Collective_modes_in_the_CFL_phase

T h e o p e n – a c c e s s j o u r n a l f o r p h y s i c s

New Journal of Physics

Collective modes in the color flavor-locked phase

Roberto Anglani1,2, Massimo Mannarelli3,4,6 andMarco Ruggieri51 Physics Division, Argonne National Laboratory, Argonne, IL 60439, USA2 Institute of Intelligent Systems for Automation, CNR, I-70126 Bari, Italy3 Departament d’Estructura i Constituents de la Matèria and Institut de Ciènciesdel Cosmos, Universitat de Barcelona, Martí i Franquès 1, 08028 Barcelona,Spain4 I.N.F.N., Laboratori Nazionali del Gran Sasso, Assergi (AQ), Italy5 Yukawa Institute for Theoretical Physics, Kyoto University, 606-8502 Kyoto,JapanE-mail: [email protected]

New Journal of Physics 13 (2011) 055002 (21pp)Received 17 January 2011Published 17 May 2011Online at http://www.njp.org/doi:10.1088/1367-2630/13/5/055002

Abstract. We study the low-energy effective action for some collective modesof the color flavor-locked (CFL) phase of QCD. This phase of matter has longbeen known to be a superfluid because by picking a phase its order parameterbreaks the quark-number U (1)B symmetry spontaneously. We consider themodes describing fluctuations in the magnitude of the condensate, namely theHiggs mode, and in the phase of the condensate, namely the Nambu–Goldstone(NG) (or Anderson–Bogoliubov) mode associated with the breaking of U (1)B .By employing as microscopic theory the Nambu–Jona-Lasinio model, wereproduce known results for the Lagrangian of the NG field to the leading orderin the chemical potential and extend such results evaluating corrections due tothe gap parameter. Moreover, we determine the interaction terms between theHiggs and the NG field. This study paves the way for a more reliable study ofvarious dissipative processes in rotating compact stars with a quark matter corein the CFL phase.

6 Author to whom any correspondence should be addressed.

New Journal of Physics 13 (2011) 0550021367-2630/11/055002+21$33.00 © IOP Publishing Ltd and Deutsche Physikalische Gesellschaft

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Contents

1. Introduction 22. The model 33. The Lagrangian for the Nambu–Goldstone (NG) boson 8

3.1. One-point function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93.2. Two-point function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103.3. Three-point function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113.4. Four-point function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

4. Higgs field and interaction terms 145. Conclusions 17Acknowledgments 18Appendix A. Interaction vertices 18Appendix B. Integrals 19References 20

1. Introduction

At extremely high densities, quantum chromodynamics (QCD) predicts that the individualnucleons that form standard hadronic matter should melt and their quark matter content shouldbe liberated [1]. At the low temperatures expected in sufficiently old compact stars, quark matteris likely to be in one of the possible color superconducting phases, whose critical temperaturesare generically of the order of tens of MeV (for reviews see [2] and [3]). Since compact startemperatures are well below these critical temperatures, for many purposes the quark matterthat may be found within compact stars can be approximated as having zero temperature, as weshall assume throughout. At asymptotic densities, where the up, down and strange quarks canbe treated on an equal footing and effects due to the strange quark mass can be neglected, quarkmatter is in the color flavor-locked (CFL) phase [2, 4]. The CFL condensate is antisymmetric incolor and flavor indices, and it involves pairing between up, down and strange quarks. The orderparameter breaks the quark-number U (1)B symmetry spontaneously, and the correspondingNambu–Goldstone (NG) boson (the Anderson–Bogoliubov mode) determines the superfluidproperties of CFL quark matter. The gapless excitation corresponds to a phase oscillation aboutthe mean-field value of the gap parameter. The fluctuation in magnitude of the condensate is as-sociated with a massive mode, which we shall refer to as Higgs mode. Both of these fluctuationsare excitations of several fermions and therefore describe collective modes of the system.

The low-energy properties of the system are completely determined by these collectivemodes [5], and their study is mandatory to understand dynamical properties that take placein compact stars with a CFL core. The actual superfluid property of the system is due to thepresence of the gapless excitations, which satisfy the Landau’s criterion for superfluidity [6].Moreover, in rotating superfluids, the interactions between NG bosons and vortices lead tothe appearance of the so-called mutual friction force between the normal and the superfluidcomponents of the system. The NG bosons are also responsible for many other propertiesof cold superfluid matter; in particular, they contribute to the thermal conductivity and to theshear and bulk viscosities [6]. A study of the shear viscosity and of the standard bulk viscosity

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coefficient due to NG bosons has been carried out in [7, 8]. A more detailed study of the bulkviscosity coefficients of the CFL phase has been undertaken in [9]. In the CFL phase, also theglobal SU (3)R × SU (3)L symmetry is spontaneously broken and the corresponding pseudo-NGbosons can contribute to the transport properties of the CFL phase. The contribution of kaons tothe bulk and shear viscosity has been studied in [10, 11]. The contribution of NG bosons to thethermal conductivity and cooling of compact stars were studied in [12–15]. Other low-energydegrees of freedom are the plasmons, which have been studied in [16, 17].

Although the CFL phase is characterized by many low-energy degrees of freedom, in thispaper we shall focus on the NG field, φ, and on the Higgs field, ρ, because we aim to build theeffective Lagrangian for the description of NG boson–vortex interaction. Vortices are describedby spatial variation of the condensate and therefore are determined by the space variation ofthe Higgs field. In principle, one could study the interaction of vortices with other low-energydegrees of freedom, but the NG boson associated with the braking of U (1)B is the only masslessmode, whereas pseudo-NG bosons associated with the breaking of chiral symmetry have a massof the order of few keV [18], and plasmons associated with the breaking of SU (3)c symmetryhave even larger masses [16, 17]. Therefore, all of these modes are thermally suppressed in coldcompact stars.

We derive the effective Lagrangian of the NG field, and of the Higgs field using asmicroscopic theory the Nambu–Jona-Lasinio (NJL) model [19, 20] with a local four-Fermiinteraction with the quantum numbers of one gluon exchange. This model mimics some aspectsof QCD at high densities [21, 22]. We derive the interaction terms of the NG bosons up to termsof the type (∂φ)4. The leading contributions to these interaction terms were obtained by Sonin [23] using symmetry arguments and the expression of the pressure of the CFL phase. To ourknowledge, the results of [23] so far have not been obtained starting from a microscopic theory.Only the free Lagrangian was obtained in [18]. We extend the results of [23] including thenext to leading corrections proportional to the gap parameter. In order to do this, we integratethe fermionic degrees of freedom employing the high-density effective theory (HDET) [3].Moreover, we determine the kinetic Lagrangian for the Higgs field and the interaction termsbetween the Higgs and the NG bosons.

An analysis similar to the one presented here was done for non-relativistic systems atunitarity in [24] and in [25, 26]. As we shall discuss in section 4, the main difference withthe non-relativistic systems at unitarity is that in the CFL phase, integrating out the Higgs fielddoes not lead to a change in the speed of sound.

This paper is organized as follows. In section 2, we present the NJL model and wedetermine the expression of the effective action of the system in the HDET approximation.In section 3, we derive the effective Lagrangian for the NG boson, neglecting the oscillations inthe magnitude of the condensate. In section 4, we derive the LO interaction terms between theHiggs mode and the NG bosons and the kinetic Lagrangian for the Higgs field. We also compareour results with the non-relativistic results of [24]. We draw our conclusions in section 5. Someinteraction vertices as well as the calculation of some integrals are reported in the appendix.

2. The model

We consider a NJL model of quark matter with a local four-Fermi interaction as a model ofQCD at large quark chemical potential µ. We assume that µ� ms and therefore we neglectu, d and s quark masses. In the absence of interactions, the Lagrangian density describing the

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system of up, down and strange quarks is given by

L0 = ψ iα(i 6∂ +µγ0)ψiα, (1)

where i, j = 1, 2, 3 are flavor indices, α, β = 1, 2, 3 are color indices and the Dirac indices havebeen suppressed. In QCD, one can show that at high densities the gluon exchange leads to theformation of a quark–quark condensate [2]. At asymptotic densities, the favored phase is theCFL phase [4], which is characterized by the condensate

〈ψ tiαCψ jβ〉 ∼1

∑I=1,2,3

εαβ Iεi j I , (2)

which locks together color and flavor rotations. Here, C = iγ 2γ 0 is the charge conjugationmatrix and εαβ I and εi j I are Levi–Civita tensors. This condensate breaks several symmetriesof QCD (see [4] for a detailed analysis). For our purposes, it is sufficient to note that the CFLcondensate breaks the U (1)B symmetry and therefore leads to the appearance of a gapless NGboson.

In the NJL model, the interaction among quarks mediated by gluons is replaced by a four-Fermi interaction analogous to the one proposed for superconductors by Bardeen, Cooper andSchrieffer (BCS). We shall consider an interaction with the same quantum numbers of one gluonexchange,

LI = −316 g ψγµλAψψγ µλAψ, (3)

where g > 0 is the coupling constant and λA with A = 1, . . . , 8 are the Gell–Mann matrices.This interaction at large chemical potentials leads to the formation of the CFL condensate.Introducing the new basis for the quark fields,

ψiα =1

√2

9∑A=1

λAiαψA, (4)

where λ9=

√2/3 × I , we have

〈ψ tACψB〉 ∼1AB, (5)

where 1AB =1AδAB , where 11 = · · · =18 =1 and 19 = −21. In this basis, the four-Fermiinteraction is given by

LI = −g

4VABC Dεabεcdψ

Aa ψ

Bb ψ

C†c ψ

D†d, (6)

where VABC D = Tr8∑

E=1

(λAλEλBλCλEλD) and a(a)= 1, 2 are the Weyl indices for L(R)

components [27].In order to study the fluctuation of the condensate, we introduce the Hubbard–Stratonovich

fields1AB(x) and1∗

AB(x), which allow us to write the partition function (normalized at the freecase for g = 0) as

ZZ0

=1

Z0

∫[dψ, dψ†][d1, d1∗] exp

{i∫

d4x

[−1AB WABC D1

∗

C D

g+L1

]}, (7)

where

WABC DVC DE F = δAEδBF and VABC DWC DE F = δAEδBF, (8)

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and the semi-bosonized Lagrangian is given by

L1 = ψ A(i 6∂ +µγ0)ψA −121AB(ψ

†ACψ∗

B)+ 121

∗

AB(ψtACψB). (9)

The fluctuations of the condensate1(x) around the mean-field value1MF can be describedby two real fields. One field describes the variation of |1(x)| while the other field is associatedwith a local phase change. Therefore, we write

1AB(x)= [1MFAB + ρAB(x)]e

2iφ(x), (10)

where ρAB(x)= ρ(x) diag(1, . . . , 1,−2)AB and φ(x) are the real fields. Hereafter, we shallsuppress the indices A, B and we shall indicate with 1 the mean-field value of the gap tosimplify the notation. We find it convenient to redefine the fermionic fields as

ψ → ψ eiφ(x), (11)

and in this way the semi-bosonized Lagrangian is given by

L1 = ψ(iγ µ∂µ + γ 0µ− γ 0∂0φ− γ i∂iφ

)ψ −

12ψ

†C(1+ ρ)ψ∗ + 12ψ

tC(1+ ρ)ψ. (12)

Writing the Lagrangian in this form, it is possible to define an ‘effective chemicalpotential’

µ= µ− ∂0φ. (13)

In other words, ∂0φ describes long-wavelength fluctuations of the chemical potential on the topof its constant value µ. We want now to clarify one point regarding the terminology used forthe field φ. Since ∂0φ describes fluctuations of the chemical potential, it is not correct to callthis field the phonon, which describes pressure oscillations. However, when one neglects theeffect of the gap 1, the Lagrangian of the phonon and of the NG boson associated with thebreaking of U (1)B symmetry coincide. The reason for this is that for vanishing values of 1,the oscillations of pressure are proportional to the oscillations of the chemical potential.

It is convenient to define a fictitious gauge field Aµ = (∂0φ,∇φ), which according toequation (12) is minimally coupled to the quark fields. Hereafter, we shall refer to Aµ asthe gauge field, although it is not related to any gauge symmetry of the system. With thissubstitution, we rewrite the Lagrangian in equation (12) as

L1 = ψ(iγ µDµ +µγ 0

)ψ −

12ψ

†C(1+ ρ)ψ∗ + 12ψ

tC(1+ ρ)ψ, (14)

where Dµ = ∂µ + iAµ.In order to simplify the calculation, we employ the HDET (see [3]). Using standard

techniques, the Lagrangian describing the kinetic terms and the interaction with the gauge fieldAµ can be written as

LI =

∫dv8π

[ψ†

+

(iV · D −

PµνDµDν

2µ+ i V · D

)ψ+ +ψ†

−

(iV · D −

PµνDµDν

2µ+ iV · D

)ψ−

], (15)

where the positive energy fields with ‘positive’ and ‘negative’ velocities are given by

ψ± ≡ ψ+(±v) (16)

and where

V µ= (1, v), V µ

= (1,−v), Pµν= gµν −

V µV ν + V ν V µ

2, (17)

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with v being the Fermi velocity. The non-local interactions that are present in equation (15) aredue to the integration of the negative energy fields [3].

In order to have compact notation, it is useful to introduce the Nambu–Gorkov spinor,

9 =1

√2

(ψ+

Cψ∗

−

), (18)

which allows us to write the semi-bosonized Lagrangian as

L1 = La +Lb, (19)

where

La =

∫dv4π9†

(iV µ∂µ − V µAµ 1+ ρ

1+ ρ iV µ∂µ + V µAµ

)9 (20)

is the sum of the kinetic term and of the local interaction term, while the non-local interactionterm is given by

Lb = −

∫dv4π

Pµν9†

−2µ+iV ·D∗

L DµDν1

L D∗

µDν

1

L DµD∗

ν2µ+iV ·D

L D∗

µD∗

ν

9, (21)

where L = (2µ+ iV · D)(−2µ+ iV · D∗)−12− iε. More details about the derivation of the

non-local interaction will be given in [28].The semi-bosonized Lagrangian is quadratic in the fermionic fields and therefore we can

integrate them. In this way, the effective action can be written in terms of the fields φ and ρ andtheir derivatives. An analogous calculation in the non-relativistic case has been done in [25].Before doing this, we consider in more detail the interaction terms between the fermionic fieldsand the fictitious gauge field determined by the non-local term. In momentum space we find that

Lb =

∫dv4π9† Pµν AµAν

(−2µ+ V · `+ V · A −1

−1 2µ+ V · `− V · A

)1

L9, (22)

where in momentum space L = (2µ+ V · `− V · A)(−2µ+ V · `+ V · A)−12− iε. The

‘residual momentum’ of the quarks, `µ, is defined as

`0 = p0, `i = pi −µvi , (23)

where pµ is the four momentum of the quarks.Expanding the denominator of the expression in equation (22) in powers of A and

considering terms up to the order A4, we find that

Lb =

∫dv4π

(9†0

µν

2 9AµAν +9†0µνρ

3 9AµAν Aρ +9†0µνρσ

4 9AµAν Aρ Aσ)

+O(A5), (24)

where the expression of the vertices 02, 03 and 04 are reported in appendix A. We also define

0µ

1 =

(−V µ 0

0 V µ

), (25)

which is the vertex that describes the minimal coupling of quarks with the gauge field inthe HDET. The various vertices are schematically depicted in figure 1, with the wavy linescorresponding to the gauge field and the full line corresponding to the fermionic field.

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Figure 1. The tree-level interaction vertices of the gauge field with the fermionsin the HDET. Full lines correspond to fermionic fields; wavy lines correspond tothe gauge fields. The vertices are named 01, 02, 03 and 04, where the subscriptsindicate the number of the external gauge fields. Their expression is reported inappendix A.

Figure 2. Interaction of the Higgs field (dashed line) with quarks (full line). Thecorresponding vertex is named 0ρ and is reported in equation (29).

Integrating the fermionic fields, the partition function turns out to be given by

ZZ0

=

∫[d1, d1∗] exp[ 1

g

∫d4x 1AB WABC D1

∗

C D] det[S−1]1/2

det[S−10 ]1/2

≡ exp[iS], (26)

where S is the action of the system and the full inverse propagator is given by

S−1≡ S−1

MF +0. (27)

The mean-field inverse propagator is given by

S−1MF =

(iV µ∂µ −1

−1 iV µ∂µ

), (28)

while 0 = 0ρ +01 +02 +03 +04. The interaction of quarks with the ρ field is described by thevertex (figure 2)

0ρ =

(0 −ρ

−ρ 0

), (29)

while the interaction of quarks with the NG bosons is given by 01 = 0µ

1 Aµ, 02 = 0µν

2 AµAν ,03 = 0

µνρ

3 AµAν Aρ and 04 = 0µνρσ

4 AµAν Aρ Aσ .We separate the mean-field action from the fluctuation, writing

S = SMF +Seff, (30)

where Seff is the effective action describing the low-energy properties of the system. The mean-field action provides the free energy of the system,

�= SMF = −i

2Tr ln[S−1

MF] −i

g

[1AB WABC D1

∗

C D

], (31)

and is a function of the quark gap parameter 1. Hereafter, Tr symbolizes the trace over theNambu–Gorkov index, the trace over color–flavor indices, the trace over spinorial indices andthe trace over a complete set of functions in space time.

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The gap parameter can be determined by the stationary condition of the mean-field action∂SMF

∂1

∣∣∣1

= 0, (32)

which in turn allows us to determine the pressure of the system by P = −�(1). In the NJLmodel, the mean-field value of the gap parameter turns out to depend on the coupling g and onthe three-momentum cutoff (see e.g. [3]). Since we do not need the numerical value of the gap,we shall treat 1 as a free parameter.

The fluctuation around the mean-field solution is described by the effective action

Seff = −i

g

∫d4x [ρAB(x)WABC DρC D(x)+ 2ρAB(x)WABC D1C D] −

i

2Tr ln (1 + SMF0), (33)

which contains the 0 expansion

Tr ln(1 + SMF0)= Tr

[∞∑

n=1

(−1)n+1

n(SMF0)

n

], (34)

and we shall evaluate terms up to the fourth order in the gauge fields. We shall also determinethe leading-order terms of the ρ field Lagrangian and the leading-order interaction terms ofthe ρ field with the NG bosons. In the expansion, we shall neglect the NLO terms of the kind(∂∂φ)n, where n is any non-zero integer.

3. The Lagrangian for the Nambu–Goldstone (NG) boson

Neglecting oscillations in the modulus of the condensate, we can determine from equations (33)and (34) the effective Lagrangian for the NG bosons. As we shall show below, for vanishingvalues of 1, we obtain the same results obtained in [23],

Lφ =3

4π 2[(µ− ∂0φ)

2− (∂iφ)

2]2. (35)

This expression relies on symmetry considerations, in particular on conformal symmetry, and onthe expression of the pressure in the CFL phase. We shall reproduce these results and evaluatethe corrections of the order (1/µ)2, which are related to the breaking of conformal symmetry.Our strategy is to first expand the Lagrangian in the Aµ fields, and write

Lφ = L1 +L2 +L3 +L4, (36)

where Lm is the term with m gauge fields. Then we expand the various terms in 1/µ. Note thatthe term Lm will be obtained by the expansion in equation (34), considering all of the terms withn 6 m.

A different way of obtaining the leading order in µ Lagrangian for terms proportional to∂0φ is the following. At the leading order in µ—neglecting terms proportional to 1—the free-energy density of the CFL phase in the absence of oscillations is given by

�=3

4π 2µ4. (37)

We have seen in section 2 that ∂0φ corresponds to a fluctuation in the chemical potential of thesystem. Therefore, including these fluctuations, the free-energy density of the CFL phase at theleading order in µ is given by

�=3

4π 2µ4, (38)

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Figure 3. One-loop diagram that contributes to the coefficient of the ∂0φ termin the effective Lagrangian L1 in equation (36). The full line corresponds to thequark field. The wavy line corresponds to the external gauge field.

where µ is defined in equation (13). Expanding the free energy, we find that the Lagrangian forthe ∂0φ field is given by

L(∂0φ)=3

4π 2µ4

−3

π 2µ3∂0φ +

9

2π 2µ2(∂0φ)

2−

3µ

π 2(∂0φ)

3 +3

4π 2(∂0φ)

4. (39)

3.1. One-point function

The evaluation of the term of the effective Lagrangian proportional to ∂φ requires thecomputation of the diagram in figure 3. One finds that

L1 = −i Tr[S0]|A, (40)

where the subscript means that in the evaluation of the trace only the terms linear in A must beincluded. We find that

L1 = −i∑∫

d`

[V · A

V · `

D∗− V · A

V · `

D

], (41)

where we have defined∑∫d` =

2

π

∑Nc,N f

∫dv4π

∫ +δ

−δ

d`‖

2π(µ+ `‖)

2

∫∞

−∞

d`0

2π, (42)

where δ is a cutoff that we shall set equal to µ (see [3]). In equation (41), we have also used7

D = V · `V · `−12 + iε, (43)

and for future convenience we define the quantity

L0 = (2µ+ V · `)(−2µ+ V · `)−12− iε. (44)

The sum over flavor and color degrees of freedom is straightforward and we obtain

L1 = −2i

π

∫dv4π

∫d2`

(2π)2(µ+ `‖)

2

[V · A

(8

V · `

D(1)∗+

V · `

D(−21)∗

)

− V · A

(8

V · `

D(1)+

V · `

D(−21)

)]. (45)

7 Here D is not the covariant derivative. Although this notation may generate confusion, we prefer to use it, as inthe review [3].

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b)a)

Figure 4. One-loop diagrams that contribute to the coefficients of the A20 term,

and of the A2 term of the effective Lagrangian L2 in equation (36). Full lines arefermionic fields. Wavy lines are gauge fields.

Employing the expressions reported in appendix B, one can carry out the integration over theresidual momentum and on the Fermi velocity. In this way, one finds that

L1 =

(−

3

π2µ3 +

6

π2µ21

)∂0φ. (46)

Note that at the leading order in µ, the expression above for the term proportional to ∂0φ agreeswith the corresponding expression reported in equation (39).

3.2. Two-point function

The Lagrangian involving two gauge fields is given by

L2 = −i(Tr[S0] −12 Tr[S0S0])|A2, (47)

where in the evaluation of the trace one has to consider only terms quadratic in A. Thecorresponding diagrams are reported in figure 4. The diagram in figure 4(a) gives

Tr[S0]∣∣∣

A2=

∑∫d` Pµν AµAν

[12 + V · `(V · `− 2µ)

L0 D+ (V → V )

], (48)

where L0 has been defined in equation (44) and (V → V ) actually means an expression that isobtained by replacing (V → V , `‖ → −`‖, ε → −ε). Hereafter, we shall always use this wayof writing to simplify the notation.

The diagram in figure 4(b) gives

Tr[S0S0]∣∣∣

A2=

∑∫d`

[((V · A)2(l0 + `‖)

2

D2−(V · AV · A)12

D2

)+ (V → V )

], (49)

and evaluating the integrals using the expressions reported in appendix B we find that

L2 '9µ2

2π 2

(1 − 2

12

µ2

)A2

0 −3µ2

2π 2

(1 − 2.1

12

µ2

)A2 +O

(12

µ2log (1/µ)

)=

1

2m2

D A20 −

1

2m2

MA2. (50)

Then the speed of the NG boson is given by cs = mM/m D and the corresponding plot is reportedin figure 5. For vanishing1, we find that cs = 1/

√3, meaning that the system is scale invariant.

The effect of a non-vanishing 1 is to increase the speed of the NG boson.

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0 0,02 0,04 0,06 0,08 0,1 ∆/µ

0,34

0,36c s

Figure 5. Speed of the NG boson associated with the breaking of U (1)B as afunction of 1/µ.

c)b)a)

Figure 6. One-loop diagrams that contribute to the effective Lagrangian L3 inequation (36). Note that the only contribution to A3

0 comes from diagram (c).Full lines are fermionic fields. Wavy lines are gauge fields.

3.3. Three-point function

The Lagrangian describing the interaction of three NG bosons is formally given by

L3 = −i(Tr[S0] −12 Tr[S0S0] + 1

3 Tr[S0S0S0])|A3, (51)

and it corresponds to the evaluation of the diagrams in figure 6. Figures 6(a) and 6(b) contributeexclusively to the term A0A2, whereas figure 6(c) gives the term proportional to A3

0. Thecontribution of figure 6(a) is given by

Tr[S0]|A3 =

∑∫d` Pµν AµAνV · A

×

[L0V · `+ (V · `− V · `− 4µ)(D + 212

− 2µV · `)

DL20

− (V → V )

]. (52)

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a)

e)d)

c)b)

Figure 7. One-loop diagrams that contribute to the effective Lagrangian L4 inequation (36). Full lines are fermionic fields. Wavy lines are gauge fields.

The contribution of the diagram in figure 6(b) is given by

Tr[S0S0]|A3 = − 2∑∫

d` Pµν AµAνV · A

×

[212V · `+ (V · `− 2µ)((V · `)2 −12)

D2L0− (V → V )

]. (53)

The contribution of the diagram in figure 6(c) is given by

Tr[S0S0S0]|A3 =

∑∫d`

[−(V · A)3(V · `)3 −12(V · A)2 V · AV · `

D3

+212(V · A)2(V · A)(V · `)

D3− (V → V )

]. (54)

Evaluating the integrals with the help of the expressions reported in appendix B, one finds that

L3 =3µ

π 2

(1 −

12

µ2

)A0A2

−3µ

π 2

(1 −

312

2µ2

)A3

0. (55)

3.4. Four-point function

The Lagrangian describing the interaction of four NG bosons is formally given by

L4 = −i(Tr[S0] −12 Tr[S0S0] + 1

3 Tr[S0S0S0] + 14 Tr[S0S0S0S0])|A4, (56)

and it corresponds to the sum of the diagrams reported in figure 7. All of these diagrams, withthe exception of figure 7(e), originate from the non-local vertices 02, 03 and 04, and thereforecan give contributions to the coefficients of the terms with A2

0A2 and A4. Figure 7(e) contributesto the coefficient of the term proportional to A4

0.

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The contribution of figure 7(a) is given by

Tr[S0]∣∣∣

A4=

∑∫d` Pµν AµAν(V · A)

{(V · A)

[−V · `(V · `+ 2µ)

DL20

+ Z(V · `− 2µ)2 + (V · `+ 2µ)2

DL30

]

+ V · A

[2Z −12

DL20

− 2Z(V · `− 2µ)(V · `+ 2µ)

DL30

]+ (V → V )

},

(57)

where Z = (D + 212− 2µV · `).

Figure 7(b) gives∑∫d` Pµν AµAν

{(V · A)2

[212 (V · `)2 + 3(V · `)2 − 3µV · `+ 5µV · `+ 4µ2

D2L20

]+ 2V · AV · A

×

[−(D + 212)2 +µ(V · `(V · `)2 −12(V · `− 3V · `))− 4(V · `)2µ2

D2L20

]

+ (V → V )

}, (58)

while figure 7(c) gives∑∫d` Pµν AµAνPαβ AαAβ

[D + 212 + 4µ2(V · `)2 − 412µ(2V · `+µ)

D2L20

+V · `(612V · `− 4µ(V · `)2 + 412µ)

D2L20

+ (V → V )

]. (59)

Both of these diagrams are determined by evaluating

Tr[S0S0]∣∣∣

A4. (60)

Figure 7(d) gives

Tr[S0S0S0]∣∣∣

A4=

∑∫d` Pµν AµAν

{(V · A)2

[3(V · `)3(V · `− 2µ)

D3L0

+ 12 2(V · `)2 + 7(V · `)2 − 2µV · `+ 4µV · `

D3L0

]

−V · AV · A

[9D + 1212 + 4µ(V · `− 2V · `)

DL30

]+ (V → V )

}. (61)

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14

Figure 8. Diagrams that contribute to the one-loop self-energy of the Higgs field.Full lines are fermionic fields. Dashed lines correspond to the Higgs field, ρ.

Finally, figure 7(e) gives

Tr[S0S0S0S0]|A4 =

∑∫d`

1

D4[(V · A)4(V · `)4 − 3(V · A)3V · A12(V · `)2

−V · A(V · A)312(V · `)2 + (V · A)2(V · A)2(2D + 312)+ (V → V )].

(62)

After a long, but straightforward, calculation, we find that at the leading order in µ,

L4 =3

4π 2A4

0 +3

4π2A4

−3

2π2A2

0A2. (63)

In this case, we restricted our calculation, for simplicity, to evaluate the leading terms in µ.

4. Higgs field and interaction terms

The fluctuations in the absolute value of the condensate are described by the Higgs field ρdefined in equation (10). The effective Lagrangian for this field, including the interaction termswith the NG bosons, can be determined by means of the same strategy employed in the previoussection.

From the 0 expansion in equation (33), we obtain the self-energy diagram in figure 8,which gives for vanishing external momentum of the Higgs field

Lρρ = −6

π2µ2ρ(x)2, (64)

which is the mass term for the ρ field. However, the actual mass of the Higgs is not proportionalto the chemical potential: as we shall show below, one needs a wave function renormalization toput the Lagrangian in the canonical form. We note that the ρ field does not carry color and flavorindices. In order to obtain equation (64) (and all the expressions below), color and flavor indiceshave been properly contracted and the resulting Lagrangian is expressed in terms of colorlessfields.

Considering the fluctuation of the chemical potential given by ∂0φ, the Lagrangian aboveturns into

Lρρ = −6

π2µ2ρ(x)2 = −

6

π 2(µ− ∂0φ)

2ρ(x)2, (65)

which automatically gives part of the interaction of the Higgs field with the NG bosons. Theremaining interaction terms of two Higgs field with two NG bosons correspond to a coupling(∂iφ)

2ρ(x)2. These interaction terms, as well as the interaction terms in equation (65), can beobtained from the diagrams in figure 9.

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15

Figure 9. Interaction of the Higgs field with two NG bosons at the one-loop level.Full lines correspond to fermionic fields, dashed lines correspond to the Higgsfield and wavy lines correspond to the gauge field Aµ = ∂µφ.

Evaluating these diagrams, one obtains

Lρρφφ = −6

π 2

[(µ− ∂0φ)

2−

1

3(∂iφ)

2

]ρ(x)2, (66)

which describes the interaction terms among the Higgs fields and two NG bosons at the leadingorder in µ. Note that the term ρ∂µφ∂

µφ describing the interaction between one Higgs field andtwo NG bosons is missing. This is basically due to the fact that one cannot have a term like µ2ρ

from the loop expansion. The fact that in our case terms linear in the ρ field are missing leadsto an interesting effect. Let us consider a Lagrangian with terms up to quadratic order in the NGbosons and Higgs fields. Since the Lagrangian is quadratic in the massive Higgs field, one canintegrate them out from the theory obtaining the actual low-energy Lagrangian of the system.In the CFL phase, this does not lead to a modification of the effective Lagrangian of the NG

bosons. In particular, the velocity of the NG bosons at the leading order in µ is cs =

√13 . This

is rather different from what happens in the non-relativistic case [24], where it was shown thatintegrating the Higgs mode one obtains the modification of the speed of sound first determinedin [29].

In order to clarify this point, we compare our Lagrangian with the quadratic non-relativisticLagrangian. Schematically, the result of [24] can be written as

LN .R.(ρ, Y )=A

2ρ2 + Bρ Y +

C

2Y 2 + D Y, (67)

where A, B,C and D are some coefficients and Y = ∂0φ + (∇φ)2

2m , with m the mass of the non-relativistic fermions. The speed of the NG bosons is given by

c2s = −

D

mC. (68)

Integrating out the ρ field, the effective Lagrangian for the φ field turns out to be

LN .R.(φ)=B2 + AC

2AY 2 + D Y, (69)

and the speed of sound is modified to

c2s = −

D A

m(B2 + AC). (70)

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16

Note that if the coupling B vanishes, then the speed of the NG bosons remains the same as onehas in equation (68).

In our case, the LO Lagrangian quadratic in the ρ field is given by equation (66) and thereis no term that couples one ρ field with two NG bosons, i.e. expanding the Lagrangian one findsthat the analogous of the coefficient B in equation (67) is missing. Therefore, integrating outthe ρ field does not change the effective Lagrangian for the NG bosons.

From the diagrams in figure 9, we can determine the kinetic terms of the effectiveLagrangian of the Higgs field. Considering soft momenta of the Higgs field, p �1, andexpanding up to the order (p/1)2, we obtain

LK (ρ)=1

2

3µ2

4π 2

1

12

[(∂0ρ)

2−

1

3(∂iρ)

2

]−

1

2

12µ2

π2ρ2. (71)

By a wave function renormalization, we can cast the above expression into canonical form andwe find that the mass of the Higgs field is given by

mρ = 41, (72)

which means that the mass of the Higgs is twice the fermionic excitation energy. This isanalogous to the result obtained in the chiral sector, where the masses of the mesons turn tobe equal to twice the effective mass of the quarks (see [20]).

From the expression of the mass of the Higgs, it seems unlikely that an NG boson couldexcite the ρ mode. The reason is that the scale of the field φ is T and in compact stars T �1.Even if thermal NG bosons in general cannot excite the Higgs, for vortex–NG boson interactionone has to properly take into account the fact that as one moves inside a vortex, the actual valueof the mass of the Higgs field should decrease. The reason is that as one moves inside a vortex,the value of the condensate becomes smaller and smaller and the mass of the ρ field shoulddecrease accordingly. Therefore, at a certain point, it should happen that mρ < T and it maybecome possible for an NG boson to excite the ρ mode. However, a vortex is a modulation ofthe ρ field itself, and therefore the discussion on the interaction between NG bosons and Higgsfield in a vortex is subtle. We postpone to future work the analysis of this situation.

We now try to give a general expression of the interaction terms between two NG bosonsand the Higgs fields. Let us first neglect space variations of the φ field. Then, expanding theeffective action, one has terms such as

L(ρ, ∂0φ)=

∑n>2

cn(µ− ∂0φ)

2ρn

1n−2, (73)

where cn are some dimensionless coefficients. This expression is simply due to the fact thatany term in the effective Lagrangian is multiplied by µ2, which comes from the phase-spaceintegration, while the denominator comes from dimensional analysis.

Now we want to derive the expression of the terms with space derivatives of the NGbosons. We know from equation (66) that for n = 2 one has to replace (µ− ∂0φ)

2 with(µ− ∂0φ)

2− 1/3(∇φ)2, where the coefficient 1/3 is precisely the square of the speed of sound.

In other words, the correct metric for the propagation of NG bosons is the acoustic metric,gµν = diag(1,−1/3,−1/3,−1/3). Then, we introduce the four vector Xµ

= (µ− ∂0φ,∇φ)

and our guess is that the Lagrangian at the leading order in µ is given by

L(ρ, Xµ)=

∑n>2

cnXµX νgµνρn

1n−2. (74)

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17

Figure 10. Interaction of three Higgs fields. Full lines correspond to fermionicfields, dashed lines correspond to the Higgs fields.

The reason is that any perturbation propagating in the medium feels the presence of thebackground, which induces the acoustic metric gµν [30, 31]. We shall investigate, in moredetail, this assumption in future work; however, it has already been shown in [31] that thefree Lagrangian of NG bosons in the CFL phase can be written employing the acoustic metric.Therefore, equation (74) seems to be an educated guess.

If the expression above is correct, it allows us to readily determine the interaction betweenany number of Higgs fields with two NG bosons in a straightforward way. As an example, wecan determine the interaction between three Higgs field and two NG bosons. We first evaluatethe diagram in figure 10, which gives the interaction among three Higgs field

Lρρρ = −2µ2

π 21ρ3, (75)

then, on replacing µ2→ XµX νgµν , we obtain the LO interaction Lagrangian,

Lρρρφφ = −2XµX νgµνπ 21

ρ3. (76)

It is not clear as to whether this reasoning can be extended to include subleading corrections ofthe order 1/µ. In principle, one would expect that the leading effect should be to perturb themetric gµν , but further investigation in this direction is needed.

5. Conclusions

The low-energy properties of cold and dense CFL quark matter are determined by the NGbosons associated with the breaking of the U (1)B symmetry. At the very low temperaturesexpected in compact stars, the NG bosons probably give the leading contribution to thetransport properties of the system. Therefore, detailed knowledge of their effective Lagrangianis important to precisely determine the transport coefficients. We have determined the effectiveLagrangian for this field starting from a microscopic theory. As a high-energy theory we haveconsidered the NJL model with a local four-Fermi interaction with the quantum numbers of onegluon exchange. Then we have gauged the U (1)B symmetry introducing a fictitious gauge field.Finally, we have integrated the fermionic degrees of freedom by means of the HDET.

We have confirmed the results of [23], which were based on symmetry arguments, andextended including next to leading terms of order (1/µ)2. These corrections are relevant

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18

because they are related to the breaking of the conformal symmetry of the system and canbe significantly large for matter at non-asymptotic densities. For this reason, this work paves theway for a more detailed calculation of the transport properties of CFL quark matter, includingthe scale breaking effects.

We have also determined the interaction of the NG bosons with the Higgs mode, i.e. withthe collective mode associated with fluctuations of |1|. These interactions are relevant in thecalculation of the interaction of the NG bosons with vortices. Indeed, vortices can be describedas space modulation of |1|. A preliminary study of the mutual friction force in the CFL phasehas been done in [32]. However, in that calculation of the mutual friction force, only the elasticscattering of NG bosons on vortices has been taken into account. Since we have determined thefull low-energy Lagrangian, we are now in a position to evaluate, in more detail, the interactionof NG bosons with vortices, including non-elastic scattering; however, we postpone the analysisof vortex–phonon interaction to future work. Indeed, the treatment of this interaction is non-trivial, moreover superfluid vortices that wind the U (1)B are topologically stable but accordingto the result of the Ginzburg–Landau analysis of [33] they are dynamically unstable.

Acknowledgments

We thank H Abuki, M Alford, C Manuel and M Nitta for comments and suggestions. Thiswork has been supported in part by the INFN-MICINN grant with reference number FPA2008-03918E. The work of RA was supported in part by the US Department of Energy, Officeof Nuclear Physics, contract no. DE-AC02-06CH11357. The work of MM was supported bythe Centro Nacional de Física de Partículas, Astropartículas y Nuclear (CPAN) and by theMinisterio de Educación y Ciencia (MEC) under grant nos FPA2007-66665 and 2009SGR502.The work of MR was supported by JSPS under contract number P09028.

Appendix A. Interaction vertices

In the HDET, fermions interact with the gauge field by the following vertices. The minimalcoupling is due to the vertex

01 =

(−V · A 0

0 V · A

). (A.1)

Non-minimal couplings are due to the expansion of the non-local interaction in equation (21).Expanding in the number of gauge fields one has the term with two gauge fields,

02 =Pµν AµAν

L0

(−2µ+ V · ` −1

−1 2µ+ V · `

), (A.2)

three gauge fields,

03 = −02

L0[V · A(2µ+ V · `)+ V · A(2µ− V · `)] −

01

L0Pµν AµAν, (A.3)

and finally the interaction with four gauge fields,

04 =03

L0[V · A(2µ+ V · `)+ V · A(2µ− V · `)]. (A.4)

In this paper, we do not consider interactions with more than four gauge fields.

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19

Appendix B. Integrals

In this appendix, we evaluate some integrals used in the evaluation of the effective Lagrangiansfor the NG boson and for the Higgs field. For the integrals in `0 and `‖, we define∫

d2`≡

∫ +µ

−µ

d`‖

∫ +∞

−∞

d`0. (B.1)

We have ∫d2`

1

D(`)n+1= (−1)n+1 iπ

n12n, for n > 1 (B.2)

∫d2`

(V · `)2

D(`)n=

∫d2`

(V · `)2

D(`)n= −iπ δn1, (B.3)

where D(`)= V · `V · `−12 + iε. We also need the integrals∫d2`

1

L0(`)= −iπ log

(µ+

õ2 +12

3µ+√

9µ2 +12

), (B.4)

∫d2`

1

L0(`)2= iπ

µ

212

(1√

µ2 +12−

3√9µ2 +12

), (B.5)

∫d2`

1

L0(`)3= −iπ

µ

812

(2µ2 + 312

(µ2 +12)3/2−

9(6µ2 +12)

(9µ2 +12)3/2

), (B.6)

where

L0 = (2µ+ V · `)(−2µ+ V · `)−12− iε. (B.7)

B.1. Angular integrals

Considering a general vector Aµ, for the integrals involving terms of order A we have

H0 =

∫dv4π

V · A = A0. (B.8)

At the order A2 we have

H1 =

∫dv4π

Pµν AµAν = −2

3A2, (B.9)

H2 =

∫dv4π(V · A)2 = A2

0 +1

3A2, (B.10)

H3 =

∫dv4π

V · AV · A = A20 −

1

3A2. (B.11)

At the order A3, we have

H4 =

∫dv4π

Pµν AµAνV · A = −2

3A0A2, (B.12)

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20

H5 =

∫dv4π(V · A)3 = A3

0 + A0A2, (B.13)

H6 =

∫dv4π(V · A)(V · A)2 = A3

0 −1

3A0A2. (B.14)

At the order A4, we have

H7 =

∫dv4π(V · A)4 = A4

0 + 2A20A2 +

1

5A4, (B.15)

H8 =

∫dv4π(V · A)(V · A)3 = A4

0 −1

5A2

0A2, (B.16)

H9 =

∫dv4π(V · A)2(V · A)2 = A4

0 −2

3A2

0A2 +1

5A4, (B.17)

H10 =

∫dv4π

Pµν AµAν(V · A)2 = −2

3A2

0A2−

2

15A4, (B.18)

H11 =

∫dv4π

Pµν AµAνPρσ Aρ Aσ =8

15A4, (B.19)

H12 =

∫dv4π

Pµν AµAν V · AV · A = −2

3A2

0A2 +2

15A4. (B.20)

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