1 Color Superconductivity: CFL and 2SC phases Introduction Hierarchies of effective lagrangians Effective theory at the Fermi surface (HDET) Symmetries.

1

Color Superconductivity: Color Superconductivity: CFL and 2SC phases CFL and 2SC phases

IntroductionIntroduction

Hierarchies of effective lagrangiansHierarchies of effective lagrangians

Effective theory at the Fermi surface Effective theory at the Fermi surface (HDET)(HDET)

Symmetries of the superconductive Symmetries of the superconductive phasesphases

2

IntroductionIntroduction Ideas about CS back in 1975 (Collins & Perry-Ideas about CS back in 1975 (Collins & Perry-1975, Barrois-1977, Frautschi-1978).1975, Barrois-1977, Frautschi-1978).

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

The phase structure of QCD at high-density The phase structure of QCD at high-density depends on the number of flavors with massdepends on the number of flavors with mass m <m <

TwoTwomost interesting cases:most interesting cases: NNf f = 2, 3= 2, 3..

Due to asymptotic freedom quarks are almost Due to asymptotic freedom quarks are almost free at high density and we expect difermion free at high density and we expect difermion condensation in the color channel condensation in the color channel 33**..

3

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

Antisymmetry in colorAntisymmetry in color ( () ) for attractionfor attraction

Antisymmetry in flavorAntisymmetry in flavor (i,j) (i,j) for Pauli for Pauli principleprinciple

α βia jb0 ψ ψ 0

Consider the possible pairings at very high Consider the possible pairings at very high densitydensity

color; color; i, j i, j flavor;flavor; a,b a,b spinspin

4

p

p

s

s

Only possible pairings Only possible pairings

LL and RRLL and RR

Favorite state forFavorite state for N Nff = 3, = 3, CFLCFL (color-(color-

flavor locking) (Alford, Rajagopal & flavor locking) (Alford, Rajagopal & Wilczek 1999)Wilczek 1999)α β α β αβC

iL 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

5

α β αβCiL jL ijC0 ψ ψ 0 Δε ε

α β αβCiR jR ijC0 ψ ψ 0 Δε ε

Why CFL?Why CFL?

6

What happens going down with What happens going down with ? If? If << m<< mss, , we we get 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 is in the intermediate region we face a situation with fermions having we face a situation with fermions having different Fermi surfaces (see later). Then different Fermi surfaces (see later). Then other phases could be important (LOFF, other phases could be important (LOFF, etc.)etc.)

7

Difficulties with lattice Difficulties with lattice calculationscalculations

Define euclidean variables:Define euclidean variables:4 i i

0 E E

4 i i0 E E

x ix , x x

, i

Dirac operator with chemical potentialDirac operator with chemical potential

4E E ED( ) D

At At †D(0) = -D(0)

Eigenvalues of Eigenvalues of D(0)D(0) pure immaginary pure immaginary

If eigenvector of D(0), If eigenvector of D(0), eigenvector with eigenvalue eigenvector with eigenvalue - -

5

5 5γ D(0)γ = -D(0)

8

det[D(0)] ( )( ) 0

For For not zeronot zero the argument does not the argument does not apply and one cannot use the sampling apply and one cannot use the sampling method for evaluating the determinant. method for evaluating the determinant. However for isospin chemical potential However for isospin chemical potential

and two degenerate flavors one can still and two degenerate flavors one can still prove the positivity.prove the positivity.

For finite baryon density no lattice For finite baryon density no lattice calculation available except for small calculation available except for small

and close to the critical line (Fodor and and close to the critical line (Fodor and Katz)Katz)

9

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

10

LLHDETHDET is the effective theory of the fermions is the effective theory of the fermions close to the Fermi surface. It corresponds to close to the Fermi surface. It corresponds to the Polchinski description. the Polchinski description. The condensation The condensation is taken into account by the introduction of a is taken into account by the introduction of a mean field corresponding to a Majorana mass. mean field corresponding to a Majorana mass. The d.o.f. are quasi-particles, holes and gauge The d.o.f. are quasi-particles, holes and gauge fields. fields. This holds for energies up to the cutoffThis holds for energies up to the cutoff..

LLGoldsGolds describesdescribes the low energy modes (the low energy modes (E<< E<< ), ), as as Goldstone bosons, ungapped fermions and Goldstone bosons, ungapped fermions and holes and massless gauge fieldsholes and massless gauge fields, depending , depending on the breaking scheme.on the breaking scheme.

11

Effective theory at the Effective theory at the Fermi surface (HDET)Fermi surface (HDET)

Starting point: Starting point: LLQCDQCD

a aQCD 0

a a a as

1L iD F F , a 1, ,8

4

D ig A T , D D , T2

at asymptotic at asymptotic QCDQCD,,

0 0( p ) (p) 0 (p ) (p) p (p)

0 2 2 0(p ) | p | p E | p |

12

Introduce the projectors:Introduce the projectors:

F F

FF

p p p p

1 v E(p) | p | ˆP , v v p2 p p

and decomposing:and decomposing:

Fv

Fp v

H

H ( 2 )

States States close to the FS close to the FS

States States decouple for large decouple for large

13

Field-theoretical version:Field-theoretical version:

4ip x

4

d p(x) e (p)

(2 )

0

p v

v (0, v), v 1, ( , )

v , ( v)v

ChoosingChoosing v p

0

4 d.of. 4 d.of. 0 , , v

Separation of light and heavy Separation of light and heavy d.o.f.d.o.f.

light d.o.f .

heavyd.o.f

| p | ,

| p | ,| p | ,. ,

14

heavyheavy

heavyheavy

lightlight

lightlight

Separation of light and heavy Separation of light and heavy d.o.f.d.o.f.

15

Momentum integration for the light Momentum integration for the light fieldsfields

4 2 2 2

04 4 2

d p dv dd d d

(2 ) (2 ) 4 (2 )

vv11

vv22

The Fourier decomposition The Fourier decomposition becomesbecomes i v x

v

dvx)

4((x) e

2 2i x

vv 2

de ( )

(2 )(x)

v ( ) (p)

For any fixed For any fixed v, v, 2-dim 2-dim theorytheory

16

In order to decouple the states In order to decouple the states corresponding to corresponding to E

i v x

2 2i x

v v2

dv(x) e (x) (x)

4

d(x) P (x) P e ( )

(2 )

substituting inside substituting inside LLQCD QCD and usingand using

2 24 † †

v v2

dv dd x (x) (x) ( ) ( )

4 (2 )

Momenta from the Fermi Momenta from the Fermi spheresphere

17

22 2 24 † F F

2 2

4 4 †v ' v

dv dv ' d d 'd x (x) (x)

4 4 (2 ) (2 )

(2 ) ' v ' v ( ') ( )

Using:Using:

4 4

2 2 22

(2 ) ' v ' v

(2 ) ' 4 v ' v

One gets easily the result.One gets easily the result.

Proof:Proof:

18

4 † †0

dvd x (iD ) iV D (2 iV D) ( iD h.c.)

4

V (1, v), V (1, v)

0( , (v )v),

†

†

V

V

0

0

iV D i D 0

(2 iV D) i D 0

Eqs. of Eqs. of motion:motion:

19

iV D 0

0

At the leading At the leading order in order in

At the same order:At the same order:

†D

dvL iV D

4

Propagator:Propagator:

1

V

00

2 20 0

0 0 †

1 (p ) p V 1( T( ) )

p (p ) | p | 2 V

V 1(1 v) P ( T( ) )

2 2

20

4 † †0

dvd x (iD ) iV D (2 iV D) ( iD h.c.)

4

Integrating out the heavy Integrating out the heavy d.o.f.d.o.f.

For the heavy d.o.f. we can formally For the heavy d.o.f. we can formally repeat the same steps leading to:repeat the same steps leading to:

Eliminating the Eliminating the EE- - fields one would get fields one would get the non-local lagrangian:the non-local lagrangian:

† †D

dv 1L iV D P D D

4 2 iV D

1P g V V V V

2

21

DecomposingDecomposingh

one getsone gets

h hD D D D

†D

h † h † hD

h h† h h† hD

L L L L

dvL iV D

4

dv 1L iV D P D D h

4 2 iV D

dv 1L iV D P D D

4 2 iV D

When integrating out the heavy fieldsWhen integrating out the heavy fields

22

This contribution from This contribution from LLhhD D gives the gives the

bare Meissner massbare Meissner mass

Contribute only if some Contribute only if some gluons are hard, but gluons are hard, but

suppressed by asymptotic suppressed by asymptotic freedomfreedom

23

HDET in the condensed HDET in the condensed phasephase

Assume Assume A BABC (A, B(A, B collective collective

indices)indices)

due to the attractive interaction:due to the attractive interaction:

A B C† D†I ab ABCD a b aab b

*ABCD CDAB ABCD BACD ABDC

GL V

4

V V , V V V

DecomposeDecompose

24

AT B AB C† D* CD*int ABCD

CD* AT B

I cond

AB C† D*cond ABCD ABCD

int

G G

GL V

L V C

C C

V C4 4

L

4

L L

CD * * CD*AB CDAB AB CDAB

G GV , V

2 2 We defineWe define

* AT B A† B*cond AB AB

1 1L C C

2 2

and neglect and neglect LLint int . . ThereforeTherefore

25

A† B A† B * AT B A† BD AB AB

AB

dv 1L iV D iV D C C

4 2

(x) ( v,x)

Nambu-Gor’kov basisNambu-Gor’kov basisA

A

A*

1

C2

AB ABA† BD *

AB AB

iV DdvL

iV D4

† †

V1S( )

(V )(V ) V

†, 0

26

* C† D*AB ABCD

GV C

2

†AB AB

1 1

D (V )(V )

2 2* *AB ABCD CE2

ED

dv d 1iGV

4 (2 ) D

From the definition:From the definition:

one derives the gap equation (e.g. via one derives the gap equation (e.g. via functional formalism)functional formalism)

27

Four-fermi interaction one-gluon exchange Four-fermi interaction one-gluon exchange inspiredinspired

a aI

3L G

16

Fierz using:Fierz using:

† †I ( i)( j)( k)( ) i j k

( i)( j)( k)( ) ik j

GL V

4V (3 )

8a a

a 1

ab dc ac bd

2( ) ( ) (3 )

3

( ) ( ) 2

(1, ), (1, )

28

( i)( j) 3 ij

2 2

2 2 2 20

dv d4iG

4 (2 )

2

22 20

Gd , 4

2

In the 2SC caseIn the 2SC case

4 pairing fermions4 pairing fermions

29

3

3 2 20

d p 11 8G

(2 ) | p | M

GG determined at determined at T = 0T = 0. . MM, constituent mass ~ , constituent mass ~ 400 400 MeVMeV

with with

ForFor 400 500 MeV, 800 MeV, M 400 MeV

2SC 33 88 MeV

Similar values for CFL.Similar values for CFL.

30

Gap equation in Gap equation in QCDQCD

2

0 02 2

02 2 2

0 0

3 1cosg 2 2(p ) dq d(cos )

12 1 cos G /(2 )

1 1cos (q )2 2

1 cos F /(2 ) q (q )

31

2DF m 2 0

D

qG m

4 | q |

0q | q | 0

2 2

2D f 2

gm N

2

Hard-loop Hard-loop approximationapproximation

electricelectricmagnetimagneticc

For small momenta For small momenta magnetic gluons are magnetic gluons are unscreened and unscreened and dominate giving a dominate giving a further logarithmic further logarithmic divergencedivergence

20

0 02 2 20 0 0 0

g b (q )(p ) dq log

18 | p q | q (q )

5/ 2

4 5

f

2b 256 g

N

32

2

0 00

3

2g0

g b(p ) sin log

p3 2

2b e

Results:Results:

from the double logfrom the double log

To be trusted only for but, To be trusted only for but, if extrapolated at 400-500 MeV, gives if extrapolated at 400-500 MeV, gives values for the gap similar to the ones found values for the gap similar to the ones found using a 4-fermi interaction.using a 4-fermi interaction.

However condensation arises at asymptotic However condensation arises at asymptotic values of values of

510 GeV

22 c / gg

1 (log( / )) ec

33

Symmetries of Symmetries of superconducting superconducting

phasesphasesConsider again the 3 flavors, Consider again the 3 flavors, u,d,su,d,s and and the group theoretical structure of the the group theoretical structure of the

two difermion condensate:two difermion condensate:

ia jb

* *c L(R ) c L(R ) S c L(R ) c L(R )[(3 ,3 ) (3 ,3 )] (3 ,3 ) (6 ,6 )

34

implying in generalimplying in general

IiL jL ijI 6 i j i j

i j i j 6 i j i j

6 i j 6 i j

( )

( ) ( )

( ) ( )

In NJL case with cutoff In NJL case with cutoff 800800 MeV, MeV, constituent mass constituent mass 400400 MeV and MeV and 400400

MeVMeV

685.3MeV, 1.3MeV

35

Original symmetry:Original symmetry:

QCD c L R B AG SU(3) SU(3) SU(3) U(1) U(1)

anomaloanomaloususbroken tobroken to

CFL c L R 2 2G SU(3) Z Z anomaloanomalousus# Goldstones# Goldstones 3 8 1 8 8 81 11 17 1

massivmassivee

88 give mass to the gluons and give mass to the gluons and 8+18+1 are true are true massless Goldstone bosonsmassless Goldstone bosons

36

Breaking of Breaking of U(1)U(1)BB makes the makes the CFL phaseCFL phase superfluidsuperfluid

The CFL condensate is not gauge The CFL condensate is not gauge invariant, but considerinvariant, but consider

Notice:Notice:

k ijk † k ijk †iL jL iR jRX ( ) , Y ( )

i † i j * ij j(Y X) (Y ) X

is gauge invariant and breaks the global part of is gauge invariant and breaks the global part of GGQCDQCD. .

AlsoAlsodet(X), det(Y) breakbreak 2 2B AU(1) U(1 Z Z)

37

AU(1) is broken by the anomaly but induced by is broken by the anomaly but induced by a 6-fermion operator irrelevant at the a 6-fermion operator irrelevant at the Fermi surface. Its contribution is Fermi surface. Its contribution is parametrically small and we expect a parametrically small and we expect a very light NGB (massless at infinite very light NGB (massless at infinite chemical potential)chemical potential)

Spectrum of the CFL Spectrum of the CFL phasephase

Choose the basis:Choose the basis:9

Ai A i

A 1

1( )

2

A

9 0

A 1, ,8 Gell Mann matrices

21

3

A B ABTr 2

38

A iA i A

i

1 1( ) Tr

2 2

Inverting:Inverting:

i j IA B ij

A B TA I B I

II

1( ) ( )

2Tr

2

I I( ) T

I II

g g Tr[g] for any 3x3 matrix for any 3x3 matrix gg

A BA AB We getWe get A

A 1, ,8A

A 9 29

2 2

A A(p) (v ) quasi-fermionsquasi-fermions

39

gluonsgluons

Expected Expected 2 2 2gm g F

NG coupling NG coupling constantconstant

but wave function renormalization effects but wave function renormalization effects important (see later)important (see later)

NG NG bosonsbosons

Acquire mass through quark masses Acquire mass through quark masses except for the one related to the except for the one related to the breaking of breaking of U(1)U(1)BB

40

NG boson masses quadratic in NG boson masses quadratic in mmqq since since

the approximate invariancethe approximate invariance

2 L 2 R L(R ) L(R )(Z ) (Z ) :

and quark mass term:and quark mass term: L RM h.c.

M M

2 L 2 R(Z ) (Z )Notice: anomaly breaks Notice: anomaly breaks through instantons, producing a through instantons, producing a chiral condensate (6 fermions -> 2), chiral condensate (6 fermions -> 2), but of order but of order ((QCDQCD//))88

41

In-medium electric In-medium electric chargecharge

aaD Qig T i A

The condensate breaks The condensate breaks U(1)U(1)em em but but leaves leaves invariant a combination ofinvariant a combination of Q Q andand T T88..

CFL vacuum:CFL vacuum:i i iX Y

Define:Define: cSU(3) 8

2 1 1 2Q T diag , , Q

3 3 33

cSU(3)Q Q 1 1 Q Q 1 1 Q leaves invariant leaves invariant the ground statethe ground state

i j jiQ X X Q Q Q 0

42

Eigs(Q) 0, 1 Integers as in the Han-Nambu Integers as in the Han-Nambu modelmodel

8

A cos sin

g

A G

A sin cosG

rotated rotated fieldsfields

new interaction:new interaction:

8s 8

ss

s

2

s

2

g g T 1 eA 1 Q eQA

2 e gtan , e ecos , g

g cos3

3T cos Q 1 sin 1 Q

2

g G T

43

The The rotated “photon” remains masslessrotated “photon” remains massless, , whereas the rotated gluons acquires a mass whereas the rotated gluons acquires a mass through the Meissner effectthrough the Meissner effect

A piece of CFL material for massless quarks A piece of CFL material for massless quarks would respond to an em field only through would respond to an em field only through

NGB:NGB:

““bosonic metal”bosonic metal”

For quarks with equal masses, no light For quarks with equal masses, no light modes:modes:

““transparent insulator”transparent insulator”

For different masses one needs non zero For different masses one needs non zero density of electrons or a kaon condensate density of electrons or a kaon condensate

leading to massless excitationsleading to massless excitations

44

In the 2SC case, new In the 2SC case, new QQ and and BB are conserved are conserved

8

8

1 2 1 1Q Q 1 1 T , 1 1 (1,2, 2)

3 3 63

2 1 1 1 1 1 2B B T , , , , (0,0,1)

3 3 3 3 3 33

1/2 0

-1/2 0

1 1

0 1

Q Bu , 1,2

d , 1,2 3u3d

45

Spectrum of the 2SC Spectrum of the 2SC phasephase

RemembeRememberr

( i)( j) 3 ij

i

3i

1,2 gapped

ungapped

c cSU(3) SU(2)

8 3 5 massive gluons

No Goldstone bosonsNo Goldstone bosons

Light modes:Light modes:

3i 3 gluons (M 0)

(equal gap)(equal gap)

46

2+1 2+1 flavorsflavors

It could happen It could happen s u dm , m ,m

Phase transition Phase transition expectedexpected

u d s

u d s

m ,m ,m

m ,m m

2 2 2 2F F FE p M p M

1 2M M

The radius of The radius of the Fermi the Fermi sphere sphere decrases with decrases with the massthe mass

47

Simple model:Simple model: 1 2

2 2F s Fp m , p

F F1 2

p p3 32 2

unpair s3 30 0

d p d p2 p m 2 | p |

(2 ) (2 )

F Fcomm comm

p p3 3 2 22 2

pair s3 3 20 0

d p d p2 p m 2 | p |

(2 ) (2 ) 4

condensation condensation energyenergy

comm

comm

2pair s

FF

m0 p

p 4

4 2 2pair unpair s2

1m 4

16

Condensation Condensation if:if: 2

sm

2

48

Notice that the Notice that the transition must be first transition must be first orderorder because for being in the pairing because for being in the pairing phasephase 2

sm0

2

(Minimal value of the gap to get (Minimal value of the gap to get condensation)condensation)

49

Effective lagrangiansEffective lagrangians

Effective lagrangian for the CFL phase Effective lagrangian for the CFL phase

Effective lagrangian for the 2SC phaseEffective lagrangian for the 2SC phase

50

Effective lagrangian for Effective lagrangian for the CFL phasethe CFL phase

NG fields as the phases of the NG fields as the phases of the condensates in the representationcondensates in the representation(3, 3)

* *k ijk k ijkiL jL iR jRX , Y

Quarks and Quarks and X, YX, Y transform as transform as

i( ) T i( ) TL c L L R c R R

i ic c L,R L,R B A

e g g , e g g

g SU(3) , g SU(3) , e U(1) , e U(1)

T 2i( ) T 2i( )c L c RX g Xg e , Y g Yg e

51

SinceSince X, Y U(3)The number of NG fields isThe number of NG fields is

# X # Y (1 8) (1 8) 18

8 of these fields give mass to the gluons. There 8 of these fields give mass to the gluons. There are onlyare only 10 physical NG bosons10 physical NG bosons corresponding corresponding

to the breaking of the global symmetry (we to the breaking of the global symmetry (we consider also the NGB associated to consider also the NGB associated to U(1)U(1)AA))

AVRL )1(U)1(U)3(SU)3(SU

22RL ZZ)3(SU

52

Better use fields belonging to SU(3). DefineBetter use fields belonging to SU(3). Define

)(i2eXX )(i2eYY andand

)(i6X e)Xdet(d )(i6

Y e)Ydet(d

transforming astransforming asTLc gXgX T

Rc gYgY

The breaking of the global symmetry can be The breaking of the global symmetry can be described by the gauge invariant order described by the gauge invariant order

parametersparametersi j ij X Y

ˆ ˆ ˆ ˆ(Y )* X Y X, d , d

53

, d, dXX, d, dYY are 10 fields describing the physical NG are 10 fields describing the physical NG bosons. Alsobosons. Also

TLR gg

shows that shows that TT transforms as the transforms as the usual chiral usual chiral field.field.

Consider the currents:Consider the currents:† † † †

X

† † † †Y

as a

ˆ ˆ ˆ ˆ ˆ ˆ ˆJ XD X X( X X g ) X X g

ˆ ˆ ˆ ˆ ˆ ˆ ˆJ YD Y Y( Y Y g ) Y Y g

g ig g T

†X,Y c X,Y cJ g J g

54

X YP : (X Y,J J , , )

Most general lagrangian up to two derivatives Most general lagrangian up to two derivatives invariant under invariant under G, the space rotation group G, the space rotation group

O(3) and ParityO(3) and Parity (R.C. & Gatto 1999) (R.C. & Gatto 1999)

2 22 2 2 20 0 0 0T T

X Y T X Y 0 0

22 2 22 22 20 0S SX Y S X Y

F F 1 1L Tr J J Tr J J

4 4 2 2

vF F vTr J J Tr J J

4 4 2 2

2 22 2† † † †T T

0 0 T 0 0 0

2 22 2† † † †S S

S

2 22 22 2

0 0

F Fˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆL Tr X X Y Y Tr X X Y Y 2g4 4

F Fˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆTr X X Y Y Tr X X Y Y 2g4 4

v1 1 v

2 2 2 2

55

Using SU(3)Using SU(3)cc gauge invariance we can gauge invariance we can choose:choose:

a aTi T / F†ˆ ˆX Y e

22

22 S

2

22 2 22 2 2a a0 0 0

T

v1 1 1 v vL

2 2 2 2 2

Fv

F

2

Expanding at the second order in the fieldsExpanding at the second order in the fields

Gluons acquire Debye and Meissner masses Gluons acquire Debye and Meissner masses (not the rest masses, see later)(not the rest masses, see later)

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

56

Low energy theory supposed to be valid at Low energy theory supposed to be valid at energies << gap. Since we will see that energies << gap. Since we will see that

gluons have masses of order gluons have masses of order they can be they can be decoupled decoupled

2 22 2† † † †T T

0 0 T 0 0 0

2 22 2† † † †S S

S

2 22 22 2

0 0

F Fˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆL Tr X X Y Y Tr X X Y Y 2g4 4

F Fˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆTr X X Y Y Tr X X Y Y 2g4 4

v1 1 v

2 2 2 2

† †1 ˆ ˆ ˆ ˆg X X Y Y2

57

22T

NGB t t

2 2 2 2 2 2t t

FL Tr v Tr

41 1

( ) v | | ( ) v | |2 2

~ ~ -lagrangian-lagrangian

we get the gauge invariant we get the gauge invariant result:result:

ˆ ˆY X