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PHYSICAL REVIEW D 68, 101701~R! ~2003!

Monopoles, antimonopoles, and vortex rings

Burkhard KleihausDepartment of Mathematical Physics, University College, Dublin, Belfield, Dublin 4, Ireland

Jutta Kunz and Yasha ShnirInstitut fur Physik, Universita¨t Oldenburg, D-26111 Oldenburg, Germany

~Received 22 July 2003; published 6 November 2003!

We present a new class of static axially symmetric solutions of SU~2! Yang-Mills-Higgs theory, where theHiggs field vanishes on rings centered around the symmetry axis. Associating a magnetic dipole moment witheach Higgs vortex ring, the dipole moments add for solutions in the trivial topological sector, whereas theycancel for magnetically charged solutions.

DOI: 10.1103/PhysRevD.68.101701 PACS number~s!: 14.80.Hv, 11.15.Kc

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Introduction. Defects, classical solutions of spontaneoubroken gauge theories, where the Higgs field vanishepoints, lines, or surfaces, are relevant in particle physicscosmology. Monopoles, for instance, represent zedimensional defects; vortex solutions or strings are assated with one-dimensional defects; domain walls represtwo-dimensional defects.

Here we present new classical solutions of SU~2! Yang-Mills-Higgs ~YMH ! theory with the Higgs field in the adjoinrepresentation, where the Higgs field vanishes either atcrete points, as in single monopoles, or at rings, as in voloops, or at rings and at a point.

Configuration space of YMH theory consists of sectocharacterized by the topological charge of the Higgs fieThe ’t Hooft–Polyakov monopole@1# carries a unit topologi-cal charge and possesses spherical symmetry. Multimopoles with higher topological charge possess at most asymmetry@2–4#, or no rotational symmetry at all@5#. Themagnetic charge of the~multi!monopoles is proportional totheir topological charge.

In the Bogomol’nyi-Prasad-Sommerfield~BPS! limit of avanishing Higgs potential, monopoles and multimonopoare obtained as solutions of the first order Bogomol’nequations@6#. The energy of these solutions satisfies exacthe lower energy bound given by the topological charSince the repulsive and attractive forces between monopexactly compensate, monopoles experience no net intetion.

As shown by Taubes@7#, each topological sector containfurther smooth, finite energy solutions, which do not satithe Bogomol’nyi equations, but only the second order EuLagrange equations. These solutions form saddlepoints oenergy functional. Their energy exceeds the Bogomol’bound.

In the topologically trivial sector the simplest such sotion is axially symmetric, and corresponds to an equilibriustate of a monopole-antimonopole pair@8,9#. Here the forcesacting on the monopole and antimonopole are balancedsulting in this~unstable! state, which carries an Abelian magnetic dipole moment. The Abelian magnetic field resembthe field of a physical dipole with magnetic charges localizon the symmetry axis at the equilibrium distance.

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Recently, more general static equilibrium solutions habeen constructed, representing chains, wherem monopolesand antimonopoles alternate along the symmetry axis@10#.m-chains in the topologically trivial sector carry a magnedipole moment and no charge, whereasm-chains in the sec-tor with topological charge one carry charge and no magndipole moment@10#.

Here we address the question, whether chains ofm mul-timonopoles and antimonopoles, each with chargen, can ex-ist in static equilibrium. The simplest such generalization,equilibrium state of a charge 2 monopole and a charge22antimonopole was obtained recently@11#. We find that, be-yond chargen52, no such equilibrium configurations of localized point charges are possible. Instead a new typeequilibrium solution appears.

Ansatz and boundary conditions. We consider SU~2!YMH theory in the BPS limit,

2L51

2Tr~FmnFmn!1

1

4Tr~DmFDmF!,

with su~2! gauge potentialAm5Ama ta/2, field strength tensor

Fmn5]mAn2]nAm1 i @Am ,An#, and covariant derivative othe Higgs fieldDmF5]mF1 i @Am ,F#. The field equationsare derived from the Lagrangian as the variational equatiwith respect to the gauge potential and the Higgs field,

DmFmn21

4i @F,DnF#50, DmDmF50. ~1!

Generalizing both the ansatz for the monopoantimonopole pairs and chains@8–11#, and the axially sym-metric multimonopole ansatz@3,12#, we parametrize thegauge potential and the Higgs field by

Amdxm5S K1

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2

2n sinuS K3

t r~n,m!

21~12K4!

tu~n,m!

2 D dw

F5F1t r~n,m!1F2tu

~n,m! ,

©2003 The American Physical Society01-1

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KLEIHAUS, KUNZ, AND SHNIR PHYSICAL REVIEW D 68, 101701~R! ~2003!

TABLE I. The dimensionless energy, the dipole moment per winding numberm/n and the coordinates of the zeros of the Higgs fieldgiven for several values ofm andn.

m/n

E@4ph# m/n (r i ,6zi)

1 2 3 4 1 2 3 4 1 2 3 4

1 1.00 2.00 3.00 4.00 0.0 0.0 0.0 0.0 ~0, 0! ~0, 0! ~0, 0! ~0, 0!

2 1.70 2.96 4.03 5.01 2.36 2.38 2.6 2.87 ~0, 2.1! ~0, 0.9! ~3.0, 0! ~4.9, 0!

3 2.44 4.17 5.62 6.96 0.0 0.0 0.0 0.0~0, 0!

~0, 4.7!~0, 0!

~0, 3.2!~0, 0!

~2.0, 1.2!

~0, 0!~3.6, 0!

~4.3, 0.8!

4 3.12 5.07 6.63 8.00 4.93 4.81 5.20 5.42~0, 2.4!~0, 7.0!

~0, 2.0!~0, 4.9!

~3.0, 3.0! ~5.4, 2.8!

5 3.78 6.11 7.96 9.59 0.0 0.0 0.0 0.0~0, 0!

~0, 4.8!~0, 9.6!

~0, 0!~0, 4.1!~0, 7.3!

~0, 0!~3.1, 5.2!

~0, 0!~5.7, 4.7!

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with su~2! matrices t r(n,m)5sin(mu)tr

(n)1cos(mu)t3, tu(n,m)

5cos(mu)tr(n)2sin(mu)t3, tw

(n)52sin(nw)t11cos(nw)t2 andtr

(n)5cos(nw)t11sin(nw)t2. We refer to the integersm andnas u winding number andw winding number, respectivelyThe profile functionsK12K4 and F1 , F2 depend on thecoordinatesr andu, only. This ansatz is consistent with thgeneral field equations~1!. The ansatz is time-independenand the time component of the gauge field vanishes. It thfore gives rise to static solutions, with vanishing~non-Abelian! electric field. The ansatz possesses a residual U~1!gauge symmetry. To fix the gauge we impose the condir ] rK12]uK250 @10,12#.

To obtain regular solutions with finite energy and enerdensity we have to impose appropriate boundary conditioRegularity at the origin requiresK15K350, K25K451,sin(mu)F11cos(mu)F250, ] r@cos(mu)F12sin(mu)F2#50.At infinity we require the solutions in the vacuum sect(m52k) to tend to a gauge transformed trivial solution,

F→UtzU†, Am→ i ]mUU†,

and the solutions in the topological chargen sector (m52k11) to tend to

F→UF`~1,n!U†, Am→UAm`

~1,n!U†1 i ]mUU†,

where

F`~1,n!5t r

~1,n! , Am`~1,n!dxm5

tw~n!

2du2n sinu

tu~1,n!

2dw

is the asymptotic solution of a chargen multimonopole, andU5exp$2ikutw

(n)%, both for even and oddm. Consequently,solutions with evenm have vanishing magnetic chargwhereas solutions with oddm possess magnetic chargen.

In terms of the functionsK12K4 , F1 , F2 these bound-ary conditions read K150, K2512m, K35„cosu2cos(mu)…/sinu for odd m and K35„12cos(mu)…/sinu forevenm, K4512sin(mu)/sinu, F151, andF250.

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Regularity on thez axis, finally, requiresK15K35F250, ]uK25]uK45]uF150, for u50 andu5p.

Defining the Abelian magnetic field via the ’t Hooft tenswith normalized Higgs fieldF @1#,

Fmn5TrH FFmn2i

2FDmFDnFJ

we note, that only solutions with evenm possess an Abelianmagnetic dipole moment@10#.

With this ansatz the general field equations reduce toPDEs in the coordinatesr and u, which are solved numeri-cally, subject to the above boundary conditions.

Results. The m-chains constructed in@10# are character-ized by u winding numberm.1 andw winding numbern51. In these solutionsm monopoles and antimonopoles alocated on the symmetry axis, with~roughly! equal distancebetween them. Their energy increases~approximately! lin-early with m, and likewise does the Abelian magnetic mment ofm-chains with evenm @10#.

Let us now consider chains consisting of multimonopowith w winding numbern52. For such chains withm<5 theenergies, magnetic moments and locations of the Higgs zare shown in Table I. Their energy increases~approximately!still linearly with m, and so does the magnetic moment of tchains with evenm.

Identifying the locations of the Higgs zeros on the symetry axis with the locations of the monopoles and anmonopoles, we observe that when each pole carries chn52, the zeros form pairs, when possible, where the distabetween the monopole and the antimonopole of a pair isthan the distance to the neighboring monopole or antimopole, belonging to the next pair.

We observe furthermore, that the equilibrium distancethe monopole-antimonopole pair composed ofn52 multi-monopoles is smaller than the equilibrium distance ofmonopole-antimonopole pair composed of single monopo

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MONOPOLES, ANTIMONOPOLES, AND VORTEX RINGS PHYSICAL REVIEW D68, 101701~R! ~2003!

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- FIG. 1. The field lines of the Abelian magnetic field are shown form52, n52 ~a!, m52,n53 ~b!, m54, n52 ~c!, m54, n53 ~d!,m53, n53 ~e! andm53, n54 ~f!.

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Thus the higher attraction between the poles of a pair wchargen52 is balanced by the repulsion only at a smalequilibrium distance.

When increasing the charge of the poles ton.2, we ex-pect this trend to continue. The monopoles and antimopoles of the pairs should approach each other further, setat a still smaller equilibrium distance.

Constructing solutions with chargen53, however, we donot find chains at all. Now there is no longer sufficient rpulsion to balance the strong attraction between3-monopoles and 3-antimonopoles. Instead of chains,now observe solutions with vortex rings, where the Higfield vanishes on closed rings around the symmetry axis

To better understand these findings let us considerphysical intermediate configurations, where we allow thewwinding numbern to continuously vary between the physicinteger values. Beginning with the simplest such solutithem52 solution, we observe, that the zeros of the solutwith winding numbern continue to approach each othwhenn is increased beyond 2, until they merge at the origHere the pole and antipole do not annihilate, however.conclude, that this is not allowed by the imposed symmetand boundary conditions. Instead the Higgs zero changecharacter completely, whenn is further increased. It turnsinto a ring with increasing radius for increasingn. The physi-cal 3-monopole-3-antimonopole solution then has a sinring of zeros of the Higgs field and no point zeros. A furthincrease ofn only increases the radius of the ring further.

Considering the magnetic moment of them52 solutions,we observe, that it is~roughly! proportional ton. The pair ofpoles on thez axis forn52 clearly gives rise to the magnetdipole moment of a physical dipole, as illustrated in F1~a!, where we show the field lines of the magnetic fieobtained from the ’t Hooft tensor. As seen in Fig. 1~b! thering of zeros also gives rise to a magnetic dipole field, whhowever looks like the field of a ring of mathematical d

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poles. This corresponds to the simple picture that the posiand negative charges have merged but not annihilated,then spread out on a ring.

The solutions with evenu winding number reside in thevacuum sector. Form52k.2 solutions it is now clear howthey evolve, when thew winding number is increased beyond n52. Starting fromk pairs of physical dipoles, thepairs merge and formk vortex rings, which carry the dipolestrength of the solutions. This is illustrated in Fig. 1~c! form54, n52, and in Fig. 1~d! for m54, n53. As seen inTable I, the total dipole moment increases~roughly! linearlyboth with m and n, since there arem/2 rings, each formedfrom charges6n.

The solutions with oddu winding number reside in thetopological sector with chargen. Form54k11 the situationis somewhat similar to the above. Here a singlen-monopoleremains at the origin, whereas all other zeros form pawhich for n.2 approach each other, merge and form rincarrying dipole strength. Since, however, a dipole onpositive axis and its respective counterpart on the negaaxis have opposite orientation, their contributions cancethe total magnetic moment. Thus the magnetic momentmains zero, as it must, because of symmetry@10#.

For m54k21, on the other hand, the situation is mocomplicated, because in this case there are forn52 always 3poles on thez axis, which cannot form pairs, such that azeros belong to a pair, symmetrically located around thegin. For the simplest case,m53, we observe, that two vortices appear in the chargen53 solution, emerging from theupper and lower unpaired zero, respectively, carrying opsite dipole strength. Forn54 a third ring appears, emerginfrom the zero at the origin, which however does not carrdipole moment. The magnetic field lines of then53 and 4solutions are shown in Figs. 1~e! and 1~f! respectively. Forn55 finally all three rings merge to form a single ring. Futher details of these solutions will be given elsewhere@13#.

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KLEIHAUS, KUNZ, AND SHNIR PHYSICAL REVIEW D 68, 101701~R! ~2003!

Concluding, we have found new static axially symmetsolutions of SU~2! YMH theory, characterized by two winding numbers,m andn. For n<2 the Higgs field vanishes om discrete points on thez axis, for n.2 it vanishes onm/2rings centered around thez axis for evenm, while for oddm,it vanishes on one or more rings and at the origin. Solutiwith evenm reside in the topologically trivial sector. Thecarry no magnetic charge but a magnetic dipole mom~roughly! proportional to the productm timesn. In contrast,solutions with oddm reside in sectors with nontrivial topology. They carry magnetic chargen and possess no magnet

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dipole moment. All these solutions correspond to sadpoints @7#. Analogous results holds for finite Higgs selcoupling @13#.

We expect that solutions of similar structure might existWeinberg-Salam theory@14#, where so far only the sphalero(m51, n51) @15#, the multisphalerons (m51, n.1) @16#,and the sphaleronS* (m52, n51) @17# are known.

Rings of vanishing or a small Higgs field are also presin Alice electrodynamics, where they carry magneCheshire charge@18#, while closed knotted vortices can arisin theories, allowing for solutions with a nontrivial Hopnumber@19#.

le

A

B

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