Longitudinal gluons and Nambu-Goldstone bosons in a two-flavor color superconductor

PHYSICAL REVIEW D 66, 054019 ~2002!

Longitudinal gluons and Nambu-Goldstone bosons in a two-flavor color superconductor

Dirk H. Rischke*Institut fur Theoretische Physik, Johann Wolfgang Goethe-Universita¨t, Robert-Mayer-Strasse 8–10, D-60054 Frankfurt/Main, Germany

Igor A. Shovkovy†

School of Physics and Astronomy, University of Minnesota, 116 Church Street S.E., Minneapolis, Minnesota 55455~Received 12 June 2002; published 30 September 2002!

In a two-flavor color superconductor, theSU(3)c gauge symmetry is spontaneously broken by diquarkcondensation. The Nambu-Goldstone excitations of the diquark condensate mix with the gluons associatedwith the broken generators of the original gauge group. It is shown how one can decouple these modes with aparticular choice of ’t Hooft gauge. We then explicitly compute the spectral density for transverse and longi-tudinal gluons of adjoint color 8. The Nambu-Goldstone excitations give rise to a singularity in the real part ofthe longitudinal gluon self-energy. This leads to a vanishing gluon spectral density for energies and momentalocated on the dispersion branch of the Nambu-Goldstone excitations.

DOI: 10.1103/PhysRevD.66.054019 PACS number~s!: 12.38.Mh, 24.85.1p

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I. INTRODUCTION

Cold, dense quark matter is a color superconductor@1#.For two massless quark flavors~say, up and down!, Cooperpairs with a total spin zero condense in the color-antitripflavor-singlet channel. In this so-called two-flavor color sperconductor, theSU(3)c gauge symmetry is spontaneousbroken toSU(2)c @2#. If we choose to orient the~anti! colorcharge of the Cooper pair along the~anti! blue direction incolor space, only red and green quarks form Cooper pawhile blue quarks remain unpaired. Then, the three gentorsT1 , T2, andT3 of the originalSU(3)c gauge group formthe generators of the residualSU(2)c symmetry. The remain-ing five generatorsT4 , . . . ,T8 are broken.@More precisely,the last broken generator is a combination ofT8 and thegenerator1 of the globalU(1) symmetry of baryon numbeconservation; for details see Ref.@3# and below.#

According to Goldstone’s theorem, this pattern of symmtry breaking gives rise to five massless bosons, the so-caNambu-Goldstone bosons, corresponding to the five brogenerators ofSU(3)c . Physically, these massless bosocorrespond to fluctuations of the order parameter, in our cthe diquark condensate, in directions in color-flavor spwhere the effective potential is flat. For gauge theor~where the local gauge symmetry cannot truly be spontaously broken!, these bosons are ‘‘eaten’’ by the gauge bosocorresponding to the broken generators of the original gagroup, i.e., in our case the gluons with adjoint colorsa54, . . . ,8. They give rise to a longitudinal degree of fredom for these gauge bosons. The appearance of a longinal degree of freedom is commonly a sign that the gaboson becomes massive.

In a dense~or hot! medium, however, evenwithout spon-taneous breaking of gauge symmetry the gauge boson

*Email address: [email protected]†On leave of absence from Bogolyubov Institute for Theoreti

Physics, 252143 Kiev, Ukraine. Email address:[email protected]

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ready have a longitudinal degree of freedom, the so-caplasmonmode@4#. Its appearance is related to the presenof gapless charged quasiparticles. Both transverse and lotudinal modes exhibit a mass gap, i.e., the gluon enep0→mg.0 for momentap→0. In quark matter withNfmassless quark flavors at zero temperatureT50, the gluonmass parameter~squared! is @4#

mg25

Nf

6p2g2m2, ~1!

whereg is the QCD coupling constant andm is the quarkchemical potential.

It is a priori unclear how the Nambu-Goldstone bosointeract with these longitudinal gluon modes. In particularis of interest to know whether coupling terms between thmodes exist and, if yes, whether these terms can be elnated by a suitable choice of~’t Hooft! gauge. The aim of thepresent work is to address these questions. We shall sthat the answer to both questions is ‘‘yes.’’ We shall thdemonstrate by focusing on the gluon of adjoint color 8 hthe Nambu-Goldstone mode affects the spectral densitythe longitudinal gluon.

Our work is partially based on and motivated by previostudies of gluons in a two-flavor color superconductor@5–7#.The gluon self-energy and the resulting spectral properhave been discussed in Ref.@7#. In that paper, however, thefluctuations of the diquark condensate have been neglecConsequently, the longitudinal degrees of freedom ofgluons corresponding to the broken generators ofSU(3)chave not been treated correctly. The gluon polarization tenwas no longer explicitly transverse~a transverse polarizationtensorPmn obeysPmPmn5PmnPn50), and it did not sat-isfy the Slavnov-Taylor identity. As a consequence, the plmon mode exhibited a certain peculiar behavior in the lomomentum limit, which cannot be physical~cf. Fig. 5~a! ofRef. @7#!. It was already realized in Ref.@7# that the reasonfor this unphysical behavior is the fact that the mixing of tgluon with the excitations of the condensate was neglec

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DIRK H. RISCHKE AND IGOR A. SHOVKOVY PHYSICAL REVIEW D66, 054019 ~2002!

It was moreover suggested in Ref.@7# that proper inclusionof this mixing would amend the shortcomings of the preous analysis. The aim of the present work is to follow thsuggestion and thus to correct the results of Ref.@7# withrespect to the longitudinal gluon. Note that in Ref.@5# fluc-tuations of the color-superconducting condensate were tainto account in the calculation of the gluon polarization tesor. As a consequence, the latter is explicitly transveHowever, the analysis was done in the vacuum, atm50, notat ~asymptotically! large chemical potential.

The outline of the present work is as follows. In Sec.we derive the transverse and longitudinal gluon propagaincluding fluctuations of the diquark condensate. In Sec.we use the resulting expressions to compute the spectralsity for the gluon of adjoint color 8. Section IV concludethis work with a summary of our results.

Our units are\5c5kB51. The metric tensor isgmn

5diag(1,2,2,2). We denote 4-vectors in energymomentum space by capital letters,Km5(k0 ,k). Absolutemagnitudes of 3-vectors are denoted ask[uku, and the unitvector in the direction ofk is k[k/k.

II. DERIVATION OF THE PROPAGATOR FORTRANSVERSE AND LONGITUDINAL GLUONS

In this section we derive the gluon propagator taking inaccount the fluctuations of the diquark condensate. A sversion of this derivation can be found in Appendix C of R@8# ~see also the original Ref.@9#!. Nevertheless, for the sakof clarity and in order to make our presentation secontained, we decide to present this once more in gredetail and in the notation of Ref.@7#. As this part is rathertechnical, the reader less interested in the details of the dvation should skip directly to our main result, Eqs.~56!, ~57!,and ~58!.

We start with the grand partition function of QCD,

Z5E DAeSAZq@A#, ~2a!

where

Zq@A#5E DcDcexpF Exc~ igm]m1mg01ggmAm

a Ta!cG~2b!

is the grand partition function for massless quarks inpresence of a gluon fieldAa

m . In Eq. ~2!, the space-timeintegration is defined as*x[*0

1/Tdt*Vd3x, whereV is thevolume of the system,gm are the Dirac matrices, andTa5la/2 are the generators ofSU(Nc). For QCD,Nc53, andla are the Gell-Mann matrices. The quark fieldsc are4NcNf-component spinors, i.e., they carry Dirac indicesa51, . . . ,4,fundamental color indicesi 51, . . . ,Nc , and fla-vor indices f 51, . . . ,Nf . The action for the gauge fieldconsists of three parts,

SA5SF21Sgf1SFPG, ~3!

where

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SF2521

4ExFa

mnFmna ~4!

is the gauge field part; here,Fmna 5]mAn

a2]nAma

1g fabcAmb An

c is the field strength tensor. The part corrsponding to gauge fixing,Sgf , and to Fadeev-Popov ghostSFPG, will be discussed later.

For fermions at finite chemical potential it is advantgeous to introduce the charge-conjugate degrees of freeexplicitly. This restores the symmetry of the theory undm→2m. Therefore, in Ref.@6#, a kind of replica methodwas applied, in which one first artificially increases the nuber of quark species, and then replaces half of these speof quark fields by charge-conjugate quark fields. More pcisely, first replace the quark partition functionZq@A# byZM@A#[$Zq@A#%M, M being some large integer numbe~SendingM→1 at the end of the calculation reproduces toriginal partition function.! Then, takeM to be an even inte-ger number and replace the quark fields by charge-conjuquark fields inM /2 of the factorsZq@A# in ZM@A#. Thisresults in

ZM@A#5E )r 51

M /2

DC rDC r

3expH (r 51

M /2 F Ex,y

C r~x!G 021~x,y!C r~y!

1ExgC r~x!Am

a ~x!GamC r~x!G J . ~5!

Here, r labels the quark species andC r , C r are8NcNf-component Nambu-Gor’kov spinors,

C r[S c r

cCrD , C r[~c r ,cCr!, ~6!

where cCr[Cc rT is the charge conjugate spinor andC

5 ig2g0 is the charge conjugation matrix. The inverse of t8NcNf38NcNf-dimensional Nambu-Gor’kov propagatofor noninteracting quarks is defined as

G 021[S @G0

1#21 0

0 @G02#21D , ~7!

where

@G06#21~x,y![2 i ~ igm]x

m6mg0!d (4)~x2y! ~8!

is the inverse propagator for noninteracting quarks~uppersign! or charge conjugate quarks~lower sign!, respectively.The Nambu-Gor’kov matrix vertex describing the interactibetween quarks and gauge fields is defined as follows:

Gam[S Ga

m 0

0 GamD , ~9!

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LONGITUDINAL GLUONS AND NAMBU-GOLDSTONE . . . PHYSICAL REVIEW D 66, 054019 ~2002!

whereGam[gmTa and Ga

m[C(gm)TC21TaT[2gmTa

T .Following Ref. @1# we now add the term

*x,ycCr(x)D1(x,y)c r(y) and the corresponding chargconjugate term *x,yc r(x)D2(x,y)cCr(y), where D2

[g0(D1)†g0, to the argument of the exponent in Eq.~5!.This defines the quark~replica! partition function in the pres-ence of the gluon fieldAa

m and the diquark source fieldsD1,D2:

ZM@A,D1,D2#[E )r 51

M /2

DC rDC r

3expH (r 51

M /2 F Ex,y

C r~x!G 21~x,y!C r~y!

1ExgC r~x!Am

a ~x!GamC r~x!G J , ~10!

where

G 21[S @G01#21 D2

D1 @G02#21D ~11!

is the inverse quasiparticle propagator.Inserting the partition function~10! into Eq.~2a!, the~rep-

lica! QCD partition function is then computed in the preence of the ~external! diquark source termsD6(x,y),Z→Z@D1,D2#. In principle, this is not the physically relevant quantity, from which one derives thermodynamic prerties of the color superconductor. The diquark condensanot an external field, but assumes a nonzero value becauan intrinsic property of the system, namely the attractgluon interaction in the color-antitriplet channel, which dstabilizes the Fermi surface.

The proper functional from which one derives thermodnamic functions is obtained by a Legendre transformationln Z@D1,D2#, in which the functional dependence on thdiquark source term is replaced by that on the corresponcanonically conjugate variable, the diquark condensate.Legendre-transformed functional is the effective actionthe diquark condensate. If the latter isconstant, the effectiveaction is, up to a factor ofV/T, identical to the effectivepotential. The effective potential is simply a function of thdiquark condensate. Its explicit form for large-density QCwas derived in Ref.@10#. The value of this function at itsmaximum determines the pressure. The maximum is demined by a Dyson-Schwinger equation for the diquark cdensate, which is identical to the standard gap equationthe color-superconducting gap. It has been solved inmean-field approximation in Refs.@11–13#. In the mean-fieldapproximation@14#,

D1~x,y!;^cCr~x!c r~y!&,

D2~x,y!;^c r~x!cCr~y!&. ~12!

In this work, we are interested in the gluon propagator, athe derivation of the pressure via a Legendre transforma

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of ln Z@D1,D2# is of no concern to us. In the following, wshall therefore continue to consider the partition functionthe presence of~external! diquark source termsD6.

The diquark source terms in the quark~replica! partitionfunction ~10! could in principle be chosen differently foeach quark species. This could be made explicit by givD6 a subscriptr, D6→D r

6 . However, as we take the limiM→1 at the end, it is not necessary to do so, as onlyD1

6

[D6 will survive anyway. In other words, we use thesamediquark sources forall quark species.

The next step is to explicitly investigate the fluctuatioof the diquark condensate around its expectation vaThese fluctuations correspond physically to the NamGoldstone excitations~loosely termed ‘‘mesons’’ in the fol-lowing! in a color superconductor. As mentioned in the Itroduction, there are five such mesons in a two-flavor cosuperconductor, corresponding to the generators ofSU(3)cwhich are broken in the color-superconducting phase. Ifcondensate is chosen to point in the~anti! blue direction infundamental color space, the broken generatorsT4 , . . . ,T7 of the originalSU(3)c group and the particulacombinationB[(11A3T8)/3 of generators of the globaU(1)B and localSU(3)c symmetry@3#.

The effective action for the diquark condensate and, csequently, for the meson fields as fluctuations of the diqucondensate, is derived via a Legendre transformationln Z@D1,D2#. In this work, we are concerned with the proerties of the gluons and thus refrain from computing teffective action explicitly. Consequently, instead of considing the physical meson fields, we consider the variablesZ@D1,D2#, which correspond to these fields. These arefluctuations of the diquark source termsD6. We choosethese fluctuations to be complex phase factors multiplythe magnitude of the source terms,

D1~x,y!5V* ~x!F1~x,y!V †~y!, ~13a!

D2~x,y!5V~x!F2~x,y!V T~y!, ~13b!

where

V~x![expF i S (a54

7

wa~x!Ta11

A3w8~x!BD G . ~14!

The extra factor 1/A3 in front of w8 as compared to thetreatment in Ref.@8# is chosen to simplify the notation in thfollowing.

Although the fieldswa are not the meson fields themselves, but external fields which, after a Legendre transmation of lnZ@D1,D2#, are replaced by the meson fieldwe nevertheless~and somewhat imprecisely! refer to them asmeson fields in the following. After having explicitly introduced the fluctuations of the diquark source terms in termphase factors, the functionsF6 are only allowed to fluctuatein magnitude. For the sake of completeness, let us menthat one could again have introduced different fieldswar foreach replicar, but this is not really necessary, as we shtake the limitM→1 at the end of the calculation anyway.

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It is advantageous to also subject the quark fieldsc r to anonlinear transformation, introducing new fieldsx r via

c r5Vx r , c r5x rV †. ~15!

Since the meson fields are real-valued and the generaT4 , . . . ,T7 andB are Hermitian, the~matrix-valued! opera-tor V is unitary, V 215V †. Therefore the measure of thGrassmann integration over quark fields in Eq.~10! remainsunchanged. From Eq.~15!, the charge-conjugate fields tranform as

cCr5V* xCr , cCr5xCrV T. ~16!

The advantage of transforming the quark fields is thatpreserves the simple structure of the terms couplingquark fields to the diquark sources,

cCr~x!D1~x,y!c r~y![xCr~x!F1~x,y!x r~y!,

c r~x!D2~x,y!cCr~y![x r~x!F2~x,y!xCr~y!. ~17!

In mean-field approximation, the diquark source termsproportional to

F1~x,y!;^xCr~x!x r~y!&,

F2~x,y!;^x r~x!xCr~y!&. ~18!

The transformation~15! has the following effect on thekinetic terms of the quarks and the term coupling quarksgluons:

c r~ igm]m1mg01ggmAamTa!c r

5x r~ igm]m1mg01gmvm!x r , ~19a!

cCr~ igm]m2mg02ggmAamTa

T!cCr

5xCr~ igm]m2mg01gmvCm!xCr , ~19b!

where

vm[V †~ i ]m1g AamTa!V ~20a!

is theNcNf3NcNf-dimensional Maurer-Cartan one-form introduced in Ref.@15# and

vCm[V T~ i ]m2g Aa

mTaT!V* ~20b!

is its charge-conjugate version. Note that the partial dertive acts only on the phase factorsV andV* on the right.

Introducing the Nambu-Gor’kov spinors

Xr[S x r

xCrD , Xr[~x r ,xCr! ~21!

and the 2NcNf32NcNf-dimensional Maurer-Cartan oneform

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Vm~x,y![2 i S vm~x! 0

0 vCm~x!

D d (4)~x2y!, ~22!

the quark~replica! partition function becomes

ZM@V,F1,F2#[E )r 51

M /2

DXrDXr

3expH (r 51

M /2 Ex,y

Xr~x!@S 21~x,y!1gmVm~x,y!#Xr~y!J ,

~23!

where

S 21[S @G01#21 F2

F1 @G02#21D . ~24!

We are interested in the properties of the gluons, and tmay integrate out the fermion fields. This integration canperformed analytically, with the result

ZM@V,F1,F2#[@det~S 211gmVm!#M /2. ~25!

The determinant is to be taken over Nambu-Gor’kov, colflavor, spin, and space-time indices. Finally, lettingM→1,we obtain the QCD partition function~in the presence ofmeson,wa , and diquark,F6, source fields!

Z@w,F1,F2#5E DA expFSA11

2Tr ln~S 211gmVm!G .

~26!

Remembering thatVm is linear inAam , cf. Eq. ~22! with Eq.

~20!, in order to derive the gluon propagator it is sufficientexpand the logarithm to second order inVm,

12 Tr ln~S 211gmVm!. 1

2 Tr lnS 211 12 Tr~SgmVm!

2 14 Tr~SgmVmSgnVn!

[S0@F1,F2#1S1@V,F1,F2#

1S2@V,F1,F2#, ~27!

with obvious definitions for theSi . The quasiparticle propagator is

S[S G1 J2

J1 G2 D , ~28!

with

G65$@G06#212S6%21, S65F7G0

7F6,

J652G07F6G6. ~29!

To make further progress, we now expandvm andvCm to

linear order in the meson fields,

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vm.gAamTa2 (

a54

7

~]mwa!Ta21

A3~]mw8!B, ~30a!

vCm.2gAa

mTaT1 (

a54

7

~]mwa!TaT1

1

A3~]mw8!BT.

~30b!

The termS1 in Eq. ~27! is simply a tadpole source term fothe gluon fields. This term does not affect the gluon progator, and thus can be ignored in the following.

The quadratic termS2 represents the contribution offermion loop to the gluon self-energy. Its computation pceeds by first taking the trace over Nambu-Gor’kov spac

S2521

4Ex,yTrc, f ,s@G1~x,y!gmvm~y!G1~y,x!gnvn~x!

1G2~x,y!gmvCm~y! G2~y,x!gnvC

n ~x!

1J1~x,y!gmvm~y! J2~y,x!gnvCn ~x!

1J2~x,y!gmvCm~y!J1~y,x!gnvn~x!#. ~31!

The remaining trace runs only over color, flavor, and sindices. Using translational invariance, the propagatorsfields are now Fourier transformed as

G6~x,y!5T

V (K

e2 iK •(x2y)G6~K !, ~32a!

J6~x,y!5T

V (K

e2 iK •(x2y)J6~K !,

~32b!

vm~x!5(P

e2 iP•xvm~P!, ~32c!

05401

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-,

nd

vCm~x!5(

Pe2 iP•xvC

m~P!. ~32d!

Inserting this into Eq.~31!, we arrive at Eq.~C16! of Ref.@8#, which in our notation reads

S2521

4 (K,P

Trc, f ,s@G1~K !gmvm~P!G1~K2P!gnvn~2P!

1G2~K !gmvCm~P!G2~K2P!gnvC

n ~2P!

1J1~K !gmvm~P!J2~K2P!gnvCn ~2P!

1J2~K !gmvCm~P!J1~K2P!gnvn~2P!#. ~33!

The remainder of the calculation is straightforward, bsomewhat tedious. First, insert the~Fourier transform of the!linearized version~30! for the fieldsvm andvC

m . This pro-duces a plethora of terms which are second order ingluon and meson fields, with coefficients that are traces ocolor, flavor, and spin. Next, perform the color and flavtraces in these coefficients. It turns out that some of themidentically zero, preventing the occurrence of terms whmix gluons of adjoint colors 1, 2, and 3@the unbrokenSU(2)c subgroup# among themselves and with the othgluon and meson fields. Furthermore, there are no terms ming the meson fieldswa ,a54, . . . 7, with w8. There aremixed terms between gluons and mesons with adjoint coindices 4, . . . ,7, andbetween the gluon fieldA8

m and themeson fieldw8.

Some of the mixed terms~those which mix gluons andmesons of adjoint colors 4 and 5, as well as 6 and 7! can beeliminated via a unitary transformation analogous to the oemployed in Ref.@6#, Eq. ~80!. Introducing the tensors

P11mn~P![P22

mn~P![P33mn~P!5

g2

2

T

V (K

Trs@gmG1~K !gnG1~K2P!1gmG2~K !gnG2~K2P!

1gmJ2~K !gnJ1~K2P!1gmJ1~K !gnJ2~K2P!#, ~34a!

cf. Eq. ~78a! of Ref. @6#,

P44mn~P![P66

mn~P!5g2

2

T

V (K

Trs@gmG01~K !gnG1~K2P!1gmG2~K !gnG0

2~K2P!#, ~34b!

cf. Eq. ~83a! of Ref. @6#,

P55mn~P![P77

mn~P!5g2

2

T

V (K

Trs@gmG1~K !gnG01~K2P!1gmG0

2~K !gnG2~K2P!#, ~34c!

cf. Eq. ~83b! of Ref. @6#, as well as

P88mn~P!5

2

3P0

mn~P!11

3Pmn~P!, ~34d!

9-5


Pmn~P!5g2

2

T

V (K

Trs@gmG1~K !gnG1~K2P!1gmG2~K !gnG2~K2P!

2gmJ2~K !gnJ1~K2P!2gmJ1~K !gnJ2~K2P!#, ~34e!

cf. Eq. ~78c! of Ref. @6#, whereP0mn is the gluon self-energy in a dense, but normal-conducting system,

P0mn~P!5

g2

2

T

V (K

Trs@gmG01~K !gnG0

1~K2P!1gmG02~K !gnG0

2~K2P!#, ~34f!

cf. Eq. ~27b! of Ref. @6#, the final result can be written in the compact form@cf. Eq. ~C19! of Ref. @8##

S2521

2

V

T (P

(a51

8 FAma ~2P!2

i

gPmwa~2P!GPaa

mn~P!FAna~P!1

i

gPnwa~P!G . ~35!

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In deriving Eq.~35!, we have made use of the transversalof the polarization tensor in the normal-conducting phaP0

mn(P)Pn5PmP0mn(P)50. Note that the tensorsPaa

mn fora51,2, and 3 are also transverse, but those fora54, . . . ,8are not. This can be seen explicitly from the expressigiven in Ref.@7#. The compact notation of Eq.~35! is madepossible by the fact thatwa[0 for a51,2,3, and because wintroduced the extra factor 1/A3 in Eq. ~14! as compared toRef. @8#.

To make further progress, it is advantageous to tendecomposePaa

mn . Various ways to do this are possible@8#;here we follow the notation of Ref.@4#. First, define a pro-jector onto the subspace parallel toPm,

Emn5PmPn

P2. ~36!

Then choose a vector orthogonal toPm, for instance

Nm[S p0p2

P2,p0

2p

P2 D [~gmn2Emn! f n , ~37!

with f m5(0,p). Note thatN252p02 p2/P2. Now define the

projectors

Bmn5NmNn

N2, Cmn5NmPn1PmNn,

Amn5gmn2Bmn2Emn. ~38!

Using the explicit form ofNm, one convinces oneself that thtensor Amn projects onto the spatially transverse subsporthogonal toPm,

A005A0i50, Ai j 52~d i j 2 pi p j !. ~39!

~Reference@4# also uses the notationPTmn for Amn.! Conse-

quently, the tensor Bmn projects onto the spatially longitudinal subspace orthogonal toPm,

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B0052p2

P2, B0i52

p0 pi

P2, Bi j 52

p02

P2pi p j . ~40!

~Reference@4# also employs the notationPLmn for Bmn.! With

these tensors, the gluon self-energy can be written inform

Paamn~P!5Paa

a ~P!Amn1Paab ~P!Bmn1Paa

c ~P!Cmn

1Paae ~P!Emn. ~41!

The polarization functionsPaaa ,Paa

b ,Paac , andPaa

e can becomputed by projecting the tensorPaa

mn onto the respectivesubspaces of the projectors~36! and ~38!. Introducing theabbreviations

Paat ~P![

1

2~d i j 2 pi p j !Paa

i j ~P!,

Paal ~P![ piPaa

i j ~P! p j , ~42!

these functions read

Paaa ~P!5

1

2Paa

mn~P!Amn52Paat ~P!, ~43a!

Paab ~P!5Paa

mn~P!Bmn

52p2

P2 FPaa00~P!12

p0

pPaa

0i ~P! pi1p0

2

p2Paa

l ~P!G ,

~43b!

Paac ~P!5

1

2N2P2Paa

mn~P!Cmn

521

P2 FPaa00~P!1

p021p2

p0 pPaa

0i ~P! pi1Paal ~P!G ,

~43c!

9-6

ofr


Paae ~P!5Paa

mn~P!Emn

51

P2@p0

2Paa00~P!12 p0 pPaa

0i ~P! pi1p2Paal ~P!#.

~43d!

For the explicitly transverse tensorP11mn , the functionsP11

c

5P11e [0. The same holds for the polarization tensorP0

mn .For the other gluon colorsa54, . . . ,8, thefunctions Paa

c

and Paae do not vanish. Note that the dimensions

Paaa ,Paa

b , and Paae are @MeV2#, while Paa

c is dimension-less.

iogThallitu

it

implfw

-s

afiofoio

05401

Now let us define the functions

A'ma ~P!5Am

n Ana~P!, Ai

a~P!5PmAm

a ~P!

P2,

ANa ~P!5

NmAma ~P!

N2. ~44!

Note that Aia(2P)52PmAm

a (2P)/P2, and ANa (2P)5

2NmAma (2P)/N2, since Nm is odd underP→2P. The

fields Aia(P) andAN

a (P) are dimensionless. With the tensodecomposition~41! and the functions~44!, Eq. ~35! becomes

S2521

2

V

T (P

(a51

8 H A'ma ~2P!Paa

a ~P!AmnA' na ~P!2AN

a ~2P!Paab ~P! N2AN

a ~P!

2FAia~2P!1

i

gwa~2P!GPaa

c ~P! N2P2ANa ~P!2AN

a ~2P!Paac ~P! N2P2FAi

a~P!1i

gwa~P!G

2FAia~2P!1

i

gwa~2P!GPaa

e ~P!P2FAia~P!1

i

gwa~P!G J . ~45!

te

In any spontaneously broken gauge theory, the excitatof the condensate mix with the gauge fields correspondinthe broken generators of the underlying gauge group.mixing occurs in the components orthogonal to the spatitransverse degrees of freedom, i.e., for the spatially longdinal fields,AN

a , and the fields parallel toPm, Aia . For the

two-flavor color superconductor, these components mix wthe meson fields for gluon colors 4, . . . ,8. Themixing isparticularly evident in Eq.~45!.

The terms mixing mesons and gauge fields can be elnated by a suitable choice of gauge. The gauge to accomthis goal is the ’t Hooft gauge. The ‘‘unmixing’’ procedure omesons and gauge fields consists of two steps. First,eliminate the terms in Eq.~45! which mix AN

a andAia . This

is achieved by substituting

Aia~P!5Ai

a~P!1Paa

c ~P!N2

Paae ~P!

ANa ~P!. ~46!

@We do not perform this substitution fora51,2,3; for thesegluon colorsPaa

c [0, such that there are no terms in Eq.~45!which mix Ai

a andANa .# This shift of the gauge field compo

nent Aia is completely innocuous for the following reason

First, the Jacobian](Ai ,AN)/](Ai ,AN) is unity, so the mea-sure of the functional integral over gauge fields is notfected. Second, the only other term in the gauge field actwhich is quadratic in the gauge fields and thus relevantthe derivation of the gluon propagator, is the free field act

nstoey-

h

i-ish

e

.

-n,r

n

SF2(0)[2

1

2

V

T (P

(a51

8

Ama ~2P!~P2gmn2PmPn!An

a~P!

[21

2

V

T (P

(a51

8

Ama ~2P!P2~Amn1Bmn!An

a~P!,

~47!

and it does not contain the parallel componentsAia(P). It is

therefore also not affected by the shift of variables~46!.After renamingAi

a→Aia , the final result forS2 reads:

S2521

2

V

T (P

(a51

8 H A'ma ~2P!Paa

a ~P!AmnA'na ~P!

2ANa ~2P!Paa

b ~P!N2ANa ~P!2FAi

a~2P!

1i

gwa~2P!GPaa

e ~P!P2FAia~P!1

i

gwa~P!G J , ~48!

where we introduced

Paab ~P![Paa

b ~P!2@Paa

c ~P!#2N2P2

Paae ~P!

. ~49!

The ’t Hooft gauge fixing term is now chosen to eliminathe mixing betweenAi

a andwa:

9-7

ica

tioioy

se

tly

o

n-eting-.e.,t

za-

ge

the

ed

nal

to

ef.er


Sgf51

2l

V

T (P

(a51

8 FP2Aia~2P!

2li

gPaa

e ~P!wa~2P!G3FP2Ai

a~P!2li

gPaa

e ~P!wa~P!G . ~50!

This gauge condition is nonlocal in coordinate space, whseems peculiar, but poses no problem in momentum spNote thatP2Ai

a(P)[PmAma (P). Therefore in various limits

the choice of gauge~50! corresponds to covariant gauge,

Scg51

2l

V

T (P

(a51

8

Ama ~2P!PmPnAn

a~P!. ~51!

The first limit we consider isT,m→0, i.e., the vacuum.Then,Paa

e [0, and Eq.~50! becomes Eq.~51!. The secondcase is the limit of large 4-momenta,P→`. As shown inRef. @7#, in this region of phase space the effects fromcolor-superconducting condensate on the gluon polarizatensor are negligible. In other words, the gluon polarizattensor approaches the one in a normal conductor. The phcal reason is that gluons with large momenta do notquark Cooper pairs as composite objects, but resolve thedividual color charges inside the pair. ConsequenPaa

e (P)P2→PmP0mn(P)Pn[0 for P→` and, for largeP,

the individual terms in the sum overP in Eqs.~50! and~51!agree. Finally, for gluon colorsa51,2,3, Paa

e [0, since theself-energyP11

mn is transverse. Thus fora51,2,3 the terms inEqs.~50! and ~51! are identical.

The decoupling of mesons and gluon degrees of freedbecomes obvious once we add Eq.~50! to Eqs.~48! and~47!,

SF2(0)

1S21Sgf521

2

V

T (P

(a51

8

3H A'ma ~2P!@P21Paa

a ~P!#AmnA' na ~P!

2ANa ~2P!@P21Paa

b ~P!#N2ANa ~P!

2Aia~2P!F1

lP21Paa

e ~P!GP2Aia~P!

1l

g2wa~2P!F1

lP21Paa

e ~P!GPaae ~P!wa~P!J . ~52!

Consequently, the inverse gluon propagator is

D21aamn~P!5@P21Paa

a ~P!#Amn

1@P21Paab ~P!#Bmn1F1

lP21Paa

e ~P!GEmn.

~53!

05401

hce.

an

nsi-e

in-,

m

Inverting this as discussed in Ref.@4#, one obtains the gluonpropagator for gluons of colora,

Daamn~P!5

1

P21Paaa ~P!

Amn11

P21Paab ~P!

Bmn

1l

P21lPaae ~P!

Emn. ~54!

For anylÞ0, the gluon propagator contains unphysical cotributions parallel toPm, which have to be cancelled by thcorresponding Faddeev-Popov ghosts when compuphysical observables. Only forl50 these contributions vanish and the gluon propagator is explicitly transverse, iPmDaa

mn(P)5Daamn(P)Pn50. Also, in this case the ghos

propagator is independent of the chemical potentialm. Thecontribution of Fadeev-Popov ghosts to the gluon polarition tensor is then;g2T2 and thus negligible atT50. Weshall therefore focus on this particular choice for the gauparameter in the following. Note that forl50, the inversemeson field propagator is

Daa21~P![Paa

e ~P!P25PmPaamn~P!Pn , ~55!

and the dispersion relation for the mesons follows fromcondition Daa

21(P)50, as demonstrated in Ref.@16# for athree-flavor color superconductor in the color-flavor-lockphase.

The gluon propagator for transverse and longitudimodes can now be read off Eq.~54! as coefficients of thecorresponding tensors Amn ~the projector onto the spatiallytransverse subspace orthogonal toPm) and Bmn ~the projec-tor onto the spatially longitudinal subspace orthogonalPm). For the transverse modes one has@4#

Daat ~P![

1

P21Paaa ~P!

51

P22Paat ~P!

, ~56!

where we used Eq.~43a!. Multiplying the coefficient of Bmn

in Eq. ~54! with the standard factor2P2/p2 @4#, one obtainsfor the longitudinal modes

Daa00~P![2

P2

p2

1

P21Paab ~P!

521

p22Paa00~P!

, ~57!

where the longitudinal gluon self-energy

Paa00~P![p2

Paa00~P! Paa

l ~P!2@Paa0i ~P! pi #

2

p02Paa

00~P!12p0 pPaa0i ~P! pi1p2Paa

l ~P!~58!

follows from the definition ofPaab , Eq. ~49!, and the rela-

tions ~43!. The longitudinal gluon propagatorDaa00 must not

be confusedwith the the 00-component ofDaamn . We deliber-

ately use this~slightly ambiguous! notation to facilitate thecomparison of our new and correct results with those of R@7#, which were partially incorrect. The results of that pap

9-8

t

atisss

uoto

.th

‘uh,lahe-

erefite

io

ohen

rshilf-

s.

erarydlyper-

on

f.ns-

on-a


were derived in Coulomb gauge, where the 00-componenthe propagator is indeedidentical to the longitudinal propa-gator ~57!. We were not able to find a ’t Hooft gauge thconverged to the Coulomb gauge in the various limits dcussed above, and consequently had to base our discuon the covariant gauge~51! as a limiting case of Eq.~50!.

To summarize this section, we have computed the glpropagator for gluons in a two-flavor color superconducBecause of condensation of quark Cooper pairs, theSU(3)cgauge symmetry is spontaneously broken toSU(2)c , lead-ing to the appearance of five Nambu-Goldstone bosonsgeneral, these bosons mix with some components ofgauge fields corresponding to the broken generators. To ‘mix’’ them we have used a form of ’t Hooft gauge whicsmoothly converges to covariant gauge in the vacuumwell as for large gluon momenta, and when the gluon poization tensor is explicitly transverse. Finally, choosing tgauge fixing parameterl50 we derived the gluon propagator for transverse, Eq.~56!, and longitudinal modes, Eq.~57!with Eq. ~58!.

III. SPECTRAL PROPERTIES OF THE EIGHTH GLUON

In this section we explicitly compute the spectral propties of the eighth gluon. We shall confirm the results of R@7# for the transverse mode and amend those for the longdinal mode, which have not been correctly computed in R@7#. In particular, we shall show that the plasmon dispersrelation now has the correct behaviorp0→mg asp→0. Fur-thermore, the longitudinal spectral density vanishes for gluenergies and momenta located on the dispersion brancthe Nambu-Goldstone bosons, i.e., for energies and momgiven by the roots of Eq.~55!. For the eighth gluon, this

condition can be written in the formPmPmn(P)Pn50@9,16#, since the hard-dense loop~HDL! self-energy is trans-verse,PmP0

mn(P)Pn[0.

A. Polarization tensor

We first compute the polarization tensor for the transveand longitudinal components of the eighth gluon. To tend, it is convenient to rewrite the longitudinal gluon seenergy~58! in the form

P8800~P![

2

3P0

00~P!11

3P00~P!, ~59!

P00~P![p2P00~P! P l~P!2@P0i~P!pi #

2

p02P00~P!12p0 pP0i~P! pi1p2P l~P!

,

~60!

with P l(P)[ piPi j (P) p j .

Let us now explicitly compute the polarization functionAs in Ref. @7# we takeT50, and we shall use the identity

1

x1 ih[P1

x2 ipd~x!, ~61!

05401

of

-ion

nr.

Ine

n-

asr-

-.u-f.n

nofta

es

whereP stands for the principal value description, in ordto decompose the polarization tensor into real and imaginparts. The imaginary parts can then be straightforwarcomputed, while the real parts are computed from the dission integral

ReP~p0 ,p![1

pPE

2`

`

dvImP~v,p!

v2p01C, ~62a!

whereC is a ~subtraction! constant. If ImP(v,p) is an oddfunction of v, ImP(2v,p)52ImP(v,p), Eq. ~62a! be-comes Eq.~39! of Ref. @7#,

ReP~p0 ,p![1

pPE

0

`

dvImPodd~v,p!

3S 1

v1p01

1

v2p0D1C, ~62b!

and if it is an even function of v, ImP(2v,p)5ImP(v,p), we have instead

ReP~p0 ,p![1

pPE

0

`

dvImPeven~v,p!

3S 1

v2p02

1

v1p0D1C. ~62c!

Since the polarization tensor for the transverse glu

modes,P88t [ 2

3 P0t 1 1

3 P t, has already been computed in Re@7#, we just cite the results. The imaginary part of the traverse HDL polarization function reads~cf. Eq. ~22b! of Ref.@7#!

ImP0t ~P!52p

3

4mg

2p0

p S 12p0

2

p2D u~p2p0!. ~63a!

The corresponding real part is computed from Eq.~62b!,with the result~cf. Eqs.~40b! and ~41! of Ref. @7#!

ReP0t ~P!5

3

2mg

2F p02

p21S 12

p02

p2D p0

2plnUp01p

p02pUG .

~63b!

We have used the fact that the value of the subtraction cstant isC0

t 5mg2 , which can be derived from comparing

direct calculation of ReP0t using Eq.~19b! of Ref. @7# with

the above computation via the dispersion formula~62b!.

The imaginary part of the tensorP t is given by~cf. Eq.~36! of Ref. @7#!

9-9


ImP t~P!52p3

4mg

2u~p022f!p0

p H u~Ep2p0!F S 12p0

2

p2~11s2!D E~ t !2s2S 122

p02

p2D K ~ t !G1u~p02Ep!F S 12

p02

p2~11s2!D E~a,t !2S 12

p02

p2D p

p0A12

4f2

p022p2

2s2S 122p0

2

p2D F~a,t !G J , ~64!

g

e-

edis

se

n-

s

f.n

eo-

t,

where f is the value of the color-superconductingap, Ep5Ap214f2, t5A124f2/p0

2, s2512t2,a5arcsin@p/(tp0)#, andF(a,t), E(a,t) are elliptic integralsof the first and second kind, whileK (t)[F(p/2,t) andE(t)[E(p/2,t) are the corresponding complete elliptic intgrals. The real part is again computed from Eq.~62b!. Theintegral has to be done numerically, see Appendix A of R@7# for details. The subtraction constant is, for reasonscussed at length in Ref.@7#, identical to the one in the HDLlimit, Ct[C0

t 5mg2 . Finally, taking the linear combination

P88t [ 2

3 P0t 1 1

3 P t completes the calculation of the transverpolarization functionP88

t .In order to compute the polarization function for the lo

gitudinal gluon, P8800, we have to know the function

P000(P), P00(P), P0i(P) pi , andP l(P). The first two func-

tions,P000(P) andP00(P), have also been computed in Re

@7#. The imaginary part of the longitudinal HDL polarizatiofunction is ~cf. Eq. ~22a! of Ref. @7#!

ImP000~P!52p

3

2mg

2p0

pu~p2p0!. ~65a!

The real part is computed from Eq.~62b!, with the result~cf.Eqs.~40a! and ~41! of Ref. @7#!

ReP000~P!523mg

2S 12p0

2plnUp01p

p02pU D . ~65b!

Here, the subtraction constant isC00050.

The imaginary part of the functionP00 is ~cf. Eq. ~35! ofRef. @7#!

ImP00~P!52p3

2mg

2u~p022f!p0

p

3H u~Ep2p0!E~ t !1u~p02Ep!

3FE~a,t !2p

p0A12

4f2

p022p2G J . ~66!

The real part is computed from Eq.~62b!, with the subtrac-tion constantC00[C0

0050. Again, the integral has to bdone numerically.

It remains to compute the functionsP0i(P) pi andP l(P).First, one performs the spin traces in Eq.~34e! to obtain Eqs.~102b! and ~102c! of Ref. @6#. Then, takingT50,

05401

f.-

P0i~P! pi5g2

2 E d3k

~2p!3

3 (e1 ,e256

~e1k1•p1e2k2•p!S j2

2e22

j1

2e1D

3S 1

p01e11e21 ih1

1

p02e12e21 ih D ,

~67a!

P l~P!52g2

2 E d3k

~2p!3 (e1 ,e256

@~12e1e2k1•k2!

12e1e2k1•pk2•p#e1e22j1j22f1f2

2e1e2

3S 1

p01e11e21 ih2

1

p02e12e21 ih D ,

~67b!

FIG. 1. Imaginary parts of the polarization tensors in a twflavor color superconductor~solid lines! as a function of gluon en-

ergy p0 for fixed gluon momentump54f. ~a! ImP1100, ~b! ImP00,

~c! ImP8800, ~d! ImP00, ~e! 2ImP0i pi , ~f! ImP l , ~g! ImP11

t , ~h!

ImP t, and~i! ImP88t . The corresponding results in the HDL limi

i.e., for f50, are shown as dotted lines.

9-10


wherek1,25k6p/2, f i[fki

ei is the gap function for quasiparticles (ei511) or quasi-antiparticles (ei521) with momentum

k i , j i[eiki2m, ande i[Aj i21f i

2.One now repeats the steps discussed in detail in Sec. II A of Ref.@7# to obtain~for p0>0):

ImP0i~P! pi5p3

2mg

2u~p022f!p0

2

p2 H u~Ep2p0!@E~ t !2s2K ~ t !#1u~p02Ep!FE~a,t !2p

p0A12

4f2

p022p2

2s2F~a,t !G J ,

~68a!

ImP l~P!52p3

2mg

2u~p022f!p0

3

p3 H u~Ep2p0!@~11s2!E~ t !22s2K ~ t !#

1u~p02Ep!F ~11s2!E~a,t !2p

p0A12

4f2

p022p2

22s2F~a,t !G J . ~68b!

e

l

tio

pu

-th

ngthin

in

c-

is-

-

rot-

n

One observes that in the limitf→0, the functions~68! ap-proach the HDL result

ImP00i~P!pi5p

3

2mg

2p0

2

p2u~p2p0!, ~69a!

ImP0l ~P!52p

3

2mg

2p0

3

p3u~p2p0!.

~69b!

Applying Eq. ~61! to Eqs.~67! we immediately see that th

imaginary part ofP0i(P) pi is even, while that ofP l(P) is

odd. Thus, in order to compute the real part ofP0i(P) pi , we

have to use Eq.~62c!, while the real part ofP l(P) has to becomputed from Eq.~62b!. When implementing the numericaprocedure discussed in Appendix A of Ref.@7# for the inte-gral in Eq. ~62c!, one has to modify Eq.~A1! of Ref. @7#appropriately.

Finally, one has to determine the values of the subtracconstantsC0i and Cl . We again use the fact thatC0i[C0

0i

and Cl[C0l , where the index ‘‘0’’ refers to the HDL limit.

The corresponding constants are determined by first com

ing ReP00i(P) pi and ReP0

l (P) from the dispersion formulas~62b! and ~62c!. The result of this calculation is then compared to that of a direct computation using, for instance,

result ~65b! for ReP000(P) and then inferring ReP0

0i(P) pi

and ReP0l (P) from the transversality ofP0

mn . The result isC0i[C0

0i50 andCl[C0l 5mg

2 .At this point, we have determined all functions enteri

the transverse and longitudinal polarization functions foreighth gluon. In Fig. 1 we show the imaginary parts andFig. 2 the real parts, for a fixed gluon momentump54f, asa function of gluon energyp0 ~in units of 2f). The units forthe imaginary parts are23mg

2/2, and for the real parts13mg

2/2. For comparison, in parts~a! and ~g! of these fig-ures, we show the results from Ref.@7# for the longitudinaland transverse polarization function of the gluon with adjo

05401

n

t-

e

e

t

color 1. In parts ~d!, ~e!, and ~f! the functions P00,

2P0i pi , andP l are shown. According to Eq.~60! these are

required to determineP00, shown in part~b!. Using Eq.~59!,this result is then combined with the HDL polarization fun

tion P000 to computeP88

00, shown in part~c!. Finally, thetransverse polarization function for gluons of color 8shown in part~i!. This function is given by the linear com

bination P88t 5 2

3 P0t 1 1

3 P t of the transverse HDL polariza

tion function P0t and the functionP t, both of which are

shown in part~h!. In all figures, the results for the two-flavocolor superconductor are drawn as solid lines, while the dted lines correspond to those in a normal conductor,f→0~the HDL limit!.

Note that parts~a!, ~d!, ~g!, ~h!, and ~i! of Figs. 1 and 2agree with parts~a!, ~b!, ~d!, ~e!, and~f! of Figs. 2 and 3 ofRef. @7#. The new results are parts~e! and~f! of Figs. 1 and2, which are used to determine the functions in parts~b! and~c!, the latter showing the correct longitudinal polarizatio

FIG. 2. The same as in Fig. 1, but for the real parts.

9-11

ow

lle

r o

th

on

d

on

-w

.sit

n-

of

he

ll

theion

ndo

arti-

e

as

onicac-

doa

ar

,ondalstin-

cita-ity.


function for the eighth gluon. In Ref.@7# this function wasnot computed correctly, as the effect from the fluctuationsthe condensate on the polarization tensor of the gluonsnot taken into account.

The singularity around a gluon energy somewhat smathanp052f visible in Figs. 2~b! and 2~c! seems peculiar. Itturns out that it arises due to a zero in the denominato

P00 in Eq. ~60!, i.e., whenPmPmn(P)Pn50. As discussedabove, this condition defines the dispersion branch ofNambu-Goldstone excitations@16#. Therefore the singularityis tied to the existence of the Nambu-Goldstone excitatiof the diquark condensate.

B. Spectral densities

Let us now determine the spectral densities for longitunal and transverse modes, defined by~cf. Eq. ~45! of Ref.@7#!

r8800~p0 ,p![

1

pImD88

00~p01 ih,p!,

r88t ~p0 ,p![

1

pImD88

t ~p01 ih,p!. ~70!

The longitudinal and transverse spectral densities for gluof color 8 are shown in Figs. 3~c! and 3~d!, for fixed gluonmomentump5mg/2 andmg58f. For comparison, the corresponding spectral densities for gluons of color 1 are shoin parts ~a! and ~b!. Parts~a!, ~b!, and ~d! are identical tothose of Fig. 6 of Ref.@7#, part ~c! is new and replaces Fig6~c! of Ref. @7#. One observes a peak in the spectral den

FIG. 3. The longitudinal~a!, ~c! and transverse~b!, ~d! spectraldensities for gluons of color 1~a!, ~b! and 8 ~c!, ~d!. The gluonmomentum isp5mg/2 and mg58f. For comparison, the dottelines represent the corresponding HDL spectral densities. The pof the spectral density corresponding to stable quasiparticlesmade visible by using a numerically small but nonzero imaginpart.

05401

fas

r

f

e

s

i-

s

n

y

aroundp05mg . This peak corresponds to the ordinary logitudinal gluon mode~the plasmon! present in a dense~orhot! medium.

Note that the longitudinal spectral density for gluonscolor 8 vanishes at an energy somewhat smaller thanp05mg/4. The reason is the singularity of the real part of tgluon self-energy seen in Figs. 2~b! and 2~c!. The location of

this point is wherePmPmn(P)Pn50, i.e., on the dispersionbranch of the Nambu-Goldstone excitations.

Finally, we show in Fig. 4 the dispersion relations for aexcitations, defined by the roots of

p22ReP8800~p0 ,p!50 ~71a!

for longitudinal gluons~cf. Eq. ~47a! of Ref. @7#!, and by theroots of

p022p22ReP88

t ~p0 ,p!50 ~71b!

for transverse gluons~cf. Eq. ~47b! of Ref. @7#!. Let us men-tion that not all excitations found via Eqs.~71! correspond totruly stable quasiparticles, i.e., the imaginary parts ofself-energies do not always vanish along the disperscurves. Nevertheless, in that case Eqs.~71! can still be usedto identify peaks in the spectral densities, which correspoto unstablemodes~which decay with a rate proportional tthe width of the peak!. As long as the width of the peak~thedecay rate of the quasiparticles! is small compared to itsheight, it makes sense to refer to these modes as quasipcles.

Figure 4 corresponds to Fig. 5 of Ref.@7#. In fact, part~b!is identical in both figures. Figure 4~a! differs from Fig. 5~a!of Ref. @7#, reflecting our new and correct results for thlongitudinal gluon self-energy. In Fig. 5~a! of Ref. @7#, thedispersion curve for the longitudinal gluon of color 8 wseen to diverge for small gluon momenta. In Ref.@7# it wasargued that this behavior was due to neglecting the mesfluctuations of the diquark condensate. Indeed, properly

lesrey

FIG. 4. Dispersion relations for~a! longitudinal and~b! trans-verse modes formg58f. The solid lines are for gluons of color 1the dashed lines for gluons of color 8. The dotted lines correspto the dispersion relations in the HDL limit. For both longitudinand transverse gluons of color 8, the dispersion curves are indiguishable from the HDL curves. The additional branch shown in~a!as a dashed-dotted line is the one for the Nambu-Goldstone extions, which appears as a zero in the longitudinal spectral dens

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so.he

lf-ng

lu

us

eth

eeb

uofo

tra

intw

ect-dingns.pre-

ex-rsef.

oriti-e ’t.R.

biaf

on-i-k

e

asE-


counting for these modes, we obtain a reasonable dispercurve, approachingp05mg as the momentum goes to zerIn Fig. 4~a! we also show the dispersion branch for tNambu-Goldstone excitations~dashed-dotted!. This isstrictly speaking not given by a root of Eq.~71!, but by thesingularity of the real part of the longitudinal gluon seenergy. However, because this singularity involves a chaof sign, a normal root-finding algorithm applied to Eq.~71!will also locate this singularity. As expected@16#, the disper-sion branch is linear,

p0.1

A3p, ~72!

for small gluon momenta, and approaches the vap052f for p→`.

IV. CONCLUSIONS

In cold, dense quark matter withNf52 massless quarkflavors, condensation of quark Cooper pairs spontaneobreaks theSU(3)c gauge symmetry toSU(2)c . This resultsin five Nambu-Goldstone excitations which mix with somof the components of the gluon fields corresponding tobroken generators. We have shown how to decouple thema particular choice of ’t Hooft gauge. The unphysical degrof freedom in the gluon propagator can be eliminatedfixing the ’t Hooft gauge parameterl50. In this way, wederived the propagator for transverse and longitudinal glmodes in a two-flavor color superconductor accountingthe effect of the Nambu-Goldstone excitations.

We then proceeded to explicitly compute the spec

,

05401

ion

e

e

ly

ebysy

nr

l

properties of transverse and longitudinal gluons of adjocolor 8. The spectral density of the longitudinal mode noexhibits a well-behaved plasmon branch with the corrlow-momentum limitp0→mg . Moreover, the spectral density vanishes for gluon energies and momenta corresponto the dispersion relation for Nambu-Goldstone excitatioWe have thus amended and corrected previous resultssented in Ref.@7#.

Our results pose one final question: using the correctpression for the longitudinal self-energy of adjoint colo4, . . . ,8, do thevalues of the Debye masses derived in R@6# change? The answer is ‘‘no.’’ In the limitp050,p→0,application of Eqs.~120!, ~124!, and~129! of Ref. @6# to Eq.

~58! yields Paa00(0)[Paa

00(0), and theresults of Ref.@6# forthe Debye masses remain valid.

ACKNOWLEDGMENTS

We thank G. Carter, D. Diakonov, and R.D. Pisarski fdiscussions. We thank R.D. Pisarski in particular for a crcal reading of the manuscript and for the suggestion to usHooft gauge to decouple meson and gluon modes. D.Hthanks the Nuclear Theory groups at BNL and ColumUniversity for their hospitality during a visit where part othis work was done. He also gratefully acknowledges ctinuing access to the computing facilities of Columbia Unversity’s Nuclear Theory group. I.A.S. would like to thanthe members of the Institut fu¨r Theoretische Physik at thJohann Wolfgang Goethe-Universita¨t for their hospitality,where part of this work was done. The work of I.A.S. wsupported by the U.S. Department of Energy Grant No. DFG02-87ER40328.

,

-

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Longitudinal gluons and Nambu-Goldstone bosons in a two-flavor color superconductor