Continuity of vortices from the hadronic to the color-flavor locked phase in dense matter
Mark G. Alford, Gordon Baym, Kenji Fukushima, Tetsuo Hatsuda, Motoi Tachibana
RRIKEN-QHP-367
Continuity of vortices from the hadronic to the color-flavor locked phasein dense matter
Mark G. Alford, Gordon Baym,
2, 3
Kenji Fukushima, Tetsuo Hatsuda,
3, 5 and Motoi Tachibana Department of Physics, Washington University, St Louis, MO 63130, USA Department of Physics, University of Illinois, 1110 W. Green Street, Urbana, IL 61801-3080, USA iTHES Research Group and iTHEMS Program, RIKEN, Wako, Saitama 351-0198, Japan Department of Physics, The University of Tokyo,7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033, Japan Nishina Center, RIKEN, Wako, Saitama 351-0198, Japan Department of Physics, Saga University, Saga 840-8502, Japan (Dated: 01 October 2018)We study how vortices in dense superfluid hadronic matter can connect to vortices in superfluidquark matter, as in rotating neutron stars, focusing on the extent to which quark-hadron continuitycan be maintained. As we show, a singly quantized vortex in three-flavor symmetric hadronicmatter can connect smoothly to a singly quantized non-Abelian vortex in three-flavor symmetricquark matter in the color-flavor locked (CFL) phase, without the necessity for boojums appearingat the transition.
I. INTRODUCTION
In a rotating neutron star, the superfluid components –the nuclear liquid at lower densities and a possible color-flavor locked (CFL) quark phase [1] at higher densitiesin the interior – carry angular momentum in the formof quantized vortices. How, we ask, are the vortices inthese two phases connected? Can one have continuityor must there be a discontinuity? How do the possibleconnections depend on the particular flavor structure ofthe matter? In the ground state of dense matter, thepicture of quark-hadron continuity [2, 3] is that as thebaryon density is increased matter undergoes a smoothcrossover from the hadronic phase to the quark phase. Bystudying how such vortices connect we can shed furtherlight on whether the notion of quark-hadron continuitycan be extended to angular momentum carrying statesof dense hadronic matter.To summarize the problem in matching hadronic withCFL vortices we note that superfluid vortices in the BCS-paired hadronic phase have quantized circulation, C B ,i.e., C B = (cid:73) C (cid:126)v · d(cid:126)(cid:96) = 2 π ν B µ B , (1)where the contour C of integration encircles the vortex, µ B is the baryon chemical potential, and ν B is an integer.We detail this result further below. (We work in units (cid:126) = c =1.) All but singly quantized vortices ( ν B = ± C A = (cid:73) C (cid:126)v · d(cid:126)(cid:96) = 2 π ν A µ q , (2)where µ q = µ B / ν A is an integer. Singly quantized U (1) B Abelianvortices in the quark phase have three times the circula-tion of singly quantized hadronic vortices. (a)
Hadronic VorticesAbelianVortex (b)
Non-AbelianVortices (c)
FIG. 1. Schematic illustrations for connecting vortices: (a) Ifangular momentum in the CFL phase is carried by AbelianCFL vortices then in the crossover to the hadronic phase a“boojum” (shaded circle) joins three hadronic vortices to asingle Abelian CFL vortex; (b) because Abelian CFL vor-tices are unstable, three hadronic vortices match onto threenon-Abelian CFL vortices through a modified boojum; or (c)each hadronic vortex matches onto a single non-Abelian CFLvortex without the need for a boojum.
Thus if one were to imagine a singly quantized hadronicvortex turning into a singly quantized Abelian CFL vor-tex, the baryon velocity would have to jump discontinu-ously by a factor of three from the hadronic to the quarkphase, eliminating any possibility of quark-hadron conti-nuity. Indeed, to make the velocity continuous one wouldhave to join three hadronic vortices to a single Abelianquark vortex, as illustrated in Fig. 1(a). Such a join isknown as a “boojum” [7].Single Abelian vortices in the CFL phase, however, areunstable against separating into three non-Abelian vor-tices [8–10], each of which has 1/3 the circulation of theAbelian vortex. Thus one might envisage a join with acontinuous baryon velocity, as shown in Fig. 1(b), where a In Ref. [8] these configurations were referred to as “semi-superfluid strings,” however we will call them “non-Abelian vor-tices” to emphasize the presence of non-Abelian color magnetic a r X i v : . [ h e p - ph ] O c t boojum connects three hadronic vortices with three non-Abelian CFL vortices [11, 12]. However, as we discuss inthis paper, one does not have to make a join involvingthree vortices in the hadronic phase, but rather one canmake a baryon-velocity conserving join between a singlehadronic vortex and a single non-Abelian vortex in theCFL phase, as shown in Fig. 1(c), without any need for aboojum. To the extent that the various flavor quantumnumbers permit a smooth transition from the hadronicto the CFL quark phase, angular momentum carryingstates remain consistent with quark-hadron continuity.To spell out this picture in detail, we first discuss moreprecisely the nature of quark-hadron continuity betweenthe hadronic and quark phases. On the deconfined quarkside the (ideal) CFL phase contains u (up), d (down),and s (strange) quarks, all with the same mass, with aFermi sea equally populated with all three flavors andall three colors of quarks. The corresponding hadronicphase, three-flavor hyperonic matter, contains all mem-bers of the light baryon flavor octet – n , p , Λ, Σ , Σ ± ,Ξ , and Ξ − – all of the same mass. In the ground stateat finite density, the particles populate a Fermi sea withall states of the octet equally present.Both phases break chiral symmetry [1] and U (1) B ,with the same symmetry breaking pattern [ SU (3) L ⊗ SU (3) R ⊗ U (1) B → SU (3) V ]. In the hadronic phase,the dibaryon condensate, which breaks U (1) B , is formedfrom two paired flavor octets, while in the CFL phase, adiquark condensate is formed, which in the unitary gaugehas the same color-flavor orientation everywhere. Also,in the hadronic phase, chiral symmetry is spontaneouslybroken by a quark-antiquark chiral condensate, produc-ing a light octet of pseudoscalar mesons, i.e., π , π ± , K , ¯ K , K ± , and η , while in the CFL phase, the di-quark condensate spontaneously breaks chiral symmetry,producing a light octet of pseudoscalar mesons [15–17].Previous studies [2, 3, 18, 19] have established the con-tinuity between the low-energy excitations of such three-flavor hadronic and three-flavor quark matter. The ninesingle-quark excitations of different colors and flavors canbe mapped, in the unitary gauge, onto the baryon octetplus a baryon singlet which is usually not mentioned indiscussions of the confined phase because it is much heav-ier than the octet baryons [3].One can further understand quark-hadron continuityin terms of the anomaly-induced coupling between thechiral and diquark condensates [21, 22]. The implica-tions of quark-hadron continuity for the QCD phase di-agram are reviewed in Ref. [23], and for neutron stars inRef. [24]. flux in the core combined with vortex-like global rotation of thequark condensate. With full three-flavor symmetry, CFL pairing is the most sta-ble [13, 14]. This continuity is an example of the complementarity betweenthe confined and Higgs phases of a non-Abelian gauge theory[20]. qqq q qq qqqq qq
FIG. 2. Schematic illustration of the smooth evolution ofa hadronic vortex into a non-Abelian CFL vortex. In thehadronic phase, the phase of the condensate correspondingto paired baryons (six quarks) increases by 2 π in windingaround the vortex core. In the CFL phase in the gauge-fixedpicture, one component of the order parameter picks up aphase 2 π in winding, as shown. In the gauge-invariant picturethe phase of the entire six-quark order parameter changes by2 π in winding. Figure 2 summarizes our results. In the confined phase(upper half of the figure) the hadronic vortex carries an-gular momentum via the circulation of a gauge-invariantdibaryon condensate which acquires a phase of 2 π whentransported around the core. This vortex can be con-tinuously connected to a non-Abelian CFL vortex [8] inthe CFL quark phase (lower half of the figure) where thevortex has the same baryon circulation, but it arises inthe unitary gauge from three diquark condensates, one ofwhich acquires a phase of 2 π when transported aroundthe core. On the other hand, in the gauge-invariant pic-ture, described in detail in Sec. III D, the phase increaseis attributed to the entire six quark order parameter.This paper is organized as follows. In Sec. II we re-view the generic properties of vortices in a superfluid.In Sec. III we discuss the vortex configurations thatexist in three-flavor hadronic and quark matter. Af-ter discussions of hadronic vortices in Sec. III A, wedescribe two different vortex configurations which havebeen constructed in three-flavor quark matter, AbelianCFL vortices in Sec. III B and non-Abelian CFL vorticesin Sec. III C, and then we show how a non-Abelian vortexcan be continuously connected with a hadronic vortex. InSec. III D we show how these non-Abelian vortices can beunderstood in a gauge-invariant description, focusing inSec. III D 2 on the continuity of flavored vortices. Finally,in Sec. IV we discuss the role of color magnetic flux. Wefocus throughout on the properties of connecting singlevortices, and leave the discussion of an array of vorticesin the CFL phase at finite rotation for the future. II. VORTEX QUANTIZATION ANDCIRCULATION
We first review the basics of vortex quantization, circu-lation, and angular momentum which are common to allthe vortices we discuss here: hadronic vortices, AbelianCFL vortices, and CFL vortices carrying non-Abeliancolor flux.Quantized vortices arise in superfluids under rotation.A superfluid can be described by a complex scalar field;the ground state expectation value Φ( (cid:126)r, t ) of the field,in the conventional description in terms of broken sym-metry, represents the condensate of bosons (or Cooperpairs of fermions) that gives rise to superfluidity. TheHamiltonian for the field is invariant under a global U (1)symmetry, so that the number of bosons or fermions isconserved by the dynamics. However, if Φ is nonzerothen the ground state of the Hamiltonian spontaneouslybreaks the U (1) symmetry.In general, the condensate can be written in terms ofits modulus and phase φ as,Φ = e iφ | Φ | . (3)In the local rest frame of the condensate, φ = − µ s t , (4)where µ s is the chemical potential of the conserved par-ticles in the ground state, namely the minimum energyrequired to add one boson or one pair of fermions to thesystem. Boosting to a frame in which the condensate isin uniform motion [25], we find φ = p ν x ν = (cid:126)p · (cid:126)r − µt , (5)where p ν p ν = − µ s and µ = γ ( v ) µ s with γ ( v ) ≡ / √ − v . The superfluid velocity is simply (cid:126)v = (cid:126)p | p | = (cid:126)pµ . (6)We can thus write the momentum carried by the unit ofconserved charge and the chemical potential as (cid:126)p = (cid:126) ∇ φ ( (cid:126)r, t ) , µ = − ∂φ ( (cid:126)r, t ) ∂t (7)for general space-time dependent φ .For a static superfluid vortex, φ ( (cid:126)r, t ) = φ ( (cid:126)r ) − µt ; thusΦ( (cid:126)r ) = e iφ ( (cid:126)r ) − iµt | Φ( (cid:126)r ) | , (8)where | Φ( (cid:126)r ) | is zero at the center of the vortex and inuniform density matter is independent of position welloutside the vortex core. Far from the vortex core theonly spatial variation is in the phase φ ( (cid:126)r ).For the mathematically simplest vortex aligned alongthe z axis, φ = νϕ , where ϕ is the azimuthal angle. Thusthe momentum per particle or pair is (cid:126)p ( r ) = (cid:126) ∇ φ = νr ˆ ϕ (9) where r is the distance from the vortex core and ˆ ϕ is aunit vector in the ϕ direction. From Eq. (6) the super-fluid velocity is v ( r ) = νµr ˆ ϕ . (10)Integrating (cid:126)p along a closed contour C surrounding thevortex we obtain the total change ∆ φ in the phase,∆ φ = (cid:73) C (cid:126)p · d(cid:126)(cid:96) = 2 π ν . (11)In a three dimensional system, the winding number ν must be an integer. From Eqs. (6) and (11) [or fromEq. (10)] the superfluid velocity obeys the circulationcondition, C = (cid:73) C (cid:126)v · d(cid:126)(cid:96) = 2 π νµ , (12)as mentioned in the introduction.Lastly we compute the angular momentum, L z , of avortex centered on the z axis. From Eq. (7) the local az-imuthal momentum density is p ϕ n where n is the particledensity (as distinguished from the condensate density),which is independent of ϕ . Thus L z = (cid:90) d r rp ϕ n ( r ) = ν (cid:90) πrdrdz n ( r ) = N ν , (13)where N is the total number of particles or pairs. Theangular momentum per particle for bosons or per fermionpair is simply ν , the winding number of the vortex. III. VORTICES IN HADRONIC AND CFLQUARK MATTER
We now consider the circulation and the angular mo-mentum associated with vortices in hadronic and CFLquark matter.
A. Hadronic vortices
In the SU (3) classification, baryon pairs can be decom-posed into irreducible representations as, ⊗ = ⊕ ⊕ (cid:124) (cid:123)(cid:122) (cid:125) sym ⊕ ⊕ ⊕ ∗ (cid:124) (cid:123)(cid:122) (cid:125) anti-sym . (14)Here and below, “sym” and “anti-sym” stand for thesymmetry under the flavor exchange of two baryons.The baryon-baryon interaction in the SU (3) limit ismost attractive in the flavor-singlet channel ( repre-sentation) [26] with a pairing gap of the form, ∆ ( )B = (cid:10) − (cid:113) [ΛΛ] sym + (cid:113) [ΣΣ] sym + (cid:113) [ N Ξ] sym (cid:11) . In theground state of three-flavor hyperonic matter, flavornon-singlet pairings in other attractive channels cancoexist with the flavor-singlet pairing, e.g., the stan-dard nucleon pairing in the spin-singlet isospin-tripletchannel, ∆ ( )B = (cid:104) [ N N ] sym (cid:105) , and the possible pairingin the spin-singlet isospin-doublet channel, ∆ ( sym )B = (cid:104)− [ N Λ] sym + (cid:113) [ N Σ] sym (cid:105) [27].In any of these pairings, the chemical potential enter-ing Eq. (12) is 2 µ B , that of a pair of baryons. Therefore,no matter whether it is flavor singlet or non-singlet, ahadronic vortex with winding number ν B has circulation2 πν B / (2 µ B ), Eq. (1). The corresponding angular mo-mentum per baryon is [see Eq. (13)] L B z N B = 12 ν B , (15)since there are N B / B. Abelian CFL Vortices
The order parameter of quark matter in the CFL phasein the unitary gauge can be written in terms of the colorand flavor triplet diquark operator [1]ˆΦ αi = N (cid:15) αβγ (cid:15) ijk q βj Cγ q γk , (16)where C = iγ γ is the charge conjugation operator, andGreek and Latin letters denote color and flavor indices,respectively; N is a normalization constant. The orderparameter is then Φ αi = (cid:104) ˆΦ αi (cid:105) . (17)The matrix Φ αi can be diagonalized by a combination ofcolor and flavor rotations, so that without loss of gener-ality we write Φ = Φ ¯ r ¯ u ¯ g ¯ d
00 0 Φ ¯ b ¯ s , (18)where r, g, b (¯ r, ¯ g, ¯ b ) denote colors (anti-colors) and u, d, s flavors; in the ground state, Φ ¯ r ¯ u = Φ ¯ g ¯ d = Φ ¯ b ¯ s = ∆ CFL .Naively one would expect the angular momentum car-rying states with lowest energy per unit of angular mo-mentum, to be global U (1) B or “Abelian CFL” vortices.In these vortices each of the three non-zero componentsof the order parameter winds around the core of the vor-tex, so for an Abelian CFL vortex aligned along the z axis the order parameter assumes the formΦ A = ∆ CFL e iν A ϕ f ( r ) 0 00 f ( r ) 00 0 f ( r ) , (19)where f ( r ) varies monotonically from zero at r = 0 tounity as r → ∞ , with ν A the winding number of theAbelian CFL vortex.The quark chemical potential is µ q = µ B /
3, and thusthe chemical potential per quark pair is 2 µ q = µ B , so from Eqs. (6) and (7) and the total momentum per quarkpair in the condensate is (cid:126)p = 23 µ B (cid:126)v , (20)where as before (cid:126)v is the superfluid velocity, so the circu-lation is C A = 3 ν A µ B (cid:73) d(cid:126)(cid:96) · (cid:126) ∇ ϕ = 3 ν A · πµ B . (21)The angular momentum per baryon of the vortex is L A z N B = 32 ν A . (22)We now ask how the vortices in hadronic matter wouldmatch on to Abelian vortices in CFL quark matter at acrossover between these phases. If the superfluid veloc-ity, and hence the circulation, Eq. (12), and angular mo-mentum per baryon, Eq. (13), do not match in the twophases, then quark-hadron continuity would be violated.By comparing Eqs. (1) and (21), or equivalently (15) and(22), we see that matching would require ν B = 3 ν A . (23)The matching relation (23) implies that three singlyquantized hadronic vortices should merge into oneAbelian CFL vortex, violating quark-hadron continuityin states with finite angular momentum. This mergingwould require a boojum [7] at the interface between thetwo phases, as sketched in Fig. 1(a). As we discuss inthe next section, the violation need not be present forthe more stable non-Abelian vortices in the CFL phase. C. Non-Abelian CFL Vortices
An Abelian CFL vortex is energetically unstableagainst formation of three “non-Abelian” vortices [8, 9].The condensate of the anti-red–anti-up (¯ r ¯ u ) non-Abelianvortex is Φ (1) = ∆ CFL e iν ϕ f ( r ) 0 00 g ( r ) 00 0 g ( r ) , (24)with corresponding gluon field A (1) ϕ = − ν g c r [1 − h ( r )] −
00 0 , (25)where g c is the QCD coupling and the boundary condi-tions are f → , g (cid:48) → , h → r → ,f → , g → , h → r → ∞ . (26)Single-valuedness of the condensate requires that ν be aninteger. Anti-green–anti-down (¯ g ¯ d ) and anti-blue–anti-strange (¯ b ¯ s ) versions, Φ (2) with ν and Φ (3) with ν , canbe obtained by permuting the diagonal elements.To obtain the superfluid velocity and angular momen-tum per baryon of the non-Abelian vortex, we rewriteEq. (24) asΦ (1) = ∆ CFL e i ν ϕ e i ν ϕ f ( r ) 0 00 e − i ν ϕ g ( r ) 00 0 e − i ν ϕ g ( r ) . (27)In this form the overall factor of e i ν ϕ is the U (1) B phase,while the phase factors within the matrix are a color ro-tation. [We note for later computation of the covariantderivative of Φ (1) that the gradients of these phases arecompensated by the color gauge field (25).]The chemical potential per quark pair is 2 µ q = µ B ,so from Eqs. (6), (7), and (9) the total momentum perquark pair is related to the superfluid velocity (cid:126)v by (cid:126)p = 13 · ν r ˆ ϕ = 23 µ B (cid:126)v . (28)The circulation around the vortex, Eq. (12), is C (1) = (cid:73) C (cid:126)v · d(cid:126)(cid:96) = πν µ B . (29)Correspondingly, the angular momentum per baryon ofthe vortex of the form (24) or (27) is L (1) z N B = 12 ν . (30)The same relations also hold for Φ (2) with ν and Φ (3) with ν .We see from Eqs. (1) and (29) and from Eqs. (15) and(30) that singly quantized ( ν B = 1) vortices in hadronicmatter can match onto singly quantized ( ν = 1, ν = 1,or ν = 1) non-Abelian vortices in CFL quark matter ata crossover between these phases, with no discontinuityin baryon velocity and angular momentum.This result can be understood intuitively as follows. Inthe hadronic vortex, the dibaryon condensate acquires aphase of 2 π as one follows it along a contour encirclingthe vortex core. Since the dibaryon can be viewed as 3diquarks, this corresponds to each diquark acquiring aphase of 2 π/
3. The non-Abelian vortex in the CFL con-densate has exactly the same circulation: each diquarkacquires a phase of 2 π/ If U (1) B were a local gauge symmetry, the vortex would becomea U (1) B flux tube. The hadronic vortex and the non-Abelianvortex would both have the same U (1) B flux in their cores. Fig. 1(b)]. As we argue below, there is no need for such aboojum: a single non-Abelian CFL vortex can smoothlyevolve into a single hadronic vortex [as in Fig. 1(c)]. Toshow this, further consideration of the flavor structureof the vortices is necessary in the hadronic and the CFLphases, as we discuss in Sec. III D.
D. Gauge-invariant description
In Sec. III we described the CFL condensate in theunitary gauge. Although such a gauge-fixed descriptionis convenient for writing down the non-Abelian vortexsolution explicitly and showing the continuity of the cir-culation and angular momentum between the hadronicphase and the CFL phase, it is not clear how the flavorstructures in the two phases are connected. To resolvethis problem, in this section we describe vortices in theCFL phase in a gauge-invariant manner [28] using di-quarks in Eqs. (16) and (17) as building blocks. We canwrite down meson-like and baryon-like gauge-invariantcombinations of diquark operators,ˆ M ji ( (cid:126)r ) ≡ ˆΦ † iα ˆΦ αj , (31)ˆΥ ijk ( (cid:126)r ) ≡ (cid:15) αβγ ˆΦ αi ˆΦ βj ˆΦ γk . (32)We will focus on ˆΥ ijk ( (cid:126)r ) for the moment and will con-sider ˆ M ji ( (cid:126)r ) later in Sec. III D 3. According to quark-hadron continuity, (cid:104) ˆΥ ijk ( (cid:126)r ) (cid:105) is nonzero in both the CFLand hadronic phases because both phases break baryonnumber, via diquark and dibaryon condensates respec-tively. In Secs. III D 1 and III D 2 below we will discussthe projection of ˆΥ ijk ( (cid:126)r ) onto specific flavor representa-tions.In the CFL phase, in the mean field approximation,Υ ijk ( (cid:126)r ) ≡ (cid:104) ˆΥ ijk ( (cid:126)r ) (cid:105) = 16 (cid:15) αβγ Φ αi Φ βj Φ γk . (33)Υ ijk ( (cid:126)r ) provides a gauge-invariant description of thenon-Abelian vortex originally defined through the gauge-dependent condensate Φ.Note that the irreducible flavor SU (3) decompositionof Υ ijk ( (cid:126)r ) is ∗ ⊗ ∗ ⊗ ∗ = ⊕ ⊕ ⊕ ∗ , (34)so that not only flavor-singlet but also flavored vor-tices can be obtained from Φ by appropriate projections.These would match to certain of the hadronic vorticesclassified in Eq. (14).According to (33) the total number of 6-quark con-densates in the CFL phase is 3 × × × ijk ( (cid:126)r ) is an order parameter for baryon number viola-tion, which is manifest with ˆΥ ijk ( (cid:126)r ) rewritten in termsof the baryon-interpolating operator, ˆ B i aj ≡ ˆΨ αi ˆ q aαj ; thespin-1 / a on q aαj . In writ-ing ˆ B i aj as interpolating operators for spin-1 / B i am can be written as a sumof flavor-singlet and flavor-octet operators asˆ B i am = ˆ B a ( δ im / √
6) + ˆ B A a ( t A ) im , (35)where the t A are the SU(3) generators ( A = 1 , . . . ,
8) inflavor space, with the normalization tr( t A ) = 1 /
2. Thenˆ B a ≡ B a ) / √ B A a ≡ t A ˆ B a ).Forming ˆ B i aj by combining the quark operator with thediquark operator written in terms of two quarks, (16), wefind the operator relationˆΥ ijk ( (cid:126)r ) = 13 (cid:15) kmn ( Cγ ) ab ˆ B i am ˆ B j bn . (36)Clearly, a dibaryon condensate (cid:104) ˆ B ˆ B (cid:105) (cid:54) = 0 in the hadronicphase, makes Υ ijk nonzero.
1. Flavor-singlet vortex
We first consider vortices in the flavor-singlet projec-tion of the gauge-invariant order parameter,ˆΥ ( (cid:126)r ) = (cid:15) ijk ˆΥ ijk ( (cid:126)r ) . (37)We can equivalently express this expectation value usingEq. (36) in terms of the baryon operators, (35),Υ ( (cid:126)r ) = 13 ( Cγ ) ab (cid:0) δ mi δ nj − δ ni δ mj (cid:1) (cid:104) ˆ B i am ˆ B j bn (cid:105) = 13 ( Cγ ) ab (cid:18) (cid:104) ˆ B a ˆ B b (cid:105) − (cid:104) ˆ B A a ˆ B A b (cid:105) (cid:19) ; (38)in hadronic language Υ ( (cid:126)r ) corresponds to a flavor-singlet condensate made with flavor-singlet and flavor-octet baryons.In the CFL phase insertion of any of Φ (1) , Φ (2) or Φ (3) gives the same formΥ = e iν q ϕ ∆ f ( r ) g ( r ) , (39)which implies that the non-Abelian vortices Φ (1 , , havea common flavor-singlet component. A singly quantized( ν q = 1) vortex has the same circulation 2 π/ µ B as asingly quantized ( ν B = 1) hadronic vortex in the flavor-singlet channel; its phase winds by 2 π on a contour en-circling the vortex core, consistent with our finding thatthese two vortices match smoothly onto each other, withquantized vortex circulation 2 π/ µ B . If, on the other hand, were we to substitute the fieldconfiguration for an Abelian vortex Φ (A) in Eq. (19) intoEq. (37), we would findΥ A = e iν A ϕ ∆ f ( r ) ; (40)the gauge-invariant form of a singly quantized Abelianvortex winds three times more (by 6 π ) on a contour en-circling the vortex core. This winding is consistent withneeding three hadronic vortices to match to one Abelianvortex [11].We now consider the vortex energy in terms of thegauge-invariant order parameter. Because of the bound-ary condition (26), the extra energy density of a vortexfar away from its core arises from the derivative terms;for a non-Abelian vortex the energy density is asymptot-ically (cid:15) (1) = tr | D Φ (1) | , (41)where the covariant derivative is D = ∇ − ig c A , and thetrace is taken with respect to color-flavor matrix indices.The gluon field (25) in D exactly cancels the derivativesof the phases in the color-flavor matrix part of Φ (1) αi inEq. (27). As a result only the derivative of the U (1) B phase contributes to the energy density at large distancefrom the vortex core, (cid:15) (1) = 3 · ν r | ∆ CFL | . (42)Calculating ∇ Υ from Eq. (39) we can write the energyin terms of the gauge-invariant order parameter as (cid:15) = 13(∆ CFL ) | ∇ Υ | . (43)This is the kinetic term of a Ginzburg-Landau theory[29] at large distance for the gauge-invariant flavor-singletorder parameter Υ .We can write the full gauge-invariant Ginzburg-Landau free energy in two-dimensions in the form: F = N (cid:90) d r (cid:18) | ∇ ˜Υ | − m | ˜Υ | + λ | ˜Υ | (cid:19) , (44)where we rescale Υ → ˜Υ to make the coefficient ofthe gradient term be unity at the mean-field level. Thefull determination of the coefficients, m and λ , fromQCD is a challenging future problem. This form of theGinzburg-Landau free energy describes the interactionbetween the flavor-singlet parts of non-Abelian vortices(see also Ref. [30]).As in simple superfluids, e.g., He, the interaction en-ergy of two non-Abelian vortices in the gauge-invariantpicture is essentially the integral of the product of thetwo vortex velocities, v · v , which is generally negativebetween two similarly quantized vortices; for two singlyquantized vortices whose cores are separated by L , as-sumed much greater than the coherence length 1 /m , theinteraction energy is F int = − πm λ ln( m L ) . (45)Here, the coefficient appears from the normalized con-densate, | ˜Υ | = m /λ in the mean-field approximation.This logarithmically diverging result (see [31]) indicatesthat the two vortices repel.
2. Flavored vortices
We now consider vortices in the flavor-octet projectionof the gauge-invariant order parameter,ˆΥ A = (cid:15) ij(cid:96) ( t A ) (cid:96)k ˆΥ ijk ( (cid:126)r ) . (46)This term vanishes in the mean field approximation, butbeyond mean field the flavor-octet part of non-AbelianCFL vortices could smoothly connect to flavor-octethadronic vortices, just as the flavor-singlet part of a non-Abelian vortex can smoothly connect to a flavor-singlethadronic vortex. As with the flavor singlet, we can ex-press Υ a ( (cid:126)r ) in terms of the baryon operators,Υ A ( (cid:126)r ) = 13 ( Cγ ) ab (cid:15) ij(cid:96) ( t A ) (cid:96)k (cid:15) kmn (cid:104) ˆ B i am ˆ B j bn (cid:105) (47)= 16 ( Cγ ) ab (cid:18) d ABC (cid:104) ˆ B Ba ˆ B Cb (cid:105) − √ (cid:104) ˆ B a ˆ B b (cid:105) (cid:19) , where the d tensor is defined by { λ A , λ B } = δ AB + d ABC λ C . Equation (47) shows how the flavor octet vor-tex Υ A can be understood as a symmetric made withtwo octet baryons [as classified in Eq. (14)].We note that the flavor structure of dibaryon pair-ings such as (cid:104) nn (cid:105) and (cid:104) pp (cid:105) in two-flavor superfluid nu-clear matter cannot be realized in the present setup forthe CFL phase. For example, a neutron pair conden-sate, (cid:104) nn (cid:105) , has an overlap with the diquark condensate, (cid:104) ud (cid:105)(cid:104) ud (cid:105)(cid:104) dd (cid:105) ; however, because (cid:104) dd (cid:105) is flavor symmetric,it must be color symmetric for a spin-singlet (antisym-metric) pair, and thus cannot be constructed out of Υ ijk given in terms of Φ αi . Such pairing is possible in thecolor sextet channel; although single gluon exchange isrepulsive for color-triplet diquarks, and such pairing ispresumably less favored, nonetheless this pairing breaksthe same symmetries and is therefore induced by color an-tisymmetric pairing [3, 32]. Another possible way to form The interaction free energy of two vortices, one at the originwith phase φ and the second with phase φ , where the φ ’sare the azimuthal angles ϕ measured from the individual vor-tex cores, is F int = (cid:82) d r ∇ ϕ · ∇ ϕ | ˜Υ | . After integrationby parts only the surface term remains, since ∇ ϕ = 0, andchoosing the branch cut in the phase along the x axis, the inte-gral becomes (cid:82) L /m dx ∂ y ϕ · ∆ ϕ | ˜Υ | . Since ∆ ϕ (except at itscore, where the order parameter vanishes), the discontinuity of ϕ along across the x axis is − π , we find Eq. (45). (cid:104) dd (cid:105) is with color-triplet and spin-triplet pairing [33, 34],which has spin one and breaks rotational symmetry. Suchstates could connect naturally to P pairing in densenuclear matter. We leave the question of vortex continu-ity between neutron P pairing and color-triplet. spin-triplet paired quark matter for the future.
3. Flavor symmetry breaking in the vortex core
At least at the level of the mean-field approximation,flavor symmetry is spontaneously broken in the core of aCFL vortex [12], SU (3) → SU (2) ⊗ U (1), which can becharacterized by the flavor-octet order parameter M ji = (cid:104) ˆ M ji (cid:105) introduced in Eq. (31). For a Φ (3) condensate, forexample, we havetr( t A M ) = − √ (cid:2) f ( r ) − g ( r ) (cid:3) δ A, . (48)Whether this prediction survives beyond mean field re-quires analysis of the fluctuation modes of a CFL vortexin (3+1) dimensions. If the core is effectively a (1+1)dimensional system, the Mermin-Wagner-Hohenberg-Coleman theorem [35–37] would imply that fluctuationsin the order parameter along the symmetry broken di-rections (the CP mode [12]) would prevent spontaneousbreaking of continuous symmetries in systems in (1+1)dimensions at T (cid:62) T > mode which requires (3+1)-dimensional analysis, the octet components of ˆΥ ijk coulddevelop an expectation value inside the hadronic vortexcore.Therefore, in either scenario, the flavor transformationproperties of the CFL vortices do not prevent continuityof vortices between the hadronic and CFL phases. IV. COLOR FLUX
In Sec. III we argued that at a crossover between thehadronic phase and the CFL phase, a hadronic vortexcan smoothly evolve into a non-Abelian CFL vortex, inkeeping with quark-hadron continuity. More generally,even if there is a first order phase transition between theCFL and hadronic phase (terminating a CFL vortex inmuch the same way as vortex terminates at a free surfacein a liquid), it is hard to avoid a hadronic vortex, sincethen one would have to have a layer of discontinuity inthe baryonic current. This raises the question of whathappens to the color magnetic flux in the non-AbelianCFL vortex. Reference [11] argued that at the quark-hadronic boundary there must be a boojum where threenon-Abelian CFL vortices with different color magneticfluxes come together so that their color fluxes cancel,and they can then connect to three hadronic vortices [seeFig. 1(b)]. However, we argue that there is no need forsuch an elaborate construction.The gauge-invariant characterization of the color mag-netic flux was recently discussed in Ref. [40] which notedthat, just as for local non-Abelian flux tubes [41, 42], thecolor magnetic flux in the non-Abelian CFL vortex canbe characterized by the Aharonov-Bohm phase acquiredby a heavy “probe” quark when transported around thevortex. This is manifest in the expectation value of thetrace of the Wilson loop operator, W ( C ) = 1 N c tr P exp (cid:18) ig c (cid:73) C ds µ A µA t A (cid:19) , (49)where N c is the number of colors (i.e., N c = 3), P de-notes path ordering, the t A are the SU (3) color gen-erators in the triplet representation, and C is a closedcontour encircling the vortex. If the contour is largeenough then the Wilson loop follows a perimeter law (cid:104) W ( C ) (cid:105) = χ C exp( − κL ( C )) in both phases, where L ( C )is the length of the contour, and κ is an effective mass.The prefactor χ C contains the Aharonov-Bohm phase forthe path C , normalized so that for a large contour C thatdoes not encircle a vortex, χ C = 1.Reference [40] emphasized that, for a non-Abelian CFLvortex, χ C is a Z phase, an element of the center ofthe color gauge group, whereas in the hadronic phase weexpect that for a contour C encircling a hadronic vortexthere will be no such phase, χ C = 1, since there is nocolor flux in the hadronic vortex. However, as we nowexplain, this does not mean that a boojum is required atthe quark-hadron boundary.One of the leading scenarios for explaining confinementis the condensation of “center vortices” [43–47]; for a re-cent review see [48]. According to this picture, the con-fining QCD vacuum is filled with flux tubes that carry Z color flux. It is therefore quite possible that whena non-Abelian CFL vortex arrives at the CFL-hadronicboundary, its color flux can leak away into the confinedhadronic phase, indistinguishable from the pre-existingcondensate of center vortices. There is no reason whymultiple CFL vortices should be constrained to convergeat a boojum before entering the hadronic phase: even ifthey are far apart their color fluxes can still cancel byconnecting with each other through the putative conden-sate of color vortices in the hadronic phase.The behavior of the color flux in the center vortexpicture is well illustrated in a spherical compact stel-lar object made of SU (3)-symmetric matter, rotating soslowly around the central z axis that it contains exactlyone vortex lying along this axis. We assume that thelower-density mantle is in the hadronic phase and thehigher-density core is in the CFL phase. The vortex has a “southern” hadronic segment, a central CFL segment,and a “northern” hadronic segment, Since such a systemcannot contain a boojum, which requires three vortices,what then is configuration of the color flux? When theCFL vortex reaches the hadronic phase, at the north poleof the core, its U (1) B circulation becomes the northernsegment of the hadronic vortex which continues upwardsalong the z axis, and in the center vortex picture, itscolor flux would become another member of the existingcondensate of center vortices in the hadronic phase. Thatcolor flux is redistributed through a chain of monopolesand antimonopoles connected by color flux tubes [48] inthe hadronic mantle and ultimately links to the southpole of the core, where it would combine with the south-ern segment of the hadronic vortex to create the CFLvortex that begins at the south pole of the core. V. CONCLUSIONS
We have argued here that singly quantized superfluidvortices in three-flavor symmetric hadronic matter cantransform smoothly into singly quantized non-Abeliansuperfluid vortices in three-flavor symmetric color-flavorlocked quark matter, without the need to include boo-jums to mark the transition at the interface between thetwo phases. One can make a one-to-one correspondencebetween vortices in the baryonic and quark phases. Wehave constructed a gauge invariant description of non-Abelian vortices. A natural next step will be to spell outthe full Ginzburg-Landau theory for non-Abelian vorticesin terms of their gauge-invariant order parameter.We have only studied the question of the connectionsof single vortices in fully SU (3) flavor symmetric mat-ter. To make our analysis applicable to more realis-tic situations in neutron stars where one does not haveeven isospin symmetry requires extending the analysis toflavor-symmetry broken states, resulting from the highermass of the strange quark (for a discussion of the rami-fications for CFL superfluid vortices see Ref. [12]). Theextension will require considering BCS pairing states inthe quark phase beyond ideal CFL with simple color,flavor, and spin asymmetry. Ultimately we would liketo determine the extent to which one can connect thehadronic and quark matter phases and their vortices in asmooth way. Furthermore, at large rotational rates oneexpects an array of vortices. While in the hadronic phasethe vortices are expected to form a triangular lattice, todetermine the optimal lattice configurations in the quarkphase requires better understanding the interactions ofnon-Abelian vortices. ACKNOWLEDGMENTS
The authors thank Aleksey Cherman, Muneto Nitta,and Srimoyee Sen for useful discussions. Researchof author GB was supported in part by NSF GrantsPHY1305891 and PHY1714042. Author TH was sup-ported by JSPS Grants-in-Aid for Scientific ResearchNo. 15H03663 and 18H05236; GB and TH were partiallysupported by the RIKEN iTHES Project and iTHEMSProgram. MGA is supported by the U.S. Departmentof Energy, Office of Science, Office of Nuclear Physicsunder Award Number [1] M. G. Alford, K. Rajagopal, and F. Wilczek, Nucl. Phys.
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