UMD-DOE/ER/40762-511

Electron shielding of vortons in high-density quark matter

Paulo F. Bedaque1,∗ Evan Berkowitz1,† Geoffrey Ji2,‡ and Nathan Ng3§1 Maryland Center for Fundamental Physics, Department of Physics,

University of Maryland, College Park, MD 20742-41112University of Maryland, College Park, MD 20742

3Montgomery Blair High School, 51 University Boulevard East, Silver Spring, MD 20901

We consider the the effect of the electron cloud about a vorton in the CFL-K0 high-density phaseby numerically solving the ultrarelativistic Thomas-Fermi equation about a toroidal charge. Includ-ing electrons removes the electric monopole contribution to the energy, and noticeably decreases theequilibrium radius of these stable vortex loops.

I. INTRODUCTION

The behavior of matter at very high densities, likethose found at the core of neutron stars, remains largelyan open problem. This is due to a lack of reliable meth-ods to extract information directly from QCD. But sincethe temperatures relevant for the physics of neutron starsis very low (as compared to the Fermi energy) a num-ber of observables of interest depend only on propertiesof the low lying excitations of the ground state. In thecase of dense matter it is believed that those excitationsare pseudo-Nambu-Goldstone (NG) bosons arising fromthe spontaneous breaking of certain approximate symme-tries. The study of these quasiparticles using the meth-ods of effective field theory opens up the possibility of amore reliable understanding of dense matter.

The breaking of symmetries in high-density QCD isrelated to quark pairing. At high density, evidence nowpoints towards color superconductivity where quarks pairin different combinations depending on the temperatureand chemical potential[1–4]. This pairing uses the BCSmechanism, just like electron pairing in conventional elec-trical superconductors. However, color superconductiv-ity is a much richer phenomenon than electrical super-conductivity because quarks carry flavor and color whileelectrons do not—different color superconducting phasesarise depending on which quarks pair. The nature of theNG bosons depends crucially on the pattern of symmetrybreaking and, consequently, on the kinds of quark pairingfound in the ground state.

At asymptotically high density, three-flavor QCD is be-lieved to pair in a pattern known as color-flavor-locking(CFL), as it pairs all quarks[5]. At lower densities, wherethe QCD coupling is not small, the problem is more com-plicated and a plethora of phases have been suggested (fora review see [6, 7]). Perhaps the most likely phase at veryhigh density is the CFL-K0 phase, which seems guaran-teed to be the energetically favored phase for some rangeof large densities[8, 9]. This range may include densi-

∗ bedaque@umd.edu† evanb@umd.edu‡ gji@umd.edu§ nang@mbhs.edu

ties relevant for compact astrophysical objects and thusmay be phenomenologically relevant for extracting pre-dictions about extreme environments, like the inner coresof neutron stars.

The NG modes of the CFL phase form an SU(3) octet,and are named in direct analogy with the in vacuo SU(3)flavor octet, but at high density the “kaons” are the light-est particles [10–14]. As they live in an SU(3) octet, theirbehavior is described by a theory resembling familiar chi-ral perturbation theory. As the chemical potential µ islowered and the quark masses become relevant some ofthe NG modes become lighter. When µ is low enough, theneutral “kaons” would have a negative mass-squared, andinstead condense with some vacuum expectation value,turning the CFL phase into CFL-K0[8, 9]. This neutralkaon condensate provides a superfluid background that isthe stage for interesting topological defects. The reasonthe neutral, as opposed to the charged, kaons condense isthe small differences in their mass induced by isospin vi-olating quark masses and electromagnetic effects. Thesedifferences are small so a condensate of charged kaonsis almost degenerate with the favored neutral kaon con-densate. This peculiar situation provides a physical re-alization of the so-called “superconducting string” solu-tions, suggested in more abstract contexts[15] and alsoappearing in some models describing beyond the Stan-dard Model physics[16–19] . Regular global vortices ap-pear due to the K−condensation. At their core, the valueof the K0 condensate vanishes, as it is the case in all vor-tices. Since there is a close competition between neutraland charged kaon condensates, the charged kaons willthen condense in the core of the K0 vortex. The con-densation of a charged particle leads to electrical super-conductivity (not color superconductivity) along the vor-tex. Loops of superconducting vortices (“vortons”) canbe stabilized by the supercurrent running around them.The mechanism, roughly speaking, is that the energy ofa vorton has two main contributions. One is the tensionalong their length, which creates an energy that scaleswith its radius R and drives a vorton to be small. How-ever, the energy of a vorton includes terms that scale likeJ2/Q2R where J is its angular momentum and Q its elec-tric charge. These terms prevent a vorton from collaps-ing. This mechanism was invoked in [20] in the contextof quark dense matter. It was subsequently pointed out

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[21, 22] that, just like in the case of cosmological vortons,the currents necessary to stabilize a vorton could be highenough that they would quench the superconductivity ofthe vortex core, destroying its supercurrent and render-ing the stability mechanism mute. The quenching effectcan, however, be counteracted by electric charge. A vor-ton with an overall electric charge, in addition, would bemore stable on account of Coulomb repulsion betweenopposong sides of the loop.

A more quantitative theory of quark matter vortonsappeared only recently. In Ref. [23], the equilibrium sizeof vortons containing charge as well current was esti-mated by making a variety of approximations and as-sumptions. It was found that stabilizing a vorton so thatits equilibrium radius was several times the thickness ofits core δ and thus could really be identified as a torus,required large electric charges (Z ∼ 104). With such alarge charge, it is possible that ambient electrons whichare in the bulk to ensure charge neutrality could be pulledclose to the vorton, shielding its electric fields, and signif-icantly reducing the effectiveness with which the electriccharge aided in stabilizing the vorton in the first place.We call these vortons with neutralizing orbiting electronsvortonium to easily distinguish them from unshielded orbare vortons.

It is the purpose of this paper to account for theseelectrons and to find the equilibrium radius of a vorto-nium “nucleus”. We use the Thomas-Fermi approxima-tion to find the electrons’ semiclassical charge distribu-tion, allowing them to shield the electric field. Second,having found the electron distribution, we can computethe energy in the electric field numerically, without mak-ing geometrical or multipole approximations. Intuitively,when including electrons we should find that vortons havea smaller equilibrium radius, because there are exactlyenough electrons to kill the monopole field. We find thatmaking these improvements decreases the equilibrium ra-dius of a vorton of given angular momentum and electriccharge significantly.

II. CFL+K0

To understand vortons, we first must understand thephase in which they appear. When the quark masses arenegligible compared to the chemical potential µ, the CFLphase is described by the quarks pairing according to[24]

⟨qaL,iCq

bL,j

⟩= −

⟨qaR,iCq

bR,j

⟩=

1

gsµ2∆ εabZεijZ (1)

where q are the quarks which carry color a, b ∈ {1, 2, 3},flavor i, j ∈ {u, d, s}, and helicity R,L indices. The di-mensionful scale ∆ is known as the gap, and gs is thestrong coupling constant. This condensate breaks the fullSU(3)C ×SU(3)L×SU(3)R×U(1)B ×U(1)A symmetrydown to SU(3)C+L+R×Z2×Z2, where the first remain-ing Z2 comes from breaking U(1)B and the second from

breaking the approximate U(1)A symmetry that Diracfermions enjoy at asymptotically high density.

Simple counting suggests that there should be 18Nambu-Goldstone mesons, but eight of these are eatenby the gluons, which gain masses ∼ gsµ, indicating thatthe phase is color superconductive[10]. The condensatebreaks the electromagnetic U(1)EM ⊂ SU(3)L×SU(3)R,but there remains an unbroken gauge symmetry thatarises from a linear combination of the eighth gluon andthe photon. This unbroken gauge symmetry, which wecall U(1)Q is “mostly” electromagnetism: when αEM issmall the contribution from the eighth gluon is small,the differences between αQ and αEM are small, and thedielectric constant εCFL differs from εvacuum by onlya small amount. All of these corrections are at leastO (αEM ) suppressed, and we henceforth refer to the re-maining U(1)Q symmetry as electromagnetism for con-venience.

After the gluons eat eight of them, ten NG modes re-main. Two describe bulk superfluidity that results frombreaking the baryon and axial symmetries. The othereight are an octet of the SU(3)C+L+R. Because theseeight “mesons” share group structure with the in vacuomesons that are described by flavor SU(3), these modesare named in an analogous manner. The theory containsa chiral field Σ which can take on a vacuum expectationvalue Σ0. The eight mesons πa are small fluctuationsabout Σ0, so that Σ = exp (iπaλa/f) Σ0, where λa arethe usual 8 Gell-Mann matrices with Tr

(λiλj

)= 2δij

and f plays a role analogous to the pion decay constant.To lowest-order, the effective theory describing the

CFL phase is described by the Lagrangian [8, 10–13]

Leff =f2

4Tr(∇0Σ† · ∇0Σ− v2DiΣ

† ·DiΣ)

+ 2ADet (M) Tr(M−1Σ + h.c.

)− 1

4F 2, (2)

where the derivatives are

DµΣ = ∂µΣ− iAµ [Q,Σ] ,

∇0Σ = D0Σ + i

[MM†

2pF,Σ

], (3)

M = diag (mu,md,ms) is the quark mass matrix, Q =e3diag (2,−1,−1) is the matrix of the quark charges underU(1)Q, Aµ is the U(1)Q gauge field, F its correspondingfield strength tensor, A, f , and v low-energy constants,and pF the Fermi momentum, which we take to be equalto the chemical potential µ.

The leading-order mass term differs from the usual chi-ral Lagrangian, which has a term linear in M. A heuristicway to understand this difference is that the CFL con-densate leaves Z2 remnants of the left- and right-SU(3)symmetries, so that left- and right-handed quark numbershould be preserved modulo 2, while terms linear in Mmix left- and right-handedness and violate this conserva-tion and are thus forbidden. The mass term includedin this Lagrangian is not the only O

(M2)

term, but

3

the other possible terms are higher-order in the power-counting scheme developed in Ref. [8].

Interestingly, the low-energy constants can be matchedto perturbative QCD calculations at asymptotic µ. Suchmatching calculations give [10, 12]

f2 =21− 8 ln 2

18

µ2

2π2, v2 =

1

3, and A =

3

4π2∆2. (4)

Armed with this information, one can easily find themasses of the CFL mesons at asymptotically large µ. Ex-panding about Σ0 = 11, one finds that the O

(M2)

termgives masses [8]

mπ± = ∓µdu +

√4A

f2(md +mu)ms,

mK± = ∓µsu +

√4A

f2md(ms +mu), (5)

mK0,K0 = ∓µsd +

√4A

f2mu(ms +md),

where µij = (m2i − m2

j )/(2µ) is the fictitious chemicalpotential which arises from the M commutator termsin the time-like derivatives. As µ comes down frominfinity, the kaons and charged pions get a fictitiousstrangeness chemical potential from the MM∗/2pF com-mutator terms in the time-like derivatives, and are incen-tivized to condense. The other mesons have even largermasses, but as they lie on the diagonal of Σ when theother fields vanish, their masses do not get contributionsfrom the commutator term and are not encouraged tocondense.

Since the neutral kaons are the lightest, they condensefirst. The charged kaons are have almost the same massas the neutral kaons—the difference is dependent on themass difference md − mu and also on electromagneticeffects. However, the interactions between mesons inthe effective theory strongly disfavors two meson speciesfrom condensing at once and, if K+ condensed, electronswould have to be sprinkled throughout the bulk with anadditional cost in energy due to the electrons’ Fermi en-ergy.

With all of the other NG modes vanishing, the expec-tation value of the neutral kaons is [8]

cos

(∣∣K0∣∣√2

f

)=m2K0

µ2sd

(6)

where mK0 is given in (5). Since cosine is restricted to bein the interval [−1, 1] we see that if m2

K0/µ2sd > 1, there

is no neutral kaon condensate at all. From the form ofthe chiral field Σ = exp (iπaλa/f) Σ0, it is clear that thevalue of the kaon expectation value is a compact variablewhose value only matters modulo 2π, and thus it is notsurprising that cosine shows up in the expression for theK0 condensate.

III. BARE VORTONS IN CFL-K0

The background condensate of neutral kaons provides astandard superfluid in which we can study global vortexstrings. As in any standard superfluid, the condensatemust vanish at a vortex’s center in order to remain single-valued. One feature of the CFL-K0 superfluid is thatthe other mesons could condense but interactions preventmore than one species from condensing. Since the centerof a superfluid vortex is evacuated, it is natural to expectthat the cores of these vortices may contain a condensateof the next lightest species—the charged kaon. Thus, thesuperfluid vortices have at their core a superconductingcondensate which may carry electrical current or hold anelectrical charge. A vorton is a vortex loop with thissuperconducting core.

A detailed analysis of the stability of a vorton withoutelectrons is given in Ref. [23]. There, an in-depth discus-sion about the stability of vortons with a fixed electricalcharge and angular momentum is provided. A vortex nat-urally has an associated tension, so a vortex loop wouldnaturally shrink to nothing and destroy itself very quicklyunless stabilized in some manner. When the inner con-densate does not break a gauge symmetry, the stabilityof a vorton is easy to understand. Requiring that thecore contain some fixed number of particles which have apreferred density can provide a pressure along the lengthof the vortex. Alternatively, the condensate can have aphase which wraps around the length of the vortex. Be-cause the condensate must be single valued, this phasemust be quantized, and is inherently topological. Thetopological winding number N enters quadratically intothe tension F⊥ as (N/R)2, while the usual terms in thetension are a constant c independent ofR (the energy costof suppressing the background condensate minus the sav-ings provided by allowing the core to condense). Thenthe energy F ∼ 2πRF⊥ ∼ N2/R + cR has a preferredequilibrium radius R0.

However, the gauged case is a bit more tricky. It wasalso noted in [23] that due to gauge invariance, the topo-logical winding number N of K+ that wraps around thelength of the vorton and the magnetic fluxoid threadedthrough a vorton ΦB , while separate observables, alwaysenter the Lagrangian and the energy functional F in justthe right combination so as to give the angular momen-tum J , so that we need not specify them separately butinstead need only specify J . In field theory, angular mo-mentum depends on one space and one time derivative,while the charge depends on a time derivative. From theform of the derivatives (2) one concludes that the timeand space derivatives are traded in the energy for Q andJ/QR.

In order to get an analytical feel for the energy func-tional F of a vorton without surrounding electrons,Ref. [23] makes several approximations. The first is thatinstead of computing the F directly for a vortex loop,instead the tension F⊥ for a straight superconductingvortex is used and simply multiplied by the length of

4

the vortex 2πR, which directly neglects curvature effects.This contribution to F accounts for all of the space ina torus whose radius and thickness are both R. To thatcontribution, the electromagnetic energy is approximatedby its leading multipole contributions–the electric fieldby the monopole contribution and the magnetic field bythe dipole contribution. These far-field energies accountfor the space outside a circumscribed sphere of radius2R around the inner torus. The apple-core-shaped spaceoutside of the torus but inside of the sphere was left un-accounted for. Artificially partitioning space and onlytaking only the leading multipole moments both reducethe accuracy of F .

The leading far-field electric part of F is easily com-puted and is given by

FE-far =

∫ ∞2R

d3r1

2

(Q

4πr2

)2

=(eZ)2

16πR(7)

The electric part of F that is outside of the vortex corebut within the torus is given by

FE-near = 2πR

∫ R

δ

d2r⊥1

2

(λ

2πr

)2

=(eZ)2

8πRlog

(R

δ

)(8)

In order improve upon the results in [23], we will mini-mize a modified F . We will minimize

F = Fold − FE-far − FE-near + FTF (9)

where Fold is the main analytical result of [23] and FTF isthe energy in the electric fields that we find numericallyby solving the Thomas-Fermi equations for a toroidalcharge of radius R, thickness δ, and charge eZ. In thenext section, we will use the Thomas-Fermi approxima-tion to find the semiclassical shape of the electron cloudsurrounding a toroidal charge. Replacing our approxima-tions with FTF accounts for electrons, avoids partitioningspace artificially for the electric energy, and avoids themultipole approximation for the electric energy. We willnot change the expressions for the angular momentum orother expressions which depend on the electric field. Fordetails on the non-Coulomb part of the vorton energy wepoint the reader to Ref. ([23]).

IV. SHIELDING BY ELECTRONS

In this section we give the method for numerically cal-culating the energy in the electric fields FTF , accountingfor electrons. The Thomas-Fermi approach approximatesthe electron density as

n =1

3π2p3F . (10)

where pF is a local fermi momentum and we work in unitswhere ~ = c = ε0 = 1. Gauss’ law,

∇2φ = en (11)

tells us how to compute the electric potential φ given afixed distribution of charges n. We also include the equi-librium condition that the marginal energy cost of mov-ing an electron from one point to another should zero,

d (pF − eφ) = 0 (12)

where we have used the ultrarelativistic approximation√p2 +m2 ≈ pF , as we will find that the electrons’ fermi

momentum is significantly higher than their mass. Thisapproximation may not be justified for electrons near theedge of the electron cloud, but we will find that most elec-trons live near the vorton, and thus are contained in a vol-ume O

(106fm3

), meaning that (for Z ∼ 105) they have

a Fermi momentum ∼ 300 MeV � me = .511 MeV andthe ultrarelativistic approximation is appropriate. Wethen know that

pF − eφ = −eφ∞ (13)

where φ∞ is a constant that describes the electric poten-tial at infinity.

Resolving (10), (11), and (13) for an equation depend-ing only on φ yields

∇2φ =16

3α2(φ− φ∞)3 (14)

Since φ∞ is a constant, ∇2φ∞ is zero and may be sub-tracted from the left-hand side, and we can absorb φ∞into φ by a simple change of variables. Equivalently,we can pick φ∞ = 0. This is a manifestation of gaugeinvariance, and picking φ∞ = 0 is a choice of gauge.Eq. (14) is the ultrarelativistic generalization of the usualThomas-Fermi equation and its solution, with appropri-ate boundary conditions, is the central theme of this pa-per. The Thomas-Fermi approximation neglects the in-teraction among the electrons but, at the high electrondensities occurring in most of the electron cloud this isa very small effect. It also neglects the energy associ-ated to gradients of the density. Again, these gradientsare negligible compared to the momenta of the electronsthemselves. By solving this nonlinear differential equa-tion with φ∞ = 0 subject to boundary conditions thataccount for the toroidal charge representing the vortonand the total charge neutrality of the whole system wefind a φ that self-consistently dictates and accounts forthe distribution of electrons.

To solve take advantage of the geometry of the situa-tion, we solved (14) in toroidal coordinates , defined interms of the usual Cartesian coordinates by

u = −2 Im arccoth(

(√x2 + y2 + iz)/a

)v = 2 Re arccoth

((√x2 + y2 + iz)/a

)(15)

ϕ = arctan(y/x)

where a is the radius of the reference circle (which doesnot exactly correspond to the radius of the vorton) and

5

Re and Im give the real and imaginary parts, respectively.We use the fundamental domain

− π < u ≤ π 0 ≤ v <∞ − π < ϕ ≤ π, (16)

where a large v corresponds to being near the referencecircle, and v near 0 corresponds to a very large circlethat comes close to the z-axis. The surface of the vortonoccurs at vmax = sinh−1

√(R/δ)2 − 1. Our expression

for the energy of a vorton already accounts for the en-ergy in the superconducting core v > vmax, so we restrictto the domain 0 ≤ v < vmax. Excluding electrons fromv > vmax creates an artificial pressure which can make δartificially smaller. Numerically we find that this effectmakes changes to δ that are ∼10%. Since our problemhas cylindrical symmetry, the ground-state electron dis-tribution will not depend on the axial coordinate ϕ andwe suppress it henceforth.

The Laplacian for these u, v toroidal coordinates is

∇2φ =1

a2(cos(u)− cosh(v))×[

(cos(u) coth(v)− csch(v))∂vφ+ sin(u)∂uφ (17)

+(cos(u)− cosh(v))(∂2u + ∂2

v

)φ].

We discretized u and v onto a 32×32 grid, and promotedφ to φ(t) where t is a fictitious time coordinate. Then, wenumerically solve the coupled ordinary differential equa-tion,

∇2φ− 16

3α3φ3 =

csch(v)

(r2 − δ2)3/2

dφ

dt(18)

so that if φ(tf ) reaches a temporal fixed point at somelate time tf , it must also automatically satisfy (14). Thefunction multiplying dφ/dt was picked to alleviate somenumerical instabilities, but the final solution of (14) isindependent of this choice as long as dφ/dt vanishes.

The toroidal coordinate u spans a range −π to π.The value u = 0 corresponds to the xy−plane withx2 + y2 > a2, while u = π corresponds to that same planebut with x2+y2 < a2. Since our physical setup has parityacross this plane, we can reduce the region where we needto solve (14) to 0 ≤ u ≤ π, with the boundary conditions

∂uφ(0, v) = ∂uφ(π, v) = 0. (19)

This reduction of the physical space allows us to get amore reliable solution while using a grid of the same size.

Axial symmetry about the z−axis (which correspondsto v = 0) entails that the solution must obey

∂vφ(u, 0) = 0. (20)

By symmetry alone we have specified the conditions onthree out of the four boundaries of the region.

The final boundary is at the wire. There, we enforcethat the electric field is that of a wire of linear chargedensity Ze/2πR and circular cross-section πδ2,

∂vφ(u, vmax) = 2

(Ze

2πR

)coth (vmax) . (21)

This condition matches the electric field to the electro-magnetic gauge field inside the vorton, whose form arefound explicitly in [23]. All the boundaries are subjectto Neumann conditions (as expected by gauge symme-try), so to numerically stabilize the solution we specifythe gauge φ(0, 0) = φ∞ = 0.

The important question now arises: how does the sys-tem know how many electrons to use to shield the vor-ton at long distances? That is, how does it come aboutthat the long-distance field behavior does not include amonopole piece? It isn’t clear that we have implementedthis physically desirable constraint. We will now showthat with the symmetries we have specified, toroidal co-ordinates implement this constraint automatically.

A straightforward manipulation shows that the spher-ical coordinate r is given by

r2 = a2 cosh v + cosu

cosh v − cosu(22)

and so large r2 corresponds to small u2 + v2. The multi-pole expansion of φ,

φ(~r) = φ∞ −Qtotal

4π

1

r−~d · r4π

1

r2+ · · · (23)

corresponds to

φ(u, v) = φ∞ −Qtotal

4π

√u2 + v2

2a−~d · r4π

u2 + v2

4a2+ · · · .

(24)when both u and v are small. We choose φ∞ to be

zero as a choice of gauge, and ~d = 0 is enforced byparity and axial symmetries. However, it is not obvi-ous that we have enforced Qtotal = 0. Part of the puz-zle is that all of the surface at infinity is mapped to asingle point, (u, v) = (0, 0). However, on the bound-aries u = 0 and v = 0 we have already recognized thatparity and axial symmetry require ∂uφ(0, v) = 0 and∂vφ(u, 0) = 0 respectively, meaning that near (0, 0) φ isflat in both directions—exactly what we need to guaran-tee Qtotal = 0. Charge neutrality is accidentally achievedby our other symmetry conditions in these coordinates,which makes implementing such a solution easy.

Unlike the usual application of the Thomas-Fermiequation to atoms with spherical symmetry which de-pend only on the charge Ze, a dimensionless number, thesolution for a toroidal charge does not have a simple scal-ing behavior, as there are length scale R, δ and not just Zand the electron mass, and the Laplacian does not scalesimply as the familiar spherical case does. Thus, for ev-ery Z, we solve the differential equation for many R andδ. It is possible to reduce this three-parameter space toa two-parameter space but for pedagogical simplicity wesimply sample the three-dimensional space. We show oneset of equipotentials from a numerical solution in Fig. 1.One can easily verify that far from the charge the equipo-tentials appear spherical, slightly closer the charge theequipotentials resemble oblate spheroids, near the charge

6

FIG. 1. (color online) Four equipotentials from a typicalnumerical solution to the Thomas-Fermi equation with atoroidal charge.

there are bialy-shaped equipotentials, and most near thetoroidal charge the equipotentials seem toroidal.

As a check on the numerical accuracy of the procedure,we integrate up the total electric charge outside of thetoroidal charge Z and hope to find −Z. That is we ask

− Z ?=

∫outside

2πdA ∇2φ(u, v) (25)

where the 2π comes from the axial integration and dAis the uv−area differential, including the appropriate Ja-cobian. This expectation was satisfied to within 1.8%for every choice of parameters; only 2% of the numeri-cal solutions had a disagreement between the left- andright-hand sides of more than 1%; only a quarter of thesolutions disagreed by more than 0.5%. Moreover, witha finer and finer grid the fulfilling of (25) is better andbetter.

From the solution φ(u, v) for a given Z, R, and δ, wecompute the energy

FTF (Z,R, δ) =

∫2πdA

1

2(−∇φ)2. (26)

For ease of comparison with the results in [23]we computed FTF for R ∈ [80, 800]fm in stepsof 20 fm, δ ∈ [1, 35]fm with steps of 1 fm, andZ ∈ {250, 500, 750} ∪ ([1250, 50000] with steps of 1250)and used these data points to construct an interpolatingfunction that worked for any set of parameters withinthose ranges and extrapolated to larger δ. Unlike theparts of F which care about the chemical potential and

other parameters which might change depending on theenvironment, FTF only cares indirectly through R andδ. Therefore, once FTF (Z,R, δ) is constructed, it can beused for any choice of the chemical potential, the gap,etc.

As a way to check the quality of the electron-free ap-proximation, we can examine the ratio of the actual en-ergy in the Thomas-Fermi calculation to the approxi-mated energy, FTF /(FE-far + FE-near). We find that forsmall radii (R . 100fm), where the electrons can’t domuch shielding, the approximation overestimates by atleast a factor of 2. At very small radiiR ∼ δ, the electron-free approximation may actually be an underestimate,but the charge ceases to have an obviously toroidal shape,and as other approximations involved in finding the equi-librium radius of a vorton breakdown, is not of physicalinterest. With larger radii, where the electrons can shieldthe Coulomb energy well, the approximation tends to beoff by roughly a factor of 5, but the ratio seems to loseany strong R dependence. This comparison is depictedin Fig. 2.

FIG. 2. The ratio between the numerically computed energyFTF and the approximation FE-far + FE-near as a function ofthe vorton radius, for Z ∈ [1250, 40 000] with steps of 1250and δ ∈ [20, 35]fm with steps of 1 fm. The exact identityof each curve is irrelevant; the generic behavior gives roughinsight.

With FTF (Z,R, δ) in hand, we can pick a (Z, J) pair,and simply minimize the energy F given in (9). Forconcreteness and ease of comparing to unshielded vor-tons, we use the same example parameters as [23], a gap∆ = 66 MeV, an overall chemical potential µ = 450 MeV,and quark masses (mu,md,ms) ≈ (2, 5, 95) MeV. Theseparameters don’t appear in FTF and thus one calcula-tion of FTF empowers us to examine any relevant densityand gap. We show R/δ as a function of (Z, J) in Fig. 3,and we compare the equilibrium radii of shielded andunshielded vortons in Fig. 4. With quick examination ofFig. 3 it is evident that the effect of electron shieldingshrinks the equilibrium radius for values of (Z, J) wherethe unshielded calculation is believable (Z & 6000), and

7

FIG. 3. The equilibrium radius R0 compared to the stringthickness δ of a shielded vorton at equilibrium as a functionof the vorton charge Ze and angular momentum J .

FIG. 4. The ratio R/δ for a vorton with electron shieldingcompared to R/δ for the same vorton with no electron shield-ing as a function of Z and J . The rough nature of the smallZ and J region is due to sampling a rough grid—a finer gridwould resolve these features.

that this effect becomes stronger for larger Z.One interesting effect that can be seen in Fig. 4 is that

shielded vortons are actually larger than their unshieldedcounterparts for small Z and J . This behavior fits withthe low-R trend seen in Fig. 2 and should be interpretedas the electron degeneracy helping prevent the vorton’scollapse. However, the results in this regime are nottrustable, as it is precisely the same region where thefictitious pressure that arises from restricting the elec-trons to v > vmax shrinks δ by 20% to 40%. To under-stand this area well, one would have to allow electronsinto the excluded regime in v, and would need to solvethe Thomas-Fermi equation (14) simultaneously with theequations of motion for the two kaon types, which is be-yond the scope of this work.

More careful examination of Fig. 3 shows that not onlyare shielded vortons smaller, but compared to unshieldedvortons they grow more slowly as one increases Z. Thisis the expected behavior, as to first order putting a posi-tive charge in the vorton is cancelled by a shielding elec-tron. If this cancellation were exact, a shielded vorton’ssize would be independent of Z. Only when one includesPauli blocking does a shielded vorton’s equilibrium radiusgrow. A related final observation is the increasing successof the electric shielding by electrons. As Z gets larger, theratio of shielded equilibrium radius and the unshieldedequilibrium radius drops. This is also expected, as largerZ corresponds to larger R, and a larger R enables elec-trons to live in the central hole of the vorton more readily,shielding the vorton’s charge more effectively.

V. SUMMARY AND OUTLOOK

In this paper we consider the effect of electron shieldingon the equilibrium radius of vortons in the CFL-K0 phaseat high density. For parameter values where unshieldedvortons were reasonably large and had a clear toroidalshape with an aspect ratio R/δ of a few (Z & 6000),the electron shielding of vortonium, as expected, shrinksboth the radius and aspect ratio for a given Z and alsodiminishes the response of R/δ to a change in Z.

By allowing that portion to be shielded with electronsand applying a relativistic Thomas-Fermi approximation,we have accounted for the electron cloud surroundingsuch a vorton and have relieved some of uncontrolledapproximations concerning the toroidal geometry of vor-tons in CFL-K0. We have shown that the energy in theelectric field considered in Ref. [23] is unrealistically op-timistic insofar as it drastically changes the equilibriumshape of the vorton as compared to the more realisticvortonium. The reduced size of vortonium may be un-derstood by realizing that the far-fields considered forunshielded vortons is predominantly a monopole piecewhile for vortonium it is a quadrupole.

Two directions seem obvious for pursuing the physicsof CFL-K0 vortons further. The first is to try to cure allof the issues we have touched on in this work: attempt to

8

find, in a toroidal geometry, the actual mean-field wave-functions of the neutral and charged kaon condensatesfor a vorton, while simultaneously incorporating electronshielding and allowing electrons to penetrate the coreand not artificially excluding them, and could accountfor curvature effects. Work in this direction will almostcertainly require intensive numerical calculations.

The other possible direction is to try to estimate thevorton production rate in the early times in the neutronstar life as it goes from a high temperature disorderedphase to a low temperature CFL-K0 phase. This num-ber is important to gauge the vorton influence on neutronstar phenomenology. Along the same lines, one mightconsider the decay rate of these objects through weakeffects, to try and estimate how long any created popu-lation might persist in these extreme environments. This

line of inquiry would help prioritize any other work onvortons at high density, and would provide the most di-rect route to understanding if vortons might contributeto observable astrophysical phenomena.

ACKNOWLEDGMENTS

P. F. B. and E. B. are supported by the U.S. Depart-ment of Energy under grant #DE-FG02-93ER-40762.E. B. also thanks Jefferson Science Associates for supportthrough the JSA/JLab Graduate Fellowship program.N.N. was encouraged by the summer research programat Montgomery Blair High School.

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