Color Superconductivity and Confinement in the ... · M. Fiolhaisb, N. Scoccolac and A. G. Grunfeldc a Depto. de F´ısica, Instituto Tecnologico de Aeron ´autica, Sao Jos˜ e dos

IntroductionPairing Lagrangian in Leading Order (LO)

Gap structureLO pairing dynamics in CFL phase and s-c gap calculation

Self-Consistent Gap equationsResults

ConclusionsAcknowledgements

Color Superconductivity and Confinement inthe Chromodieletric Model

M. Malheiroa, B. V. Carlsona, T. Fredericoa, S. Martinsa andM. Fiolhaisb, N. Scoccolac and A. G. Grunfeldc

a Depto. de Fısica, Instituto Tecnologico de Aeronautica, Sao Jose dosCampos/SP, Brazil

b Dep. Fısica and Centro de Fısica Computacional, Univ. Coimbra, Portugalc Physics Department, Comision National de Energia Atomica, Argentina

13 de Julho de 2009

M. Malheiro a , B. V. Carlson a, T. Frederico a , S. Martins a and M. Fiolhais b , N. Scoccola c and A. G. Grunfeld cColor Superconductivity and Confinement in the Chromodiele tric





1 Introduction

2 Pairing Lagrangian in Leading Order (LO)

3 Gap structure

4 LO pairing dynamics in CFL phase and s-c gap calculation

5 Self-Consistent Gap equations

6 Results

7 Conclusions






Summary

1 Introduction


3 Gap structure



6 Results

7 Conclusions






Introduction

The Chromodielectric model (CDM) provides a reasonableframework to study baryons (such as the nucleon, delta, Roper)at low densities and to study strange quark matter at very highdensities. In this regime, two different phases show up: a chiralbroken and a chiral symmetric phase, the latter not necessarilyabsolutely stable. At high densities, the abundance of quarks u,d and s are the same in the chiral symmetric phase and thereare no electrons. These two properties are also obtained in anew phase which is expected to occur in QCD at very highdensities, known as color flavor locked (CFL) phase.






Introduction

This suggests that strange matter may undergo a transition tothe CFL phase, with an energy lowering due to the quark BCSpairing. It is now generally believed that the CFL state (at leastfor asymptotic densities) is likely to be the ground state forstrongly interacting matter. In this work we report on a study inan extended version of the Chromodielectric model (CDM) withthe BCS quark pairing implemented. Pairing corrections arecarried using the methods developed for nuclear matter [1].






Chromodielectric Model

Lagrangian of the CDM model

L = Ψ (ı/∂ − m)Ψ +12

(∂µσ∂µσ + ∂µ~π · ∂µ~π) − W (σ, ~π)

+G(χ)

fπΨ (σ + ı~τ · ~π) Ψ + Gs(χ)ΨsΨs +

12∂µχ∂µχ− U(χ),

where G(χ) = −gfπ/χ







W (σ, ~π) =m2

σ

8f 2π

(

σ2 + π2 − f 2π

)2,

U(χ) =12

m2χχ

2

[

1 +

(

8η4

γ2 − 2)(

χ

γmχ

)

+

(

1 −6η4

γ2

)(

χ

γmχ

)2]







Model parameters: g = 0.023 GeV, fπ = 0.093 GeV,mχ = 1.7 GeV, η = 0.1, γ = 0.2.






Summary

1 Introduction


3 Gap structure



6 Results

7 Conclusions






Pairing Lagrangian in Leading Order (LO)

Non-linear Lagrangian for quarks interacting with a scalar field(only the confine sector) with a quadratic potential:

L = Ψ (/∂ − m)Ψ + G(χ)ΨΨ − U(χ) +12∂µχ∂µχ, (1)







where G(χ) = gfπ/χ and U(χ) = 12m2

χχ2.

Assuming infinite matter χ = m−2χ G′(χ)〈ΨΨ〉,

the scalar density ρS = 〈Ψ(x)Ψ(x)〉

χ = −m−2χ

gfπχ2 ρS (2)







Assuming the pairing effect to be enough small

χ = χ+ δχ

the Lagrangian around the mean field χ up to order O(

δχ2)

is

L = Ψ (/∂ − m)Ψ + G(χ)ΨΨ −12

m2χχ

2 +12

(

G′(χ)ΨΨ + m2χχ)

×

∫

d4x ′G(x , x ′)(

G′(χ)Ψ(x ′)Ψ(x ′) − m2χχ)

. (3)







Expanding up to order (ΨΨ)2 and constructing a pairingLagrangian keeping terms up to the order (ΨΨ − 〈ΨΨ〉)2.

The mass of the χ field gains a self-energy contribution givenby:

M2

= m2χ − G′′(χ)〈ΨΨ〉 . (4)







The expansion for the Lagrangian just to leading order in(ΨΨ − 〈ΨΨ〉) reads

LLO = Ψ (/∂ − m)Ψ + G(χ)ΨΨ −12

M2χ2

+1

2M2 (G′(χ))2 ((ΨΨ) − 〈ΨΨ〉

)2, (5)

where the term neglected corresponds to NLO in the expansionof the Lagrangian in terms of (ΨΨ − 〈ΨΨ〉).







Thus pairing of quarks are generated by

LI,LO = hΨa,iα Ψa,i

α Ψb,jβ Ψb,j

β , (6)

the quartic fermionic term in the LO Lagrangian, with thecoupling,

h =12

(

G′(χ)

M

)2

. (7)







In the complete CDM version

M = U ′′ (χ) − U ′ (χ)G′′(χ)

G′(χ). (8)

M = m2χ

(

3 + 12(

4η4

γ2 − 1)(

χ

γmχ

)

+ 10(

1 −6η4

γ2

)(

χ

γmχ

)2)

. (9)






Summary

1 Introduction


3 Gap structure



6 Results

7 Conclusions






Gap structure

The color indices are a,b, flavor i , j and Dirac α,beta. Now weintroduce the conjugate fields ΨT = BΨ

⊤and ΨT = Ψ⊤B with

B = γ5C and the charge conjugation operator C = iγ2γ0. Theproperties are valid B2 = −1, B† = B⊤ = −B = B−1. Theoperator Ψc∆Ψ has to obey the following consistence relation

ΨT∆Ψ = −Ψ⊤∆⊤Ψ⊤

T = −Ψ⊤BB∆⊤BBΨ⊤

T = −ΨTB∆⊤BΨ , (10)

expressed as ∆ = −B∆⊤B, because Ψ is a Grassmannvariable.






Gap structure

We classify the possible invariants that satisfy the consistencyrelation ∆ = −B∆⊤B. The color and flavor (u,d,s) irreducibletensors belong to the representations of SU(3): 3 ⊗ 3 = 3 ⊕ 6.The 3 representation corresponds to antisymmetric tensorsgiven by totally antisymmetric tensor εkij for flavor and εcab forcolor. The 6-symmetric irreducible representation can beassociated with the identity and Gellmann matrices for thegenerators of SU(3) corresponding to λ1, λ3, λ4, λ6, λ8 and√

23 I, with the trace of products being zero for different matrices

and 2 otherwise.






Gap structure

In table I, the non-vanishing structures for the contactinteraction Lagrangian of Eq.(6) are presented. The structuresimplifies because the gap has no momentum dependence andsome structures are not allowed.

∆ 3c ⊗ 3f 6c ⊗ 6f 3c ⊗ 6f 6c ⊗ 3f

I × × - -γµ × × - -γ5 × × - -γµγ5 - - × ×

σµν - - × ×






Gap structure

The gap function has the general form allowed by theinteraction (6) and, for infinite quark matter it is given by:

∆a,i ;b,j = εcabεkij

(

∆3⊗3S;c;k + ∆3⊗3

0;c;kγ0 + ∆3⊗3

PS;c;kγ5)

+ScabSkij

(

∆6⊗6S;c;k + ∆6⊗6

0;c;kγ0 + ∆6⊗6

PS;c;kγ5)

+Scabεkij∆3⊗6PV ;c;kγ

0γ5 + εcabSkij∆6⊗3PV ;c;kγ

0γ5 . (11)

where the six symmetric tensors are represented by Scab.






Gap structure

The color-flavor locked phase corresponds to take into accountonly the gap structure 3 ⊗ 3 terms yielding a ground statesymmetric under color and flavor rotations. The gap reduces to:

∆a,i ;b,jCFL =

(

δaiδbj − δbiδaj)

(

∆S + ∆0γ0 + ∆PSγ

5)

, (12)

where we have dropped the notation for the group irreduciblerepresentation, and we have used εcabεcij = δaiδbj − δbiδaj .






Summary

1 Introduction


3 Gap structure



6 Results

7 Conclusions






LO pairing dynamics in CFL phase and self-consistent gap calculation

Pairing of quarks will be dynamically generated by

LI,LO = hΨa,iT,αΨa,i

T,αΨ

b,jβ Ψb,j

β where we have make use of the

properties of the Grassmann variables and B2 = −1. The gapis the expectation value of the operator

∆a,i ;b,jCFL = 2h〈Ψa,i

α Ψb,jT,β〉, (13)

and equating it to the gap decomposed in its three terms fromEq.(12). To make the decomposition of the different gap termsthe following contraction is useful:

∑

a,b,i ,j

δaiδbj(


=∑

i ,j

(

δiiδjj − δijδij)

= 6. (14)







Therefore, one has(

∆S + ∆0γ0 + ∆PSγ

5)

αβ= 1

12〈Ψi ,iα Ψ

j ,jT,β〉.

Performing the traces with appropriate spinor operators the gapequations are found:

∆X =h48

Tr[

ΓX〈Ψi ,iΨj ,jT 〉]

, (15)

with X=S, 0 and PS, with corresponding ΓX = I, γ0 and γ5,respectively.







We now present the formalism proposed by Gorkov [2] in orderto arrive at the Dirac-Gorkov equations of motion. We use theextended form of the time–reversed states ψ⊤, already definedby ψ⊤ = Ψc = BΨ

⊤, such that now we have an ansatz for the

effective single–particle Lagrangian

Seff =

∫

dt Leff =

∫

d 4x{

ψ(x) [i /∂ − M + γ0µ]ψ(x) − ψ(x)Σψ(x)

+12ψ(x)∆ψT(x) +

12ψT(x)∆ψ(x)

}

,

where µ is the chemical potential to be used as a Lagrangemultiplier to fix the average number of particles.







The symmetries of the effective mean–field Lagrangian undertransposition and Hermitian conjugation, yield the followingproperties of the mean fields: ∆ = −B ∆⊤ B and∆ = −B ∆

⊤B; Σ = γ0Σ

†γ0 and ∆ = γ0∆†γ0 where

∆ = γ0∆†γ0.

The Dirac–Gorkov equations are the following coupledequations of motion for the fields ψ and ψ⊤

(

(i /∂ − M + γ0µ) − Σ ∆∆ (i /∂ + M − γ0µ) + ΣT

)(

ψ(x)ψT(x)

)

= 0 .







and one obtains a generalized quark (quasiparticle) propagator

S(x) =

(

G(x) F (x)

F (x) G(x)

)

= −i⟨(

ψ(x)ψT(x)

)

(ψ(x) , ψT(x))

⟩

,

where, by 〈· · · 〉, we mean the timer-ordered expectation valuein the interacting quark matter ground state, 〈0|T (· · · )|0〉.The Dirac-Gorkov equations are in fact given in term of theinverse Green-function written in terms of the generalizedquark propagator as G−1 = −iS−1, such that

G−1(

ψ(x)ψ⊤(x)

)

= G−1 Ψ(x) = 0, (16)







We observe that in S , G(x) is the usual quark propagator,while G(x) describes the propagation of quarks intime-reversed states. The off-diagonal terms of S(x) describethe propagation of correlated quarks and are just the relativisticgeneralization of the anomalous propagators defined by Gorkov[2]. Thus we need to find G inverting G−1 in order to obtain theself consistent equations for the quark self-energy and quarkpairing that can be express in terms of the two–fermion vacuumexpectation values as

Σ = 2h⟨

ψ(x)ψ(x)

⟩

, and ∆ = 2h⟨

ψ(x)ψT(x)

⟩

,







where the equation for ∆ can be obtained using the Hermiticitycondition. The quark mass is M = m − G(χ) and one may usethe form

∆a,i ;b,j

=(


(

∆∗S + ∆∗

0γ0 + ∆∗

PSγ5)

. (17)






Summary

1 Introduction


3 Gap structure



6 Results

7 Conclusions






Self-Consistent Gap equations

To invert G−1 one expands it in a SU(2) isospin matrix basis and expand thecolor-flavor matrices in their eigenvector basis 〈ai |s〉 = v s

ai using therespective eigenvalues, λs. From the solution of the expansion for thecolor-flavor matrix Ca,i;b,j we find nine eigenvalues, λs: 2, -1 and 1, withdegeneracies n(s). The eigenvalue 2 appears once; the −1 appears in twoisolated times; there are three pairs of −1 and 1. If we write the matrix∆

(s)αβ = 2n(s)h〈ψ⊤αψβ〉s, after a straightforward calculation we arrive at the

following gap equation

∆(s)αβ = 2n(s)hi

∑

λ

∫

d4k(2π)4

(

v (s)λ

)

α

(

u(s)λ

)

β

k0 − ǫ(s)λ (k) + iδλ

. (18)

The formalism has been developed, and preliminary results for the solution of

the gap equation were obtained.






Summary

1 Introduction


3 Gap structure



6 Results

7 Conclusions






Results

Parameters: g = 0.023 GeV mχ = 1.0 GeV, γ = 0.2 and η = 0.1They imply in no gap in Phase II.

0 0,5 1 1,5

ρ [fm-3]

0,1

0,12

0,14

χ/γ

mχ

singlettriplet

Confine field

0 0,5 1 1,5

ρ [fm-3]

80

100

120

140

Mq

[M

ev]

singlettriplet

Quark Mass






Results

0 0,5 1 1,5

ρ [fm-3 ]

0

10

20

30

40

50

60

70

∆ [M

eV]

∆S

∆V

Gap - Singlet Channel

0,12 0,14 0,16 0,18 0,2 0,22

ρ [fm-3]

0,05

0,1

0,15

0,2

0,25

∆ [M

eV]

∆S

∆V

Gap -Triplet Channel






Results

0,5 1 1,5 ρ [MeV]

5

10

15

H [

GeV

-2]

singlettriplet

Effective Coupling Constant






Results

0,5 1 1,5

ρ [fm-3]

-4

-2

0

2

4

6

E/N

-M [

fm-1

]

singlettriplet

Energy per particle






Summary

1 Introduction


3 Gap structure



6 Results

7 Conclusions






Conclusions

It is possible to obtain color superconductivity in a selfconsistent way in the CDM. The confinement affects directly the

pairing coupling constant.

h =12

(

G′(χ)

M

)2

. (19)

M = m2χ

(

3 + 12(

4η4

γ2 − 1)(

χ

γmχ

)

+ 10(

1 −6η4

γ2

)(

χ

γmχ

)2)

.(20)






Conclusions

The singlet channel gap is much stronger than in the triplet.The gluon mass mχ needs to be smaller at high density around0.5GeV to increase the triplet channel. How to implement Color

Superconductivity on the Light Cone?






Acknowledgements

This work was supported in part by the Portuguese-BrazilianFCT-CAPES Program, project 183/07 , CNPq and FAPESP.






Acknowledgements

I would like to thanks in the name of the local organizingcommittee the presence and the talks and posters of all theparticipants and we hope that you had a pleasant and fruitfulLC2009 workshop.

We hope to see all of you in Valencia, Spain next year!!!


Documents

Color Superconductivity and Confinement in the ... · M. Fiolhaisb, N. Scoccolac and A. G. Grunfeldc a Depto. de F´ısica, Instituto Tecnologico de Aeron ´autica, Sao Jos˜ e dos