1 color superconductivity: cfl and 2sc phases introduction hierarchies of effective lagrangians ...
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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
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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**..
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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
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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
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α β αβCiL jL ijC0 ψ ψ 0 Δε ε
α β αβCiR jR ijC0 ψ ψ 0 Δε ε
Why CFL?Why CFL?
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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.)
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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 - -
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5 5γ D(0)γ = -D(0)
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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)
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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
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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.
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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 |
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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
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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 | ,. ,
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heavyheavy
heavyheavy
lightlight
lightlight
Separation of light and heavy Separation of light and heavy d.o.f.d.o.f.
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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
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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
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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:
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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:
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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.)
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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
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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
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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
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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
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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
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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
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* 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)
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Four-fermi interaction one-gluon exchange Four-fermi interaction one-gluon exchange inspiredinspired
a aI
3L G
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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, )
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( 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
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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.
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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 )
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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
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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
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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
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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 )
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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
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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
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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)
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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
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3
A B ABTr 2
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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
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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
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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
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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
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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
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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
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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
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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)
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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
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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
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Condensation Condensation if:if: 2
sm
2
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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)
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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
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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
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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
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, 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
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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
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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
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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