Pseudoscalar Goldstone bosons in the color-flavor locked phase at moderate densities

Pseudoscalar Goldstone bosons in the color-flavor locked phase at moderate densities

Verena Kleinhaus,1 Michael Buballa,1 Dominik Nickel,1 and Micaela Oertel21Institut fur Kernphysik, Technische Universitat Darmstadt, Germany

2LUTH, Observatoire de Paris, CNRS, Universite Paris Diderot, 5 place Jules Janssen, 92195 Meudon, France(Received 17 July 2007; published 18 October 2007)

The properties of the pseudoscalar Goldstone bosons in the color-flavor locked phase at moderatedensities are studied within a model of the Nambu-Jona-Lasinio type. The Goldstone bosons areconstructed explicitly by solving the Bethe-Salpeter equation for quark-quark scattering in random phaseapproximation. The main focuses of our investigations are (i) the weak decay constant in the chiral limit,(ii) the masses of the flavored (pseudo)Goldstone bosons for nonzero but equal quark masses, (iii) theirmasses and effective chemical potentials for nonequal quark masses, and (iv) the onset of kaoncondensation. We compare our results with the predictions of the low-energy effective field theory.The deviations from results obtained in the weak-coupling limit are discussed in detail.

DOI: 10.1103/PhysRevD.76.074024 PACS numbers: 12.39.�x, 11.10.St, 11.30.Qc, 12.38.�t

I. INTRODUCTION

Much effort has recently been devoted to the study ofstrongly interacting matter at nonzero baryon density. Inparticular, the rich phase structure of color superconduct-ing quark matter has attracted much interest. (For reviewson color superconductivity see, e.g., Refs. [1–8].) In na-ture, quark matter phases might be realized in compactstars [9–11]. It is therefore natural to ask whether quarkpairing has interesting phenomenological consequencesfor compact star physics. In this context the energeticallylowest lying degrees of freedom are relevant for manydynamic properties of quark matter.

At low temperatures and very high densities, the pre-ferred state is most probably the color-flavor locked (CFL)phase where up, down, and strange quarks are paired in aparticularly symmetric way [12]. This can be shown fromfirst principles within a weak-coupling expansion [13–15].Although this expansion is not valid at ‘‘moderate’’ den-sities which could be reached in compact stars, recentDyson-Schwinger studies indicate that the CFL phasemight be the preferred phase all the way down to thehadronic phase [16].

In the CFL phase, all quark flavors and colors participatein a condensate. As a consequence, all fermionic modes aregapped and do not appear in the low-energy excitationspectrum. The diquark condensates break the originalU�1�baryon � SU�3�color � SU�3�L � SU�3�R symmetry ofthree-flavor QCD (in the chiral limit) down to a residualZ2 � SU�3�color�V , corresponding to a simultaneous(‘‘locked’’) rotation in color and flavor space. Because ofthe breaking of the color symmetry, all eight gluons receivea mass, while the breaking of baryon number and chiralsymmetry leads to the emergence of one scalar and eightpseudoscalar Goldstone bosons. In addition, there is a ninthpseudoscalar Goldstone boson related to the spontaneousbreaking of UA�1� which is a symmetry of QCD at veryhigh density [17,18]. In the presence of quark masses chiralsymmetry is broken explicitly and the pseudoscalar

Goldstone bosons acquire a mass, while the scalarGoldstone boson remains massless. Since, with all quarksbeing gapped, the Goldstone bosons are the lowest lyingexcitations, they play an important role for the thermody-namic and transport properties of strongly interacting mat-ter, relevant for compact star phenomenology (cf., e.g.,Refs. [19–21]).

The symmetry breaking pattern is the basis for theconstruction of the low-energy effective theory (LEET)[22–28], which describes the Goldstone-boson dynamicsand is valid for energies much smaller than the super-conducting gap. At very high densities, the interaction isweak and the constants for the LEET can be calculatedfrom QCD using high density effective theory (HDET)[26,29–31]. For instance, in the weak-coupling limit, pseu-doscalar meson masses and decay constants have beeninvestigated [23,30,32–35]. It was also shown that thestress imposed by the strange quark mass on the CFLCooper pairs acts as an effective strangeness chemicalpotential, which may eventually lead to kaon condensation[24,26,27].

At intermediate densities, relevant for compact star phe-nomenology, the interaction becomes nonperturbative andit is difficult to study the Goldstone-boson dynamics fromfirst principles. The leading-order predictions, however,are often universal, in the sense that they do not dependon the interaction, but should hold in any model exhibitingthe same symmetry pattern. One such model is theNambu–Jona-Lasinio (NJL) model [36], often used inthe intermediate-density regime to study at least qualita-tively the main features. (For reviews, see, e.g.,Refs. [5,37–39].) This model has already been applied tostudy kaon condensation in the CFL phase at nonzerostrange quark masses [40– 42]. However, this was doneby focusing on the ground state properties, i.e., withoutexplicit construction of the Goldstone bosons. InRefs. [43,44], on the other hand, meson and diquark prop-erties in the CFL phase have been studied explicitly, butthis investigation was restricted to the chiral limit. (Mesons

PHYSICAL REVIEW D 76, 074024 (2007)

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and diquarks in the 2SC phase have been discussed inRefs. [45,46].)

In the present paper we discuss a detailed analysis ofproperties of pseudoscalar mesons1 in the CFL phase in anNJL-type model including the cases of equal and unequalnonzero quark masses. Emphasis is put on a comparisonwith the weak-coupling results.

The paper is organized as follows. In Sec. II we intro-duce our model and discuss how to construct the mesonicexcitations. Section III is devoted to some general resultswhich can be obtained in the limit of equal quark massesbased on chiral Ward-Takahashi identities. In Sec. IV nu-merical results will be presented. Within that section, weinvestigate the pion decay constant in the chiral limit aswell as meson masses for the cases of equal and unequalquark masses. In this context we also discuss the onset ofkaon condensation in the CFL phase. Our results aresummarized in Sec. V.

II. FORMALISM

A. Model Lagrangian

We consider an NJL-type Lagrangian,

L � �q�i@6 � m�q�Lqq; (1)

where q is a quark field with three flavor and three colordegrees of freedom, m � diagf�mu;md;ms� is the massmatrix, and

L qq � HX

A;A0�2;5;7

�� qi�5�A�A0C �qT��qTCi�5�A�A0q�

� � �q�A�A0C �qT��qTC�A�A0q�� (2)

describes an SU�3�color �U�3�L �U�3�R symmetric four-point interaction with a dimensionful coupling constant H.C � i�2�0 is the matrix of charge conjugation, and � and �denote Gell-Mann matrices acting in flavor space and colorspace, respectively. In this article, we follow the conven-tion that the indices A and A0 are used for the antisym-metric Gell-Mann matrices only, i.e., A, A0 2 f2; 5; 7g,whereas arbitrary Gell-Mann matrices will be denoted bysmall Latin letters, e.g., �a, a � 1; . . . ; 8.

The first term in Eq. (2) corresponds to a scalar quark-quark interaction in the color and flavor antitriplet channel,just as needed for giving rise to the diquark condensates inthe CFL phase [see Eq. (12) below]. The second term is thecorresponding pseudoscalar interaction and is required bychiral symmetry. This will be the essential term for thepseudoscalar Goldstone excitations we want to study.

For simplicity, we restrict ourselves to quark-quark in-teractions. The effect of quark-antiquark interactions,which give rise to normal self-energies and thereby to

dynamical quark masses, will be investigated in a futurepublication.

B. Operators in Nambu-Gorkov space

Introducing Nambu-Gorkov bispinors,

� �1��2p

qC �qT

� �; (3)

Eq. (2) can be rewritten as

L qq � 4HX

A;A0�2;5;7

f ��s"AA0��s#AA0�

� ��p"AA0��p#AA0�g; (4)

with 18 scalar operators,

�s"AA0 �0 i�5�A�A00 0

� �; �s#AA0 �

0 0i�5�A�A0 0

� �;

(5)

and 18 pseudoscalar operators,

�p"AA0 �0 �A�A00 0

� �; �p#AA0 �

0 0�A�A0 0

� �: (6)

From these expressions we obtain the scattering kernel

K � �iKij ��j; (7)

where �i are the 36 operators defined above,

�� i � �0�yi �0; (8)

and

Kij � 4H�ij: (9)

Repeated operator indices are summed over, unless statedotherwise.

Vertices describing the coupling of an external source toa bare quark are generalized to Nambu-Gorkov (NG) spacein the following way:

�! ��NG � 00 �C�TC

!: (10)

This guarantees that quark-antiquark bilinears remain un-changed, ��NG� � �q � q.

C. CFL ground state

Before we construct the mesonic excitations, we have todetermine the ground state of the system. For equal quarkmasses the CFL phase can be characterized by the equalityof three scalar diquark condensates in the color and flavorantitriplet channel,

s22 � s55 � s77; (11)

where

sAA0 � hqTC�5�A�A0qi: (12)1In this article, we often use the words ‘‘mesons’’ and ‘‘di-

quarks’’ synonymously, see Sec. II E for more details.

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In general, these condensates are accompanied by inducedcolor-flavor sextet condensates. These are, however, smalland can be neglected. If the SU�3�-flavor symmetry isexplicitly broken by unequal quark masses, Eq. (11) doesno longer hold exactly, but the three condensates maydiffer from each other.

To obtain the ground state, we must minimize the ther-modynamic potential (per volume V),

��T; f�ig� � �TV

lnZ�T; f�ig�; (13)

where Z�T; f�ig� is the grand partition function at tem-perature T and a given set of chemical potentials f�ig. For� equilibrated matter, these can be expressed in terms ofthe quark number chemical potential �, the electric chargechemical potential �Q, and two color chemical potentials�3 and �8 [47].

For the CFL phase in mean-field approximation, � isgiven by

��T; f�ig� � �TXn

Z d3p

�2��31

2Tr ln

�1

TS�1�i!n; ~p�

�

�1

4H

XA�2;5;7

j�Aj2; (14)

where the gap parameters �A are related to the diquarkcondensates,

�A � �2HsAA: (15)

The inverse dressed propagator reads

S�1�p� �p6 � ��0 � m

PA�2;5;7

�A�5�A�A

�P

A�2;5;7�A�5�A�A p6 � ��0 � m

0B@1CA:(16)

Here � denotes the diagonal matrix in color and flavorspace which is given by the set of chemical potentials f�ig.

In order to determine the ground state, � must beminimized with respect to the gap parameters �A, leadingto three gap equations:

@�

@�A� 0; A � 2; 5; 7: (17)

Furthermore, we require the solutions to be color andelectrically neutral in the presence of leptons. In general,this leads to three additional equations:

ni �@�tot

@�i� 0; i � Q; 3; 8; (18)

where �tot � ��leptons is the sum of the quark part,Eq. (14), and the contribution of the leptons. Thus, alto-gether we have a set of six coupled equations for �A and�iwhich must be solved simultaneously. This has been donemany times before, and we can refer to the literature fortechnical details, e.g., Refs. [47–50]. In the present article,we restrict ourselves to the (fully gapped) CFL phase atzero temperature. In this case, color neutral quark matter iselectrically neutral without leptons [51] and �Q � 0.Moreover, we consider isospin symmetry, mu � md, sothat �3 vanishes as well.

The gap equations (17) can be derived from the Dysonequation for the dressed quark propagator, too, diagram-matically shown in Fig. 1. This is well known, but somedetails are useful in our later discussion. Therefore, wepresent this derivation in Appendix A.

D. Axial transformations

In the chiral limit, mu � md � ms � 0, the Lagrangian,Eq. (1), is invariant under SU�3�color �U�3�L �U�3�Rtransformations. In the CFL phase, this symmetry is spon-taneously broken to the diagonal vector subgroup,SU�3�color�V . This is reflected by nonzero values for thecondensates, Eq. (11). Axial transformations,

q! q0 � exp�i�a�5ta�q; (19)

then connect a continuous set of degenerate ground statesfor the CFL phase. Here ta �

�a2 , a � 0; . . . ; 8, where

�1; . . . ; �8 denote the eight Gell-Mann matrices in flavorspace, as before, and �0 �

��2=3

p1f. These transformations

can be parametrized by nine pseudoscalar Goldstone bo-sons. In order to identify the operators corresponding toGoldstone-boson excitations, we inspect the effect of anaxial transformation with specified quantum numbers onthe condensates, Eq. (11). In Nambu-Gorkov formalismthis can be described as follows:

h ��i�i ! h ��0�i�0i h ��0i�i; (20)

for �i � �s"AA and �s#AA. Considering infinitesimal transfor-mations, we find

�0i � �i � i�a��i;a; (21)

with

FIG. 1. Dyson equation for the dressed Nambu-Gorkov quark propagator (thick line). The thin line indicates to the bare propagator.

PSEUDOSCALAR GOLDSTONE BOSONS IN THE COLOR- . . . PHYSICAL REVIEW D 76, 074024 (2007)

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��i;a � f��5ta�NG;�ig; (22)

where fA;Bg � AB� BA denotes the anticommutator.The Goldstone bosons are coupled to the quarks by

vertices related to ��i;a (see Ref. [32] for a detailed dis-cussion). Thus, by evaluating Eq. (22) for �i � �s"AA and�s#AA, we can determine the operators which contribute tothe vertex of a given Goldstone boson. It turns out that theassignment is most simple if we work in the ‘‘particlebasis’’ instead of using Hermitian flavor operators. In thiscase all flavored mesons are coupled to only two Nambu-Gorkov operators, while the hidden-flavor mesons arecoupled to four (�0) or six �8; 0� operators. The resultsare summarized in Table I. As we will see below, thisassignment remains basically unchanged when we includefinite quark masses.

E. Mesonic excitations

Iterating the scattering kernel as illustrated in Fig. 2leads to the RPA equation for the T-matrix in Nambu-Gorkov space,

T � K � K J T : (23)

Since the interaction is separable, this operator equationcan be converted into a matrix equation by using Eq. (7) forthe scattering kernel K, an analogous expression for T,

T � �iTij ��j; (24)

and defining

�� iJ�j � Jij: (25)

One finds

T � K � KJT; (26)

with the solution

T�q� � �1� KJ�q��1K ��

1

4H� J�q�

��1; (27)

where the last equality follows from Eq. (9). The polariza-tion matrix, corresponding to the loop in Fig. 2, is given by

Jij�q� � iZ d4k

�2��41

2Tr� ��iS�k� q��jS�k��: (28)

Here we have introduced a ‘‘vacuumlike’’ notation forbrevity. In medium we should replace

q!i!m

~q

� �; k!

i!n~k

� �; (29)

and

iZ d4k

�2��4! �T

Xn

Z d3k

�2��3(30)

with bosonic Matsubara frequencies i!m and fermionicMatsubara frequencies i!n. In the end, the result shouldbe analytically continued to real external energies. We willuse this notation throughout this paper. For our numericalresults, we will introduce a 3-momentum cutoff � toregularize the divergent integrals.

The matrices T and J are 36� 36 matrices in operatorspace, corresponding to the 36 operators defined in Eqs. (5)and (6). It turns out, however, that J and, thus, T are blockdiagonal, i.e., only certain combinations of operators oc-cur. More precisely, scalar operators do not mix withpseudoscalar ones, and each of the resulting 18� 18blocks can be decomposed further into six 2� 2 blocksand one 6� 6 block. These blocks carry different quantumnumbers and reflect the assignment to different mesonmodes, as given in Table I.2 In particular, the 2� 2 blockscorrespond to the flavored mesons. The 6� 6 blocks, onthe other hand, describe the hidden-flavor mesons, i.e., �0,8, and 0 in the pseudoscalar case. As in vacuum, thesestates are mixed for unequal quark masses.

This assignment can be tested by coupling the T-matrixto an external meson source, as illustrated in Fig. 3. To thatend, we evaluate the loop integral

Iij�q� � iZ d4k

�2��41

2Tr� ��iS�k� q��ext

j �NGS�k��: (31)

where �extj denotes the vertex of the external source to the

TABLE I. Pseudoscalar Goldstone modes, corresponding fla-vor operators tj, and Nambu-Gorkov operators �j contributing tothe quark-meson vertex. The �j are obtained by evaluatingEq. (22) for ta � tj and all possible �i � �s"AA or �s#AA.

‘‘meson’’ tj �j

�� 1�i�2

2��2p �p"57, �p#75

�� 1�i�2

2��2p �p"75, �p#57

K� �4�i�5

2��2p �p"27, �p#72

K� �4�i�5

2��2p �p"72, �p#27

K0 �6�i�7

2��2p �p"25, �p#52

�K0 �6�i�7

2��2p �p"52, �p#25

�0 �3

2 �p"55, �p"77, �p#55, �p#77

8�8

2�0

2

� ��p"22; �p"55; �p"77

�p#22; �p#55; �p#77

�0

FIG. 2. RPA equation for the T-matrix in Nambu-Gorkovspace.

2In Table I we have listed the pseudoscalar mesons only, butthe scalar sector is completely analogous.


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quark. It is given by �extj � tj for scalar sources and �ext

j �

i�5tj for pseudoscalar sources. Here tj is one of the gen-erators in flavor space listed in Table I, and ��ext

j �NG is thegeneralization of this vertex to Nambu-Gorkov space, asdefined in Eq. (10).

The blocks can be diagonalized by unitary transforma-tions,

J0 � WJWy; (32)

with W being a (in general 4-momentum dependent) uni-tary matrix and

J0ij � J�i��ij: (33)

Defining new operators

�0j � �iWyij; (34)

J0 can be rewritten as

J0ij�q� � iZ d4k

�2��41

2Tr� ��0iS�k� q��

0jS�k��: (35)

Since the scattering kernel remains diagonal in this newbasis,

K � 4H�i ��i � 4H�0i��0i; (36)

the T-matrix becomes diagonal as well

T � �0iT�i� ��0i; (37)

with

T�i��q� �1

14H� J

�i��q�: (38)

In the vicinity of a pole, we can parametrize these modeslike a free boson with mass mi in the presence of achemical potential �i corresponding to this particular bo-son,

T�i��q� ��g2

i

�q0 ��i�2 � c2

i ~q2 �m2

i

: (39)

Here gi can be interpreted as a coupling constant of theboson to an external quark, and ci denotes the in-mediumgroup velocity. In this article we restrict ourselves to ~q � 0in order to keep artifacts of the 3-momentum cutoff assmall as possible.

The various modes T�i� describe bosonic excitations ofthe CFL ground state. Because of the formal quark-

antiquark structure of the polarization loops, we will callthese excitations mesons. However, it should be kept inmind that the propagators and vertices entering the polar-ization loops live in Nambu-Gorkov space and therefore inprinciple describe quark-antiquark as well as diquark andantidiquark (or di-hole) excitations. In vacuum or in anormal-conducting medium, these are independent modes,protected by the conserved baryon number. In the CFLphase, however, baryon number is broken, and quark-antiquark, diquark, and antidiquark states can mix. Asour model Lagrangian does not contain quark-antiquarkinteractions, our mesons are in fact superpositions of di-quarks and antidiquarks or, more important, di-holes.

In vacuum we have nine scalar and nine pseudoscalardiquarks and nine scalar and nine pseudoscalar antidi-quarks. Since the total number of states does not changewhen the states are mixed, there must be 36 meson states inthe CFL phase, 18 scalars and 18 pseudoscalars. (If weincluded quark-antiquark interactions in our model, wewould obtain 27 scalars and 27 pseudoscalars.) As wewill see below, nine pseudoscalars are massless in thechiral limit, while the others stay heavy. This has beenfound in Ref. [44], too, within a similar model.

F. Pseudoscalar meson decay constants

As in vacuum, the pseudoscalar mesons can decayweakly. The decay amplitudes are related to the loopintegrals

F0�ij �q� � iZ d4k

�2��41

2Tr� ��0iS�k� q��

��5tj�NGS�k��;

(40)

describing the coupling of the meson i (i.e., the one whichcorresponds to the operator �0i) to an external axial currentA�5j.

3 In the following, we are mostly interested in flavoredmesons, which are the main focus of our studies. Eachflavored meson i couples to only one tj, as listed inTable I.4 For simplicity, we denote the flavor operatorwhich fits to the meson i by ti.

The meson decay constants are related to the on-shellvalues of these amplitudes,

fiq� � igiF

0�ii �q�jon-shell; (41)

with no summation over the index i on the right-hand side(rhs). The coupling constant gi has been defined implicitlyin Eq. (39).

FIG. 3. Coupling the T-matrix (double line) to an externalmeson source (wavy line).

3Strictly speaking, Eq. (40) describes the time-reversed pro-cess, i.e., the production of a meson by an incoming axialcurrent. This choice was made for later convenience inSec. III B. There we apply chiral Ward identities, which areformulated for incoming axial currents, see Eq. (52).

4Note that the opposite is not true: Since we have 18 pseudo-scalars, there are in general two meson states i which couple to agiven flavor operator tj.


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In general, there are different values for the timelike(� � 0) and the spacelike (� � 1, 2, 3) decay constants,which differ by the group velocity of the Goldstone modes.However, since we only consider mesons with vanishing 3-momenta in this article, we are restricted to the timelikedecay constants.

III. EQUAL QUARK MASSES

It is rather instructive to investigate the simplified caseof equal quark masses, mu � md � ms m. In this case,we have only one gap parameter �2 � �5 � �7 � andone common chemical potential � in electrically and colorneutral CFL matter. Moreover, the set of the 18 pseudo-scalar meson states consists of two SU�3� octets and twosinglets, with all mesons in the same multiplet beingdegenerate.5 Finally, there is no stress caused by quarkmass or chemical potential differences, which could actas an effective meson chemical potential as in Eq. (39).Hence, for vanishing 3-momenta, the pole approximationfor the T-matrix, Eq. (39), becomes

T�i��q0� ��g2

i

q20 �m

2i

: (42)

This yields for the decay constants

fiq0 � igiF00ii �q�jq0�mi; ~q�0: (43)

We will often refer to the states of the lowest pseudoscalaroctet as ‘‘pions’’ and denote their masses, couplings, anddecay constants by m�, g�, and f�, respectively.

A. Dressed vertex functions

Attaching an external axial current to both sides of theDyson equation for the dressed quark propagator (seeupper line of Fig. 4), we obtain a self-consistency equationfor the dressed axial vertex,

��5j�p; q� � ��5tj�NG � 4iH�iZ d4k

�2��4

�1

2Tr� ��iS�k� q��

�5j�k; q�S�k��: (44)

The first term on the rhs corresponds to the bare vertex,while the second term contains the dressed vertex again.Thus, iterating the equation, the dressed vertex can bewritten as a Born series of quark-antiquark loops (secondline of Fig. 4), with the bare vertex attached to the last loop.Employing the quark-antiquark T-matrix, this could berewritten (last line of Fig. 4) as

��5j�p; q� � ��5tj�NG � T�i��q��0iF0�ij �q�; (45)

with F0�ij as defined in Eq. (40). We have seen earlier that agiven flavor operator tj couples to only two meson modes iin flavored channels. We may call these modes �j and �0j.Then, for ~q � 0 and q0 approaching m�, we obtain

�05j�p; q0 ! m�� i

g�f�q0

q20 �m

2�

�0�j � nonsingular terms:

(46)

Similarly, we have an equation for the dressed pseudo-scalar vertex:

�5j�p; q� � �i�5tj�NG � 4iH�iZ d4k

�2��4

�1

2Tr� ��iS�k� q��5j�k; q�S�k��; (47)

which could be rewritten as

�5j�p; q� � �i�5tj�NG � T�i��q��0iI0ij�q�; (48)

with

I0ij�q� � iZ d4k

�2��41

2Tr� ��0iS�k� q��i�5tj�NGS�k��: (49)

Thus, approaching the pole, we find

�5j�p; q0 ! m�� g2�I��

q20 �m

2�

�0�j � nonsingular terms;

(50)

with

I�� I0��q20 � m2

��: (51)

B. Chiral Ward-Takahashi identity

It can be shown on general grounds that the exact vertexfunctions and inverse propagators must satisfy the follow-ing axial Ward-Takahashi identity:

q��5j�p; q� � 2mi�5;j�p; q� � S�1�p� q��5tj�NG

� ��5tj�NGS�1�p�: (52)

In Appendix B we show by using the gap equation that thevertex functions defined above and the inverse propagator(16) are consistent with this relation.

We now evaluate this equation for ~q � 0 and q0 ! m�.In the chiral limit,m � 0, only the axial vertex contributes,and we find from Eq. (46) that the left-hand side (lhs) isgiven by

q��5j � iq2

0

q20 �m

2�g�f��0�j � nonsingular terms: (53)

In general, this has a singularity at q0 � m�. On the otherhand, one can easily see that the rhs of Eq. (52) remains

5The degeneracy in the octets is due to the residualSU�3�color�V symmetry of the CFL phase. Thus, even thoughwe start from a U�3� invariant Lagrangian in our model, there isno unbroken symmetry which relates the singlet states to theoctets.


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finite. We thus conclude that the singularity on the lhs mustbe suppressed, i.e., either m� � 0 or g�f� � 0. In fact,from the symmetry breaking pattern we expect nine pseu-doscalar Goldstone bosons. Therefore, both scenariosshould be realized, i.e., nine pseudoscalar mesons (oneoctet and one singlet) are massless and the others aremassive.

For the massless solution, we can now evaluate Eq. (52)directly at q � 0. This yields

ig�f��0�0��j � f��5tj�NG; S�1�p�g: (54)

Inserting Eq. (16) for the inverse propagator with m � 0and taking the freedom to choose the gap parameter � to bereal, we find the solution

g�f� � � (55)

with the assignment [cf. Eq. (22)]

�0�0��j � �X

A�2;5;7

f��5tj�NG; ��s"AA � �s#AA�g: (56)

Equation (55) may be viewed as a generalization of thewell-known Goldberger-Treiman relation in vacuum [32].

Explicit evaluation of Eq. (56) yields

�0�0�� i��

2p ��p"57 � �p#75�; �0�0��

�i��2p ��p"75 � �p#57�;

�0�0�K� �i��2p ��p"27 � �p#72�; �0�0�K� �

i��2p ��p"72 � �p#27�;

�0�0�K0 �

�i��2p ��p"25 � �p#52�; �0�0��K0 �

�i��2p ��p"52 � �p#25�;

(57)

for the flavored mesons, and

�0�0��0 � �

i2��p"55 � �p"77 � �p#55 � �p#77�;

�0�0�8 ��i

2��3p �2�p"22 � �p"55 � �p"77 � 2�p#22 � �p#55 � �p#77�;

�0�0�0 � �i

��2

3

s��p"22 � �p"55 � �p"77 � �p#22 � �p#55 � �p#77�;

(58)

for the mesons with hidden flavor. Thus, antiparticle modesare related to each other as

�0�0��j ��0�0��j ; (59)

(with �0, 8, and 0 being their own antiparticles) as itshould be.

The results above are consistent with the operatorsidentified in Ref. [32] following a similar reasoning.Except for an arbitrary phase, Eqs. (57) and (58) also agreewith the vertex structure of nine of the 18 pseudoscalarmodes obtained by diagonalizing the meson polarizationfunction Jij for vanishing quark masses (see Sec. II E). Itfollows that the nine remaining pseudoscalar modes do notcorrespond to Goldstone bosons, but to massive modes, asanticipated before.

We should keep in mind that we have evaluated themeson vertices at the pole only, when we derivedEqs. (57) and (58). In general, the diagonalization of theT-matrix leads to 4-momentum dependent vertex functionswith 4-momentum dependent weights for the contributingoperators. For instance, the �� vertex can be written as

�0��q� � �i�sin’�q��p"57 � cos’�q��p#75�; (60)

with some function ’�q� and an arbitrary overall phasewhich we have chosen to be purely imaginary to complywith Eq. (57). For equal quark masses, ’�0� � �

4 and we

recover �0�j�0� � �0�0��j , Eq. (57).

FIG. 4. Vertex function for an external axial current (wavy line) coupled to a quark. The shaded circles (black dots) indicate dressed(bare) vertices. The open circles correspond to quark-quark vertices and the double line to the T-matrix, as in Fig. 2.


074024-7

Next, we evaluate Eq. (52) for nonvanishing (but stillequal) quark massesm. In this case the pseudoscalar vertexcontributes a pole at ~q � 0 and q0 ! m�, which is presentfor any choice of m� [see Eq. (50)]. Since the rhs ofEq. (52) remains nonsingular, this pole must be canceledby the pole in the axial part (unless g2

�I�� vanishes). This

means that the formerly massless mesons receive a masswhich is determined by the requirement that the residues ofthe axial and the pseudoscalar pole cancel each other. Onefinds

m2�f2

� � 2mg�f�I��: (61)

This is an exact relation, valid for arbitrary values ofm. Wecan now perform a chiral expansion of this equation toleading order. This amounts to replace g�f� by � and toevaluate

I�� iZ d4k

�2��41

2Tr� ��0��q�S�k� q�

� �i�5t��NGS�k��jq0�m�; ~q�0 (62)

to leading nontrivial order in the quark mass. It turns outthat I�� vanishes in the chiral limit and the leading order islinear in m.6 Note that, in principle, I�� depends explicitly(via the quark propagators) and implicitly (via m�) on m.One can show, however, that the implicit contributionsvanish in leading order. It is therefore consistent to evaluatethe integral at q0 � 0, and we obtain

m2�f

2� � 8Am2 � higher orders; (63)

with

A ��

4iZ d4k

�2��4

�ddm

1

2Tr� ��0�0�� S�k��i�5t��NGS�k��

��m�0

(64)

and �0�0�� 0��0� as given in Eq. (57). This expression canbe evaluated exactly. The result is given in Eq. (E1) in theAppendix. Expanding that formula in �, we find

A ��2

8�2

�� lny2 � 2�

4

3ln2� ln�x2 � 1�

�� ;

(65)

where we have introduced the abbreviations

x ��

�; y �

�

�; (66)

with � being the 3-momentum cutoff. This should becompared with the weak-coupling result [23],

Awc �3�2

4�2 : (67)

An important difference is the logarithmic term in Eq. (65),which does not exist in Eq. (67). In fact, in the beginning,there was a controversy about the correct weak-couplinglimit, and similar logarithmic terms have also been dis-cussed in the literature [26,30,32–34,52]. To be precise, itwas found that the leading contribution to A in weakcoupling is given by [30]

A� ��

8�2 lny2; (68)

where �� is the antiparticle gap. In the NJL model, the gapfunction is energy independent and, hence, �� .Therefore, our result, Eq. (65), is consistent withEq. (68). Moreover, if we introduce particle and antipar-ticle gaps by hand as independent constants in the quarkpropagator, we recover Eq. (68) in leading order.

On the other hand, as argued in the erratum of Ref. [23]and confirmed in Ref. [53], in a gauge invariant treatmentof QCD at weak coupling, the antiparticle gap contribu-tions are canceled by other terms, and the logarithmicterms drop out. This finally leads to Eq. (67). Of course,gauge invariance is not an issue in the NJL model.Nevertheless, the logarithmic behavior at very weak cou-pling should be viewed as a model artifact.

IV. NUMERICAL RESULTS

In the following we present our results for T � 0 and afixed quark chemical potential � � 500 MeV. For theseparameters we are in the fully gapped CFL phase for allquark masses used in our calculations.

A. Pion decay constant in the chiral limit

We begin with a discussion of the pion decay constant inthe chiral limit, i.e., for vanishing quark masses. In Fig. 5our results are displayed as functions of the gap parameter� for two different values of the cutoff �. The pointscorrespond to the numerical results for � � 600 MeV(asterisks) and � � 700 MeV (crosses). To be precise,these calculations have been performed with mu � md �ms � 0:1 MeV for technical reasons.

The weak-coupling limit, �! 0, of f� has been derivedin Ref. [23] from an effective theory involving only fermi-onic modes in the vicinity of the Fermi surface. The result,

f2� !

21� 8 ln2

18

�2

2�2 ; (69)

is universal and should hold in any model exhibiting thesame symmetry pattern.7 Indeed, our results converge to

6If we had taken into account dynamical quark masses itwould be linear in the constituent quark mass M. The leadingterm on the rhs of Eq. (63) would then read 8AmM.

7Strictly speaking, the chemical potential � should be re-placed by the Fermi momentum pF because in general theFermi velocity can differ from the speed of light.


074024-8

this limit for �! 0. In the general case we find, however,deviations from Eq. (69), e.g., about 10% for � �600 MeV and � � 80 MeV. Moreover, these deviationsdepend rather strongly on the cutoff.

In order to understand this behavior and to confirm thecorrect weak-coupling limit, we employ the semianalyticalformula derived in Appendix D,

f2� � ~f2

� � �f2�: (70)

~f2� thereby describes the contribution to f� arising for a

constant pion vertex function, i.e., neglecting the energydependence of the mixing angle ’ in Eq. (60). This part isgiven in a closed analytical form in Eq. (D5). �f2

�, on theother hand, incorporates the effect of the energy depen-dence of the vertex function and is given in Eq. (D11). It isproportional to the derivative d’=dq0, which is evaluatednumerically.

Expanding ~f2� and the analytical factor of �f2

� for smallvalues of � yields

~f 2� �

�2

36�2 f�21� 8 ln2� � 9y2 lny2 � c2y2 � � � �g

(71)

and

�f2� �

�2

36�2 f�36y2 lny2 � d2y2 � � � �g�d’dq0

��q�0;

(72)

where

c2 �81

4� 18 ln2� 9 ln�x2 � 1� �

45x2 � 27

�x2 � 1�2;

d2 � 102� 56 ln2� 36 ln�x2 � 1� �36

x2 � 1;

(73)

and x and y are defined in Eq. (66).We see that the weak-coupling limit, Eq. (69), is cor-

rectly reproduced by the leading term in ~f2�, while �f2

�does not contribute to this order, provided d’=dq0 does notdiverge as strongly as �y2 lny2��1 for �! 0. We thereforeexpect that ~f2

� gives the main contribution to f�.As already mentioned, the weak-coupling limit is uni-

versal and therefore must be cutoff independent. In thedetailed calculations, this results from the fact that in thelimit �! 0 the integrand of the 3-momentum integral inEq. (D3) becomes proportional to �-functions at the Fermisurface. While the y2 lny2-term in Eq. (71) is cutoff inde-pendent as well, the quadratic term is not universal anddepends on � via the variable x. This is also the case for thequadratic term in Eq. (72). Anyway, the situation for �f2

� ismore complicated because the derivative d’=dq0 dependson the cutoff as well.

The results of the semianalytical formula for f� areindicated by the thick lines in Fig. 5. The solid line corre-sponds to � � 600 MeV and the dashed line to � �700 MeV. We have employed the exact formulas,Eqs. (D5) and (D11), for ~f2

� and �f2�, respectively, with

d’=dq0 being computed numerically. Obviously, the re-sults for f� are in perfect agreement with the numericalcomputations.

In order to analyze the influence of the momentumdependence of the vertex function, we display the function~f�, too (thin lines). Since ~f� contains the leading term,Eq. (69), it is not surprising that it is the dominant con-tribution to f�. However, for a correct description of thedeviations from the weak-coupling limit, �f2

� can be quiteimportant: Whereas for � � 600 MeV, we find that f� iswell reproduced by ~f�, this is not the case for � �700 MeV. This indicates that the correction due to thecontributions arising from the momentum dependence ofthe vertex function is rather small for � � 600 MeV andconsiderably larger for � � 700 MeV.

This result becomes clear if we look at the derivatived’=dq0 at q � 0, which is shown in Fig. 6. For � �700 MeV (dashed line), the derivative is nowhere smallin the shown region. Therefore, �f2

� can never be ne-glected, in agreement with our findings in Fig. 5. On theother hand, for � � 600 MeV (solid line) we find that thederivative is rather small for � * 40 MeV, explaining why�f2

� is negligible in this regime. For smaller values of � thecorrection becomes larger, in agreement with the devia-tions we found in Fig. 5. In fact, for �! 0, the derivativeeven seems to diverge. If it was stronger than �y2 lny2��1,the divergence could affect the weak-coupling limit.However, at least numerically, we find that d’=dq0 grows

90

92

94

96

98

100

102

104

106

108

0 20 40 60 80 100

f π [

MeV

]

∆ [MeV]

FIG. 5. Pion decay constant f� in the chiral limit as a functionof the gap parameter �: numerical results (points) in comparisonwith the semianalytical formula Eq. (70), neglecting (thin lines)and including (thick lines) the momentum dependence of thevertex function. Asterisks and solid lines: � � 600 MeV;crosses and dashed lines: � � 700 MeV; dotted line: weak-coupling limit, Eq. (69).


074024-9

much slower than �y2 lny2��1, and we therefore concludethat the weak-coupling limit is safe.

B. Equal quark masses

Next, we study the effect of explicit chiral symmetrybreaking through nonvanishing, but equal quark massesmu � md � ms m. For the cutoff, we choose � �600 MeV and keep this value fixed in the remainder ofthis article.

Before turning to our main focus, i.e., the masses of theGoldstone bosons, we briefly investigate the dependence ofthe pion decay constant and of the gap parameter � on m.The results are displayed in Fig. 7. We find that bothquantities depend only very weakly on the quark mass.In the plotted range they vary less than 0.2%. This is muchweaker than the m dependence of f� in vacuum or of the

constituent quark mass in comparable models. In the fol-lowing discussion we will therefore neglect the distinctionbetween the m dependent gap parameter � and its chirallimit ��0� and often use � in order to characterize thecoupling strength.

From Eq. (63), we expect that, to leading order, themasses mM of the Goldstone bosons in the octet dependlinearly on m:

mM �

��8A

f2�

sm: (74)

This is confirmed by our numerical calculations. In Fig. 8the values of mM are displayed as functions of the quarkmass for three different couplingsH�2 � 0:6, 1.0, and 1.4,corresponding to CFL gaps of 12.5 MeV, 43.2 MeV, and79.1 MeV, respectively. As one can see, the results show analmost perfect linear behavior.

The slopes of the straight lines are decreasing withdecreasing coupling strength H, i.e., with decreasing �.This is also expected from the QCD weak-coupling limit,Eqs. (67) and (69) inserted in Eq. (74). However, as dis-cussed in Sec. III B, we expect that the �-dependence ofthe slopes in the NJL model and in weak-coupling QCD israther different. To study this aspect in detail, we determinethe slopes a of the functions mM�m� � am for differentvalues of � and use our (� dependent) results for f� toobtain A � 1

8a2f2

�.The result, divided by the weak-coupling limit Awc, is

displayed in Fig. 9 as a function of �. The ratios obtainedfrom the fit to the numerically determined meson massesare indicated by the crosses. The solid line corresponds toEq. (E1), i.e., to the exact analytical solution of Eq. (64);the dashed line indicates the approximate formula Eq. (65).

0

2

4

6

8

10

12

0 5 10 15 20 25 30

mM

[MeV

]

m [MeV]

FIG. 8. Masses of the flavored (pseudo)Goldstone bosons asfunctions of a common quark mass m for three different cou-plings. Solid: H�2 � 1:4 (corresponding to � � 79:1 MeV);dashed: H�2 � 1:0 (� � 43:2 MeV); dotted: H�2 � 0:6 (� �12:5 MeV).

0.99

0.995

1

1.005

1.01

0 5 10 15 20 25 30

f π/f

π(0) , ∆

/∆(0

)

m [MeV]

FIG. 7. f� (solid line) and � (dashed line) divided by theirrespective chiral limit values f�0�� and ��0� as functions of thecommon quark mass m. The calculations have been performedfor H�2 � 1:4, corresponding to ��0� � 79:1 MeV.

-0.5

0

0.5

1

1.5

2

2.5

3

3.5

4

0 20 40 60 80 100

µ dϕ

/dq 0

∆ [MeV]

FIG. 6. Derivative of the mixing angle ’ as a function of thegap parameter �. Solid: � � 600 MeV; dashed: � � 700 MeV.


074024-10

The former is again in perfect agreement with the numeri-cal results.

We see that for small values of �, A is larger than Awc.This is due to the logarithmic term discussed in Sec. III B[see Eq. (65) and the subsequent discussion]. For largecouplings, on the other hand, the NJL-model value of Ais considerably smaller than Awc, leading to even smallerGoldstone-boson masses than predicted in weak coupling.This has also been found in Ref. [40] within a ratherdifferent approach.

C. Unequal quark masses

Finally, we study the effect of unequal quark masses. Inthe upper panel of Fig. 10 the pole positions of the flavored(pseudo)Goldstone modes are displayed as functions of thestrange quark mass, keeping mu and md fixed. Our numeri-cal results are indicated by the points. For practical rea-sons, we have chosen a relatively strong diquark couplingH�2 � 1:4 (corresponding to � � 79:1 MeV) and a rela-tively large value of 30 MeV for mu and md, in order tohave not too small meson masses. As one can see, the poleswhich are degenerate for equal masses split into threebranches, corresponding to different strangeness, i.e., pions(S � 0), kaons (S � �1), and antikaons (S � �1). On theother hand, since we have chosen mu � md, the differentisospin states of these modes, i.e., �� and ��, K� and K0,and K� and �K0, remain degenerate.

Our results can be analyzed in terms of the parametri-zation given in Eq. (39). Each mode T�i� has two poles,which for ~q � 0 are located at

q0 � mi ��i ! i : (75)

We can thus extract the meson masses and chemical po-tentials as

mi �12�!

�i �!

�i �; �i � �

12�!

�i �!

�i �: (76)

The resulting functions are displayed in the two lowerpanels of Fig. 10. The masses are moderately rising with

-50

-40

-30

-20

-10

0

10

20

30

40

50

40 60 80 100 120 140 160

ωi± [

MeV

]

ms [MeV]

π±

K-, K0

K+, K0

10

12

14

16

18

20

22

24

26

28

40 60 80 100 120 140 160

mi [

MeV

]

ms [MeV]

-25

-20

-15

-10

-5

0

5

10

15

20

25

40 60 80 100 120 140 160

µ i [

MeV

]

ms [MeV]

FIG. 10. Properties of the flavored Goldstone modes as func-tions of the strange quark mass for mu � md � 30 MeV andH�2 � 1:4 (� � 79:1 MeV): pole positions of the T-matrix at~q � 0 (upper panel), meson masses (center), and meson chemi-cal potentials (lower panel). The various points indicate thenumerical results: � (circles), K� and K0 (triangles), and K�

and �K0 (squares). The lines correspond to the predictions fromEqs. (75), (77), and (78).

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

0 10 20 30 40 50 60 70 80

A/A

wc

∆ [MeV]

FIG. 9. Ratio of A and Awc as a function of the gap parameter�. Points: numerical results; solid line: exact analytical resultwith A from Eq. (E1); dashed line: approximate result with Afrom Eq. (65).


074024-11

ms and exhibit an ‘‘inverse ordering’’ (mK <m�), as pre-dicted first in Ref. [23]. Also note that kaons and antikaonshave the same masses. For the meson chemical potentials,on the other hand, we find�K � �� K, as it should be, and�� 0.

These results can be compared with those derived inRefs. [23,26,27] in an effective field theory (EFT) ap-proach. For mu � md mq, they read

m� �

��8A

f2�msmq

s; mK � m �K �

��4A

f2�mq�mq �ms�

s;

(77)

and

�� 0; �K � �K0; �K0 � m2s �m

2q

2�: (78)

For A and f� we insert the NJL-model values obtained inSecs. IVA and IV B in the limit of vanishing quark masses.The resulting functions are indicated by the lines in Fig. 10.Obviously, they agree almost perfectly with the numericalcalculations. Note, however, that the agreement would notbe good if we had used the weak-coupling results for A andf�.

When �i reaches the value of mi,

�i�mq;mcrits � � mi�mq;m

crits �; (79)

!�i vanishes and meson condensation sets in. For thechosen parameters, this occurs in the kaon branch (K�

and K0) at a critical strange quark mass mcrits �

145 MeV. For higher strange quark masses, the CFL phaseis no longer the correct ground state and the shown resultshave no physical meaning.

Inserting the EFT expressions Eqs. (77) and (78) intoEq. (79), one finds that the critical strange quark mass forkaon condensation is approximately given by [26]

mcrits �

�16�2A

f2�

�1=3m1=3q ; (80)

which becomes exact in the limit mq ! 0. For mq � 0 thisimplies that kaon condensation is favored for arbitrarilysmall strange quark masses. One might expect that this isalso the case in our model, after we found good agreementwith the EFT predictions in Fig. 10.

On the other hand, this would contradict an earlier NJL-model study [40], where a nonzero critical strange quarkmass was found, even for mq � 0. This was concluded,without explicit construction of the Goldstone bosons, bystudying the stability of the CFL ground state againstpartially rotating the scalar diquark condensates into pseu-doscalar ones. Thus, with our present approach, we cancheck this result from a different perspective. Since inRef. [40] indications were found that the deviations fromthe EFT predictions are due to terms of higher order in theinteraction, we perform our analysis using a rather strong

diquark coupling, H�2 � 1:7, corresponding to � �107:9 MeV.

In Fig. 11, the critical strange quark mass for kaoncondensation is displayed as a function of the third rootof mq. The NJL-model results are indicated by the crosses.We also show the solution of Eq. (79) for the EFT massesand chemical potentials [Eqs. (77) and (78) (solid line)]and the approximate solution [Eq. (80) (dashed line)].

We find that the NJL points are always above the EFTpredictions (solid line). However, while the deviations aresmall for mq * 1 MeV, they become essential for smallervalues of mq. In particular, we confirm that mcrit

s does notvanish at mq � 0 but goes to a finite value, which is about21 MeV in our example.

To understand this behavior, we analyze the poles of theT-matrix in the kaon channel for mq � 0 as functions ofms. In Fig. 12 the NJL-model results (crosses) are com-pared with the EFT predictions (solid lines). As before, thepole positions (upper panel) can be interpreted in terms ofkaon masses (center) and chemical potentials (lowerpanel). It turns out that the latter are in fair agreementwith Eq. (78). On the other hand, while Eq. (77) predictsthe kaon masses to be zero for mq � 0, we find that mK isin general nonzero and rises linearly with ms. As a con-sequence, the pole position !�K is not degenerate with !�K[cf. Eq. (75)] and first rises with ms. Hence, kaon conden-sation does not occur at arbitrarily small values of ms butonly for ms * 21 MeV, as already seen in Fig. 11.

It should be noted, however, that, although the linear riseof mK with ms is qualitatively different from Eq. (77), theslope is very small on a quantitative scale. In our example,

0

20

40

60

80

100

120

140

0 0.5 1 1.5 2 2.5

mscr

it [M

eV]

mq1/3 [MeV1/3]

FIG. 11. Critical strange quark mass mcrits for kaon condensa-

tion as a function of m1=3q , where mq mu � md: NJL-model

results (crosses), EFT predictions using Eqs. (77) and (78) (solidline), and approximate solution Eq. (80) (dashed line). Thedotted line is based on Eq. (81) with as � 0:0203. The NJL-model calculations have been performed with H�2 � 1:7, cor-responding to � � 107:9 MeV.


074024-12

a linear fit mK � asms yields as � 0:0203, which is anorder of magnitude smaller than the slope in Eq. (77),��

4A=f2�

p� 0:347. Moreover, by varying the coupling

strength, we found numerically that as depends quadrati-cally on �. This is consistent with our expectation that the

effect corresponds to a higher-order correction in the in-teraction and only becomes visible when the leading order,Eq. (77), is artificially suppressed by choosing very smallvalues of mq. In fact, for any realistic value of mu and md

the correction is quite irrelevant.Also note that our results are somewhat complementary

to Ref. [52], where corrections to the effective kaon chemi-cal potential have been discussed. However, it was foundthere that these corrections are of the order of �m2

s=2��2,which is completely negligible in all of our examples.

We may finally combine Eq. (77) with our numericalfindings for mq � 0 by parametrizing

mK �

��4A

f2�mq�mq �ms� � a

2sm

2s

s: (81)

In fact, such a form is allowed in the LEET approach (formq � ms), but a2

s vanishes in the weak-coupling limit[53]. Equating this with the kaon chemical potential,Eq. (78), we obtain the dotted line in Fig. 11 for the criticalstrange quark mass. It is obviously in good agreement withour numerical results. The tiny deviations for small valuesof mq can be explained by the fact that the kaon chemicalpotential in Fig. 12 is slightly overestimated by Eq. (78). Asimilar effect could also play a role at larger mq, but theremight be other terms as well.

V. SUMMARY AND CONCLUSIONS

We have studied the properties of pseudoscalarGoldstone-boson excitations in the color-flavor lockedphase within an NJL-type model. To that end, we solvedthe Bethe-Salpeter equation in RPA. So far we have onlyincluded quark-quark interactions such that our Goldstone-boson states are in fact superpositions of diquark and di-hole states.

Our results are consistent with the model independentpredictions of the low-energy effective theory, i.e., withthose predictions which only depend on the symmetrybreaking pattern. We found, however, deviations from thevalues for the constants appearing in the LEET, as forinstance the pion decay constant f�, obtained in theweak-coupling limit. In fact, it was the main motivationof this paper to locate such deviations and to understandtheir origins. In several cases this could even be doneanalytically. Although these model results are in generalnot universal, they may give important hints about to whatextent the weak-coupling results can be trusted in theintermediate-density regime and where to expect majordeviations.

The weak-coupling limit for f� [23] is correctly repro-duced in the chiral limit in zeroth order in the gap parame-ter �. This must be the case since this result is universal,i.e., independent of the specific choice of the interaction.For � not being small we found deviations, typically of theorder of a few percent. We have shown that this is an effect

-1.2

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0 5 10 15 20 25

ωK± [

MeV

]

ms [MeV]

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0 5 10 15 20 25

mK

[M

eV]

ms [MeV]

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0 5 10 15 20 25

µ K [

MeV

]

ms [MeV]

FIG. 12. Kaon (K� and K0) properties as functions of thestrange quark mass for mu � md � 0 and H�2 � 1:7 (� �107:9 MeV): pole positions of the T-matrix at ~q � 0 (upperpanel), kaon masses (center), and kaon chemical potentials(lower panel). The crosses indicate the numerical results, thelines correspond to the predictions from Eqs. (75), (77), and (78).Note that from Eq. (77) mK is expected to vanish.


074024-13

of higher order in the gap parameter. In this context, themomentum dependence of the dressed vertex functionplays an interesting role.

Next, we discussed the masses of the Goldstone bosonsin the limit of equal quark masses. In agreement with theLEET prediction, we found that the meson masses behavelinearly in the quark masses. However, the correspondingcoefficient A does not agree with the weak-coupling resultobtained in HDET [23]. In the limit �! 0, this should beviewed as an artifact of the NJL model. Probably morerelevant are the deviations at large values of � where wefound the meson masses to be considerably smaller thanpredicted by the weak-coupling formula.

Finally, we have studied the case of unequal quarkmasses. In general, we found a very good agreement withthe LEET prediction for the meson masses and effectivechemical potentials, Eqs. (77) and (78). In particular, wefound mK <m�, i.e., an inverse meson mass ordering aspredicted in Ref. [23]. We also confirmed that the strangequark mass acts as an effective strangeness chemical po-tential, eventually leading to kaon condensation at suffi-ciently large values of ms (Refs. [26,27]). However, evenquantitatively, our model results are in an almost perfectagreement with Eqs. (77) and (78) if the constants A and f�entering Eq. (77) are taken from our preceding NJL-modelstudies in the limit of vanishing quark masses. Since thesevalues do not agree with the weak-coupling limit, as dis-cussed above, our meson masses are in general smallerthan those obtained with the weak-coupling coefficients.

In the limit of vanishing light quark masses, we found aqualitative difference. The critical strange quark mass forthe onset of kaon condensation does not vanish with thecubic root of light quark masses but attains a nonzerovalue. We identified this numerically as a higher-ordereffect on the kaon mass, adding a �2ms dependence atlow light quark masses.

This paper should be seen as the basis for further studiesof the Goldstone-boson dynamics in cold dense quarkmatter at nonasymptotic densities where deviations fromthe weak-coupling limit become visible. As already men-tioned, our simple Lagrangian does not include quark-antiquark interactions, which, although subdominant, cangive important corrections to the pseudoscalar mesonmasses. Moreover, we miss instanton effects, which havebeen shown to be important [28,34]. The ultimate goal is toinclude the backreaction of the Goldstone bosons on thephase structure of color superconducting quark matter [54].In the intermediate-density regime, the ratio of the gapparameter and the Fermi energy is of the order of 0.25, suchthat the Goldstone-boson excitations can have a significanteffect on the ground state properties.

ACKNOWLEDGMENTS

We thank M. Ruggieri and T. Schafer for useful com-ments. This work has been supported in part by the BMBF

under Contract No. 06DA123 and by the Helmholtz-University Young Investigator Grant No. VH-NG-332.

APPENDIX A: GAP EQUATIONS

The Dyson equation shown in Fig. 1 reads

S�p� � S0�p� � S0�p��S�p�: (A1)

Solving for the self-energy, one obtains

� � S�10 �p� � S

�1�p�; (A2)

where

S�10 �p� �

p6 � ��0 � m 00 p6 � ��0 � m

� �(A3)

is the inverse bare quark propagator, while S�1 is theinverse dressed propagator defined in Eq. (16). Thus,

� �

0 �P

A�2;5;7�A�5�A�AP

A�2;5;7�A�5�A�A 0

0B@1CA: (A4)

On the other hand, � can be evaluated diagrammatically. InHartree approximation, it corresponds to the quark loop inFig. 1 and is given by

� � 4iH�iZ d4k

�2��41

2Tr� ��iS�k��: (A5)

Comparing the two expressions for �, we can read off thefollowing gap equations:

�A � 4HZ d4k

�2��41

2Tr��s#AAS�k��: (A6)

We also see that the contributions of scalar vertices withA � A0 and of pseudoscalar vertices should vanish to beconsistent. Using the explicit expression for the dressedpropagator (see Appendix C), it can be shown that this isindeed the case.

APPENDIX B: CHIRAL WARD-TAKAHASHIIDENTITY

In this Appendix, we demonstrate that the dressed vertexfunctions and the dressed quark propagator are consistentwith the chiral Ward-Takahashi identity (WTI) in thesense that, if the WTI holds, we recover the gap equation.As in Sec. III, we restrict ourselves to the case of equalquark masses.


074024-14

From Eqs. (44) and (47) we obtain

q��5j�p;q�� 2mi�5;j�p;q�

� ��q6 � 2m��5tj�NG� 4iH�iZ d4k

�2��4

�1

2Tr� ��iS�k�q��q��5j�k;q�� 2mi�5;j�k;q��S�k��:

(B1)

Hence, imposing the WTI, Eq. (52), one gets

S�1�p� q��5tj�NG � ��5tj�NGS�1�p�

� ��q6 � 2m��5tj�NG � 4iH�iZ d4k

�2��4

�1

2Tr�� i��5tj�NG � ��5tj�NG

��i�S�k��: (B2)

Using Eq. (16) with equal masses and gap parameters, onefinds that the diagonal Nambu-Gorkov components areequal to the first term on the rhs, and one is left with

0 �P

A�2;5;7��At

Tj � tj�A��A

��P

A�2;5;7��Atj � tTj �A��A 0

0B@1CA

� 4iH�iZ d4k

�2��41

2Tr�� i��5tj�NG � ��5tj�NG

��i�S�k��:

(B3)

Next we compute the sums over A on the lhs for any of theflavor operators tj. For instance, for tj � t��

�1�i�2

2��2p we

find that the lhs is equal to 1��2p ��p"57 ��p#75�. We con-

clude that the rhs must vanish for all �i, except for �i �

�p"57 and �i � �p#75. In the first case, we have ��i��5t��NG �

��5t��NG��i � �

i��2p �s#77. Thus by comparison with the lhs

we obtain

� � 4HZ d4k

�2��41

2Tr��s#77S�k��; (B4)

in agreement with one of the gap equations (A6) for thecase of equal masses. The two other equations can bederived analogously, if we evaluate Eq. (B3) for tj � tK�or tj � tK0 . Moreover, the fact that most �i must notcontribute to the rhs for a given tj can be used to showthat scalar operators with A � A0 and pseudoscalar opera-tors do not contribute to the gap equation.

APPENDIX C: DRESSED QUARK PROPAGATOR

1. General case

The dressed quark propagator S�p� is the inverse of theinverse quark propagator, defined in Eq. (16). Followingstandard methods (see, e.g., Refs. [47–50]), we write

S�1�p� S�1�p0; ~p� � �0�p0 � A� ~p��; (C1)

where A� ~p� is a Hermitian 72� 72 matrix, which does notdepend on p0. Thus A can always be diagonalized, i.e., wecan find a unitary matrix U� ~p�, so that

A� ~p� � Uy� ~p�D� ~p�U� ~p�; (C2)

with

D� ~p� �

"1� ~p� 0

. ..

0 "72� ~p�

0BB@

1CCA (C3)

being a diagonal matrix with eigenvalues "1; . . . ; "72. It canbe shown that all eigenvalues are twofold degenerate, andfor each eigenvalue "i, there is a counterpart �"i in thespectrum. This means, there are basically 18 independenteigenvalues. Moreover, part of the diagonalization is trivialbecause the matrix A can be brought into block diagonalform by reordering of lines and columns. The remainingblocks are in general diagonalized numerically.

Combining Eqs. (C1)–(C3), the propagator is finallygiven by

S�p� � Uy� ~p�

1p0�"1� ~p�

0

. ..

0 1p0�"72� ~p�

0BBB@

1CCCAU� ~p��0: (C4)

2. Equal quark masses

In the limit of an exact SU�3� symmetry, we can give aclosed expression for the quark propagator.Straightforward inversion of Eq. (16) for mu � md �ms � m and �22 � �55 � �77 � � yields

S �S11 S12

S21 S22

� �; (C5)

with

S21 � �p6 � �mx�

�5

XA�2;5;7

�A�AS11 (C6)

and

S11 �

�p6 � �m�

j�j2

x��p6 � �m�

XA;A0�2;5;7

�A�A0�A�A0��1;

(C7)

where we have introduced the notations

p� � p� �g�0; x � p2 �m

2: (C8)

S22 and S12 are obtained from S11 and S21, respectively,under the exchange �$ �� and �$ ��.

The matrices Sij are 36� 36 matrices representing thenormal (i � j) and anomalous (i � j) Nambu-Gorkovcomponents of S. S11 can explicitly be written as


074024-15

S11 � S� �1

6�T� � S��

X8

a�0

�a�Ta ; (C9)

with

S �x �p6 � �m� � j�j2�p6 �m�

�p20 � E

�28 ��p

20 � E

�28 �

; (C10)

corresponding to the eigenvalue � of the gap matrix, and

T �x �p6 � �m� � 4j�j2�p6 �m�

�p20 � E

�21 ��p

20 � E

�21 �

(C11)

corresponding to the eigenvalue 2� of the gap matrix. Theoctet and singlet dispersion relations for particles (� ) andantiparticles (� ) are given by

E�8 �

��~p2 �m2

q��2 � j�j2

r(C12)

and

E�1 �

��~p2 �m2

q��2 � 4j�j2

r; (C13)

respectively.

APPENDIX D: PION DECAY CONSTANT IN THECHIRAL LIMIT

In this Appendix we derive a semianalytical expressionfor the pion decay constant in the chiral limit, which is usedin Sec. IVA to discuss the deviations from the weak-coupling limit. The 0 decay constant can be obtained ina similar way but we do not discuss this here.

The starting point is Eq. (43) for the (timelike) decayconstant in the case of equal quark masses. In the chirallimit, we have to evaluate this formula at mi � 0. Sinceboth sides of this equation vanish in this limit, this meansthat the decay constant is given by the derivative of the rhs

with respect to q0. Moreover, we can employ the‘‘Goldberger-Treiman relation,’’ Eq. (55), to eliminatethe coupling constant gi. We then obtain

f2i � ��

�ddq0

Z d4k

�2��41

2Tr� ��0i�q�S�k� q�

� ��0�5ti�NGS�k��q�0

: (D1)

Here we have indicated explicitly that the vertex function�0i depends on the momentum [see Eq. (60)] and is there-fore subject to the derivative. We may thus write

f2i �

~f2i � �f

2i ; (D2)

where

~f2i � ��

Z d4k

�2��41

2Tr�

��0i�0�dS�k� q�dq0

��q�0

� ��0�5ti�NGS�k��

(D3)

corresponds to the contribution where the derivative actson the propagator, while

�f2i � ��

Z d4k

�2��41

2Tr�d ��0i�q�dq0

��q�0S�k�

� ��0�5ti�NGS�k��

(D4)

corresponds to the contribution where the derivative actson the vertex function.

The evaluation of ~f2i is tedious, but straightforward.

Inserting �0i�0� � �0�0�i from Eqs. (57) or (58), respectively,ti from Table I as well as the expressions for the quarkpropagator given in Appendix C 2 in the chiral limit, theresult for the octet mesons (pions) reads

~f2� �

�2

216�2

�2

y2 ��1 � �

�8 ��x� 1��3x2 � 2x� 1� � ��1 � �

�8 ��x� 1��3x2 � 2x� 1�� 8 �9x� 31�

� ��8 �9x� 31� � 16��1 � 16��1 � 45�� 24�ln��1 � x� 1� � ln��1 � x� 1� � ln��8 � x� 1�

� ln��8 � x� 1�� 45�x� 1

��8�x� 1

��8�

1

�

�� 48 ln2� 54y2 lny2 � 54y2�ln��8 � x� 1�

� ln��8 � x� 1�� 45y2

�x� 1

��8�x� 1

��8�

1

�

��; (D5)

where the 3-momentum integral has been regularized by acutoff � (as in the numerical calculations) and we haveintroduced the abbreviations

� 1 ��x 1�2 � 4y2

q; � 8 �

��x 1�2 � y2

q;

(D6)

and

� ��1� y2

q: (D7)

Moreover, x � �� and y � �

� , as defined in Eq. (66).The evaluation of �f2

i is more difficult because we needto know the derivative of the vertex function. For the


074024-16

flavored mesons this is encoded in the mixing angle ’�q�,cf. Eq. (60). We can therefore write

d ��0i�q�dq0

��q�0�d’�q�dq0

��q�0

��0?i ; (D8)

where

�0?i �@�0i@’

��’��=4(D9)

is the vertex of the orthogonal state with the same quantum

numbers as the meson i (e.g., �0?�� i��2p ��p"57 � �p#75�).

Hence,

�f2i � ��

Z d4k

�2��41

2Tr� ��0?i S�k��

0�5ti�NGS�k��

�d’�q�dq0

��q�0: (D10)

Evaluating this part for the octet mesons, one finally ob-tains

�f2� �

�2

108�2

��108y2 lny2 � 48y2 ln2� 2��1 � �

�8 ��x

2 � x� 1� � 2��1 � ��8 ��x

2 � x� 1� � 4y2��8 � ��8 �

� 16y2��1 � ��1 � � 12y2�2 ln��1 � x� 1� � 2 ln��1 � x� 1� � 7 ln��8 � x� 1� � 7 ln��8 � x� 1��

� 45y2

�x2 � 4x� 2

��8�x2 � 4x� 2

��8

�� 90y4

�1

��8�

1

��8

��d’�q�dq0

��q�0:

(D11)

One could also try to derive an analytical expression for the derivative d’�q�dq0jq�0 from the polarization-loop matrix

Eq. (28), but this is beyond the scope of this paper.

APPENDIX E: EXACT FORMULA FOR A

The exact result of Eq. (64) is given by

A ��2

384�2

��48 lny2 � 32 ln2� 4��1 � �

�1 � � 44��8 � �

�8 � � �16� 3y2��ln��1 � x� 1� � ln��1 � x� 1��

� �32� 3y2��ln��8 � x� 1� � ln��8 � x� 1�� 6y2 ln2� 9y2 �2

�3�ln�4�x� 1� � ��1 �2�3 � �3x� 13�y2�

� ln�4�x� 1� � ��1 �2�3 � �3x� 13�y2� � ln�4�x� 1� � ��8 �2�3 � �3x� 7�y2�

� ln�4�x� 1� � ��8 �2�3 � �3x� 7�y2��

�; (E1)

with � i as defined in Eq. (D6),

�2 ��4� y2

q; �3 �

��4� 9y2

q; (E2)

and x � �� and y � �

� , as defined in Eq. (66).

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Pseudoscalar Goldstone bosons in the color-flavor locked phase at moderate densities