Roberto Casalbuoni Department of Physics, Sezione INFN,

1

Roberto CasalbuoniRoberto CasalbuoniDepartment of Physics, Sezione INFN,Department of Physics, Sezione INFN,

G. Galilei Institute for Theoretical Physics (GGI)G. Galilei Institute for Theoretical Physics (GGI)Florence - ItalyFlorence - Italy

Leuven, February 18, 2010Leuven, February 18, 2010

[email protected]@fi.infn.it

The ground state of QCD at The ground state of QCD at finite density and T = 0finite density and T = 0

Firenze 23/2/07 R. Casalbuoni, The ground state of…

2

SummarSummaryy

Introduction Color Superconductivity: CFL and 2SC phases The case of pairing at different Fermi surfaces LOFF (or crystalline) phase Conclusions


3

Motivations for the study of high-density QCD

Study of the QCD phase in a region difficult for lattice calculations

Understanding the interior of CSO’s

Asymptotic region in fairly well understood: existence of a CS phase. .

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

(~5-6 0)?

Introduction


4

CFL and S2C phasesCFL and S2C phases● Ideas about CS back in 1975 (Collins & Perry-1975, Ideas about CS back in 1975 (Collins & Perry-1975, Barrois-1977, Frautschi-1978).Barrois-1977, Frautschi-1978).

● Only in 1998 (Alford, Rajagopal & Wilczek; Rapp, Schafer, Only in 1998 (Alford, Rajagopal & Wilczek; Rapp, Schafer, Schuryak & Velkovsky) a real progress. Schuryak & Velkovsky) a real progress.

● Why CS? Why CS? For an arbitrary attractive interaction it is energetically convenient the formation of Cooper pairs (difermions)

●Due to asymptotic freedom quarks are almost free at high density and we expect diquark condensation in the color attractive channel 3* (the channel 6 is repulsive).


5

In matter In matter SC SC only under particular conditionsonly under particular conditions ((phonon interaction should overcome the phonon interaction should overcome the

Coulomb forceCoulomb force))

430 54

0c 1010

K1010K 101

E(electr.)(electr.)T

SCSC much more efficient imuch more efficient in n QCDQCD

In QCDIn QCD attractive attractive interactioninteraction (antitriplet (antitriplet

channel)channel) cT (quarks) 50 MeV 1E(quarks) 100 MeV


6

Antisymmetry in spin (a,b) for better use of Antisymmetry in spin (a,b) for better use of the Fermi surfacethe Fermi surface

Antisymmetry in color (a, b) for attractionAntisymmetry in color (a, b) for attraction

Antisymmetry in flavor (i,j) due to Pauli Antisymmetry in flavor (i,j) due to Pauli principleprinciple

α βia jb0 ψ ψ 0

The symmetry breaking pattern of QCD at high-The symmetry breaking pattern of QCD at high-density depends on the number of flavors with density depends on the number of flavors with mass mass m < m < . Two most interesting cases:. Two most interesting cases: N Nf f = 2, 3 = 2, 3. . Consider the possible pairings at very high densityConsider the possible pairings at very high density

color; i, j flavor; a,b spin


7

p

p

s

s

Only possible pairings Only possible pairings

LL and RRLL and RR

Favorite state for Nf = 3, CFL (color-flavor locking) (Alford, Rajagopal & Wilczek 1999)

α β α β αβCiL jL iR jR ijC0 ψ ψ 0 = - 0 ψ ψ 0 Δε ε

Symmetry breaking patternSymmetry breaking pattern

c L R c+L+RSU(3) SU(3) SU(3) SU(3)

For For mmuu, m, mdd, m, mss


8

α β αβCiL jL ijC0 ψ ψ 0 Δε ε

α β αβCiR jR ijC0 ψ ψ 0 Δε ε

Why CFL?Why CFL?


9

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)

α β αβ3iL jL ij0 ψ ψ 0 = Δε ε

c L R c L RSU(3) SU(2) SU(2) SU(2) SU(2) SU(2)

However, if However, if is in the intermediate region we is in the intermediate region we face a situation with quarks belonging to face a situation with quarks belonging to different Fermi surfaces (see later). Then other different Fermi surfaces (see later). Then other phases could be important (LOFF, etc.)phases could be important (LOFF, etc.)


10

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)!


11

Consider 2 fermions with m1 = M, m2 = 0 at the same chemical potential . The Fermi momenta are

Effective chemical potential for the massive quark

Mismatch:Mismatch:

pF q

¹ ¡ M pF ¹

¹ q

¹ ¡ M ¼ ¹ ¡ M

¹

±¹ ¼ M

¹M2/2 effective

chemical potential


12

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

From this: and

N.B. e is not a free parameter, it is fixed by the neutrality condition:

¹ i ¹ Q i ¹ Q

¹ e ¡ ¹ QQ ¡ @V

@¹ e

d ! ueº ) ¹ d ¡ ¹ u ¹ e

Neutrality and Neutrality and ββ equilibrium equilibrium


13

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

¹ d;s ¹ u ¹ e

The case of non interacting quarks


14

● 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 ¼ pdF ¡ puF ¼ puF ¡ psF ¼ M s = ¹

pdF ¡ psF ¼ ¹ e; ¹ e M s ¹


15

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


16

The problem of two fermions with different chemical potentials:

u d

u d u d

,

,2 2

can be described by an interaction hamiltonian

†I 3H

In normal SC:


17

u,d (ε(p,Δ)±δμ)/T

1n =e +1

ForFor T T 003

3

g d p 11 = (1- θ(-ε - δμ) - θ(-ε + δμ))2 (2π) ε(p,Δ)

ε <| δμ |blocking blocking regionregion

The blocking region reduces the gap:

BCS

3

u d3

g d p 11 = (1- n - n )2 (2π) ε(p,Δ) Gap equation:

2 2ε p,Δ = p -μ + Δ


18

In a simple model without supplementary conditions, with two fermions at chemical potentials and2, the system becomes normal at the Chandrasekhar - Clogston point. Another unstable phase exists.

±¹ B CSp

±¹ B C S

0.2 0.4 0.6 0.8 1 1.2

0.2

0.4

0.6

0.8

1

1.2

BCS

BCS

Normal unstable phase

unstable phase

0.2 0.4 0.8 1 1.2

-0.4

-0.2

0.2

0.4

CC

CC0

0.6

Sarma phase, tangential to the BCS phase at =


19

Phase diagram


20

● If neutrality constraints are present the situation is different.

● For || > (= e/2) 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.

g2SCg2SC ((Huang & Shovkovy, 2003))


21

0

e

Q = 0

e = const.

solutions of thegap equation

V

Normal state

200

150

100

50

00 100 200 300 400

(MeV)

e (MeV)

gap equation

neutrality line

e2 =

20 40 60 80 100 120( MeV )

- 0.087

- 0.086

- 0.085

- 0.084

- 0.083

- 0.082

- 0.081V ( GeV / fm 3 )

=148 MeVe

neutrality line

Normal state

eμδμ =2


22

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

For For , the gaps are, the gaps are andand

2dm Energy cost for pairing

2D Energy gained in pairing

begins to unpair

E

p

gapless m odes

blocking region

E p , p ¹ §q

±¹ ¡

±¹ >

E p j § ±¹ q p ¡ ¹ j


23

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 jÃ ®aL Ã ¯bL j i ²®¯ ²ab ²®¯ ²ab ²®¯ ²ab

(Alford, Kouvaris & Rajagopal, 2005)


24

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


25

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


26

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


27

● 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)


28

4,5,6,78

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).

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

mg ¹ g¼


29

● 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.

Proposals for solving the chromomagnetic

instability


30

● 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.


31

● 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).


32

● 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).


33

● 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, for a review R.C. & Nardulli, 2003)

LOFF phase


34

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 hÃ x Ã x i e i~q¢~x

hÃ x Ã x i X

m me i~qm¢~x

~p ~p ~q

j~qj

~q =j~q j

rings


35

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 >

hÃ xÃ x i X

~qie i~qi ¢~x

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

¹ ±¹ ¡ ~vF ¢~q


36

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


37

Gap equationGap equation

Propagator expansion

Insert in the gap equation

General strategy


38

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


39

22normalLOFF )(44.

)2(4

2BCS

2normalBCS

)(15.1 2LOFF

2/BCS1 BCS2 754.0

Small window. Opens up in QCD (Leibovich, Rajagopal & Shuster 2001)


40

Single plane wave

Critical line from

0q

,0

Along the critical line

)2.1q,0Tat( 2


41

Bowers and Bowers and Rajagopal (2002)Rajagopal (2002)

Preferred structure: face-centered cube


42

Crystalline Crystalline structures in LOFFstructures in LOFF


43

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.

hÃ®aL Ã ¯bL i

X

I I ~x ²®̄ I ²abI ; I ~x I e i~qI ¢~x


44

A further simplifications is to assume only the following geometrical configurations for the vectors qI, i = 1,2,3

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 ¡

¼ ¹ e


45

¯ I qI ; ±¹ I ¹ ¼

qI ¡ ±¹ I

®I qI ; ±¹ I ¡ ¹

¼Ã ¡ ±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

´ B CS ; ¹ u ¹ ¡

¹ e; ¹ d ¹

¹ e; ¹ s ¹

¹ e¡ M s ¹


46

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).


47

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.


48

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)

;


49

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


50

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


51

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


52


53

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.


54

PhononsPhononsIn the LOFF phase translations and rotations are brokenIn the LOFF phase translations and rotations are broken

phononsphononsPhonon 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 1 φ(x) = Φ(x) - 2q xf


55

2 2 22 2 2

2 2 2

12phonon llL v v

x y z

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 It is possible to evaluate the parameters of LLphononphonon (R.C., Gatto, Mannarelli & Nardulli 2002)(R.C., Gatto, Mannarelli & Nardulli 2002)

153.0|q|

121v

22

694.0

|q|v

22||

1,0,0q

++


56

Cubic structureCubic structure

i

)i(i

i

iik

;3,2,1i

)x(i

;3,2,1i

x|q|i28

1k

xqi2 eee)x(

3 scalar fields 3 scalar fields (i)(i)(x)(x)

i)i( x|q|2)x(

i)i()i( x|q|2)x()x(

f1


57

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

2( ) 2( )

1,2,3 1,2,3

2( ) ( ) ( )

1,2,3 1,2,3

1 | |2 2

2

ii

phononi i

i i ji i j

i i j

aLt

b c


58

we get for the coefficientswe get for the coefficients

121a 0b

1

|q|3

121c

2

One can evaluate the effective lagrangian for the gluons in One can evaluate the effective lagrangian for the gluons in tha anisotropic medium. In the case of the cube one findstha anisotropic medium. In the case of 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!

Documents

Roberto Casalbuoni Department of Physics, Sezione INFN,