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



Longitudinal gluons and Nambu-Goldstone bosons in a two-?avor color superconductor
Dirk H. Rischke
Institut f¨ ur Theoretische Physik, Johann Wolfgang Goethe-Universit¨ at Robert-Ma

yer-Str. 8–10, D-60054 Frankfurt/Main, Germany E-mail: drischke@th.physik.uni-frankfurt.de

Igor A. Shovkovy?
School of Physics and Astronomy, University of Minnesota 116 Church Street S.E., Minneapolis, MN 55455, U.S.A. E-mail: shovkovy@physics.umn.edu (February 8, 2008)

arXiv:nucl-th/0205080v2 12 Jun 2002

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

I. INTRODUCTION

Cold, dense quark matter is a color superconductor [1]. For two massless quark ?avors (say, up and down), Cooper pairs with total spin zero condense in the color-antitriplet, ?avor-singlet channel. In this so-called two-?avor color superconductor, the SU (3)c gauge symmetry is spontaneously broken to SU (2)c [2]. If we choose to orient the (anti-) color charge of the Cooper pair along the (anti-) blue direction in color space, only red and green quarks form Cooper pairs, while blue quarks remain unpaired. Then, the three generators T1 , T2 , and T3 of the original SU (3)c gauge group form the generators of the residual SU (2)c symmetry. The remaining ?ve generators T4 , . . . , T8 are broken. (More precisely, the last broken generator is a combination of T8 and the generator 1 of the global U (1) symmetry of baryon number conservation, for details see Ref. [3] and below). According to Goldstone’s theorem, this pattern of symmetry breaking gives rise to ?ve massless bosons, the socalled Nambu-Goldstone bosons, corresponding to the ?ve broken generators of SU (3)c . Physically, these massless bosons correspond to ?uctuations of the order parameter, in our case the diquark condensate, in directions in color?avor space where the e?ective potential is ?at. For gauge theories (where the local gauge symmetry cannot truly be spontaneously broken), these bosons are “eaten” by the gauge bosons corresponding to the broken generators of the original gauge group, i.e., in our case the gluons with adjoint colors a = 4, . . . , 8. They give rise to a longitudinal degree of freedom for these gauge bosons. The appearance of a longitudinal degree of freedom is commonly a sign that the gauge boson becomes massive. In a dense (or hot) medium, however, even without spontaneous breaking of the gauge symmetry the gauge bosons already have a longitudinal degree of freedom, the so-called plasmon mode [4]. Its appearance is related to the presence of gapless charged quasiparticles. Both transverse and longitudinal modes exhibit a mass gap, i.e., the gluon energy p0 → mg > 0 for momenta p → 0. In quark matter with Nf massless quark ?avors at zero temperature T = 0, the gluon mass parameter (squared) is [4] m2 g = Nf 2 2 g ? , 6 π2 (1)

where g is the QCD coupling constant and ? is the quark chemical potential. It is a priori unclear how the Nambu-Goldstone bosons interact with these longitudinal gluon modes. In particular, it is of interest to know whether coupling terms between these modes exist and, if yes, whether these terms can be

?

On leave of absence from Bogolyubov Institute for Theoretical Physics, 252143 Kiev, Ukraine.

1

eliminated by a suitable choice of (’t Hooft) gauge. The aim of the present work is to address these questions. We shall show that the answer to both questions is “yes”. We shall then demonstrate by focussing on the gluon of adjoint color 8, how the Nambu-Goldstone mode a?ects the spectral density of the longitudinal gluon. Our work is partially based on and motivated by previous studies of gluons in a two-?avor color superconductor [5–7]. The gluon self-energy and the resulting spectral properties have been discussed in Ref. [7]. In that paper, however, the ?uctuations of the diquark condensate have been neglected. Consequently, the longitudinal degrees of freedom of the gluons corresponding to the broken generators of SU (3)c have not been treated correctly. The gluon polarization tensor was no longer explicitly transverse (a transverse polarization tensor Π?ν obeys P? Π?ν = Π?ν Pν = 0), and it did not satisfy the Slavnov-Taylor identity. As a consequence, the plasmon mode exhibited a certain peculiar behavior in the low-momentum limit, which cannot be physical (cf. Fig. 5 (a) of Ref. [7]). It was already realized in Ref. [7] that the reason for this unphysical behavior is the fact that the mixing of the gluon with the excitations of the condensate was neglected. It was moreover suggested in Ref. [7] that proper inclusion of this mixing would amend the shortcomings of the previous analysis. The aim of the present work is to follow this suggestion and thus to correct the results of Ref. [7] with respect to the longitudinal gluon. Note that in Ref. [5] ?uctuations of the color-superconducting condensate were taken into account in the calculation of the gluon polarization tensor. As a consequence, the latter is explicitly transverse. However, the analysis was done in the vacuum, at ? = 0, not at (asymptotically) large chemical potential. The outline of the present work is as follows. In Section II we derive the transverse and longitudinal gluon propagators including ?uctuations of the diquark condensate. In Section III we use the resulting expressions to compute the spectral density for the gluon of adjoint color 8. Section IV concludes this work with a summary of our results. Our units are h ? = c = kB = 1. The metric tensor is g ?ν = diag (+, ?, ?, ?). We denote 4-vectors in energymomentum space by capital letters, K ? = (k0 , k). Absolute magnitudes of 3-vectors are denoted as k ≡ |k|, and the ? ≡ k/k . unit vector in the direction of k is k
II. DERIVATION OF THE PROPAGATOR FOR TRANSVERSE AND LONGITUDINAL GLUONS

In this section, we derive the gluon propagator taking into account the ?uctuations of the diquark condensate. A short version of this derivation can be found in Appendix C of Ref. [8] [see also the original Ref. [9]]. Nevertheless, for the sake of clarity and in order to make our presentation self-contained, we decide to present this once more in greater detail and in the notation of Ref. [7]. As this part is rather technical, the reader less interested in the details of the derivation should skip directly to our main result, Eqs. (56), (57), and (58). We start with the grand partition function of QCD, Z= where Zq [A] = ? Dψ exp Dψ ? iγ ? ?? + ?γ0 + gγ ? Aa Ta ψ . ψ ? (2b) DA eSA Zq [A] , (2a)

x

is the grand partition function for massless quarks in the presence of a gluon ?eld A? a . In Eq. (2), the space-time 1/T 3 integration is de?ned as x ≡ 0 dτ V d x , where V is the volume of the system, γ ? are the Dirac matrices, and Ta = λa /2 are the generators of SU (Nc ). For QCD, Nc = 3, and λa are the Gell-Mann matrices. The quark ?elds ψ are 4Nc Nf -component spinors, i.e., they carry Dirac indices α = 1, . . . , 4, fundamental color indices i = 1, . . . , Nc , and ?avor indices f = 1, . . . , Nf . The action for the gauge ?elds consists of three parts, SA = SF 2 + Sgf + SFPG , where SF 2 = ? 1 4
?ν a Fa F?ν

(3)

(4)

x

a a abc b c is the gauge ?eld part; here, F?ν = ?? Aa A? Aν is the ?eld strength tensor. The part corresponding ν ? ?ν A? + gf to gauge ?xing, Sgf , and to Fadeev-Popov ghosts, SFPG , will be discussed later.

2

For fermions at ?nite chemical potential it is advantageous to introduce the charge-conjugate degrees of freedom explicitly. This restores the symmetry of the theory under ? → ??. Therefore, in Ref. [6], a kind of replica method was applied, in which one ?rst arti?cially increases the number of quark species, and then replaces half of these species of quark ?elds by charge-conjugate quark ?elds. More precisely, ?rst replace the quark partition function Zq [A] by ZM [A] ≡ {Zq [A]}M , M being some large integer number. (Sending M → 1 at the end of the calculation reproduces the original partition function.) Then, take M to be an even integer number, and replace the quark ?elds by charge-conjugate quark ?elds in M/2 of the factors Zq [A] in ZM [A]. This results in ? ? M/2 ?M/2 ? ? ? r DΨr exp ? r (x) G ?1 (x, y ) Ψr (y ) + g Ψ ? r (x) Aa ? DΨ Ψ ( x ) Γ Ψ ( x ) ZM [A] = . (5) r ? a 0 ? ? x,y x
r =1 r =1

? r are 8Nc Nf -component Nambu-Gor’kov spinors, Here, r labels the quark species and Ψr , Ψ Ψr ≡ ψr ψCr ?r , ψ ?Cr ) , ? r ≡ (ψ , Ψ

(6)

?T is the charge conjugate spinor and C = iγ 2 γ0 is the charge conjugation matrix. The inverse of the where ψCr ≡ C ψ r 8Nc Nf × 8Nc Nf -dimensional Nambu-Gor’kov propagator for non-interacting quarks is de?ned as
?1 G0 ≡ ?1 [G+ 0 0] ?1 0 [G? 0 ]

,

(7)

where
?1 ? [G± (x, y ) ≡ ?i (iγ? ?x ± ?γ0 ) δ (4) (x ? y ) 0]

(8)

is the inverse propagator for non-interacting quarks (upper sign) or charge conjugate quarks (lower sign), respectively. The Nambu-Gor’kov matrix vertex describing the interaction between quarks and gauge ?elds is de?ned as follows: ?? ≡ Γ a Γ? a 0 ?? 0 Γ a , (9)

where

? ? T ?1 T T ?? where Γ? T a ≡ ?γ ? T a . a ≡ γ Ta and Γa ≡ C (γ ) C ? Following Ref. [1] we now add the term x,y ψCr (x) ?+ (x, y ) ψr (y ) and the corresponding charge-conjugate term ? ? + ? ? x,y ψr (x) ? (x, y ) ψCr (y ), where ? ≡ γ0 (? ) γ0 , to the argument of the exponent in Eq. (5). This de?nes the + ? quark (replica) partition function in the presence of the gluon ?eld A? a and the diquark source ?elds ? , ? : ? ? M/2 ?M/2 ? ? r DΨr exp ? r (x) G ?1 (x, y ) Ψr (y ) + g Ψ ? r (x) Aa ?? DΨ ZM [A, ?+ , ?? ] ≡ Ψ , (10) ? (x) Γa Ψr (x) ? ? x,y x r =1 r =1

G ?1 ≡

?1 [G+ ?? 0] + ?1 ? [G? 0]

(11)

is the inverse quasiparticle propagator. Inserting the partition function (10) into Eq. (2a), the (replica) QCD partition function is then computed in the presence of the (external) diquark source terms ?± (x, y ), Z → Z [?+ , ?? ]. In principle, this is not the physically relevant quantity, from which one derives thermodynamic properties of the color superconductor. The diquark condensate is not an external ?eld, but assumes a nonzero value because of an intrinsic property of the system, namely the attractive gluon interaction in the color-antitriplet channel, which destabilizes the Fermi surface. The proper functional from which one derives thermodynamic functions is obtained by a Legendre transformation of ln Z [?+ , ?? ], in which the functional dependence on the diquark source term is replaced by that on the corresponding canonically conjugate variable, the diquark condensate. The Legendre-transformed functional is the e?ective action for the diquark condensate. If the latter is constant, the e?ective action is, up to a factor of V /T , identical to the e?ective potential. The e?ective potential is simply a function of the diquark condensate. Its explicit form for large-density QCD was derived in Ref. [10]. The value of this function at its maximum determines the pressure. The 3

maximum is determined by a Dyson-Schwinger equation for the diquark condensate, which is identical to the standard gap equation for the color-superconducting gap. It has been solved in the mean-?eld approximation in Refs. [11–13]. In the mean-?eld approximation [14], ?r (y ) ?+ (x, y ) ? ψCr (x) ψ ?Cr (y ) . , ?? (x, y ) ? ψr (x) ψ (12)

In this work, we are interested in the gluon propagator, and the derivation of the pressure via a Legendre transformation of ln Z [?+ , ?? ] is of no concern to us. In the following, we shall therefore continue to consider the partition function in the presence of (external) diquark source terms ?± . The diquark source terms in the quark (replica) partition function (10) could in principle be chosen di?erently for each quark species. This could be made explicit by giving ?± a subscript r, ?± → ?± r . However, as we take the ± limit M → 1 at the end, it is not necessary to do so, as only ?± 1 ≡ ? will survive anyway. In other words, we use the same diquark sources for all quark species. The next step is to explicitly investigate the ?uctuations of the diquark condensate around its expectation value. These ?uctuations correspond physically to the Nambu-Goldstone excitations (loosely termed “mesons” in the following) in a color superconductor. As mentioned in the introduction, there are ?ve such mesons in a two-?avor color superconductor, corresponding to the generators of SU (3)c which are broken in the color-superconducting phase. If the condensate is chosen to point in the (anti-) blue direction in fundamental color √ space, the broken generators are T4 , . . . , T7 of the original SU (3)c group and the particular combination B ≡ (1 + 3T8 )/3 of generators of the global U (1)B and local SU (3)c symmetry [3]. The e?ective action for the diquark condensate and, consequently, for the meson ?elds as ?uctuations of the diquark condensate, is derived via a Legendre transformation of ln Z [?+ , ?? ]. In this work, we are concerned with the properties of the gluons and thus refrain from computing this e?ective action explicitly. Consequently, instead of considering the physical meson ?elds, we consider the variables in Z [?+ , ?? ], which correspond to these ?elds. These are the ?uctuations of the diquark source terms ?± . We choose these ?uctuations to be complex phase factors multiplying the magnitude of the source terms, ?+ (x, y ) = V ? (x) Φ+ (x, y ) V ? (y ) , ?? (x, y ) = V (x) Φ? (x, y ) V T (y ) , where V (x) ≡ exp i 1 ?a (x)Ta + √ ?8 (x)B 3 a=4
7

(13a) (13b)

.

(14)

√ The extra factor 1/ 3 in front of ?8 as compared to the treatment in Ref. [8] is chosen to simplify the notation in the following. Although the ?elds ?a are not the meson ?elds themselves, but external ?elds which, after a Legendre transformation of ln Z [?+ , ?? ], are replaced by the meson ?elds, we nevertheless (and somewhat imprecisely) refer to them as meson ?elds in the following. After having explicitly introduced the ?uctuations of the diquark source terms in terms of phase factors, the functions Φ± are only allowed to ?uctuate in magnitude. For the sake of completeness, let us mention that one could again have introduced di?erent ?elds ?ar for each replica r, but this is not really necessary, as we shall take the limit M → 1 at the end of the calculation anyway. It is advantageous to also subject the quark ?elds ψr to a nonlinear transformation, introducing new ?elds χr via ?r = χ ψr = V χr , ψ ?r V ? . (15)

Since the meson ?elds are real-valued and the generators T4 , . . . , T7 and B are hermitian, the (matrix-valued) operator V is unitary, V ?1 = V ? . Therefore, the measure of the Grassmann integration over quark ?elds in Eq. (10) remains unchanged. From Eq. (15), the charge-conjugate ?elds transform as ψCr = V ? χCr , ?Cr = χ ψ ?Cr V T , (16)

The advantage of transforming the quark ?elds is that this preserves the simple structure of the terms coupling the quark ?elds to the diquark sources, ?Cr (x) ?+ (x, y ) ψr (y ) ≡ χ ?r (x) ?? (x, y ) ψCr (y ) ≡ χ ψ ?Cr (x) Φ+ (x, y ) χr (y ) , ψ ?r (x) Φ? (x, y ) χCr (y ) . In mean-?eld approximation, the diquark source terms are proportional to 4 (17)

Φ+ (x, y ) ? χCr (x) χ ?r (y )

, Φ? (x, y ) ? χr (x) χ ?Cr (y ) .

(18)

The transformation (15) has the following e?ect on the kinetic terms of the quarks and the term coupling quarks to gluons: ?r (i γ ? ?? + ? γ0 + g γ? A? Ta ) ψr = χ ψ ?r (i γ ? ?? + ? γ0 + γ? ω ? ) χr , a
? ?Cr i γ ? ?? ? ? γ0 ? g γ? A? T T ψCr = χ ?Cr (i γ ? ?? ? ? γ0 + γ? ωC ) χCr , ψ a a

(19a) (19b)

where ω ? ≡ V ? (i ? ? + g A? a Ta ) V is the Nc Nf × Nc Nf -dimensional Maurer-Cartan one-form introduced in Ref. [15] and
? T ωC ≡ V T i ? ? ? g A? V? a Ta

(20a)

(20b)

is its charge-conjugate version. Note that the partial derivative acts only on the phase factors V and V ? on the right. Introducing the Nambu-Gor’kov spinors Xr ≡ χr χCr ? r ≡ (χ , X ?r , χ ?Cr ) (21)

and the 2Nc Nf × 2Nc Nf -dimensional Maurer-Cartan one-form ?? (x, y ) ≡ ?i the quark (replica) partition function becomes
M/2

ω ? (x) 0 ? (x) 0 ωC

δ (4) (x ? y ) ,

(22)

ZM [?, Φ+ , Φ? ] ≡ where

r =1

? r DXr exp DX

? ?M/2 ?
r =1

? ? ? r (x) S ?1 (x, y ) + γ? ?? (x, y ) Xr (y ) , X ? x,y .

(23)

S ?1 ≡

?1 [G+ Φ? 0] + ?1 Φ [G? 0]

(24)

We are interested in the properties of the gluons, and thus may integrate out the fermion ?elds. This integration can be performed analytically, with the result ZM [?, Φ+ , Φ? ] ≡ det S ?1 + γ? ??
M/2

.

(25)

The determinant is to be taken over Nambu-Gor’kov, color, ?avor, spin, and space-time indices. Finally, letting M → 1, we obtain the QCD partition function (in the presence of meson, ?a , and diquark, Φ± , source ?elds) Z [?, Φ+ , Φ? ] = DA exp SA + 1 Tr ln S ?1 + γ? ?? 2 . (26)

Remembering that ?? is linear in A? a , cf. Eq. (22) with (20), in order to derive the gluon propagator it is su?cient to expand the logarithm to second order in ?? , 1 Tr ln S ?1 + γ? ?? 2 1 1 1 Tr ln S ?1 + Tr (S γ? ?? ) ? Tr (S γ? ?? S γν ?ν ) 2 2 4 ≡ S0 [Φ+ , Φ? ] + S1 [?, Φ+ , Φ? ] + S2 [?, Φ+ , Φ? ] , ? G+ Ξ? Ξ+ G? 5

(27)

with obvious de?nitions for the Si . The quasiparticle propagator is S≡ , (28)

with
?1 G± = [G± ? Σ± 0] ?1

,

± ± ± Σ± = Φ? G? , Ξ± = ?G? 0 Φ 0 Φ G .

(29)

? To make further progress, we now expand ω ? and ωC to linear order in the meson ?elds,

ω ? ? g A? a Ta ?

1 (? ? ?a ) Ta ? √ (? ? ?8 ) B , 3 a=4 1 T (? ? ?a ) Ta + √ (? ? ?8 ) B T . 3 a=4
7

7

(30a) (30b)

? T ωC ? ?g A? a Ta +

The term S1 in Eq. (27) is simply a tadpole source term for the gluon ?elds. This term does not a?ect the gluon propagator, and thus can be ignored in the following. The quadratic term S2 represents the contribution of a fermion loop to the gluon self-energy. Its computation proceeds by ?rst taking the trace over Nambu-Gor’kov space, S2 = ? 1 4
? ν (y ) G? (y, x) γν ωC (x) Trc,f,s G+ (x, y ) γ? ω ? (y ) G+ (y, x) γν ω ν (x) + G? (x, y ) γ? ωC ? ν + Ξ+ (x, y ) γ? ω ? (y ) Ξ? (y, x) γν ωC (x) + Ξ? (x, y ) γ? ωC (y ) Ξ+ (y, x) γν ω ν (x) .

x,y

(31)

The remaining trace runs only over color, ?avor, and spin indices. Using translational invariance, the propagators and ?elds are now Fourier-transformed as G± (x, y ) = Ξ± (x, y ) = ω ? (x) =
P ? ωC (x) = P ? e?iP ·x ωC (P ) .

T V T V

e?iK ·(x?y) G± (K ) ,
K

(32a) (32b) (32c) (32d)

e?iK ·(x?y) Ξ± (K ) ,
K

e?iP ·x ω ? (P ) ,

Inserting this into Eq. (31), we arrive at Eq. (C16) of Ref. [8], which in our notation reads S2 = ? 1 4
? ν (P ) G? (K ? P ) γν ωC (?P ) Trc,f,s G+ (K ) γ? ω ? (P ) G+ (K ? P ) γν ω ν (?P ) + G? (K ) γ? ωC ? ν + Ξ+ (K ) γ? ω ? (P ) Ξ? (K ? P ) γν ωC (?P ) + Ξ? (K ) γ? ωC (P ) Ξ+ (K ? P ) γν ω ν (?P ) .

K,P

(33)

The remainder of the calculation is straightforward, but somewhat tedious. First, insert the (Fourier-transform of ? the) linearized version (30) for the ?elds ω ? and ωC . This produces a plethora of terms which are second order in the gluon and meson ?elds, with coe?cients that are traces over color, ?avor, and spin. Next, perform the color and ?avor traces in these coe?cients. It turns out that some of them are identically zero, preventing the occurrence of terms which mix gluons of adjoint colors 1, 2, and 3 (the unbroken SU (2)c subgroup) among themselves and with the other gluon and meson ?elds. Furthermore, there are no terms mixing the meson ?elds ?a , a = 4, . . . 7, with ?8 . There are mixed terms between gluons and mesons with adjoint color indices 4, . . . , 7, and between the gluon ?eld A? 8 and the meson ?eld ?8 . Some of the mixed terms (those which mix gluons and mesons of adjoint colors 4 and 5, as well as 6 and 7) can be eliminated via a unitary transformation analogous to the one employed in Ref. [6], Eq. (80). Introducing the tensors
?ν ?ν Π?ν 11 (P ) ≡ Π22 (P ) ≡ Π33 (P ) =

g2 T 2 V

K

Trs γ ? G+ (K ) γ ν G+ (K ? P ) + γ ? G? (K ) γ ν G? (K ? P ) + γ ? Ξ? (K ) γ ν Ξ+ (K ? P ) + γ ? Ξ+ (K ) γ ν Ξ? (K ? P ) , (34a)

cf. Eq. (78a) of Ref. [6], 6

?ν Π?ν 44 (P ) ≡ Π66 (P ) =

g2 T 2 V

K

ν + ? ? ν ? Trs γ ? G+ 0 (K ) γ G (K ? P ) + γ G (K ) γ G0 (K ? P ) ,

(34b)

cf. Eq. (83a) of Ref. [6],
?ν Π?ν 55 (P ) ≡ Π77 (P ) =

g2 T 2 V

K

? ? ν ? Trs γ ? G+ (K ) γ ν G+ 0 (K ? P ) + γ G0 (K ) γ G (K ? P ) .

(34c)

cf. Eq. (83b) of Ref. [6], as well as 1 ? ?ν 2 ?ν (P ) , Π (P ) + Π 3 0 3 2 ? ?ν (P ) = g T Trs γ ? G+ (K ) γ ν G+ (K ? P ) + γ ? G? (K ) γ ν G? (K ? P ) Π 2 V Π?ν 88 (P ) =
K

(34d)

? γ ? Ξ? (K ) γ ν Ξ+ (K ? P ) ? γ ? Ξ+ (K ) γ ν Ξ? (K ? P ) , cf. Eq. (78c) of Ref. [6], where Π?ν 0 is the gluon self-energy in a dense, but normal-conducting system, Π?ν 0 (P ) = g2 T 2 V
ν + ? ? ν ? Trs γ ? G+ 0 (K ) γ G0 (K ? P ) + γ G0 (K ) γ G0 (K ? P ) ,

(34e)

(34f)

K

cf. Eq. (27b) of Ref. [6], the ?nal result can be written in the compact form (cf. Eq. (C19) of Ref. [8]) 1V S2 = ? 2 T
8 P a=1

Aa ? (?P ) ?

i i a P? ?a (?P ) Π?ν Pν ?a (P ) aa (P ) Aν (P ) + g g

.

(35)

In deriving Eq. (35), we have made use of the transversality of the polarization tensor in the normal-conducting ?ν ?ν phase, Π?ν 0 (P ) Pν = P? Π0 (P ) = 0. Note that the tensors Πaa for a = 1, 2, and 3 are also transverse, but those for a = 4, . . . , 8 are not. This can be seen explicitly from the expressions given in Ref. [7]. The compact notation √ of Eq. (35) is made possible by the fact that ?a ≡ 0 for a = 1, 2, 3, and because we introduced the extra factor 1/ 3 in Eq. (14) as compared to Ref. [8]. To make further progress, it is advantageous to tensor-decompose Π?ν aa . Various ways to do this are possible [8]; here we follow the notation of Ref. [4]. First, de?ne a projector onto the subspace parallel to P ? , E?ν = Then choose a vector orthogonal to P ? , for instance N? ≡ p0 p2 p2 p , 0 P2 P2 ≡ (g ?ν ? E?ν ) fν , (37) P? Pν . P2 (36)

2 2 with f ? = (0, p). Note that N 2 = ?p2 0 p /P . Now de?ne the projectors

B?ν =

N? Nν , C?ν = N ? P ν + P ? N ν , A?ν = g ?ν ? B?ν ? E?ν . N2

(38)

Using the explicit form of N ? , one convinces oneself that the tensor A?ν projects onto the spatially transverse subspace orthogonal to P ? , A00 = A0i = 0 , Aij = ? δ ij ? p ?i p ?j . (39)

?ν (Reference [4] also uses the notation PT for A?ν .) Consequently, the tensor B?ν projects onto the spatially longitu? dinal subspace orthogonal to P ,

B00 = ?

p0 pi p2 i j p2 , B0i = ? 2 , Bij = ? 0 p ? p ? . 2 P P P2

(40)

7

?ν (Reference [4] also employs the notation PL for B?ν .) With these tensors, the gluon self-energy can be written in the form a ?ν ?ν ?ν ?ν Π?ν + Πb + Πc + Πe . aa (P ) = Πaa (P ) A aa (P ) B aa (P ) C aa (P ) E

(41)

b c e ?ν The polarization functions Πa aa , Πaa , Πaa , and Πaa can be computed by projecting the tensor Πaa onto the respective subspaces of the projectors (36) and (38). Introducing the abbreviations

Πt aa (P ) ≡ these functions read

1 ij ? δ ?p ?i p ?j Πij ?i Πij ?j . aa (P ) , Πaa (P ) ≡ p aa (P ) p 2

(42)

1 ?ν Π (P ) A?ν = ?Πt aa (P ) , 2 aa p0 0i p2 ? p2 00 ?ν Π (P ) , Πaa (P ) p ?i + 0 Πb aa (P ) = Πaa (P ) B?ν = ? 2 Πaa (P ) + 2 P p p2 aa 2 1 p2 1 0+p 00 i Πc Π?ν Π0 ?i + Π? aa (P ) = aa (P ) C?ν = ? 2 Πaa (P ) + aa (P ) p aa (P ) 2 2 2N P P p0 p 1 i ?ν p2 Π00 (P ) + 2 p0 p Π0 ?i + p2 Π? Πe aa (P ) p aa (P ) . aa (P ) = Πaa (P ) E?ν = P 2 0 aa Πa aa (P ) =

(43a) (43b) , (43c) (43d)

c e For the explicitly transverse tensor Π?ν 11 , the functions Π11 = Π11 ≡ 0. The same holds for the HDL polarization tensor ?ν c Π0 . For the other gluon colors a = 4, . . . , 8, the functions Πaa and Πe aa do not vanish. Note that the dimensions of 2 b e c Πa , Π , and Π are [MeV ], while Π is dimensionless. aa aa aa aa Now let us de?ne the functions ν a a Aa ⊥ ? (P ) = A? Aν (P ) , A (P ) =

N ? Aa P ? Aa ? (P ) ? (P ) a , A ( P ) = . N 2 P N2

(44)

2 a ? a 2 ? Note that Aa (?P ) = ?P ? Aa ? (?P )/P , and AN (?P ) = ?N A? (?P )/N , since N is odd under P → ?P . The ?elds Aa (P ) and Aa N (P ) are dimensionless. With the tensor decomposition (41) and the functions (44), Eq. (35) becomes

S2 = ?

1 V 2 T

8 P a=1 a ?ν a b 2 a Aa A⊥ ν (P ) ? Aa ⊥ ? (?P ) Πaa (P ) A N (?P ) Πaa (P ) N AN (P )

? Aa (?P ) +

i a 2 2 a a c 2 2 ? (?P ) Πc aa (P ) N P AN (P ) ? AN (?P ) Πaa (P ) N P g i i 2 ? Aa (?P ) + ?a (?P ) Πe Aa (P ) + ?a (P ) . aa (P ) P g g

Aa (P ) +

i a ? (P ) g (45)

In any spontaneously broken gauge theory, the excitations of the condensate mix with the gauge ?elds corresponding to the broken generators of the underlying gauge group. The mixing occurs in the components orthogonal to the ? spatially transverse degrees of freedom, i.e., for the spatially longitudinal ?elds, Aa N , and the ?elds parallel to P , a A . For the two-?avor color superconductor, these components mix with the meson ?elds for gluon colors 4, . . . , 8. The mixing is particularly evident in Eq. (45). The terms mixing mesons and gauge ?elds can be eliminated by a suitable choice of gauge. The gauge to accomplish this goal is the ’t Hooft gauge. The “unmixing” procedure of mesons and gauge ?elds consists of two steps. First, we a eliminate the terms in Eq. (45) which mix Aa N and A . This is achieved by substituting
c 2 ?a (P ) = Aa (P ) + Πaa (P ) N Aa A N (P ) . e Πaa (P )

(46)

(We do not perform this substitution for a = 1, 2, 3; for these gluon colors Πc aa ≡ 0, such that there are no terms in a Eq. (45) which mix Aa and Aa N ). This shift of the gauge ?eld component A is completely innocuous for the following ? , AN )/? (A , AN ) is unity, so the measure of the functional integral over gauge ?elds reasons. First, the Jacobian ? (A is not a?ected. Second, the only other term in the gauge ?eld action, which is quadratic in the gauge ?elds and thus relevant for the derivation of the gluon propagator, is the free ?eld action 8

SF 2 ≡ ?

(0)

1 V 2 T

8 P a=1 2 ?ν Aa ? P ? P ν Aa ν (P ) ≡ ? ? (?P ) P g

1 V 2 T

8 P a=1 2 ?ν Aa + B?ν ) Aa ? (?P ) P (A ν (P ) ,

(47)

and it does not contain the parallel components Aa (P ). It is therefore also not a?ected by the shift of variables (46). ?a → Aa , the ?nal result for S2 reads: After renaming A S2 = ? 1 V 2 T
8 P a=1 a ?ν a 2 a ?b Aa A⊥ ν (P ) ? Aa ⊥ ? (?P ) Πaa (P ) A N (?P ) Πaa (P ) N AN (P )

? Aa (?P ) + where we introduced

i a i 2 Aa (P ) + ?a (P ) ? (?P ) Πe aa (P ) P g g

,

(48)

c 2 2 ? b (P ) ≡ Πb (P ) ? [Πaa (P )] N P . Π aa aa Πe aa (P )

2

(49)

The ’t Hooft gauge ?xing term is now chosen to eliminate the mixing between Aa and ?a : Sgf = 1 V 2λ T
8 P a=1

P 2 Aa (?P ) ? λ

i e Π (P ) ?a (?P ) g aa

P 2 Aa (P ) ? λ

i e Π (P ) ?a (P ) g aa

.

(50)

This gauge condition is non-local in coordinate space, which seems peculiar, but poses no problem in momentum space. Note that P 2 Aa (P ) ≡ P ? Aa ? (P ). Therefore, in various limits the choice of gauge (50) corresponds to covariant gauge, Scg = 1 V 2λ T
8 P a=1 ? ν a Aa ? (?P ) P P Aν (P ) .

(51)

The ?rst limit we consider is T, ? → 0, i.e. the vacuum. Then, Πe aa ≡ 0, and Eq. (50) becomes (51). The second case is the limit of large 4-momenta, P → ∞. As shown in Ref. [7], in this region of phase space the e?ects from a color-superconducting condensate on the gluon polarization tensor are negligible. In other words, the gluon polarization tensor approaches the HDL limit. The physical reason is that gluons with large momenta do not see quark Cooper pairs as composite objects, but resolve the individual color charges inside the pair. Consequently, ?ν 2 Πe aa (P ) P → P? Π0 (P ) Pν ≡ 0 for P → ∞ and, for large P , the individual terms in the sum over P in Eqs. (50) and ?ν (51) agree. Finally, for gluon colors a = 1, 2, 3, Πe aa ≡ 0, since the self-energy Π11 is transverse. Thus, for a = 1, 2, 3 the terms in Eqs. (50) and (51) are identical. The decoupling of mesons and gluon degrees of freedom becomes obvious once we add (50) to (48) and (47),
(0) SF 2

+ S2 + Sgf

1V =? 2 T

8 P a=1 2 a ?ν a Aa A⊥ ν (P ) ⊥ ? (?P ) P + Πaa (P ) A 2 2 a ?b ? Aa N (?P ) P + Πaa (P ) N AN (P )

? Aa (?P )

1 2 2 a P + Πe aa (P ) P A (P ) λ 1 2 λ e a P + Πe + 2 ?a (?P ) aa (P ) Πaa (P ) ? (P ) g λ

.

(52)

Consequently, the inverse gluon propagator is
?ν ? b (P ) B?ν + ??1 aa (P ) = P 2 + Πa + P2 + Π aa (P ) A aa ?ν

1 2 ?ν P + Πe . aa (P ) E λ

(53)

Inverting this as discussed in Ref. [4], one obtains the gluon propagator for gluons of color a, ??ν aa (P ) = λ 1 1 B?ν + 2 A?ν + E?ν . 2 b ? P 2 + Πa ( P ) P + λ Πe P + Πaa (P ) aa aa (P ) 9 (54)

For any λ = 0, the gluon propagator contains unphysical contributions parallel to P ? , which have to be cancelled by the corresponding Faddeev-Popov ghosts when computing physical observables. Only for λ = 0 these contributions ?ν vanish and the gluon propagator is explicitly transverse, i.e., P? ??ν aa (P ) = ?aa (P ) Pν = 0. Also, in this case the ghost propagator is independent of the chemical potential ?. The contribution of Fadeev-Popov ghosts to the gluon polarization tensor is then ? g 2 T 2 and thus negligible at T = 0. We shall therefore focus on this particular choice for the gauge parameter in the following. Note that for λ = 0, the inverse meson ?eld propagator is
?1 2 ?ν Daa (P ) ≡ Πe aa (P ) P = P? Πaa (P ) Pν ,

(55)

?1 and the dispersion relation for the mesons follows from the condition Daa (P ) = 0, as demonstrated in Ref. [16] for a three-?avor color superconductor in the color-?avor-locked phase. The gluon propagator for transverse and longitudinal modes can now be read o? Eq. (54) as coe?cients of the corresponding tensors A?ν (the projector onto the spatially transverse subspace orthogonal to P ? ) and B?ν (the projector onto the spatially longitudinal subspace orthogonal to P ? ). For the transverse modes one has [4]

?t aa (P ) ≡

P2

1 1 = 2 , a + Πaa (P ) P ? Πt aa (P )

(56)

where we used Eq. (43a). Multiplying the coe?cient of B?ν in Eq. (54) with the standard factor ?P 2 /p2 [4], one obtains for the longitudinal modes
2 1 1 ? 00 (P ) ≡ ? P =? , ? aa 2 2 b 2 ? aa (P ) ? 00 p P +Π p ?Π aa (P )

(57)

where the longitudinal gluon self-energy
2 ? 00 Π aa (P ) ≡ p ? 0i ?i Π00 aa (P ) Πaa (P ) ? Πaa (P ) p 2 00 0 i p0 Πaa (P ) + 2 p0 p Πaa (P ) p ?i + p2 Π? aa (P ) 2

(58)

? b , Eq. (49), and the relations (43). The longitudinal gluon propagator ? ? 00 must not follows from the de?nition of Π aa aa ?ν be confused with the the 00-component of ?aa . We deliberately use this (slightly ambiguous) notation to facilitate the comparison of our new and correct results with those of Ref. [7], which were partially incorrect. The results of that paper were derived in Coulomb gauge, where the 00-component of the propagator is indeed identical to the longitudinal propagator (57). We were not able to ?nd a ’t Hooft gauge that converged to the Coulomb gauge in the various limits discussed above, and consequently had to base our discussion on the covariant gauge (51) as limiting case of Eq. (50). To summarize this section, we have computed the gluon propagator for gluons in a two-?avor color superconductor. Due to condensation of quark Cooper pairs, the SU (3)c gauge symmetry is spontaneously broken to SU (2)c , leading to the appearance of ?ve Nambu-Goldstone bosons. In general, these bosons mix with some components of the gauge ?elds corresponding to the broken generators. To “unmix” them we have used a form of ’t Hooft gauge which smoothly converges to covariant gauge in the vacuum, as well as for large gluon momenta, and when the gluon polarization tensor is explicitly transverse. Finally, choosing the gauge ?xing parameter λ = 0 we derived the gluon propagator for transverse, Eq. (56), and longitudinal modes, Eq. (57) with (58).
III. SPECTRAL PROPERTIES OF THE EIGHTH GLUON

In this section, we explicitly compute the spectral properties of the eighth gluon. We shall con?rm the results of Ref. [7] for the transverse mode and amend those for the longitudinal mode, which have not been correctly computed in Ref. [7]. In particular, we shall show that the plasmon dispersion relation now has the correct behavior p0 → mg as p → 0. Furthermore, the longitudinal spectral density vanishes for gluon energies and momenta located on the dispersion branch of the Nambu-Goldstone bosons, i.e., for energies and momenta given by the roots of Eq. (55). For ? ?ν (P ) Pν = 0 [9,16], since the HDL self-energy is the eighth gluon, this condition can be written in the form P? Π ?ν transverse, P? Π0 (P ) Pν ≡ 0.

10

A. Polarization tensor

We ?rst compute the polarization tensor for the transverse and longitudinal components of the eighth gluon. To this end, it is convenient to rewrite the longitudinal gluon self-energy (58) in the form 1 ? 00 2 00 ? 00 (P ) , Π (P ) + Π Π 88 (P ) ≡ 3 0 3 2 ? 00 (P ) Π ? ? (P ) ? Π ? 0i (P ) p Π ?i 00 2 ? (P ) ≡ p , Π ? ? (P ) ? 00 (P ) + 2 p0 p Π ? 0i (P ) p ?i + p2 Π p2 Π
0

(59) (60)

? ij (P ) p ? ? (P ) ≡ p ?j . with Π ?i Π Let us now explicitly compute the polarization functions. As in Ref. [7] we take T = 0, and we shall use the identity 1 1 ≡ P ? iπ δ (x) , x + iη x (61)

where P stands for the principal value description, in order to decompose the polarization tensor into real and imaginary parts. The imaginary parts can then be straightforwardly computed, while the real parts are computed from the dispersion integral Re Π(p0 , p) ≡ 1 P π



?∞

Im Π(ω, p) +C , ω ? p0

(62a)

where C is a (subtraction) constant. If Im Π(ω, p) is an odd function of ω , Im Π(?ω, p) = ?Im Π(ω, p), Eq. (62a) becomes Eq. (39) of Ref. [7], Re Π(p0 , p) ≡ 1 P π


dω Im Πodd (ω, p)
0

1 1 + ω + p0 ω ? p0 1 1 ? ω ? p0 ω + p0

+C ,

(62b)

and if it is an even function of ω , Im Π(?ω, p) = Im Π(ω, p), we have instead Re Π(p0 , p) ≡ 1 P π


dω Im Πeven (ω, p)
0

+C ,

(62c)

2 t 1 ?t Since the polarization tensor for the transverse gluon modes, Πt 88 ≡ 3 Π0 + 3 Π , has already been computed in Ref. [7], we just cite the results. The imaginary part of the transverse HDL polarization function reads (cf. Eq. (22b) of Ref. [7])

Im Πt 0 (P ) = ?π

3 2 p0 m 4 g p

1?

p2 0 p2

θ(p ? p0 ) .

(63a)

The corresponding real part is computed from Eq. (62b), with the result (cf. Eqs. (40b) and (41) of Ref. [7]) Re Πt 0 (P ) = 3 2 m 2 g p2 p2 0 0 + 1 ? p2 p2 p0 p0 + p ln 2p p0 ? p . (63b)

t We have used the fact that the value of the subtraction constant is C0 = m2 g , which can be derived from comparing a t direct calculation of Re Π0 using Eq. (19b) of Ref. [7] with the above computation via the dispersion formula (62b). ? t is given by (cf. Eq. (36) of Ref. [7]) The imaginary part of the tensor Π

? t (P ) = ?π Im Π + θ(p0 ? Ep )

p0 3 2 m θ(p0 ? 2 φ) 4 g p 1? p2 0 (1 + s2 ) p2

θ(Ep ? p0 ) p2 0 p2

1?

p2 0 (1 + s2 ) p2 1? p2 0

E(t) ? s2

1?2 1?2

p2 0 p2

K(t) F (α, t) , (64)

E (α, t) ? 1 ?

p p0

4 φ2 ? s2 ? p2

p2 0 p2

2 2 p2 + 4φ2 , t = 1 ? 4φ2 /p2 where φ is the value of the color-superconducting gap, Ep = 0, s = 1 ? t , α = arcsin[p/(tp0 )], and F (α, t), E (α, t) are elliptic integrals of the ?rst and second kind, while K(t) ≡ F (π/2, t) and

11

E(t) ≡ E (π/2, t) are the corresponding complete elliptic integrals. The real part is again computed from Eq. (62b). The integral has to be done numerically, see Appendix A of Ref. [7] for details. The subtraction constant is, for t reasons discussed at length in Ref. [7], identical to the one in the HDL limit, C t ≡ C0 = m2 g . Finally, taking the linear 1 ?t 2 t t combination Π88 ≡ 3 Π0 + 3 Π completes the calculation of the transverse polarization function Πt 88 . 00 ? In order to compute the polarization function for the longitudinal gluon, Π88 , we have to know the functions Π00 0 (P ), 00 ? ? 0i (P ) p ? ? (P ). The ?rst two functions, Π00 ? 00 Π (P ), Π ?i , and Π 0 (P ) and Π (P ) have also been computed in Ref. [7]. The imaginary part of the longitudinal HDL polarization function is (cf. Eq. (22a) of Ref. [7]) Im Π00 0 (P ) = ?π 3 2 p0 m θ(p ? p0 ) . 2 g p (65a)

The real part is computed from Eq. (62b), with the result (cf. Eqs. (40a) and (41) of Ref. [7])
2 Re Π00 0 (P ) = ?3 mg

1?

p0 + p p0 ln 2p p0 ? p

.

(65b)

00 Here, the subtraction constant is C0 = 0. ? 00 is (cf. Eq. (35) of Ref. [7]) The imaginary part of the function Π

? 00 (P ) = ?π Im Π

3 2 p0 mg θ(p0 ? 2 φ) 2 p

θ(Ep ? p0 ) E(t) + θ(p0 ? Ep ) E (α, t) ?

p p0

1?

4 φ2 2 p2 0?p

.

(66)

00 The real part is computed from Eq. (62b), with the subtraction constant C 00 ≡ C0 = 0. Again, the integral has to be done numerically. ? 0i (P ) p ? ? (P ). First, one performs the spin traces in Eq. (34e) to obtain It remains to compute the functions Π ?i and Π Eqs. (102b) and (102c) of Ref. [6]. Then, taking T = 0,

g2 ? 0i (P ) p Π ?i = 2

d3 k (2π )3

e1 ,e2 =±

?2 · p ? 1 · p + e2 k e1 k

ξ1 ξ2 ? 2 ?2 2 ?1 , (67a)

× ? ? (P ) = ? g Π 2
2

d3 k (2π )3

1 1 + p0 + ?1 + ?2 + iη p0 ? ?1 ? ?2 + iη

e1 ,e2 =±

?1 · p k ? 2 · p ?1 ?2 ? ξ1 ξ2 ? φ1 φ2 ? 1 · k2 + 2 e1 e2 k 1 ? e1 e2 k 2 ?1 ?2 , (67b)

×

i where k1,2 = k ± p/2, φi ≡ φe ki is the gap function for quasiparticles (ei = +1) or quasi-antiparticles (ei = ?1) with 2 + φ2 . momentum ki , ξi ≡ ei ki ? ?, and ?i ≡ ξi i One now repeats the steps discussed in detail in Section II.A of Ref. [7] to obtain (for p0 ≥ 0)

1 1 ? p0 + ?1 + ?2 + iη p0 ? ?1 ? ?2 + iη

? 0i (P ) p Im Π ?i = π

p2 3 2 mg θ(p0 ? 2 φ) 0 2 p2

θ(Ep ? p0 ) E(t) ? s2 K(t) + θ(p0 ? Ep ) E (α, t) ? p p0 1? p2 0 4 φ2 ? s2 F (α, t) ? p2 , (68a)

? ? (P ) = ?π Im Π

p3 3 2 mg θ(p0 ? 2 φ) 0 2 p3

θ(Ep ? p0 ) (1 + s2 ) E(t) ? 2 s2 K(t) p p0 1? 4 φ2 ? 2 s2 F (α, t) 2 p2 0?p . (68b)

+ θ(p0 ? Ep ) (1 + s2 ) E (α, t) ?

One observes that in the limit φ → 0, the functions (68) approach the HDL result
i Im Π0 ?i = π 0 (P ) p

3 2 p2 m 0 θ(p ? p0 ) , 2 g p2 3 2 p3 m 0 θ(p ? p0 ) . Im Π? ( P ) = ? π 0 2 g p3 12

(69a) (69b)

? 0i (P ) p ? ? (P ) Applying Eq. (61) to Eqs. (67) we immediately see that the imaginary part of Π ?i is even, while that of Π 0 i ? (P ) p ? ? (P ) is odd. Thus, in order to compute the real part of Π ?i , we have to use Eq. (62c), while the real part of Π has to be computed from Eq. (62b). When implementing the numerical procedure discussed in Appendix A of Ref. [7] for the integral in Eq. (62c), one has to modify Eq. (A1) of Ref. [7] appropriately. Finally, one has to determine the values of the subtraction constants C 0i and C ? . We again use the fact that 0i 0i ? C ≡ C0 and C ? ≡ C0 , where the index “0” refers to the HDL limit. The corresponding constants are determined by 0i ?rst computing Re Π0 (P ) p ?i and Re Π? 0 (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, the result (65b) for Re Π00 0 (P ) and then inferring ?ν i ? 0i 0i ? ? Re Π0 ( P ) p ? and Re Π ( P ) from the transversality of Π . The result is C ≡ C = 0 and C ≡ C0 = m2 i 0 0 0 g. 0 At this point, we have determined all functions entering the transverse and longitudinal polarization functions for the eighth gluon. In Fig. 1 we show the imaginary parts and in Fig. 2 the real parts, for a ?xed gluon momentum p = 4 φ, as a function of gluon energy p0 (in units of 2 φ). The units for the imaginary parts are ?3 m2 g /2, and for the real parts +3 m2 / 2. For comparison, in parts (a) and (g) of these ?gures, we show the results from Ref. [7] for g the longitudinal and transverse polarization function of the gluon with adjoint color 1. In parts (d), (e), and (f) the ? 00 , ?Π ? 0i p ? ? are shown. According to Eq. (60) these are required to determine Π ? 00 , shown in part functions Π ?i , and Π 00 ? 00 (b). Using Eq. (59), this result is then combined with the HDL polarization function Π0 to compute Π 88 , shown in part (c). Finally, the transverse polarization function for gluons of color 8 is shown in part (i). This function is given 1 ?t 2 t t ?t by the linear combination Πt 88 = 3 Π0 + 3 Π of the transverse HDL polarization function Π0 and the function Π , both of which are shown in part (h). In all ?gures, the results for the two-?avor color superconductor are drawn as solid lines, while the dotted lines correspond to those in a normal conductor, φ → 0 (the HDL limit). Note that parts (a), (d), (g), (h), and (i) of Figs. 1 and 2 agree with parts (a), (b), (d), (e), and (f) of Figs. 2 and 3 of Ref. [7]. The new results are parts (e) and (f) of Figs. 1 and 2, which are used to determine the functions in parts (b) and (c), the latter showing the correct longitudinal polarization function for the eighth gluon. In Ref. [7], this function was not computed correctly, as the e?ect from the ?uctuations of the condensate on the polarization tensor of the gluons was not taken into account.
p = 4φ 4 (a) Π0 00 Π11
00

[?3mg /2]

3 2 1 0 4

(b)

Π0 ^ 00 Π

00

(c)

Π0 ^ 00 Π88

00

2

(d)

[?3mg /2]

Π0 ~ 00 Π

00

(e)

?Π0 pi ~ 0i ^ ?Π pi

0i ^

(f)

Π0 ~l Π

l

2

3 2 1 0 (g) Π0 t Π11
t

(h)

[?3mg /2]

1.0 0.5 0.0

Π0 ~t Π

t

(i)

Π0 t Π88

t

2

0

1

2

p0/2φ

3

4

0

1

2

p0/2φ

3

4

0

1

2

p0/2φ

3

4

5

FIG. 1. Imaginary parts of the polarization tensors in a two-?avor color superconductor (solid lines) as a function of gluon ? 00 ? 00 ? 00 ? 0i ?i , (f) Im Π ? ? , (g) energy p0 for ?xed gluon momentum p = 4 φ. (a) Im Π00 11 , (b) Im Π , (c) Im Π88 , (d) Im Π , (e) ?Im Π p t t t ? Im Π11 , (h) Im Π , (i) Im Π88 . The corresponding results in the HDL limit, i.e. for φ = 0, are shown as dotted lines.

13

The singularity around a gluon energy somewhat smaller than p0 = 2 φ visible in Figs. 2 (b) and (c) seems peculiar. ? ?ν (P ) Pν = 0. As ? 00 in Eq. (60), i.e., when P? Π It turns out that it arises due to a zero in the denominator of Π discussed above, this condition de?nes the dispersion branch of the Nambu-Goldstone excitations [16]. Therefore, the singularity is tied to the existence of the Nambu-Goldstone excitations of the diquark condensate.
p = 4φ 8 6 4 2 0 ?2 ?4 ?6 6 4 2 0 ?2 ?4 ?6 1.0 0.5 0.0 ?0.5 ?1.0 ?1.5 ?2.0 (a) Π0 00 Π11
00

(b)

Π0 ^ 00 Π

00

(c)

[3mg /2]

Π0 ^ 00 Π88

00

2

(d)

[3mg /2]

Π0 ~ 00 Π

00

(e)

^ ?Π0 p i ~ 0i ^ ?Π pi

0i

(f)

Π0 ~l Π

l

2

(g)

(h)

(i)

[3mg /2]

2

Π0 t Π11 0 1 2 3 4 0 1 2 3

t

Π0 ~t Π

t

Π0 t Π88 0 1 2

t

p0/2φ

p0/2φ

4

p0/2φ

3

4

5

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

B. Spectral densities

Let us now determine the spectral densities for longitudinal and transverse modes, de?ned by (cf. Eq. (45) of Ref. [7]) ρ00 88 (p0 , p) ≡ 1 1 t ? 00 Im ? Im ?t 88 (p0 + iη, p) , ρ88 (p0 , p) ≡ 88 (p0 + iη, p) π π (70)

The longitudinal and transverse spectral densities for gluons of color 8 are shown in Figs. 3 (c) and (d), for ?xed gluon momentum p = mg /2 and mg = 8 φ. For comparison, the corresponding spectral densities for gluons of color 1 are shown in parts (a) and (b). Parts (a), (b), and (d) are identical to those of Fig. 6 of Ref. [7], part (c) is new and replaces Fig. 6 (c) of Ref. [7]. One observes a peak in the spectral density around p0 = mg . This peak corresponds to the ordinary longitudinal gluon mode (the plasmon) present in a dense (or hot) medium. Note that the longitudinal spectral density for gluons of color 8 vanishes at an energy somewhat smaller than p0 = mg /4. The reason is the singularity of the real part of the gluon self-energy seen in Figs. 2 (b) and (c). The ? ?ν (P ) Pν = 0, i.e., on the dispersion branch of the Nambu-Goldstone excitations. location of this point is where P? Π Finally, we show in Fig. 4 the dispersion relations for all excitations, de?ned by the roots of ? 00 p2 ? Re Π 88 (p0 , p) = 0 for longitudinal gluons (cf. Eq. (47a) of Ref. [7]), and by the roots of
2 t p2 0 ? p ? Re Π88 (p0 , p) = 0

(71a)

(71b)

14

for transverse gluons (cf. Eq. (47b) of Ref. [7]). Let us mention that not all excitations found via Eqs. (71) correspond to truly stable quasiparticles, i.e., the imaginary parts of the self-energies do not always vanish along the dispersion curves. Nevertheless, in that case Eqs. (71) can still be used to identify peaks in the spectral densities, which correspond to unstable modes (which decay with a rate proportional to the width of the peak). As long as the width of the peak (the decay rate of the quasiparticles) is small compared to its height, it makes sense to refer to these modes as quasiparticles.

3 2.5
00

longitudinal (a)
CSC HDL

transverse

p=0.5mg=4φ (b)

3

?πmg ρ11

2

1.5 1 0.5 0 2.5 (c) (d) 0 2.5 1

00

?πmg ρ88

2

1.5 1 0.5 0 0 0.5 1 1.5 0 0.5 1 1.5 2

1.5 1 0.5 0

p0/mg

p0/mg

FIG. 3. The longitudinal (a), (c) and transverse (b), (d) spectral densities for gluons of color 1 (a), (b) and 8 (c), (d). The gluon momentum is p = mg /2 and mg = 8φ. For comparison, the dotted lines represent the corresponding HDL spectral densities. The poles of the spectral density corresponding to stable quasiparticles are made visible by using a numerically small but nonzero imaginary part.
longitudinal 2.5 (a) 2 1.5 1 0.5 0 (b) transverse mg=8φ 1 8 HDL NG

p0/mg

0

0.5

1

1.5

0

0.5

1

1.5

2

p/mg

p/mg

FIG. 4. Dispersion relations for (a) longitudinal and (b) transverse modes for mg = 8 φ. The solid lines are for gluons of color 1, the dashed lines for gluons of color 8. The dotted lines correspond to the dispersion relations in the HDL limit. For both longitudinal and transverse gluons of color 8, the dispersion curves are indistinguishable from the HDL curves. The additional branch shown in (a) as dashed-dotted line is the one for the Nambu-Goldstone excitations, which appears as a zero in the longitudinal spectral density.

15

?πmg ρ88

2

2

2

t

?πmg ρ11

2

2

2

t

Fig. 4 corresponds to Fig. 5 of Ref. [7]. In fact, part (b) is identical in both ?gures. Fig. 4 (a) di?ers from Fig. 5 (a) of Ref. [7], re?ecting our new and correct results for the longitudinal gluon self-energy. In Fig. 5 (a) of Ref. [7], the dispersion curve for the longitudinal gluon of color 8 was seen to diverge for small gluon momenta. In Ref. [7] it was argued that this behavior was due to neglecting the mesonic ?uctuations of the diquark condensate. Indeed, properly accounting for these modes, we obtain a reasonable dispersion curve, approaching p0 = mg as the momentum goes to zero. In Fig. 4 (a) we also show the dispersion branch for the Nambu-Goldstone excitations (dash-dotted). This is strictly speaking not given by a root of Eq. (71), but by the singularity of the real part of the longitudinal gluon self-energy. However, because this singularity involves a change of sign, a normal root-?nding algorithm applied to Eq. (71) will also locate this singularity. As expected [16], the dispersion branch is linear, 1 p0 ? √ p , 3 for small gluon momenta, and approaches the value p0 = 2 φ for p → ∞.
IV. CONCLUSIONS

(72)

In cold, dense quark matter with Nf = 2 massless quark ?avors, condensation of quark Cooper pairs spontaneously breaks the SU (3)c gauge symmetry to SU (2)c . This results in ?ve Nambu-Goldstone excitations which mix with some of the components of the gluon ?elds corresponding to the broken generators. We have shown how to decouple them by a particular choice of ’t Hooft gauge. The unphysical degrees of freedom in the gluon propagator can be eliminated by ?xing the ’t Hooft gauge parameter λ = 0. In this way, we derived the propagator for transverse and longitudinal gluon modes in a two-?avor color superconductor accounting for the e?ect of the Nambu-Goldstone excitations. We then proceeded to explicitly compute the spectral properties of transverse and longitudinal gluons of adjoint color 8. The spectral density of the longitudinal mode now exhibits a well-behaved plasmon branch with the correct low-momentum limit p0 → mg . Moreover, the spectral density vanishes for gluon energies and momenta corresponding to the dispersion relation for Nambu-Goldstone excitations. We have thus amended and corrected previous results presented in Ref. [7]. Our results pose one ?nal question: using the correct expression for the longitudinal self-energy of adjoint colors 4, . . . , 8, do the values of the Debye masses derived in Ref. [6] change? The answer is “no”. In the limit p0 = 0, p → 0, ? 00 (0) ≡ Π00 (0), and the results of Ref. [6] application of Eqs. (120), (124), and (129) of Ref. [6] to Eq. (58) yields Π aa aa for the Debye masses remain valid.

ACKNOWLEDGMENTS

We thank G. Carter, D. Diakonov, and R.D. Pisarski for discussions. We thank R.D. Pisarski in particular for a critical reading of the manuscript and for the suggestion to use ’t Hooft gauge to decouple meson and gluon modes. D.H.R. thanks the Nuclear Theory groups at BNL and Columbia University for their hospitality during a visit where part of this work was done. He also gratefully acknowledges continuing access to the computing facilities of Columbia University’s Nuclear Theory group. I.A.S. would like to thank the members of the Institut f¨ ur Theoretische Physik at the Johann Wolfgang Goethe-Universit¨ at for their hospitality, where part of this work was done. The work of I.A.S. was supported by the U.S. Department of Energy Grant No. DE-FG02-87ER40328.

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