Anisotropic Admixture in Color-Superconducting Quark Matter

P H Y S I C A L R E V I E W L E T T E R S week ending9 MAY 2003VOLUME 90, NUMBER 18

Anisotropic Admixture in Color-Superconducting Quark Matter

Michael Buballa,1,2 Jirı Hosek,3 and Micaela Oertel41Institut fur Kernphysik, Schlossgartenstraße 9, D-64289 Darmstadt, Germany

2Gesellschaft fur Schwerionenforschung, Planckstraße 1, D-64291 Darmstadt, Germany3Department of Theoretical Physics, Nuclear Physics Institute, 25068 Rez (Prague), Czech Republic

4IPN-Lyon, 43 boulevard du 11 Novembre 1918, 69622 Villeurbanne CEDEX, France(Received 6 May 2002; published 8 May 2003)

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The analysis of color-superconducting two-flavor deconfined quark matter at moderate densities isextended to include a particular spin-1 Cooper pairing of those quarks which do not participate in thestandard spin-0 diquark condensate. (i) The relativistic spin-1 gap �0 implies spontaneous breakdown ofrotation invariance manifested in the form of the quasifermion dispersion law. (ii) The criticaltemperature of the anisotropic component is approximately given by the relation T0

c ’ �0�T � 0�=3.(iii) For massless fermions the gas of anisotropic Bogolyubov-Valatin quasiquarks becomes effectivelygapless and two dimensional. Consequently, its specific heat depends quadratically on temperature.(iv) All collective Nambu-Goldstone excitations of the anisotropic phase have a linear dispersion lawand the whole system remains a superfluid. (v) The system exhibits an electromagnetic Meissner effect.

DOI: 10.1103/PhysRevLett.90.182002 PACS numbers: 12.39.Ki, 11.30.Qc, 12.38.Aw

a self-consistency equation [8]. For each flavor there is an the projector on color 3. � is a ground-state expectation

Recent investigations suggest that the phase structureof QCD is very rich [1,2]. At low temperatures and highdensities, strongly interacting matter is expected to be acolor superconductor [3]. At asymptotically high den-sities, where the QCD coupling constant becomes small,this can be analyzed starting from first principles [4,5],whereas at more moderate densities, present (presumably)in the interiors of neutron stars, these methods are nolonger justified. In this region, the low-energy dynamicsof deconfined quark matter is often studied employ-ing effective Lagrangians Leff which contain local ornonlocal four-fermion interactions, most importantly in-teractions derived from instantons or on a more phenome-nological basis [6–8]. The nonconfining gluon SU�3�cgauge fields are then treated as weak external perturba-tions, and neglected in lowest approximation.

In this Letter, we consider the case of two flavors whichis most likely relevant at chemical potentials just abovethe deconfinement phase transition. On physical grounds,it is then natural to assume that Leff favors the sponta-neous formation of spin-0 isospin-singlet Cooper paircondensates [1,9] � � h TC�5�22 i, where is a quarkfield, C the matrix of charge conjugation, �2 a Paulimatrix which acts in flavor space, and 2 a Gell-Mannmatrix which acts in color space. Because of the latter,SU�3�c is broken down to SU�2�c. This has the followingconsequences for the physical excitations of the system.

(i) Corresponding to the mixing of the colors 1 and 2,there are two Bogolyubov-Valatin quasiquarks for eachflavor with the dispersion law E�

1 � ~pp� � E�2 � ~pp�

E�� ~pp� �� p ��2 j�j2

q. The energy gap � is the

solution of a self-consistent gap equation and is foundto be typically of the order of �100 MeV in modelcalculations [6–8]. p �

��~pp2 M2

p, where M is an effec-

tive Dirac mass, related to the chiral condensate h i via

0031-9007=03=90(18)=182002(4)$20.00

unpaired quark of color 3 with the dispersion law �3 � ~pp� � p � �.

(ii) Because of the spontaneous breaking of SU�3�cdown to SU�2�c five of the eight gluons receive a mass(Meissner effect), whereas three remain massless [10].Since no global symmetry is spontaneously broken, thereare no massless Goldstone bosons.

(iii) The condensate � is invariant under a local U�1�transformation generated by ~QQ � Q �1=�2

��3

p��8,

where Q is the electromagnetic charge operator, and 8is a Gell-Mann matrix in color space. As long as thissymmetry is not broken by other condensates, there is a‘‘new’’ photon (a linear combination of the ‘‘normal’’photon and the eighth gluon) which remains massless.This means there is no electromagnetic Meissner effect.

According to Cooper’s theorem, any attractive interac-tion leads to an instability at the Fermi surface. It istherefore rather unlikely that the Fermi sea of color-3quarks stays intact. As only quarks of a single color areinvolved, the pairing must take place in a channel whichis symmetric in color. Assuming s-wave condensation inan isospin-singlet channel, a possible candidate is a spin-1 condensate [6]. Although the size of the correspondinggap was estimated to be much smaller than � [6], itsexistence can have important astrophysical consequen-ces. For example, if all quarks are gapped, the specificheat of a potential quark core of a neutron star (and,hence, the cooling of the star) is governed by the sizeof the smallest gap [1]. The same is true for other trans-port properties, such as neutrino emissivity or viscosity.

This Letter is devoted to a quantitative analysis of thispossibility. To this end, we consider the condensate

�0 � h TC�03�2PP�c�3 i; (1)

where �� i=2 ��; �� and PP�c�3 � 1=3 1=

��3

p8 is

0

2003 The American Physical Society 182002-1


value of a complex vector order parameter �0n �ndescribing spin-1 and breaking spontaneously the rota-tional invariance of the system. There are well-knownexamples for spin-1 pairing in condensed matter physics,e.g., superfluid 3He, where some phases are also aniso-tropic [11]. In relativistic systems this is certainly not avery frequent phenomenon. It is possible only at finitechemical potential, which itself breaks Lorentz invari-ance explicitly. (Relativistic Cooper pairing into spin-1with nonzero angular momentum was considered else-where, e.g., [9,12].) Another example of nonisotropicquark pairing are crystalline color superconductors [13].The role of spin-1 condensates as an alternative to crys-tals in single color or single flavor systems has also beendiscussed in Ref. [14].

Since rotational invariance is a global symmetry of theprimary Lagrangian, an arbitrary small gap of the aniso-tropic phase implies specific gapless collective excitationswith given Landau critical velocity crucial for the super-fluid behavior of the system. We will briefly discuss this atthe end of this Letter. It is also interesting to note that �0 isnot neutral with respect to the ‘‘rotated’’ electric charge~QQ, and there is no generalized electric charge for whichboth � and �0 are neutral. This means, if both � and �0

were present in a neutron star, there would be an electro-

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magnetic Meissner effect, which would strongly influencethe magnetic field. Recently, similar effects have beendiscussed in Ref. [15]. The detailed evaluation of theMeissner masses for our case remains to be done.

For the quantitative analysis, we have to specify theinteraction. Guided by the structure of instanton-inducedinteractions (see, e.g., [7]), we consider a quark-antiquarkterm and a quark-quark term,

Lq qq �Gf� �2 � ~�� 2 � i�5 �2 � i�5 ~�� 2g;

Lqq � Hs

XO��5;1

� OC�2A T�� TCO�2A T�

Ht� ��C�2S T�� TC��2S T�;

(2)

where A and S are the antisymmetric and symmetriccolor generators, respectively. For instanton-induced in-teractions, the coupling constants fulfill the relationG:Hs:Ht � 1: 34 :

316 , but for the moment we will treat

them as arbitrary parameters. As long as they stay posi-tive, the interaction is attractive in the channels givingrise to the diquark condensates � and �0 as well as to thechiral condensate h i. It is straightforward to calculatethe mean-field thermodynamic potential ��T;�� in thepresence of these condensates:

��T;�� 4X3i�1

X

Z d3p

�2!�3

�E�i

2 T ln�1 e E

�i =T�

�

1

4G�M m�2

1

4Hsj�j2

1

16Htj�0j2; (3)

where m is the bare quark mass, M � m 2Gh i, � � 2Hs�, and �0 � 4Ht�

0. The dispersion law for quarksof color 3 reads

E�3 � ~pp� �

��

��M2

eff ~pp2q

��eff�2 j�0

effj2

r; (4)

where �2eff � �2 j�0j2sin2$, Meff � M�=�eff , and

j�0eff j

2 � j�0j2 �cos2$M2=�2eff sin

2$�. Here cos$ �p3=j ~ppj. Thus, as expected, for �0 � 0, E�

3 �~p�p� is an an-

isotropic function of ~pp, clearly exhibiting the spontaneousbreakdown of rotational invariance. For M � 0, the gap�0

eff vanishes at $ � !=2. In general, its minimal valueis given by �0

0 � Mj�0j=��2 j�0j2

p. Expanding E

3around its minimum, the low-lying quasiparticlespectrum takes the form E

3 �p?; p3� ��02

0 v2?�p? p0�2 v23p

23

q, where v? � f1

��020 =�M�02��2g1=2, v3 � �0

0=M, p0 �v?v3j�0j, and p2

? �p21 p2

2. This leads to a density of states linear in energy:

N�E� �1

2!�2 j�0j2

j�0jE$�E �0

0�: (5)

The actual values for �, �0, and M follow from thecondition that the stable solutions correspond to the ab-solute minimum of � with respect to these quantities.Imposing @�=@�0� � 0 leads to the following gap equa-tion for �0:

�0 � 16Ht�0X

Z d3p

�2!�3

1�

~pp2?

s

1

E�3

tanhE�3

2T; (6)

where s � �eff� ~pp2 M2

eff�1=2. Similarly, one can derive

gap equations for � and M by the requirements@�=@�� 0 and @�=@M � 0, respectively. Togetherwith Eq. (6), they form a set of three coupled equations,which have to be solved simultaneously. However, theequations for � and �0 are not directly coupled, butonly through their dependence on M.

In our numerical calculations, we use a sharp three-momentum cutoff � to regularize the integrals. We thenhave five parameters: m, �, G, Hs, and Ht. We choosem � 5 MeV, � � 600 MeV, andG�2 � 2:4—leading toreasonable vacuum properties, M � 393 MeV andh uuui � � 244 MeV�3—and the instanton relation to fixHs and Ht (‘‘parameter set 1’’). The resulting values ofM,�, and �0 as functions of � at T � 0 are displayed inFig. 1. The chemical potentials correspond to baryondensities of about 4–7 times nuclear matter density.In agreement with earlier expectations [6], �0 is smallcompared with �. However, its value rises strongly with�. Being a solution of a self-consistency problem, �0

is also extremely sensitive to the coupling constant Ht.If we double the value of Ht (‘‘parameter set 2’’), wearrive at the dash-dotted line for �0, which is then com-parable to �. As a consequence of the factor �1 ~pp2

?=s�

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0.1

1

10

100

400 450 500

M, ∆

, ∆′ [

MeV

]

µ [MeV]

FIG. 1. M (dotted line), � (dashed line), and �0 (solid line) atT � 0 as functions of the quark chemical potential � usingparameter set 1 (see text). The dash-dotted line indicates theresult for �0 taking parameter set 2.

0

0.5

1

0 0.2 0.4 0.6

∆ i/∆

i(T=

0)

T /∆i(T=0)

FIG. 2. �i=�i (T � 0) as a function of T=�i (T � 0). Dashedline: �i � �. Solid line: �i � �0. The calculations have beenperformed at � � 450 MeV for parameter set 1.


in the gap Eq. (6), �0 is very sensitive to value and theform of the cutoff.

With increasing temperature both condensates, � and�0, are reduced and eventually vanish in second-orderphase transitions at critical temperatures Tc and T0

c, re-spectively. It has been shown [5] that Tc is approximatelygiven by the well-known BCS relation Tc ’ 0:57�(T � 0). In order to derive a similar relation for T0

c,we inspect the gap Eq. (6) at T � 0 and in the limitT ! T0

c. Neglecting M (since M � � this is valid up tohigher orders in M2=�2) and antiparticle contributions,one gets

Z d3p

�2!�3

��1

~pp2?

s

1

E 3 � ~pp�

��0�T�0�

1

~pp2?

�j ~ppj

1

j� j ~ppjjtanh

j� j ~ppjj2T0

c

�� 0: (7)

Since the integrand is strongly peaked near the Fermisurface, the j ~ppj integrand must approximately vanish atj ~ppj � �, after the angular integration has been per-formed. From this condition, one finds to lowest orderin �0=�: T0

c=�0�T � 0� � 1=3. The analogous stepswould lead to Tc=��T � 0� � 1=2 instead of the textbookvalue of 0.57. This gives a rough idea about the quality ofthe approximation. Note that there are other examples ofdiquark condensates, where Tc � 0:57� (T � 0) [16].This is also the case for crystalline superconductors [17].

Numerical results for ��T� and �0�T� are shown inFig. 2. The quantities have been rescaled in order tofacilitate a comparison with the above relations for Tcand T0

c. Our results are in reasonable agreement with ourestimates. These findings turn out to be insensitive to theactual choice of parameters.

The specific heat is given by cv � T@2�=@T2 [18].For T � Tc, it is completely dominated by quarks of color3, since the contribution of the first two colors is sup-pressed by a factor e �=T . Neglecting the T dependence ofM and �0, and employing Eq. (5), one finds

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cv �12

!�2 j�0j2

j�0jT2e �0

0=TX3n�0

1

n!

�0

0

T

n; (8)

which should be valid for T � T0c. In this regime, cv

depends quadratically on T for T * �00, and is exponen-

tially suppressed at lower temperatures. To test this rela-tion, we evaluate cv�T� explicitly using Eq. (3). Theresults for fixed � � 450 MeV are displayed in Fig. 3.For numerical convenience, we choose parameter set 2,leading to a relatively large �0�T � 0� � 30:8 MeV. Thecritical temperature is T0

c ’ 0:40�0 (T � 0). For the en-ergy gap, we find �0

0 � 0:074T0c. It turns out that Eq. (8),

evaluated with constant values of �0 and M (dash-dottedline) is in almost perfect agreement with the numericalresult (solid line) up to T � T0

c=2. The phase transition,causing the discontinuity of cv at T � T0

c, is of courseoutside the range of validity of Eq. (8). We also display cvfor M � 0 (dashed line). Since �0

0 vanishes in this case,there is no exponential suppression, and cv is proportionalto T2 down to arbitrarily low T. However, even whenM isincluded, the exponential suppression is partially can-celed by the sum on the right-hand side of Eq. (8). Forcomparison, we also show cv for a system with �0 � 0,which exhibits a linear T dependence at low temperatures(dotted line).

Our results show that, even though the magnitude ofthe gap parameter �0 is strongly model dependent, itsrelations to the critical temperature and the specific heatare quite robust. Thus, if we had empirical data, e.g., forthe specific heat of dense quark matter, they could be usedto extract information about the existence and the size of�0. In this context, neutron stars and their cooling proper-ties are the natural candidates to look at. In Ref. [1], it wassuggested that the exponential suppression of cv related tothe potential pairing of quarks of color 3 might haveobservable consequences for the neutrino emission of aneutron star. This argument has to be somewhat refinedsince, as seen above, cv�T� first behaves as T2 and theexponential suppression sets in only at T <�0

0. The rele-vance of cv and the possible effect of diquark condensates

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1 x 10

0.1

1

0.01 0.1 1

c v [

fm-3

]

T / Tc’

−2

1 x 10−3

1 x 10−4

1 x 10−5

FIG. 3. Specific heat for parameter set 2 at � � 450 MeV asa function of T=T0

c. Solid line: full calculation; dashed line:result for M � 0; dotted line: without spin-1 condensate. Thedash-dotted line indicates the result of Eq. (8).


on neutron star cooling was also discussed in Ref. [20].On the other hand, it has recently been argued [21] thatthe constraints imposed by charge and color neutralitymight completely prohibit the existence of two-flavorcolor-superconducting matter in neutron stars.

Because of the spontaneously broken U�1� �O�3�symmetry in Eq. (1), for �0 � 0 there should be collec-tive Nambu-Goldstone (NG) excitations in the spectrum.However, due to the Lorentz noninvariance of the system,there can be subtleties [22–24]. The NG spectrum can beanalyzed within an underlying effective Higgs potential,

V�� a2�yn�n

121��

yn�n�

2 122�

yn�

yn�m�m;

for the complex order parameter �n [23], with 12 > 0 for stability. For 2 < 0, the ground state is char-acterized by ��1�

vac � �a2

1�1=2�0; 0; 1� which corresponds to

our ansatz Eq. (1) for the BCS-type diquark condensate�0. This solution has the property h ~SSi2 � ��1�y

vac~SS��1�

vac�2 �

0. The spectrum of small oscillations above ��1�vac consists

of 1 2 NG bosons, all with linear dispersion law: onezero-sound phonon and two spin waves [23]. Implying afinite Landau critical velocity, this fact is crucial for amacroscopic superfluid behavior of the system [24].

Note that for 2 > 0 there is a different solution��2�vac �

� a22�12�

�1=2�1; i; 0� with h ~SSi2 � 1. In this case, the NGspectrum above ��2�

vac consists of one phonon with lineardispersion law and one spin wave whose energy tends tozero with momentum squared [23]. The quasiquark dis-persion law corresponding to ��2�

vac has recently been dis-cussed in Ref. [14]. A detailed analysis of the transportproperties implied by this type of pairing seems of par-ticular interest for the phenomenology of neutron starssince the dispersion law allows for gapless excitationseven for massive quarks [14].

We thank D. Blaschke, D. Litim, K. Rajagopal, I. A.Shovkovy, and E.V. Shuryak for useful discussions. J. H.thanks J. Wambach and IKP TU Darmstadt for generoushospitality and support. We acknowledge financial sup-port by ECT� during its 2001 collaboration meeting on

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color superconductivity. This work was supported in partby Grant No. GACR 202/02/0847. M. O. acknowledgessupport from the Alexander von Humboldt foundation.

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Anisotropic Admixture in Color-Superconducting Quark Matter