1

NGB and their parametersNGB and their parameters

Gradient expansion: parameters of the NGB’s

Masses of the NGB’s

The role of the chemical potential for scalar fields: BE condensation

Dispersion relations for the gluons

2

Hierarchies of effective Hierarchies of effective lagrangianslagrangians

Integrating out Integrating out heavy degrees of heavy degrees of freedom we have freedom we have

two scales. The gap two scales. The gap and a cutoff, and a cutoff, above which we above which we integrate out. integrate out.

Therefore: Therefore: two two different effective different effective theories, theories, LLHDETHDET and and

LLGoldsGolds

3

Gradient Gradient expansion: NGB’s expansion: NGB’s

parametersparameters

AB ABA† BD *

AB AB

iV DdvL (L R)

iV D4

Recall from HDET that in the CFL Recall from HDET that in the CFL phasephase

9A

i A iA 1

1( )

2

and in the basis

AB A AB A

A 1, ,8A

A 9 29

4

AA† AD

A

iVdvL (L R)

iV4

AABAB 2

A A

VS

V V V

Propagator

Coupling to the Coupling to the U(1)U(1) NGB: NGB:i / f i / fU e , V e

f , f

i( ) i( )L L R R

i i

e , e

U e U, V e V

†L RUV , UV Invariant Invariant

couplingscouplings

5

†2A† AA

D 2A

iV UdvL

4 U iV

Consider now the case of the U(1)B NGB. The invariant Lagrangian is:

At the lowest order in 2

2A† A

A 2

2

2i 20

f fdvL

4 2i 20

f f

generates 3-linear and 4-linear couplings

6

Generating functional:†i A( )†Z[ ] D D e

21 A A

0 0 12

0 1

2i 2A( ) S

f f

0 1 0 1,

1 0 1 0

A10

A

VS

V

1Tr[log A( )]1/ 2 2Z[ ] (det[A( )]) e

eff

iS [ ] Tr[log A( )]

2

7

21

0 12

n2n 1 21

0 12n 1

2i 2iTr[log A( )] iTr log S 1 S S

f f

( 1) 2i 2iTr logS i iS i iS i

n f f

At the lowest order:

eff 0 0

2

12

i iS(y,x)2i (x) iS(x, y)2i (y)S dxdyTr i i

4 f f

i iS(x,x)2 (x)dxTr i

2 f

8

Feynman rules

For each fermionic internal line

AABAB AB 2

A A

ViiS i S(p)

V V V

For each vertex a term iLiLintint

For each internal momentum not For each internal momentum not constrained by momentum conservation:constrained by momentum conservation:

2 22

04 3

4d d d

(2 ) 4

Factor 2x(-1) from Fermi statistics and spin. A factor 1/2 from replica trick.

A statistical factor when needed.

9

+2 2

Aeff I II 3 2

A

2 2 22 A

A A A

1 dviL iL (p) iL (p)

2 4 f

V ( p) V V ( p) V 2 2d

D ( p)D ( ) D ( )

2A AD ( ) V V i

I IIL (0) L (0) 0 Goldstone theorem:

Expanding in p/

10

2

eff 2 2

9 1 dvL (x) (V ) (x)(V ) (x)

f 2 4

1 0 0 0

10 0 0

3dvV V 1

4 0 0 03

10 0 0

3

2 22 2

eff 02 2

22 2

2

1 9L (x) v

2 f

1 9v , f

3

CFL

11

For the V NGB same result in CFL, whereas in 2SC

22 2

2

1 4v , f

3

2SC

With an analogous calculation:

2 8a 2 2 a 2

eff 02 2a 1

(21 8log 2) 1L ( ) v | |

36 F 2

22 2 2

T 2

1 (21 8log 2)v , F F

3 36

12

Dispersion relation for the NGB’s

1E | p |

3

Different way of computing:

a b ab 0 10 | J | iF p , p p , p

3

Current conservation:

2 21p p E | p | 0

3

13

Masses of the Masses of the NGB’sNGB’s

QCD mass term: L RM h.c.

2 LZM M

† T 4i T(Y X) e

1 † 2masses

† †

L c det[M]Tr[M h.c. c ' det( )Tr[(M ) ]

c" Tr[M ]Tr[M ]

14

Calculation of the coefficients from QCD

Mass insertion in QCD

Effective 4-fermi

Contribution to the vacuum energy

2

2

3c , c ' 0, c" 0

2

15

Consider:

a aQCD 0 L R R L

1L (iD ) M M G G

4

Solving for as in HDET,L

,L 0 ,L 0 ,R

1i D M

2

† † 2D ,L ,L ,L ,L

†

†1L (iV D) ( D )

2

L

MM

R, M M

like chemical potential

16

Consider fermions at finite density:

0L gi i

as a gauge field A0

Invariant under: i ( t )e , (t)

Define: † †L R

1 1X MM , X M M

2 2

Invariance under:

,L ,L ,R ,R

† †L L 0

† †R R 0

L(t) , R(t) ,

X L(t)X L (t) iL(t) L (t),

X R(t)X R (t) iR(t) R (t)

17

The same symmetry should hold at the level of the effective theory for the CFL phase (NGB’s), implying that

T T† †

0 0 0

MM M Mi i

2 2

The generic term in the derivative expansion of the NGB effective lagrangian has the form mn p† 2

2 2 q †r0NGB 2

iMM / 2 ML F

F

18

mn p† 22 2 q †r0

NGB 2

iMM / 2 ML F

F

Compare the two contribution to quark masses:

kinetic term4 4

2 22 2 2 2

m 1 mF

F

mass insertion2 2 2

2 22 2 2

m 1 mF

F F F

Same order of magnitude for since

m F

19

The role of the chemical The role of the chemical potential for scalar fields: potential for scalar fields:

Bose-Einstein condensationBose-Einstein condensation

A conserved current may be coupled to the a gauge field.

Chemical potential is coupled to a conserved charge.

The chemical potential must enter as the fourth component of a gauge field.

20

Complex scalar field:

2† † 2 † †0 0

2† 2 † †0 0

2 † †

L i i m

im

negative mass term

breaks C

Mass spectrum:

2 2 20

2 2 2

p (m ) 2 Qp 0 (Q 1)

(E Q) m | p |

P,Pm m For < mm

21

At = m = m, second order phase transition. Formation of a condensate obtained from:

22 2 † †V m

2 2

† mm

2

† 2 2V2 m

Charg

e densit

yGround state = Bose-Einstein condensate

22

2 22v m

, v2

i (x ) / v1

(x) v h(x) e2

2 22 0

1 1L h h v h 2 h

2 2

Mass spectrum2 2

2

p 2 v 2i Edet 0

2i E p

At zero momentum

2 2 2 2M M 2 v 4 0

2

2 2 2

M 0

M 6 2m

23

At small momentum

2 2

NGB 2 2

2 22 2 2

massive 2 2

mE | p |

3 m

9 mE 6 2m | p |

6 m

2 22NGB 2 2

m

m 33

1v

24

Back to CFL. From the structure P,Pm m

0 0

2 2d u

u d s2

2 2s u

u s d2K

2 2s d

d s u2K ,K

m m 2cm (m m )m ,

2 F

m m 2cm (m m )m ,

2 F

m m 2cm (m m )m

2 F

First term from “chemical potential” like kinetic term, the second from mass insertions

†MM

2

2

3c

2

22

2

(21 8log 2)F

36

25

For large values of ms:

0 0

u d s2

2 2s s

s d s u2 2K K ,K

2cm (m m )m ,

F

m 2c m 2cm m m , m m m

2 F 2 F

and the masses of K+ and K0 are pushed down. For the critical value

1/ 322 23 3

s u,d u,d2 2crit

s crit

12m m 3.03 m ,

F

m 40 110 MeV

masses vanish

26

For larger values of ms these modes become unstable. Signal of condensation. Look for a kaon condensate of the type:

4i 24 4e 1 (cos 1) i sin

(In the CFL vacuum, = 1) and substitute inside the effective lagrangian

22 2s s

2

1 m 2cm mV( ) F sin (1 cos )

2 2 F

negative contribution from the “chemical potential”

positive contribution from mass insertion

27

Defining

2

20s seff K 2

m 2cm m, m

2 F

2 2 2 0 2eff K

1V( ) F sin (m ) (1 cos )

2

with solution 20

K 0eff K2

eff

mcos , m

and hypercharge density

40K2

Y eff 4eff eff

mVn F 1

28

Mass terms break original SU(3)c+L+R to

SU(2)IxU(1)Y. Kaon condensation breaks

this to U(1)

3 8

1 1Q , [Q, ] 0

2 3

I Y

0 0 0SU(2) U(1)

( , ) (K , K ) (K , K ) ( )

breaking through the doublet as in the SM

Only 2 NGB’s from K0, K+ instead of expected 3 (see Chada & Nielsen 1976)

29

Chada and Nielsen theorem: The number of NGB’s depends on their dispersion relationI. If E is linear in k, one NGB for any

broken symmetry

II. If E is quadratic in k, one NGB for any two broken generators

In relativistic case always of type I, in the non-relativistic case both

possibilities arise, for instance in the ferromagnet there is a NGB of type II, whereas for the antiferromagnet there

are to NGB’s of type I

30

Dispersion relations for the Dispersion relations for the gluonsgluons

The bare Meissner mass

The heavy field contribution comes from the term

2

† †h h h h

D DDP

2 iV D 2 iV D

1P g V V V V

2

31

Notice that the first quantized hamiltonian is:

2

2 20 0

gH p gA eA | p | gA gv A | A | (v A)

2 | p |

Since the zero momentum propagator is the density one gets

3 22 2

f 3 2|p|

d p 1 (p A)g 2 N Tr A

(2 ) 2 | p | | p |

spin

2 2a a 2 a a

f BM2a a

2 22BM f 2

g 1 1N A A m A A ,

6 2 2

gm N

6

32

Gluons self-energy

Vertices froma

aigA J

Consider first 2SC for the unbroken gluons:

00 2ab ab

kl kl,self klab ab ab

2 2 2 2 2

2

kl kl kl 20ab ab ab 02 2 2

0

2

2 2

2 2

2 2

k2 2

2a 20 k

b ab

(p) | p | ,

(p) (p) (p)

g p g1 p ,

3 6 3

(p)

g

18

g

18

p pg

18

2BMfrom m

33

Bare Meissner mass cancels out the constant contribution from the s.e.

All the components of the vacuum polarization have the same wave function renormalization

a a b a a a a a aa ab i i i i i i

2 2

2 2

1 1 1 kL F F A A E E B B E E

4 2 2 2

gk

18

Dielectric constant = k+1, and magnetic permeability =1 1

vg

34

Broken gluons

a (0) - ij(0)

1-3 0 0

4-7 3mg2/2 mg

2/2

8 3mg2 mg

2/3

2 22g 2

gm

3

35

But physical masses depend on the wave function renormalization

2 2

2

g

Rest mass defined as the energy at zero momentum:

R

R

m 2 , a 4,5,6,7

gm , a 8

The expansion in p/ cannot be trusted, but numerically

Rm 0.9 , a 4,5,6,7

36

In the CFL case one finds:2 2

2 2 2D 2

2 2 22 DM 2

gm (21 8log 2) g F

36

g 11 2 1 mm log 2

36 27 2 3

Recall that from the effective lagrangian we got:

2 2 2 2 2 2 2D T T M S Tm g F , m v g F

from bare Meissner mass

implying and fixing all the parameters.

S T 1

37

We find:2 2

DR 1 2 2

1

R

m g 16m , 7 log 2

216 33

m 1.70

Numerically Rm 1.36

38

LOFF phaseLOFF phase

Different quark masses

LOFF phase

Phonons

39

Different quark massesDifferent quark masses

We have seen that for one massless flavors and a massive one (ms), the condensate may be disrupted for 2

sm

2

The radii of the Fermi spheres are:

1 2

22 2 s

F s F

mp m , p

2

As if the two quarks had different chemical potential (ms

2/2)

40

Simulate the problem with two massless quarks with different chemical potentials:

u d

u d u d

,

,2 2

Can be described by an interaction hamiltonian

†I 3H

Lot of attention in normal SC.

41

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

The impurities produce a constant exchangeThe impurities produce a constant exchange fieldfield acting upon the electron spins giving rise acting upon the electron spins giving rise to anto an effective difference in the chemical effective difference in the chemical potentials of the opposite spins.potentials of the opposite spins.

Very difficult experimentally but claims of Very difficult experimentally but claims of observations in heavy fermion observations in heavy fermion superconductorssuperconductors ( (Gloos & al 1993Gloos & al 1993) ) and in quasi-and in quasi-two dimensional layered organic two dimensional layered organic superconductorssuperconductors ( (Nam & al. 1999, Manalo & Klein 2000Nam & al. 1999, Manalo & Klein 2000))

42

HI changes the inverse propagator

10 *

3

3

VS

V

and the gap equation (for spin up and down fermions):

2 2

2 2 2 20

dv dig

4 (2 ) ( )

This has two solutions:

2 20 0 0a) b, ): : 2

43

Grand potential:

0

0

22

2 00

0( 0)

H dg| |

g g g

d

2

02

0

d 2 dg

g

Also:

20 0( ) (0)

2

Favored solution

0

44

Also: 2 20 0( ) 2

4

First order transition to the normal state at

01

2

For constant Ginzburg-Landau expanding up to

45

LOFF phaseLOFF phase

In 1964 Larkin, Ovchinnikov and Fulde, Ferrel, argued the possibility that close to the first order-line a new phase could take place.

According LOFF possible condensation with non zero total

momentum of the pair1p k q

2p k q

xqi2e)x()x(

xqi2

mm

mec)x()x(More generallyMore generally

46

q2pp 21

|q|

|q|/q

fixed variationallyfixed variationally

chosen chosen spontaneouslyspontaneously

)qk(E)p(E

qvF

Gap equation: ),p(

nn1

)2(

pd

2

g1 du

3

3

du nn

Non zero total

momentum

47

1e

1n

T/)),p((d,u

ForFor T T 00

))()(1(),p(

1

)2(

pd

2

g1

3

3

||blocking blocking regionregion

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

BCSLOFF

48

Possibility of a crystalline structure (Larkin

& Ovchinnikov 1964, Bowers & Rajagopal 2002)

xqi2

2.1|q|iq

i

i

e)x()x(

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

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

grand potential

see latersee later

49

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) ))

50

Gap equationGap equation

Propagator expansion

Insert in the gap equation

General strategy

51

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

52

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)

53

Single plane wave

Critical line from

0q

,0

Along the critical line

)2.1q,0Tat( 2

54

Preferred structure:

face-centered

cube

Bowers and Bowers and Rajagopal Rajagopal

(2002)(2002)

55

In the LOFF phase translations and rotations are broken

phonons

Phonon 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(f

1

PhononPhononss

56

2

22||2

2

2

222

phonon zv

yxv

2

1L

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|

12

1v

2

2

694.0

|q|v

2

2||

++

57

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

58

0)x(

4)x(

4)x(

59

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

bc

2

60

we get for the coefficients

12

1a 0b

1

|q|3

12

1c

2

One can evaluate the effective lagrangian for the gluons in tha

anisotropic medium. For the cube one finds

Isotropic propagationIsotropic propagation

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

are the same!

61

Compact stellar objectsCompact stellar objects

62

Compact stellar objectsCompact stellar objects

High density core of a compact star, a good lab for testing QCD at high density.

63

Some features of a compact starFor simplicity consider a gas of free massless fermions.

3

f p p3

4

f 2

d p2N V ( V) ( )

2N

1) 2(

Grand potential:

Density:3

f 2VN

3

Eq. of state:f

4 / 34

2P VN

12P K

64

For a non-relativistic fermion:3/ 2 3/ 2

5/ 2 3/ 2f f2 2

5/ 3

8 m 4 mP VN , VN

15 32

P K

2

More generally assumed P K

For high densities inverse beta decay becomes importante p n

At the equilibrium

e p n

65

1/ 3 1/ 3 1/ 3e p n e p n

From charge neutrality e p

p

n

1

8

Neutron star

Radius of a neutron star (Landau 1932)

66

N fermions in a box of volume V. Number density

3

Nn

R

Position uncertainty 1/ 31/ 3

Rn

N

Uncertainty principle1/ 3 1/ 3

/ 3F

1F

Np n

R

NE c

R

Gravitational energy per baryon

2B

G

GNmE

R

67

1/ 3 2B

G F

cN GNmE E E

R R

E > 0 otherwise not bounded. This condition gives

3/ 2

571/

m xB

3 2B

a 2

cN 2 10

Gm

cN GNm0 N

R R

Maximum mass

maxM 1.4M Chandrasekhar limit

max max BM N n 1.5M

68

1/ 2

2 1/ 3F max 2

B

c c cE mc N

R R Gm

1/ 28

e52

nB

c 5 10 cm (m m )R

3 10 cm (m m )mc Gm

Typical neutron star density

15 3 15 3Nnm

1.3ferminm

m10 g / cm , 0.15 10 g / cm5 6

V

min

16 3max

R

M2.5 10 gr / cm

V Density (f or a neutron star)

69

Neutron stars are a good laboratory to test hadronic matter at high density and

zero temperature

70

In neutron stars CS can be studied at T = 0 (TTnsns~10~105 5 KK)

)K10MeV1(

100)MeV(201010T

10

BCS76

BCS

ns

Orders of magnitude from a crude model: 3 free quarks

0M,0MM sdu

Consider the LOFF state. From BCSBCS14 (MeV) 70

71

s,d,ui

iqeqees,d,ui

ii NNQNNN

0Qe

2

s,d,ui

iF2B )p(

3

1

3

1

Weak equilibrium:

2s

2s

sFes

ddFed

uuFeu

Mp,3

1

p,3

1

p,3

2

Electrical neutrality:

72

n.m.n.m. is the saturation nuclear density ~ .15x10~ .15x1015 15 g/cmg/cm

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

65.m.n

B Choosing ~ 400 ~ 400

MeVMeVMs = 200 pF = 25

Ms = 300 pF = 50Right

ballpark (14 - 70

MeV)

73

)10Ω/Ω( 6

Glitches: discontinuity in the period of the pulsars Standard explanation: metallic crust

+ neutron superfluide inside

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

dipole emission

74

In the superfluid phase there are vortices pinned to the crust. When the star slows down the vortices do not participate in the motion until an instability is produced. Then there is a release of angular momentum to the crust seen as a jump in the rotational frequency.

The presence of the LOFF phase might avoid the main objection against the existence of strange stars (made of u,d,s quarks in equal ratios) since they cannot have a crust.

75

ConclusionsConclusions

SC almost 100 years old, but still actual

Important technological applications

Source of inspiration for other physical theories (SM as an example)

Deep implications in QCD at very high density: very rich phase structure

Possible applications for compact stellar objects

Unvaluable theoretical laboratory