ ¹  Q i ¹ u  ¡ ¹ Q ¹ e 

N.B.

m

e is not a free parameter: neutrality requires:

Q  ¡ @V @¹ e  

JHW Budapest, August 1-3, 2005 R. Casalbuoni:Superconductivity in high density QCD

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

Example 2SC: normal BCS pairing when

¹ u  ¹ 

But neutral matter for

d ) n u  n d n d ¼  n u ) ¹ d ¼    ¹ u ) ¹ e  ¹ d ¡ ¹ u ¼   ¹ u

Mismatch

±¹  p d F ¡ p u F   ¹ d ¡ ¹  u  ¹  e ¼ ¹ u 

JHW Budapest, August 1-3, 2005 R. Casalbuoni:Superconductivity in high density QCD

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As long as

dm

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



Notice that there is also a color neutrality condition

@V @¹

 

T



@V @¹

 

T

  

JHW Budapest, August 1-3, 2005 R. Casalbuoni:Superconductivity in high density QCD

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In a simple model without supplementary conditions, with two fermions at chemical potentials

m 1

and

m

2 , the system becomes normal at the Chandrasekhar - Clogston point.

Another unstable phase exists.

   1.2

1 BCS CC ±¹   B CS 0.8

0.6

0.4

0.2

unstable phase 0.2

0.4

0.6

0.8

1 1.2

dm   r    0.4

Normal unstable phase 0.2

0 -0.2

0.2

0.4

BCS 0.6

CC 0.8

1 1.2

dm  

±¹

  

p

B CS  -0.4

JHW Budapest, August 1-3, 2005 R. Casalbuoni:Superconductivity in high density QCD

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

The point

|dm| = 

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

E   j±¹ § E q blocking region   j dm =  |dm| =  |dm|   For |dm| <  , the gaps are and  + dm  - dm For |dm| =  , an unpairing (blocking) region opens up and gapless modes are present p gapless modes 2 dm Energy cost for pairing E    , begins to unpair p  ¹ § q ±¹  ¡ 

±¹ >

   2 D Energy gained in pairing

JHW Budapest, August 1-3, 2005 R. Casalbuoni:Superconductivity in high density QCD

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g2SC

Same structure of condensates as in 2SC 4x3 fermions:



2 quarks ungapped q ub , q db

(

Huang & Shovkovy, 2003

) h  ® aL à ¯ bL j i   ² ®¯  ² ab  

4 quarks gapped q ur , q ug , q dr , q dg General strategy (NJL model):

 

Write the free energy Solve

V

 

; ¹



; ¹

e

;

 

Neutrality Gap equation

@V @¹ e  @V @¹   @V @¹    @V  

JHW Budapest, August 1-3, 2005 R. Casalbuoni:Superconductivity in high density QCD

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●

For

|dm|   ( dm = m

e

/ )

2 gapped quarks become gapless. The gapped quarks begin to unpair destroying the BCS solution. But a new stable phase exists, the gapless 2SC (g2SC) phase.

●

It is the unstable phase (Sarma phase) which becomes stable in this case (and in gCFL, see later) when charge neutrality is required.

JHW Budapest, August 1-3, 2005 R. Casalbuoni:Superconductivity in high density QCD

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Q = 0 g2SC V m e = const.

(

Huang & Shovkovy, 2003

) Normal state  (MeV) 200 solutions of the gap equation  150 neutrality line m e 2 =  V ( GeV /fm 3 ) - 0.081

- 0.082

Normal state 0 gap equation neutrality line 100 - 0.083

- 0.084

- 0.085

50 - 0.086

- 0.087

m e =148 MeV 0 0 100 200 300

JHW Budapest, August 1-3, 2005

400 m e (MeV) 20 40 60 80

R. Casalbuoni:Superconductivity in high density QCD

m e 100 120  ( MeV ) 13



However, evaluation of the gluon masses (5 out of 8 become massive) shows an instability of the g2SC phase. Some of the gluon masses are imaginary ( Huang and Shovkovy 2004 ). Connected with gapless modes? ( Alford & Wang, 2005 ).

4,5,6,7 8 m  g  ¹  g   ¼  

But this instability occurs also in the 2SC phase (no gapless modes)

p   < ±¹ < 

JHW Budapest, August 1-3, 2005 R. Casalbuoni:Superconductivity in high density QCD

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Generalization to 3 flavors (Alford, Kouvaris & Rajagopal, 2005)

gCFL h  ® aL à ¯ bL j i    ®¯  ab   ² ®¯  ² ab    ² ®¯  ² ab 

Different phases are characterized by different values for the gaps. For instance (but many other possibilities exist)

CFL : g2SC :

g CF L :

D = D = D

1 2 3

= D D № D = D =

3

0,

1 2

0 D > D

3 2

> D

1

JHW Budapest, August 1-3, 2005 R. Casalbuoni:Superconductivity in high density QCD

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0 0 0 -1 +1 -1 +1 0 0 ru gd bs rd ru gd bs rd gu rs bu gs bd

D D 2 3 D 3 D 1 D D 1 2 - D 3

gu

- D 3

rs bu

- D 2 - D 2

gs

- D 1

bd

- D 1

JHW Budapest, August 1-3, 2005 R. Casalbuoni:Superconductivity in high density QCD

Gaps in gCFL

 1  2  3 : ds : us : ud pairin g pairing pairing 16

Strange quark mass effects:

●

Shift of the chemical potential for the strange quarks:

¹ ®s ) ¹ ®s ¡ M  ¹ s  ●

Color and electric neutrality in CFL requires

¹   ¡ M  ¹  s ; ¹   ¹ e   ●

The transition CFL to gCFL starts with the unpairing of the pair gs-bd having (close to the transition)

±¹ ds  M  ¹ s 

JHW Budapest, August 1-3, 2005 R. Casalbuoni:Superconductivity in high density QCD

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M

It follows:

 s

Energy cost for pairing

¹ 

Energy gained in pairing

begins to unpair M s  ¹ > 

Again, by using NJL model (modelled on one-gluon exchange):

●

Write the free energy:

V



; ¹



; ¹

e

;

  ●

Solve: Neutrality

@V @¹ e  @V @¹   @V @¹   

Gap equations JHW Budapest, August 1-3, 2005

@V   i

R. Casalbuoni:Superconductivity in high density QCD

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10 g2SC unpaired ~  0 ●

CFL

#

gCFL 2 nd order transition at M s 2

/m ~ 

, when the pairing gs bd starts breaking

-10 -20 -30 2SC gCFL -40 CFL ~  30  3 -50 0 25 50 M 2 s / 75 m [MeV] 100 125 25 20  2 15

(Alford, Kouvaris & Rajagopal, 2005)

10

(



0 = 25 MeV,

m

= 500 MeV)

5 ~   1 0 0 25 50 M S 2 / m 75 [MeV] 100

JHW Budapest, August 1-3, 2005

125 150

R. Casalbuoni:Superconductivity in high density QCD

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●

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

m (M ) M s m (0) M 1 0 -1 -2 -3 -4 0 m (M ) M s m (0) M 1.25

1 0.75

0.5

0.25

0 -0.25

0 3 1,2 8 20 40 60 80 100 120 M s 2 m 4,5 6,7 20 40 60 80 100 120 M s 2 m

JHW Budapest, August 1-3, 2005 R. Casalbuoni:Superconductivity in high density QCD

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Instability cured by gluon condensation? Assuming artificially

m

3 > or

m

8 > not zero (of order 10 MeV) this can be done. See also a very recent paper ( Gorbar, Hashimoto & Miransky, 2005) about a gluonic phase curing the chromomagnetic instability in 2SC.

Three recent results obtained by Giannakis & Ren:



Chromomagnetic instability of g2SC makes the crystalline phase (LOFF) with two flavors energetically favored ( Giannakis & Ren 2004 )



LOFF with two flavors without requiring electrical neutrality has no magnetic instability although it has gapless modes ( Giannakis & Ren 2005 )



Last week the same result obtained requiring color and electric neutrality in the weak coupling limit ( Giannakis, Hou & Ren 2005 ) JHW Budapest, August 1-3, 2005 R. Casalbuoni:Superconductivity in high density QCD

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LOFF phase

●

LOFF

(

Larkin, Ovchinnikov, Fulde & Ferrel, 1964

):

ferromagnetic alloy with paramagnetic impurities

.

●

The impurities produce a constant exchange field to an acting upon the electron spins giving rise effective difference in the chemical potentials of the opposite spins producing a

mismatch

of the Fermi momenta

●

Studied also in the QCD context (

Alford, Bowers & Rajagopal, 2000

)

JHW Budapest, August 1-3, 2005 R. Casalbuoni:Superconductivity in high density QCD

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

~    ~ p 

More generally

  ~ hà    i X hà    i  m   e  i ~  m e  i ~ m ¢~

~

 

p

  

~

fixed variationally chosen spontaneously JHW Budapest, August 1-3, 2005 R. Casalbuoni:Superconductivity in high density QCD

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E  ¡ ¹ )

Single plane wave:

q E  p  ~ ¡ ¹ ¨ ±¹ ¼   ¨ ¹  ¹   ±¹ ¡ ~ F

Also in this case, for

¹

 

±¹ ¡ ~

F 

an unpairing (blocking) region opens up and gapless modes are present More general possibilities include a crystalline structure

hà (

Larkin & Ovchinnikov 1964, Bowers & Rajagopal 2002

) X   Ã x  i   e  i ~ i ¢~ ~ i

The q i ’s define the crystal pointing at its vertices.

JHW Budapest, August 1-3, 2005 R. Casalbuoni:Superconductivity in high density QCD

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 BCS  normal = r 4 (  2 BCS 2 dm 2 )  LOFF  normal = - .

44 r ( dm dm 2 ) 2  LOFF  1 .

15 ( dm 2 dm ) dm 1 dm 2 =   BCS / 2 0 .

754  BCS  

±¹

Small window. Opens up in QCD? ( Leibovich, Rajagopal & Shuster 2001; Giannakis, Liu & Ren 2002

)

JHW Budapest, August 1-3, 2005 R. Casalbuoni:Superconductivity in high density QCD

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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

m

3 taken equal to zero).

and

m

8



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

hà ® aL à ¯ bL i  I X   I   ² ®¯I ² abI ;  I   

JHW Budapest, August 1-3, 2005 R. Casalbuoni:Superconductivity in high density QCD

I e  i ~ I 26



A further simplifications is to assume only the following geometrical configurations for the vectors q I , I=1,2,3 (a more general angular dependence will be considered in future work) 1 2 3 4



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

 ¡  norm al  I 0 X  @ ® I    I   ¯ I   I   1 X I 6 J ¯ I J   I    J A  O    

with coefficients

 I,  I

and



IJ calculable from an effective NJL four-fermi interaction simulating one-gluon exchange JHW Budapest, August 1-3, 2005 R. Casalbuoni:Superconductivity in high density QCD

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  ´  B CS ; ¹ u  ¹ ¡   ¹ e ; ¹ d  ¹    ¹ e ; ¹ s  ¹    ¹ e ¡ M  ¹  s ® I  I ; ±¹ I  ¡  ¹ ¼   Ã ±¹  q I I  ¯ ¯ q I q I  ±¹ ¡ ±¹ I I ¯ ¯ ¡ ¯ I  I ; ±¹ I  ¹  ¼  q I   ¡ ±¹ I    ¯ ¯  I  ¡ ±¹    I   ¯ ¯ !

¯   ¡  ¹ ¼   Z d n  ¼  q 1 ¢ n  ¹ s ¡ ¹  d  q 2 ¢ n  ¹ s ¡ ¹ u     !

!

 ; ¹ s $ ¹ d  ; ¹ s $ ¹ u

JHW Budapest, August 1-3, 2005 R. Casalbuoni:Superconductivity in high density QCD

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 ¡  norm al  I 0 X  @ ® I    I   ¯ I   I   1 X I 6 J ¯ I J   I    J A  O   

We require:

@  I  @q I  @  @¹ e  

At the lowest order in



I

@    ) @q I @® @q I I  

since



I depends only on q I and

dm

i we get the same result as in the usual LOFF case:

j~

I

j

 

±¹

I

JHW Budapest, August 1-3, 2005 R. Casalbuoni:Superconductivity in high density QCD

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-2 -4 -6 -8 2 0 2 1 0.8

0.6

0.4

0.2

25   1  2 50 1 3 4 75 M  s 3 2 / 100 m [MeV] 125 150 175

Structure 4 dominates starting from about 30 MeV (we have assumed the same parameters as in



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

m

= 500 MeV)

   3 1 2

: ds : us : ud

-

pairin g pairing pairing

25 50 75 M 2 / s 100 m [MeV] 125

JHW Budapest, August 1-3, 2005

150 175

R. Casalbuoni:Superconductivity in high density QCD

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Comparison with other phases:



If no chromo-magnetic instability LOFF takes over gCFL at about 120 MeV and goes over to the normal phase at about 150 MeV (both first order transitions)

10 g2SC unpaired ~  0 -10 LOFF -20 2SC gCFL -30 -40 CFL ~  -50 0 25

JHW Budapest, August 1-3, 2005

50 M s 2 / 75 m [MeV] 100 125 150

R. Casalbuoni:Superconductivity in high density QCD

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

Otherwise it should go over to the CFL phase at 75 MeV

10 0 -10 -20 -30 -40 -50 0 CFL 25 unpaired 2SC ~  50 ~  gCFL M s 2 / 75 m [MeV] 100 g2SC LOFF 125 150

JHW Budapest, August 1-3, 2005 R. Casalbuoni:Superconductivity in high density QCD

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m e [MeV] 40 30 20 10 50 75 M 2 / s 100 125 m [MeV] 150 175

The behaviour of

m

e in LOFF is pretty similar to the one in gCFL. If the same is true for

m

3 and

m

8 our assumption of their vanishing is not too bad for M s 2 /

m

in the region where LOFF is the favored phase.

50 40 30 20 10 0 -10 -20 -30 0 CFL m e , m 3 m 8 gCFL m e m 8 m 3 25 50 M S 2 / 75 m [MeV] 100 125 150

JHW Budapest, August 1-3, 2005 R. Casalbuoni:Superconductivity in high density QCD

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Conclusions

●

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

●

Various phases are competing, many of them having gapless modes. However, when such modes are present a chromomagnetic instability arises (but this happens also under different conditions, see 2SC).

●

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

●

Our recent study of the LOFF phase with three flavors seems 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. JHW Budapest, August 1-3, 2005 R. Casalbuoni:Superconductivity in high density QCD

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