Surface effects in color superconducting strange-quark matter

Surface effects in color superconducting strange-quark matter

Micaela OertelLUTH, Observatoire de Paris, CNRS, Universite Paris Diderot, 5 Place Jules Janssen, 92195 Meudon, France

Michael UrbanIPN Orsay, CNRS-IN2P3 and Universite Paris-Sud, 91406 Orsay Cedex, France

(Received 15 January 2008; revised manuscript received 21 February 2008; published 21 April 2008)

Surface effects in strange-quark matter play an important role for certain observables which have beenproposed in order to identify strange stars, and color superconductivity can strongly modify these effects.We study the surface of color superconducting strange-quark matter by solving the Hartree-Fock-Bogoliubov equations for finite systems (‘‘strangelets’’) within the MIT bag model, supplemented witha pairing interaction. Because of the bag-model boundary condition, the strange-quark density issuppressed at the surface. This leads to a positive surface charge, concentrated in a layer of �1 fmbelow the surface, even in the color-flavor locked (CFL) phase. However, since in the CFL phase allquarks are paired, this positive charge is compensated by a negative charge, which turns out to be situatedin a layer of a few tens of fm below the surface, and the total charge of CFL strangelets is zero. We alsostudy the surface and curvature contributions to the total energy. Because of the strong pairing, the energyas a function of the mass number is very well reproduced by a liquid-drop type formula with curvatureterm.

DOI: 10.1103/PhysRevD.77.074015 PACS numbers: 21.65.Qr, 12.39.Ba, 26.60.�c

I. INTRODUCTION

From rather general arguments it is expected that at lowtemperatures and high densities quark matter is in a colorsuperconducting state [1]. More recently [2,3] it has beensuggested that the diquark pairing gaps for quark matter atdensities of several times nuclear matter saturation densitycould be of the order of �100 MeV. Since this could haveimportant phenomenological consequences, in particular,for the interior of compact stars, this has triggered muchwork on color superconductivity in dense quark matter (forreviews, see, e.g., Ref. [4]). These investigations of theQCD phase diagram have revealed a very rich phase struc-ture with many different possible pairing patterns, depend-ing on external conditions such as, for instance, electricalneutrality or quark masses. The largest diquark pairinggaps arise from scalar condensates, leading either to thetwo-flavor color superconducting (2SC) phase or to thecolor-flavor-locked (CFL) phase [5,6]. The latter pairingpattern involves strange (s) quarks, in addition to the twolight-quark flavors, up (u) and down (d).

If color superconducting quark matter exists in nature,the most likely place to find it is the interior of compactstars because matter is compressed there to densities muchhigher than nuclear matter saturation density. However, ithas been argued that strange-quark matter (SQM) might beabsolutely stable [7]. Under this hypothesis, even purestrange stars should exist [8], i.e., stars entirely composedof SQM. Also small lumps of SQM, called ‘‘strangelets,’’might be stable. Because of their low charge to baryonnumber ratio Z=A, strangelets have been proposed topopulate ultrahigh energy cosmic rays [9].

In SQM without pairing, the density of strange quarks issupposed to be smaller than that of light quarks because oftheir higher mass. Consequently, SQM and strangelets arepositively charged and the charge neutrality of strange starshas to be achieved via the presence of electrons. At thesurface an atmosphere of electrons forms [8] which canpotentially be detected [10,11] via the emission ofelectron-positron pairs from an extremely strong electricfield at the surface.

Recently another possible picture of the surface of astrange star has been proposed [12]: there could be a‘‘crust’’ composed of strangelets immersed in an electrongas. Similar to an ordinary neutron star, there could be aninterface between the crust and the interior in form of thefamous ‘‘pasta phases.’’ Within this scenario the electricfield at the surface would be strongly reduced. Obviously,surface effects for the strangelets play an important role forthe description of this scenario. For instance, there is acritical surface tension deciding whether a homogeneousphase or the droplet phase is favored [13]. Another ques-tion for which surface effects should be considered is theformation of a strange star in a supernova explosion.Before the explosion the original star contains hadronicmatter. During the formation of the star, nucleation ofstrangelets sets in, leading then to a conversion of the entirestar to SQM. For the nucleation process the properties ofsmall strangelets are important.

Pairing tends to reduce the differences in density ofdifferent quark species. For bulk quark matter in the CFLstate, requiring color neutrality, all quarks are paired. Thedensities are thus equal and CFL quark matter is electri-cally neutral on its own, i.e., without any electrons [14].

PHYSICAL REVIEW D 77, 074015 (2008)

1550-7998=2008=77(7)=074015(13) 074015-1 © 2008 The American Physical Society

http://dx.doi.org/10.1103/PhysRevD.77.074015

This would suggest dramatic changes in the properties ofstrangelets and SQM inside compact stars. For instance,the electrosphere at the surface of a strange star couldcompletely disappear. But, the presence of the surfacecan modify this picture since it can lead to a nonzerosurface charge which remains even for large objects. Forexample, the boundary condition of the MIT bag modelsuppresses the density of the massive strange quarks at thesurface, resulting in a positive surface charge [15]. Withinthis scenario, the total charge of a strangelet, followingroughly Z � 0:3A2=3, is drastically reduced with respect to‘‘normal’’ strangelets. For strange stars, this requires thepresence of electrons [16]. However, pairing has not beentreated self-consistently in previous work (see, e.g.,Ref. [15]). In this paper we will therefore reinvestigatefinite-size strangelets with pairing by considering quarkmatter in a color superconducting spherical bag, solvingthe Hartree-Fock-Bogoliubov (HFB) equations. We willshow, in particular, that there exist CFL-type solutionswhere all quarks are paired and the total charge of thestrangelet strictly vanishes.

The outline of the paper is as follows. In Sec. II we willpresent our model for treating color superconducting quarkmatter in a finite volume. In Sec. III we will show numeri-cal results. In Sec. III A we discuss the possibility ofqualitatively different configurations. In Sec. III B we con-centrate on the charge-density distributions of the CFL-likesolutions. In Sec. III C we discuss a liquid-drop-like massformula for the CFL-like solutions and calculate the sur-face tension. Finally, in Sec. IV we will summarize ourresults.

II. MODEL

A. Lagrangian

Since it is not possible to describe strangelets or SQMwith a surface from first principles (QCD), we will use aquark model which allows to describe finite-size objects.For this purpose we will use here the MIT bag model [17].The idea of this model is that confinement can be simulatedby the existence of a ‘‘bag’’ which consists of a ‘‘hole’’ inthe nonperturbative QCD vacuum. Inside this bag, thevacuum is supposed to be perturbative, i.e., inside thebag the interactions of the quarks can be treated perturba-tively. To create this hole in the nonperturbative QCDvacuum, an energy per volume, B, is necessary. In thepresent work we will consider a static spherical bag withradius R. On the surface of the bag, the quark field has tosatisfy an appropriate boundary condition. In the simplestversion of the MIT bag model, the boundary conditionreads

� ier � � � jr�R; (1)

which ensures that there is no particle flux across thesurface. By r � jrj we denote the radial coordinate, mea-sured from the center of the bag, and er � r=r is the radial

unit vector. The boundary condition (1) leads to a suppres-sion of the wave functions of massive particles at thesurface. This means that the strange-quark density will apriori be suppressed at the surface with respect to the light-quark densities.

The MIT bag model can be expressed in terms of aLagrangian density as follows [18]:

Lbag � � � �i��@� �m� � B��R� r�

�1

2� ��R� r�; (2)

where m is the matrix of quark masses. Because of thesecond term, the boundary condition (1) follows immedi-ately from the Euler-Lagrange equation for the quark field[18].

In order to include pairing, we will supplement the bagmodel with a pairing interaction. In principle, perturbativeone-gluon exchange generates an attractive pairing inter-action in certain channels, in particular, in the scalar colorantitriplet channel. For simplicity, we will use here a four-point pairing interaction acting only in this dominantchannel. The corresponding Lagrangian reads (see any ofthe standard review articles on color superconductivity [4])

L pair � HXA;A0� � i�5�A�A0C � T�� TCi�5�A�A0 �; (3)

where H is a dimensionful coupling constant, C denotesthe charge conjugation matrix, and �A, �A0 represent SU�3�matrices in flavor and color space, respectively. We followthe convention that capital letters A, A0 indicate that we arerestricting �A and �A0 to be antisymmetric, i.e., in terms ofthe Gell-Mann matrices, A, A0 2 f2; 5; 7g.

In addition to the strong interaction, the quarks willexhibit electromagnetic interactions which, due to theirlong range, become particularly important for large ob-jects. The corresponding Lagrangian reads

L e:m: � �1

4F��F

�� e � QA�� ; (4)

where F�� @�A� � @�A� and A� denote, respectively,the electromagnetic field strength tensor and four-potential, and Q is the matrix of quark charges in units ofe, Qu � 2=3, Qd � Qs � �1=3.

It would be in the spirit of the bag model to include alsothe gluon exchange in a perturbative way, i.e., in the sameway as the photon. However, this goes beyond the scope ofthe present paper and will be postponed to a future study.

B. Solution in the framework of HFB theory

The model described by L � Lbag Lpair Le:m: willbe treated in the framework of HFB theory. By minimizingthe energy in mean-field approximation (for more details,see Appendix B and Ref. [19] where the ‘‘Dirac-Hartree-Bogoliubov’’ approximation was developed for finite nu-clei), one obtains the following HFB equations:

MICAELA OERTEL AND MICHAEL URBAN PHYSICAL REVIEW D 77, 074015 (2008)

074015-2

h �� h

� �U��r��0V��r�

� ��

U��r��0V��r�

� �: (5)

The single-particle Hamiltonian

h � �i� � rm�0 �� (6)

includes besides the free Dirac Hamiltonian the quark self-energy � (in our case due to Coulomb interaction) and thematrix of chemical potentials � which depend on flavorf 2 fu; d; sg and color c 2 fr; g; bg (we will denote thethree colors by red, green, and blue). � denotes the pairingfield (gap). The spinors U� and V� describe the particle-and hole-like components of the quark fields, respectively[see Eq. (B3)], where � is a multi-index containing allquantum numbers characterizing a single-particle state(see Appendix A). In writing Eq. (5), we implicitly as-sumed that the pairing field � can be chosen real, which isthe case for the pairing pattern we consider, and that theself-energy � is local, which is the case since we neglectthe exchange (Fock) term (see below).

The pairing field � and the self-energy � depend them-selves on the wave functionsU and V, such that we have tosolve a self-consistency problem. To be specific, the pair-ing field � depends on the diquark condensates

sAA0 �x� � �h � T�x��A�A0 �x�i; (7)

where T denotes the time-reversed conjugate of ,

T � �5C � T: (8)

The diquark condensates can be expressed in terms of theU and V functions as

sAA0 �r� � �X

;<0

�V�r��A�A0U�r� (9)

(since we are dealing with a static problem, the conden-sates do not depend on time, and due to spherical symme-try, they depend only on the radial coordinate r). We willlimit our investigations here to diagonal condensates, i.e.,only condensates with A � A0 are nonzero.1 In uniforminfinite matter and for an exact SU�3� flavor symmetry, theCFL phase is characterized by nonzero values s22 � s55 �s77, whereas the 2SC state has only s22 � 0. The relationbetween the condensates sAA and the pairing field � reads

��r� �X

A�2;5;7

�A�r��A�A; (10)

�A�r� � 2HsAA�r�: (11)

In practice, the expression (9) is divergent, and it is neces-sary to introduce a cutoff in order to obtain a finite result.

Since in a finite system the levels are discrete, a sharpcutoff would generate discontinuities as a function of thesystem’s size. We therefore introduce a smooth cutofffunction f�p=�� (see Appendix C for details). Anotherpractical problem arises from antiparticle contributions.However, since the chemical potentials �fc are large andpositive and pairing involves mostly the states near theFermi surface, we assume that the antiparticle contribu-tions are not important and can be neglected. We checkedthis approximation (analogous to the ‘‘no-sea approxima-tion’’ in nuclear physics [19]) in infinite matter and foundthat the effect of antiparticle states can be absorbed in areadjustment of the coupling constant by �20%.

For the normal self-energy � we employ the Hartreeapproximation, i.e., we neglect the Coulomb exchange(Fock) term as well as exchange contributions from themagnetic field. We also disregard the contribution of Lpair

to the normal self-energy. Hence, the self-energy is simplyproportional to the static Coulomb potential

��r� � eQA0�r��0: (12)

The Coulomb potential is related to the quark densities by

A0�r� � eZd3r0

�ch�r0�jr� r0j

; (13)

where

�ch�r� �Xf

Qf�f�r� (14)

is the charge density (divided by e), �f being the numberdensity of quarks of flavor f. As it was the case for thediquark condensates, the quark number densities can beexpressed in terms of the U and V functions. Denoting by~ all single-particle quantum numbers except flavor, wecan write the number density of quarks of flavor f as

�f�r� �X

~;f ~<0

Uyf ~�r�Uf ~�r�: (15)

Let us now summarize the procedure concerning howthe HFB equations are solved. We start with an initial guessfor the pairing fields �A�r� and for the Coulomb potentialA0�r�. Then we solve the eigenvalue problem (5) in order tofind the U and V functions. From these functions thediquark condensates sAA�r� and the quark densities �f�r�are computed according to Eqs. (9) and (15), which arethen used to update the pairing fields �A and the Coulombfield A0 according to Eqs. (11) and (13). These steps areiterated until convergence (i.e., self-consistency) isreached.

The crucial difference to the BCS formalism in homo-geneous infinite matter is that in our case the wave func-tions adapt themselves to the pairing field and to theCoulomb potential, whereas in the case of homogeneousinfinite matter the wave functions always stay plane waves,

1In uniform infinite matter it can be shown [5] that for theenergetically favored solution the arbitrary orientation in colorcan be chosen in such a way that only the diagonal condensateswith A � A0 are nonzero.

SURFACE EFFECTS IN COLOR SUPERCONDUCTING . . . PHYSICAL REVIEW D 77, 074015 (2008)

074015-3

and the U and V factors are just coefficients multiplyingthem.

C. Determination of chemical potentials and bag radius

In Sec. II B we described how the HFB equations aresolved for given values of the chemical potentials �fc andof the bag radius R. However, in reality, only one quantityis given, namely, the baryon number A. Even the fractionsof different quark flavors cannot be fixed, unless one allowsfor unstable strangelets. Let us now describe how wedetermine the chemical potentials�fc and the bag radius Rfor given baryon number A.

The first step consists in fixing the quark numbers, Nfc,for each flavor f and color c, and to adjust the chemicalpotentials �fc in order to obtain these quark numbers.Before we address the question of how the nine quarknumbers Nfc are determined, let us discuss the issue ofthe bag radius R. Until now, the radius was imposed fromoutside, but in reality the system will choose its radius suchthat it minimizes its total energy for given quark numbersNfc. Within the bag model, the total energy is given by

E � Eq BV; (16)

where V � 4�R3=3 is the volume of the bag. By Eq wedenote the energy of the quarks inside the bag, includingthe interaction energy, which in our case comes frompairing and Coulomb interactions. It can be obtainedfrom the solution of the HFB equations as follows [19]:

Eq �Zr<R

d3rX

;<0

�Uy�r��U�r�

1

2� �U�r��r�V�r� �U

y�r�eQA0�r�U�r�

�: (17)

Minimizing the total energy E is of course completelyequivalent to saying that the quark pressure in the bag iscounterbalanced by the bag pressure B, i.e.,

dEqdV

��N� �B: (18)

This equation determines the radius of the strangelet forgiven bag pressure B, interaction strength H, and quarknumbers Nfc. In practice, however, we find it more conve-nient to minimize E rather than solve Eq. (18).

Let us now turn to the determination of the quark num-bers. The nine quark numbers Nfc cannot be chosen arbi-trarily, but they have to fulfill certain requirements.Imposing the total baryon number A and color neutrality,i.e., equal numbers of quarks for each color, we have tosatisfy the constraint

Xf

Nfc � A for all c: (19)

Of course, these three equations are not sufficient for

determining all the nine quark numbers. In order to getunique values for the Nfc, it is necessary to impose stability, as we will describe now.

In an infinite homogeneous system the condition for equilibrium gives just a relation between the chemicalpotentials2

�dc � �sc � �uc �e for all c: (20)

In a small system this is slightly different. First, even ifthere are electrons (i.e., if the strangelet is charged), theyare not localized inside the strangelet, but they form a largecloud like in ordinary atoms and hence their chemicalpotential �e is approximately equal to the electron massand can be neglected. Without pairing, it has been esti-mated in Ref. [20] that this may be still the case forstrangelets with charge Z & 1000, corresponding roughlyto A & 106. The second difference to bulk matter comesfrom the fact that, due to the discrete levels, particlenumbers are discontinuous functions of the chemical po-tentials. The term equilibrium should now be replaced by stability, which means that the system does not gainenergy by performing a decay, inverse decay, orelectron capture, i.e., transforming an up into a down orstrange quark, or vice versa, accompanied by the corre-sponding leptons.

To achieve stability, we therefore compare the ener-gies of adjacent strangelets with the same total quarknumber per color, differing only in the number of up,down, and strange quarks, respectively, in order to findthe configuration with the lowest total energy E. Of course,in the case of large particle numbers, the minimum-energyconfiguration fulfills approximately the condition (20).

D. Choice of the model parameters

Besides the quark masses, which we take asmu � md �0 and ms � 120 MeV, our model contains three parame-ters: the bag constant B, the coupling constant of thepairing interaction, H, and the cutoff � which is necessaryto avoid the divergence of the gap Eq. (9), see belowEq. (11). In fact, a change of the cutoff in reasonable limitscan to very good approximation be compensated by achange of the coupling constant. We therefore chooserather arbitrarily � � 600 MeV and give instead of thedimensionful coupling constant H the dimensionless com-bination H�2. So we are left with two parameters, B andH�2.

We can get an idea of the value of the bag pressure bylooking at the stability of bulk quark matter. Non-strange-quark matter should be energetically less favored thannormal hadronic matter, whereas SQM should be stableif for some baryon number A> Ac strangelets becomestable and consequently strange stars can exist. This means

2Here we assume that neutrinos are not trapped, i.e., they canfreely leave the system.


074015-4

that we want the energy per baryon of SQM to be less than931 MeV, the energy per baryon of the most stable nucleus,57Fe. On the other hand, the energy per baryon of non-strange-quark matter should be larger than the nucleonmass. Without interaction the window for the values ofthe bag constant is then 148 MeV<B1=4 < 157 MeV.These values change as a function of the interactionstrength H. To better compare the results, we will readjustfor each coupling strength the bag constant in order to getE=A � 900 MeV. The corresponding values are listed inTable I, together with other properties of infinite matter.Non-strange-quark matter is unstable with these parametervalues. Note that for the weakest nonvanishing couplingconstant given in Table I, SQM is in the 2SC phase and notin the CFL phase. For the larger coupling constants, theCFL phase is preferred. Note that, due to the mass differ-ence of light and strange quarks, the flavor SU�3� symme-try is not exact and the gap �2 is different from �5 and �7.

However, since the CFL phase is electrically neutral, andwe have mu � md � 0, the isospin SU�2� symmetry in theup- and down-quark sector is exact and therefore �5 � �7.

III. RESULTS

A. Different types of solutions

We will first discuss the qualitatively different configu-rations we find. Let us start by discussing a small strangelet(A � 108, Z � 24) without any pairing interaction(H�2 � 0). The mass number has been chosen such thatthe minimum-energy configuration is a closed-shell con-figuration. The quark numbers and other relevant informa-tion are listed in Table II. Because of the finite size of thebag, the energy per baryon (E=A � 932:5 MeV, including1.0 MeV due to Coulomb) is much higher than that of colorneutral infinite matter with �uc � �dc � �sc

3 (E=A �899:5 MeV). This effect will be discussed in more detailin Sec. III C. The density profiles of light and strangequarks are shown in Fig. 1. As expected, due to theboundary condition, the strange-quark density is stronglysuppressed at the surface, contrary to the densities of thelight quarks. For comparison we mention that for the samevalue of the bag constant, the densities in color neutralinfinite matter with �uc � �dc � �sc are �u � �d �0:355 fm�3, �s � 0:274 fm�3. We see that not only thestrange-quark density but also the densities of the lightquarks are quite different from these values and dependstrongly on r because of the existence of discrete levels inthe bag. Let us mention that, due to the Coulomb potential,the density profiles of up and down quarks are slightlydifferent, but the difference is too small to be visible inFig. 1.

TABLE I. Values of the bag constants for different values ofthe coupling constant H, resulting in color and electricallyneutral SQM with electrons in equilibrium with an energyper baryon of E=A � 900 MeV. The corresponding baryondensities �B, electron densities �e, and pairing gaps in infinitematter are also displayed.

H�2 B1=4�MeV��B�fm�3� �e�fm

�3� �2�MeV��5 � �7�MeV�

0 152.03 0.329 7:3� 10�6 0 01.5 152.44 0.339 9:7� 10�5 27.7 01.75 153.97 0.367 0 35.1 34.52 156.26 0.395 0 50.6 49.72.25 159.46 0.427 0 67.2 66.02.5 163.46 0.463 0 84.6 83.1

TABLE II. Parameters and properties of the strangelets discussed in Sec. III A: B � bag constant, H �coupling constant of the pairing interaction, A � baryon number, Z � charge, Nfc � number of quarks of flavor f and color c,E=A � energy per baryon, R � radius of the bag, �A�0� � value of the gap at r � 0.

B1=4�MeV� H�2�MeV� A Z Nur Nug NubNdr Ndg NdbNsr Nsg Nsb

0B@1CA E=A�MeV� R�fm� �2�0��MeV� �5�0��MeV� �7�0��MeV�

152.03 0 108 2444 44 4444 44 4420 20 20

0@ 1A 932.5 4.36 0 0 0

152.44 1.5 108 2444 44 4444 44 4420 20 20

0@ 1A 930.7 4.31 32.9 0 0

153.97 1.75 108 2444 44 4444 44 4420 20 20

0@ 1A 934.0 4.24 49.5 0 0

153.97 1.75 108 1038 39 4139 38 4131 31 26

0@

1A 934.8 4.21 41.6 24.9 24.9

153.97 1.75 108 038 35 3535 38 3535 35 38

0@ 1A 934.8 4.17 33.9 37.0 36.9


074015-5

Now we switch on the pairing interaction. In the case ofH�2 � 1:5, SQM is in the 2SC phase, i.e., only up anddown quarks of two colors (red and green in our notation)are paired. This is also true in a finite strangelet. Thereforeit is clear that the strange-quark density profile remains thesame as without pairing. The oscillations of the densities ofthe light quarks, however, are much weaker now than in thecase without pairing, since pairing washes out the occupa-tion numbers. This can be seen in Fig. 2. In this 2SC-likesolution, only one of the gaps, �2, is nonzero. Since �2

involves only the wave functions of up and down quarks,which are not suppressed at the surface, it extends up to thesurface of the bag, as shown in Fig. 3. As a function of r, itis almost constant and quite close to the correspondingvalue in infinite matter with �uc � �dc � �sc, which is�2 � 29:2 MeV.

If we increase the coupling constant to H�2 � 1:75, weobtain three qualitatively different solutions which havecomparable energies. The most stable one is still of the

2SC type, although in infinite matter the CFL phase ispreferred. In this case, the strangelet still has Z � 24 andthe density profiles are almost identical to those shown inFig. 2. The main difference is that now the value of the gapis larger.

In the two other solutions, also strange quarks partici-pate in pairing (�5 � �7 � 0—note that �5 and �7 arenot exactly equal because the isospin symmetry is brokenby the Coulomb interaction). These two solutions havecharge Z � 10 and Z � 0, respectively. Let us first discussthe case Z � 10. In this case, there are a couple of up anddown quarks which remain unpaired. The wave function ofthe unpaired level is mainly localized near the surface ofthe bag, as can be seen in Fig. 4, where the density profilesare shown. In the inner part, the densities of up, down, andstrange quarks are almost equal, while near the surface,where the strange-quark density is suppressed due to theboundary condition, there is an excess of up and downquarks. This excess is due to the unpaired quarks. The fact

0

10

20

30

40

50

60

0 1 2 3 4 5

∆ (

MeV

)

r (fm)

∆2

FIG. 3. Gap �2�r� of the strangelet A � 108, Z � 24 in thecase of H�2 � 1:5 and B1=4 � 152:44 MeV.

0

0.1

0.2

0.3

0.4

0.5

0 1 2 3 4 5

ρ (

fm-3

)

r (fm)

ρu, ρd

ρs

FIG. 4. Density profiles of the strangelet A � 108, Z � 10 inthe case of H�2 � 1:75 and B1=4 � 153:97 MeV.

0

0.1

0.2

0.3

0.4

0.5

0 1 2 3 4 5

ρ (

fm-3

)

r (fm)

ρu, ρd

ρs

FIG. 2. Quark number density profiles of the strangelet A �108, Z � 24 in the case of H�2 � 1:5 and B1=4 � 152:44 MeV.

0

0.1

0.2

0.3

0.4

0.5

0 1 2 3 4 5

ρ (

fm-3

)

r (fm)

ρu, ρd

ρs

FIG. 1. Quark number density profiles of the strangelet A �108, Z � 24 in the case of vanishing pairing interaction (freequarks in a bag) and B1=4 � 152:03 MeV.


074015-6

that one level of up and down quarks (in the present casethe 1g9=2 level, i.e., the lowest level with j � 9=2, � �5in the notation of Appendix A) does not participate inpairing means that the occupation number of this level isequal to 1. At the same time, the corresponding level of thestrange quarks has an occupation number equal to 0. In acertain sense this situation is analogous to the ‘‘breachedpairing’’ phase of infinite matter [21]. The charge Z isequal to the degeneracy 2j 1 of the unpaired level. Thegaps �A as functions of r corresponding to this solution aredisplayed in Fig. 5.

In the third solution, all quarks are paired. As a conse-quence, the numbers of up, down, and strange quarks areequal, and the total charge is Z � 0. This is analogous tothe CFL phase in the infinite system. Since the strange-quark density is suppressed near the surface, but the num-ber of strange quarks is equal to that of up and downquarks, it is clear that the strange-quark density must belarger than the up- and down-quark densities in some otherpart of the system. This is indeed the case, as can be seen in

Fig. 6. We also see that the excess of the light-quarkdensities over the strange-quark density is reduced ascompared with the case Z � 10 discussed above(cf. Fig. 4). We will discuss the charge-density distributionin detail in Sec. III B. The gaps, shown in Fig. 7, are muchcloser to the gaps in infinite matter (cf. Table I) than in thecase Z � 10.

For the larger values of the coupling constant we con-sidered (H�2 � 2, 2.25, 2.5), it is always the CFL-typesolution (Z � 0) which has the lowest energy. We do notshow any figures because in all these cases the results areanalogous to those shown in Figs. 6 and 7 (just the valuesof the gaps change; they are close to those given in Table Ifor infinite matter).

It should be mentioned that the fully paired solutionswith Z � 0 are very robust as soon as the coupling constantis sufficiently large, i.e., we find this type of solution forarbitrary numbers of quarks.4 This solution is in contrast toprevious findings (see, e.g., Ref. [15]), where it was sup-posed that the CFL matter should be neutral in the bulkwith just a thin positively charged surface layer with anexcess of up and down quarks because of the boundarycondition. In fact, this idea corresponds roughly to oursolution with unpaired up and down quarks near the sur-face. This solution is, however, very fragile and exists onlyfor certain values of parameters and mass numbers, since itrequires the existence of suitable levels of light and strangequarks near the respective Fermi surfaces which can serveas unpaired levels.

0

0.1

0.2

0.3

0.4

0.5

0 1 2 3 4 5

ρ (

fm-3

)

r (fm)

ρu, ρd

ρs

FIG. 6. Density profiles of the strangelet A � 108, Z � 0 inthe case of H�2 � 1:75 and B1=4 � 153:97 MeV.

0

10

20

30

40

50

60

0 1 2 3 4 5

∆ (

MeV

)

r (fm)

∆2

∆5, ∆7

FIG. 7. Gaps �A as functions of r for the strangelet A � 108,Z � 0 in the case of H�2 � 1:75 and B1=4 � 153:97 MeV.

0

10

20

30

40

50

60

0 1 2 3 4 5

∆ (

MeV

)

r (fm)

∆2

∆5, ∆7

FIG. 5. Gaps �A as functions of r for the strangelet A � 108,Z � 10 in the case of H�2 � 1:75 and B1=4 � 153:97 MeV.

3As discussed below Eq. (20), it is more appropriate tocompare a small strangelet with this kind of matter rather thanelectrically neutral matter with electrons in equilibrium.

4If the number of quarks is odd, it is impossible to pair allquarks, and one or several state(s) should be ‘‘blocked’’ by theunpaired quark(s). At present, we have not included this effect inour calculation, and we restrict ourselves to even quark numbers


074015-7

B. Charge-density distribution

We have seen in Sec. III A that in all cases except the2SC phase, pairing drastically reduces the total charge Z.Because of surface effects, the local charge density does,however, not vanish, even within the CFL-type solutionwhich has Z � 0. Because of the suppression of thestrange-quark wave function at the surface, a positivelycharged surface layer remains with an extension of�1 fm,as has already been pointed out in Ref. [15].

Within the configuration with some unpaired lightquarks at the surface, the total charge of the strangeletresults from this positive surface charge, the interior ofthe strangelet has almost zero charge density. The totalcharge is here reduced compared with a strangelet withoutpairing, for example, the A � 108 strangelet has Z � 10within this paired configuration, whereas the correspond-ing unpaired strangelet has Z � 24. A systematic study ofthe total charge of strangelets in this configuration will notbe discussed here since this configuration is rather fragilewith respect to the details of the single-particle spectra andthus difficult to realize for many different particle numbers.

Let us therefore concentrate on the CFL-type solution,which exists for arbitrary particle numbers. We considerdifferent mass numbers A from A � 108 to A � 90000, forone particular value of the coupling constant, H�2 � 2. Inorder to reduce the considerable numerical effort, we usefor the large strangelets (starting from A � 15000) thecondition (20) with �e � 0 (as a consequence, the quarknumbers for each flavor and color are not integers) insteadof looking for the true energy minimum with respect to decay. In addition, we do not minimize the energy withrespect to the radius, but we simply estimate the volume ofthe bag by dividing the mass number A by the baryondensity �Bbulk of infinite matter. These two approximationsare very accurate for such large strangelets. Already in thecase of A � 3000, the quark numbers and the radius arevery well reproduced within these approximations: the fullminimization results in quark numbers Nur � Ndg �Nsb � 1052, Nug � Ndr � Nub � Nsr � Ndb � Nsg �974, and a radius R � 12:23 fm, while the approximationslead to Nur � 1051:8, Ndg � 1051:7, Nsb � 1051:1,Nug � Ndr � 973:8, Nub � Nsr � 974:4, Ndb � Nsg �974:5, and R � 12:19 fm. Our results for the charge den-sities for A � 108, 3000, 15000, 45000, and 90000 areshown in Fig. 8.

Since all quarks are paired, we have equal numbers ofup, down, and strange quarks such that the total charge ofthese strangelets is zero. The positive surface charge ismostly compensated by an excess of negative charge con-centrated at around 1–3 fm below the surface. We stressthat this concentration of negative charge in a thin layer is aconsequence of pairing and the effect persists if Coulombinteraction is switched off. In fact, since also the strangequarks are paired, the ‘‘missing’’ strange-quark density atthe surface must be compensated by an ‘‘overshooting’’ of

the strange-quark density within a distance correspondingto the size of the Cooper pairs, i.e., the coherence length �.Because of the strong gap, the coherence length is verysmall: Using the estimate �� 1=��, one finds that it isof the same order as the Fermi wavelength and, strictlyspeaking, one might therefore question that mean-fieldresults are quantitatively correct [22]. The smallness of �explains why the compensation of the negative surfacecharge is mostly concentrated in such a thin layer at asmall distance from the surface.

Below this strongly negatively charged layer, the chargedensity stays negative but much smaller. Because ofCoulomb interaction, which tries to push the charge to-wards the surface, this negative charge density decreaseswith increasing distance from the surface, especially forlarge strangelets. Actually, if Coulomb interaction isswitched off, the remaining charge is distributed more orless homogeneously over the whole volume.

The behavior of the charge density far away from thesurface in the presence of Coulomb interaction can easilybe interpreted in terms of Debye screening (similar con-siderations can be found in Ref. [23] for the case of hadron-quark mixed phases): We know that in a uniform mediumwith Debye screening the Laplace equation for theCoulomb potential is replaced by

�r2 �

1

�2

�A0 � 0; (21)

where � is the screening length, which can be obtainedfrom the limit �00�q0 � 0;q! 0�, where ��q� is thepolarization tensor in the uniform system. This is equiva-lent to the expression [23]

1

�2� 4�e2

Xfc

Qf@�ch

@�fc: (22)

Computing numerically this derivative within our model

-0.02

0

0.02

0.04

0.06

0 10 20 30 40

ρ ch

(fm

-3)

r (fm)

FIG. 8. Charge-density profiles of the fully paired (H�2 � 2)strangelets A � 108, 3000, 15000, 45000, and 90000 (from leftto right).


074015-8

for the case of bulk CFL matter with B � 156:26 MeV andH�2 � 2, we obtain � � 7:74 fm.

Taking the Laplacian of Eq. (21), we see that the chargedensity obeys the analogous equation

�r2 �

1

�2

��ch � 0: (23)

In the case of half-infinite matter with a surface at z � 0,the solution of this equation shows that the charge densitygoes to zero as �ch / exp�z=�� if one goes away from thesurface (z! �1). In the case of a sphere, the correspond-ing solution reads

�ch /sinh�r=��r=�

: (24)

Far away from the surface, the charge densities which weobtain are very well described by Eq. (24). To show this,we display in Fig. 9 the same charge densities as in Fig. 8,but divided by their value at r � 0. Far away from thesurface, all curves follow exactly Eq. (24) with the value� � 7:74 fm calculated for bulk CFL matter. Near thesurface, i.e., at distances which are of the order of a coupleof Fermi wavelengths, there are strong deviations from thisbehavior due to Friedel-type oscillations [24]. This isbecause Eq. (21) is not exact, but it is only valid in auniform medium and in the long-wavelength limit.

It is interesting to notice that the value of the Debyescreening length we obtain is in reasonable agreement withthe photon Debye mass calculated from perturbative QCD,which reads, for the CFL phase, m2

D;�� 1=�2 �

4 21�8 ln254 e2Nf�2=�6�2� [25] (mD;�� denotes the Debye

mass without gluon-photon mixing, see below). For typicalvalues of the chemical potential this gives �� 10 fm.

In principle, in color superconducting phases, the photoncan mix with one of the gluons. In the CFL phase, in bulk

matter, one linear combination of photon and gluon staysmassless. This means that at large distances d� �, theDebye screening for the ‘‘rotated’’ photon [26,27] does notwork, since the Cooper pairs are neutral with respect to therotated charge ~Q. Within the simple model we use for themoment, there are no gluons, such that the mixing cannotbe studied. It could be taken into account, as mentioned atthe end of Sec. II A, by including the gluons in the sameway as the photon, i.e., on the Hartree level. We expect thatif we included the gluons in this way, we would find aneven faster decrease of the charge if we go away from thesurface, since in addition to the electromagnetic force wewould have the color forces, which try to push the colorcharges to the surface, and in the CFL phase color neutral-ity goes hand in hand with electrical neutrality. Therefore,this is not in contradiction with the fact that the rotatedphoton is massless, but it is just a consequence of the factthat the combination of photon and gluon which is orthogo-nal to the rotated photon is massive (in fact, it is evenheavier than the other gluons [27]). This means that in alarge object, like a strange star, all the negative charge willbe concentrated within a layer of a thickness of at most afew tens of fm below the surface. However, before drawingany firm conclusion, one should study this problem in moredetail. This will be left for future work.

C. Liquid-drop type expansion

The advantage of the present approach is that finite-sizeeffects are correctly implemented. For large numbers ofparticles, this becomes, however, rather cumbersome andasymptotic expansions such as a liquid-drop type approachcan be very useful. We will discuss here the determinationof the parameters, such as the surface tension, of a liquid-drop type formula for the energy per baryon as a functionof the baryon number A, including a surface and a curva-ture term,

EA�

�EA

�bulk

aSA1=3

aCA2=3

; (25)

from our results. As in Sec. III B, we will restrict ourdiscussion to the CFL-type solutions with Z � 0, suchthat we do not need to include a Coulomb term / Z=A1=3.

As explained after Eq. (20), �E=A�bulk should be theenergy per baryon of infinite matter with �e � 0 ratherthan that of stable infinite matter. However, since weconsider only the CFL-type solution, this distinction isirrelevant. Hence, for our chosen parameter sets, we have�E=A�bulk � 900 MeV. Since for the neutral strangelets theCoulomb interaction has only a negligible effect on thetotal energies (for example, in the case of the strangeletsconsidered in Sec. III B, the Coulomb interaction changesthe total energy per baryon by less than 5 keV) it will beneglected here in order to reduce the numerical effort. Theresult of the fitted coefficients aS and aC for the differentparameter sets are listed in Table III. As an example, in

0

2

4

6

8

10

0 10 20 30 40

ρ ch(

r) /

ρ ch(

0)

r (fm)

FIG. 9. Zoom into the part of Fig. 8 where the charge densitybehaves as given by Eq. (24). For a better visibility, the chargedensities have been divided by their respective values at r � 0.The thin dotted curve corresponds to Eq. (24) with � � 7:74 fm.


074015-9

order to show the accuracy of the asymptotic expansion,we display in Fig. 10 some results for the energy perbaryon together with the liquid-drop formula, Eq. (25).The dashed line corresponds to the liquid-drop formulawithout the curvature term (aC � 0). From this figure itbecomes clear that the liquid-drop formula with curvatureterm works extremely well, much better than in the casewithout pairing [28]. The reason is that shell effects arecompletely washed out because, contrary to the situation inordinary nuclei, the pairing gap is much larger than thespacing between neighboring shells. Another interestingobservation is that the curvature term is very important,even for rather large mass numbers A.

The coefficient aS is closely related to a very interestingquantity, namely, the surface tension. As explained inRef. [29], the surface tension is obtained as

� �ES

4�R20

; (26)

where

ES � E� A�EA

�bulk

(27)

is the energy excess due to the surface and R0 is aneffective radius defined by

A � �B;bulk4�R3

0

3; (28)

which is actually very close to R for not-too-small strange-lets. On the one hand, using the liquid-drop formula (25)for E in Eq. (26), one would obtain a surface tension whichdepends on A because of the curvature term. Therefore it isclear that one has to use Eq. (26) in the limit A! 1, wherethe curvature term vanishes, i.e.,

� �aS�

2=3B;bulk

�36��1=3: (29)

The corresponding numbers are given in the last column ofTable III. They are of the same order of magnitude as theestimate �� 70 MeV�3 � 8:8 MeV=fm2 for SQM with-out color superconductivity [20]. On the other hand, thefact that the curvature term is very strong implies that theknowledge of the surface tension alone might not be suffi-cient in order to determine, e.g., the possibility of mixedphases, the size of droplets, etc.

Before we conclude, let us comment on the physicalmeaning of the surface tension we obtain. In the MIT bagmodel, it is supposed that the energy needed to create a bagwith volume V is simply given by BV. In principle onecould imagine that there is an explicit dependence of thebag energy on, e.g., the surface or the curvature of the bagboundary. In Ref. [20], this contribution to the surfacetension was called ‘‘intrinsic surface tension,’’ �I, and itwas argued that it should be small. What we calculate hereis the ‘‘dynamical surface tension,’’ �D, which has itsorigin in the change of the level density of the quarksinside the bag as a function of the bag geometry.

IV. SUMMARY

In this paper we have investigated finite lumps of colorsuperconducting SQM. To that end we have treated theMIT bag model, supplemented with a pairing interaction,in the framework of HFB theory. This allows us to cor-rectly include finite-size effects for pairing, too. The cal-culation is numerically rather involved, since in addition tosolving self-consistently the HFB equations, we have todetermine the bag radius and the fractions of the differentquark species by minimizing the total energy of the system.

As expected from previous MIT bag-model studies, wefind a suppression of the strange-quark densities at thesurface, resulting in a positive surface charge. Our mainresult is that, in spite of this surface charge, the total chargeof the CFL-type solution is zero due to pairing, as in bulkmatter. Most of the positive surface charge is compensatedin a negatively charged layer situated at about 1–3 fmbelow the surface. The origin of this concentration of thenegative charge is pairing: Since all quarks are paired, thepositive surface charge must be compensated on a lengthscale corresponding to the coherence length. The remain-ing negative charge, which is necessary to compensate all

900

910

920

930

940

950

0 1000 2000 3000 4000 5000

E/A

(M

eV)

A

FIG. 10. Energy per baryon as a function of baryon number forH�2 � 2 and B1=4 � 156:26 MeV. The exact results are indi-cated by the crosses, the fitted liquid-drop formula by the solidline. The dashed line corresponds to the liquid-drop formulawithout the curvature term.

TABLE III. Fitted liquid-drop parameters for the CFL-typeneutral strangelets (Z � 0). The surface tension � correspondingto the fitted value of aS is also given.

B1=4�MeV� H�2�MeV� aS�MeV� aC�MeV� ��MeV=fm2�

156.26 2 107 289 11.9159.46 2.25 109 297 12.8163.46 2.5 112 306 13.9


074015-10

of the positive surface charge, is situated below this layer.With increasing distance from the surface, the chargedensity decreases on a length scale of �8 fm, correspond-ing to the Debye screening length. This number willprobably be strongly decreased if the gluons are includedin a perturbative way similar to the photon. In any case, inthe biggest part of a large object, such as a strange star, onefinds vanishing charge density if one goes more than a fewtens of fm away from the surface. It remains to be inves-tigated in which way our results change the traditionalpicture of the surface of a strange star and the detectabilityof smaller strangelets in current experiments such as AMS-02 or LSSS [30].

We have also compared our results for the energy perbaryon of finite strangelets with a liquid-drop-like formula.We obtain a surface tension of the order of 12–14 MeV, inreasonable agreement with previous studies where colorsuperconductivity was not considered, and a strong curva-ture term which is crucial to reproduce the correct energiesup to baryon numbers of several thousands. An interestingresult is that, in the presence of color superconductivity, theliquid-drop formula describes very accurately the totalenergies even for A & 100, at least for strangelets witheven baryon number. The reason is that, since the gap � ismuch larger than the spacing between the energy levels,shell effects are strongly suppressed.

ACKNOWLEDGMENTS

We thank Michael Buballa for useful discussions and forthe critical reading of the first version of this manuscript.

APPENDIX A: SPINORS IN A SPHERICAL CAVITY

In this appendix we recall basic properties of free Diracspinors in a spherical cavity (see, e.g., Ref. [18]). They canbe written as

fj mn�r� �gfj n�r�Y

mjl��

iffj n�r�Ymjl0 ��

!; (A1)

where Y are spinor spherical harmonics [31]. We have thefollowing relations between the angular momentum quan-tum numbers:

� j1

2! l � j

1

2; l0 � j�

1

2

� ��j

1

2

�! l � j�

1

2; l0 � j

1

2:

(A2)

For the solutions of the free Dirac equation, the functions fand g are given as follows in terms of the spherical Bessel

functions (�fj n ��p2fj n m

2f

q):

gfj n�r� � Cfj njl�pfj nr�

ffj n�r� � Cfj nsgn� n�

��fj n �mf

�fj n mf

sjl0 �pfj nr�;

(A3)

where the Cfj n are normalization coefficients which canbe determined from the normalization

Z R

0drr2

Zd� y�r� �r� � 1: (A4)

The momenta pfj n are obtained from the boundary con-dition. The boundary condition of the MIT bag model,Eq. (1), translates into the following equation:

ffj n�R� � �gfj n�R�; (A5)

or, explicitly,

jl�pfj nR� � sgn� n�

��fj n �mf

�fj n mf

sjl0 �pfj nR�; (A6)

where we number by n > 0 the positive-energy (particle)states and by n < 0 the negative-energy (antiparticle)states. In practice, we will keep only the states with posi-tive eigenvalues and neglect the antiparticle contributions.The latter can approximately be absorbed into a redefini-tion of the coupling constant.

APPENDIX B: HFB EQUATIONS

In this appendix we will give some more details aboutthe HFB equations. Their derivation is analogous to thederivation of the Dirac-Hartree-Bogoliubov equations infinite nuclei, which is given in Ref. [19].

The HFB equations are derived from the Lagrangian byminimizing the energy in the mean-field approximation,i.e., linearizing the interaction under the assumption ofnonzero expectation values for the condensates sAA0 �x�,Eq. (7). Because of the inhomogeneities of a finite system,the Green’s functions become nondiagonal in momentum.In the stationary case, it is convenient to work in r space forthe spatial coordinates but to perform the Fourier trans-formation for the time variable. Then the Green’s func-tions,

S�x; y� � �ihT��x� ��y��i; (B1)

with

��x� � �x� T�x�

� �(B2)

take the following general form in Nambu-Gorkov space:


074015-11

S�r; r0;!� �G�r; r0;!� F�r; r0;!�~F�r; r0;!� ~G�r; r0;!�

!

�X

��>0�

U��r�

V��r�

!1

!� � i�

� � �U��r0�; �V��r0��

X

�<0�

U�r�

V�r�

!1

! � i�

� � �U�r0�; �V�r0��; (B3)

where G, ~G and F, ~F are normal and anomalous Green’sfunctions, respectively. The spinors U�; and V�; corre-spond to the particle- and hole-like components,respectively.

The energy in mean-field approximation can now bewritten as [19]

Eq �Zd3x

�iTr��i� � r�m�G�x; x� �

i2

�Zd4yTr��x; y�G�y; x� � ��x; y� ~F�y; x�

�;

(B4)

where the derivative in the first term acts only on x and noton x, and x means the four-vector �x0 t;x� in the limitt! 0. In our case, the normal and anomalous self-energies � and � are local and time-independent:��x; y� � eQA0�x��0��x� y� and ��x; y� � ��x��x�y�, and Eq. (B4) can be reduced to Eq. (17).

As mentioned in Sec. II B, the expectation values (likecondensates, densities, etc.) which are needed for calculat-ing self-consistently the self-energy � and the pairing field� can be expressed in terms of the U and V functions. Tothat end, it is sufficient to express them in terms of theGreen’s functions, e.g.,

sAA0 � �h � T�x��A�A0 �x�i � iTrF�x; x��A�A; (B5)

which leads to Eq. (9).By minimizing the total energy with respect to theU and

V functions, one obtains the HFB equations, [see Eq. (5)]:

H W� � �W�; (B6)

with

W� �

�U�

�0V�

�and H being the matrix on the left-hand side of Eq. (5).

For homogeneous infinite systems the matrix elementsof H are diagonal in momentum space and solutions to theHFB equations are known for many cases. For finite sys-tems, in general, these equations are solved numerically bydiagonalizing the matrix H in some conveniently chosenbasis. Here, we are working in the basis which diagonalizes

the Dirac Hamiltonian (i.e., hfc without the Coulombpotential), see Appendix A, and the eigenvectors U��r�and V��r� are developed within this basis.

The matrix elements of the pairing fields �A�r� and ofthe Coulomb field A0�r� are computed in the usual way. Forillustration, we give here the explicit expression for thematrix elements of �2�r�, which connects up and downquarks, in the basis described in Appendix A:

��2�j nn0 �Zr<R

d3r yuj mn�r��2�r� dj mn0 �r�

�Z R

0drr2�2�r��guj n�r�gdj n0 �r�

fuj n�r�fdj n0 �r��: (B7)

Note that, due to spherical symmetry, all matrices arediagonal in j and and proportional to the unit matrixwith respect to m.

In spite of the spherical symmetry, the matrix to bediagonalized is still huge, limiting the baryon numberwhich can be calculated with reasonable computationaleffort. It is therefore important to reduce the size of theactual matrix to be diagonalized. By means of an orthogo-nal transformation

~H � SHST; ~W � SW; SST � 1 (B8)

in color, flavor, and Nambu-Gorkov space, the matrix canactually be block-diagonalized (see, e.g., Ref. [32]) con-taining seven blocks. Six of them, ~H B;...G, are 2� 2matrices in Nambu-Gorkov space, describing mutual pair-ing of two particles, such as, e.g., red down quarks (dr)with green up quarks (ug):

~H B �hug �2

�2 �hdr

� �; (B9)

where hfc is the single-particle Hamiltonian for flavor fand color c. The second and third 2� 2 blocks are

~H C �hub �5�5 �hsr

� �; ~HD �

hdb �7�7 �hsg

� �:

(B10)

Since we have in addition the pairwise relations ~H E;F;G �

� ~H B;C;D, only three of the six 2� 2 blocks have to bediagonalized in practice. The seventh block, ~H A, is 6� 6in Nambu-Gorkov space and describes pairing between redup, green down, and blue strange quarks

~H A �

hur 0 0 0 �2 �50 hdg 0 �2 0 �7

0 0 hsb �5 �7 00 �2 �5 �hur 0 0

�2 0 �7 0 �hdg 0�5 �7 0 0 0 �hsb

0BBBBBB@

1CCCCCCA:

(B11)


074015-12

APPENDIX C: CUTOFF FOR THE GAP EQUATION

As mentioned in Sec. II B, the divergent gap equation isregularized with the help of a smooth cutoff function

f�p=�� 1

1 c1 exp�c2a�p=�� 1��; (C1)

where c1 ��2p� 1, c2 � 1=�4� 2

��2p�, and a � 22:58

have been chosen such that f2�p=�� approximates thecutoff function g�p=�� used in Ref. [33], but our functionhas the advantage to fall off more rapidly at very highmomenta, which allows us to truncate the basis at a lowerenergy.

This function is used as a form factor multiplying eachof the four legs of the four-point vertex. In practice, thismeans that the form factor is used in two places: First,when calculating sAA�r�, and second, when calculating thematrix elements of �A�r� in the basis of the spinors definedin Appendix A. It should be noted that the diagonalizationof the HFB matrix does not directly provide us with theeigenfunctions U��r� and V��r�, but with their respectiveexpansion coefficients in the basis of the spinors defined inAppendix A. When calculating sAA�r� according to Eq. (9),the coefficients have to be multiplied with the correspond-ing basis functions, and in this step the factor f�pfj n=�� isattached to each basis function. Second, when calculatingthe matrix elements of the gap �A, we again attach a factorf�pfj n=�� to each basis function.

[1] J. C. Collins and M. J. Perry, Phys. Rev. Lett. 34, 1353(1975); B. C. Barrois, Nucl. Phys. B129, 390 (1977); S. C.Frautschi, in Proc. of the Workshop on Hadronic Matter atExtreme Energy Density, Erice, 1978, edited by N.Cabibbo (Plenum Press, New York, 1980); D. Bailin andA. Love, Phys. Rep. 107, 325 (1984).

[2] M. G. Alford, K. Rajagopal, and F. Wilczek, Phys. Lett. B422, 247 (1998).

[3] R. Rapp, T. Schafer, E. V. Shuryak, and M. Velkovsky,Phys. Rev. Lett. 81, 53 (1998).

[4] K. Rajagopal and F. Wilczek, in At the Frontier of ParticlePhysics, Handbook of QCD, Boris Ioffe Festschrift, editedby M. Shifman (World Scientific, Singapore, 2001),Vol. 3, p. 2061; M. Alford, Annu. Rev. Nucl. Part. Sci.51, 131 (2001); T. Schafer, arXiv:hep-ph/0304281; D. H.Rischke, Prog. Part. Nucl. Phys. 52, 197 (2004); M.Buballa, Phys. Rep. 407, 205 (2005); H.-C. Ren,arXiv:hep-ph/0404074; M. Huang, Int. J. Mod. Phys. E14, 675 (2005); I. A. Shovkovy, Found. Phys. 35, 1309(2005).

[5] M. G. Alford, K. Rajagopal, and F. Wilczek, Nucl. Phys.B537, 443 (1999).

[6] I. A. Shovkovy and L. C. R. Wijewardhana, Phys. Lett. B470, 189 (1999); T. Schafer, Nucl. Phys. B575, 269(2000); N. J. Evans, J. Hormuzdiar, S. D. H. Hsu, and M.Schwetz, Nucl. Phys. B581, 391 (2000).

[7] A. R. Bodmer, Phys. Rev. D 4, 1601 (1971); E. Witten,Phys. Rev. D 30, 272 (1984).

[8] C. Alcock, E. Farhi, and A. Olinto, Astrophys. J. 310, 261(1986); P. Haensel, J. L. Zdunik, and R. Schaeffer, Astron.Astrophys. 160, 121 (1986).

[9] J. Madsen and J. M. Larsen, Phys. Rev. Lett. 90, 121102(2003); M. Rybczynski, Z. Wlodarczyk, and G. Wilk,Nucl. Phys. B, Proc. Suppl. 151, 341 (2006).

[10] V. V. Usov, Phys. Rev. Lett. 80, 230 (1998).[11] D. Page and V. V. Usov, Phys. Rev. Lett. 89, 131101

(2002).[12] P. Jaikumar, S. Reddy, and A. W. Steiner, Phys. Rev. Lett.

96, 041101 (2006).

[13] M. G. Alford, K. Rajagopal, S. Reddy, and A. W. Steiner,Phys. Rev. D 73, 114016 (2006).

[14] K. Rajagopal and F. Wilczek, Phys. Rev. Lett. 86, 3492(2001).

[15] J. Madsen, Phys. Rev. Lett. 87, 172003 (2001).[16] V. V. Usov, Phys. Rev. D 70, 067301 (2004).[17] A. Chodos, R. L. Jaffe, K. Johnson, C. B. Thorn, and V. F.

Weisskopf, Phys. Rev. D 9, 3471 (1974).[18] R. K. Bhaduri, Models of the Nucleon: From Quarks to

Soliton (Addison-Wesley, Redwood City, 1988).[19] B. V. Carlson and D. Hirata, Phys. Rev. C 62, 054310

(2000).[20] E. Farhi and R. L. Jaffe, Phys. Rev. D 30, 2379 (1984).[21] W. V. Liu and F. Wilczek, Phys. Rev. Lett. 90, 047002

(2003).[22] H. Abuki, T. Hatsuda, and K. Itakura, Phys. Rev. D 65,

074014 (2002).[23] T. Tatsumi, M. Yasuhira, and D. N. Voskresensky, Nucl.

Phys. A718, 359 (2003).[24] F. Garcıa-Moliner and F. Flores, Introduction to the

Theory of Solid Surfaces (Cambridge University Press,Cambridge, 1979).

[25] A. Schmitt, Q. Wang, and D. H. Rischke, Phys. Rev. D 69,094017 (2004).

[26] M. G. Alford, J. Berges, and K. Rajagopal, Nucl. Phys.B571, 269 (2000).

[27] D. F. Litim and C. Manuel, Phys. Rev. D 64, 094013(2001).

[28] E. P. Gilson and R. L. Jaffe, Phys. Rev. Lett. 71, 332(1993).

[29] B. C. Parija, Phys. Rev. C 48, 2483 (1993).[30] J. Madsen, arXiv:astro-ph/0612784.[31] D. A. Varshalovich, A. N. Moskalev, and V. K.

Khersonskii, Quantum Theory of Angular Momentum(World Scientific, Singapore, 1988).

[32] M. G. Alford, J. Berges, and K. Rajagopal, Nucl. Phys.B558, 219 (1999).

[33] S. Yasui and A. Hosaka, Phys. Rev. D 74, 054036 (2006).


074015-13

Documents

Surface effects in color superconducting strange-quark matter