EXACT RESULTS IN THE THEORYOF NON-ABELIAN MAGNETIC MONOPOLES

Paolo ROSSI

ScuoloNormale Superiore,Pisa. Italy

1NORTH-HOLLAND PUBLISHING COMPANY-AMSTERDAM

PHYSICS REPORTS (Review Sectionof PhysicsLetters)86, No. 6 (1982)3 17—362. North-HollandPublishingCompany

EXACT RESULTS IN THE THEORY OF NON-ABELIAN MAGNETIC MONOPOLES

PaoloROSSIScuola Nor,nale Superiore, Pisa, Italy

Received20 February1982

Contents:

1. Introduction 319 6. Bosonicandfermionic zeromodes 3432. The theoryof pointlike magneticmonopoles 319 7. Multimonopole solutions 3473. Monopolesin non-Abeliangaugetheories 325 8. Supersymmetricmonopoles 3574. The Prasad—Sommerfieldlimit 336 References 364)5. The small fluctuationsand theconstructionof propagators 339

Abstract:

The theoryof Abelianand non-Abelianmagneticmonopolesis reviewedwith specialfocuson theexactintegrability propertiesof suchsystems.The limit of vanishingHiggs potential (Prasad—Sommerfieldlimit) is analyzedin detail. At the classical level, the constructionof all static

multimonopolesolutionsis presented,with emphasison the explicit axially symmetricstates.At the semiclassicallevel, the problemsof smallfluctuations,bosonicandfermionic zeromodesandtheconstructionof staticpropagatorsarediscussed.

Finally we consider thepossibility of embeddingmonopolesin supersymmetrictheoriesin orderto obtain models with strongerconvergencepropertiesandpossiblyfull quantummechanicalintegrability.

Singleordersfor this issue

PHYSICS REPORTS (Review Sectionof PhysicsLetters)86, No. 6 (1982) 317—362.

Copies of this issue may be obtainedat theprice given below. All ordersshould besentdirectly to the Publisher.Ordersmust beaccompaniedby check.

Single issuepriceDfl. 24.00, postageincluded.

0 370-1573/82/0000—0000/$11.50 © 1982 North-Holland PublishingCompany

P. Rossi,Exact results in the theory of non-Abelian magneticmonopoles 319

1. Introduction

The theory of magneticmonopoleshasseenan upsurgeof interestamongfield theoristsin the lastfew years,after the discovery,dueto 0. ‘t Hooft and A.A. Polyakov,that somespontaneouslybrokennon-Abeliangaugetheoriesadmit perfectly regular,static,finite energysolutionsof their classical fieldequations.Thesefield configurationshavethe propertiesof magneticmonopoles,i.e. theyact as sourcesfor the magneticpart of the long range,unbrokenelectromagneticcomponentof the gaugefield.

Due to the non-Abelianstructureof the gaugegroup no string singularity appears,in contrastwiththeAbelian theory of monopoles,developedoriginally by Dirac in 1931,wherethe string was requiredin order to preservethe magneticflux. We review the Abelian theory of monopolesin section2 anddescribethe non-Abelianresultsin section3.

Soon after the discovery of these classical solutions it was observedby M.K. Prasadand C.Sommerfieldthat, at leastin somespecialcase,the solutionscan be presentedin analytic, closedform.This limit andits propertiesare consideredin section4.

The recent successfulapplication of exact methodsin (two dimensional) field theory stimulatedfurther effort in the quest for exact monopole solutionsand, more generally, in the study of exactpropertiesof the monopolesystem.

This topic turned out to be deeplyrelatedto the formal aspectsof the theory of instantons,wheremodern advancedmathematicalmethodshave been applied up to the successfulidentification of apresumablygeneralclass of solutions. As a result of this effort we now possessa ratherexhaustiveanalysisof the classicalsectorof the theory.

The problem of fluctuations,as well as that of interactionwith external test particles (bosonicandfermionic) has beenextensivelystudiedin the linearizedversion of the time-dependentfield equations,anda vastamountof informationon theformal andphysicalpropertiesof the systemis now available.Sections5 and6 are devotedto the presentationof theseproperties.

Moreover, all staticmultimonopolesolutionsseemto havebeenidentified and,at leastwhen somesymmetryis imposedthe solutionsarestill ratherexplicit, althoughharderandharderto deal with whenthe numberof monopolechargesincreases.The statusof our presentknowledgeon multimonopolesolutionsis reviewedin section7.

Neverthelesswe are still far from an exact solutionof the quantummonopole-gaugefield system.Yetis is intriguing to observethat at least in somevery specialcase,when further symmetry properties(supersymmetry)are assumed,the exactmassspectrumis known, thereis a hint of full self-duality (i.e.completeinterchangeabilitybetweenthe magneticandelectriccomponents)andthe systemappearsasthe best available candidate to full quantum mechanicalsolvability in presentday three-plus-onedimensionalquantumfield theory. Section8 is devotedto this topic.

2. The theory of pointlike magnetic monopoles

The classicalmagnetic monopole is describedas a pointlike sourceof magneticflux, generatingaradialmagneticfield characterizedby a hr2 behavioranalogousto that of the electric field of an electricpointlike charge:

B=curlA=—-~---~ (2.1)r

320 P. Rossi. Exactresults in the theory of non-Abelianmagneticmonopoles

whereg is a measureof the strengthof the magneticcharge. The electromagneticpotential A has anarbitrary gaugefreedom. However it was found by Dirac [32] that in no gauge A may he completelysingularity free outsidethe origin. If we require the singularity of A to be alongthe negative z axis wefind:

A=A~ (2.2)

A gcosOl (_)~)

~ 4ir rsinO

and the corresponding(singular)magneticfield is indeed:

B= —~— [~+4~O(_z)~(x)~(Y)] (2.4)

correspondingto a “string’~of magnetic field along the z axis that accountsfor magnetic fluxconservation.The only way to haveno quantum mechanicallyobservableeffectsfrom the string is toimposethe so-calledDirac chargequantizationcondition:

eg = 4rrn (h = c = 1) (2.5)

where e is the electricchargeandn is any integeror half integer.Let us now consider the motion of a classical chargedparticle in the field of a monopole. This

problem turns out to be completelysolvable:its solvability is linked to the large, dynamicsymmetrygroupunderwhich the system turns out to be invariant [45,82].

First of all, the invarianceunder rotationsinsuresus abouttheexistenceof a conservedtotal angularmomentum.Its functionaldependenceon the dynamicalvariablesis howevernon-conventional166]:

L=M+~-i= ri\~p—eA)+n,~. (2.6)

The anomalous,radial contributionto the angularmomentummay be checkedto find its origin in thespace-integrationof the e.m.field angularmomentumdensity:

J d3rIrA(BAE)I. (2.7)

Another obviouslyconservedquantity is the Hamiltonian, that turns out to coincide with the kineticenergy:

H = (p -eA)2= ~mi~2 (2.8)

as it alwayshappensof particlesin purelymagneticfields. Moreover,the absenceof an intrinsic scaleinthe systemleavesthepossibilityof scaletransformations,that areassociatedwith a conservedgenerator

P. Rossi~Exact results in the theoryof non-Abelian magnetic monopoles 321

of time dilatations(or = tr — r/2)

D=tH—~m(rr+r.r) (2.9)

anda conservedgeneratorof the specialconformaltransformations(Or = (t)2i~— tr/2):

K —t2H+2tD+~mr2. (2.10)

The (quantum)commutationrelationsof theseoperatorsare [45]:

[D, H] = —iH (2.lla)

[D,K1=iK (2.llb)

[H,K]=2iD (2.llc)

that is, they generate an 0(2, 1) conformal group, so that the full symmetry of the system is0(2,1) X 0(3) and the magneticmonopolesystembecomesanalogousto the Coulomb—Keplersystem.For the sakeof completeness,let us observethat similar symmetrypropertiesarisewhen we generalizethe Hamiltonian by introducing hr2 potentialsor combinationsof hr2 and hr potentials.This lastpossibility is relevantto the study of dyon properties,the dyon beinga pointlike magneticand electriccharge.

We would alsolike to stressthat, while our discussiondealswith an infinitely heavy,fixed monopole,it could be repeatedby introducinga finite massand dynamical monopole variablesand using theappropriateversionof the Lorentz forcesbetweenmoving charges.It is easyto show that separationofthe center-of-massmotion and reducedmasstechniquesturn this problem into the potentialproblemwe havediscussed.

Let usnow briefly discussthe propertiesof the classicalsolution. The condition (following from eq.(2.6)):

Lr=n (2.12)

implies that the trajectory lies on the surfaceof a conewhose axis is the direction of L and whosehalf-angleis

= cost(n/~L~). (2.13)

Moreover,from the conservationof energy,we find that the absolutevalueof the speedv is aconstantof the motion. Fromthe conservationof L2 it is thenimmediateto find out that the motion is nothingbut a uniform rectilinearmotion on the conesurface,with impactparameterb = \/(L2 — n2)/2mE.

Let us now turn to the quantummechanicalversionof theproblem.Let us first considerthe non-relativisticSchrodingerequation[83,971:

(p_eA)2~E~, (2.14)

322 P. Rossi,Exact results in the theory of non-Abelian magneticnionopoles

andnotice that:

(p— eA)2= ~(r .p)(p . r)+~(L2_n2]. (2.15)

The eigenfunctionsof the Hamiltonianmay be classifiedaccordingto the eigenvaluesof H, L~andL~.andseparationof variablesoccurs,suchthat

t1f=REl(r)Y~,,,,(6,çc) (2.16)

where

(2.17)

L2Y,~= 1(1+ l)Y~,~1 (2.l8a)

~ = mY~,,,,. (2.18h)

It is immediateto find that

REI(r)= —\/k/rJM(kr)=j(.(kr), k = \/2mE (2.19)

where JM is the Besselfunction of order M = \/(l + ~)2 — n2. The generalizedsphericalharmonicsY,~

1,,,are relatedto the representationfunctionsof the rotationgroup:

~ ~) = ~ 0, ~) = (I. n~e ~ e~eC~~3JI.m). (2.20)

They are expressiblein terms of Jacobi polynomials in (cos6) with phasefactors c” ~ and are

subject to the constraints:

l�n. m~l. (2.21)

The Klein—Gordon equation for a relativistic scalar particle is substantially the same as the non-relativistic equation.with the replacement2rnE—~w—m~.

More interestingis the caseof relativistic spinorparticles 122. 52, 781. whose Dirac equationis:

I(peA)+3mltJi=wtIi (2.22)

with conservedtotal angularmomentum:

J=L+~. (2.23)

Again separationof variablesis possible.dueto the presenceof a conservedoperator1781:

P. Rossi, Exactresults in the theoryof non-Abelian magneticmonopoles 323

yo(~.M+1) (2.24)

whereM = r A (p — eA).

Let us first considerthe angulareigenstatesof J2, J~,L2, u L:1/j+m\U2 fj~\I/2

Q

1n0—t/2)m = R 2j ) Yj_ti2n rn—1/2, ~2j ) YI_t/2.n.m÷I/2] (2.25a)

+ m + 1\h/2I /j—m+1)i/2

~jnU±1/2)m = L~2j + 2 t/2,n,m- 1/2, ( 2j + 2 ) ~ 1/2.n.m±1/2] (2.25b)

andbuild up the eigenstatesof u M:

= ~ V1+~-~)fljnti~ti

2im +~(lJ1+~T±j1~_r)12jnU±I/2)m (2.26)1+2 J+sj

satisfying

(o~M)f1~= ~1~M’ fl~ , M_V(j+~)2_n2 (2.27)jnm ~. — J US/nm

andrelatedby

O~i~f2~m= f2~m. (2.28)

Let us now observethat the quadraticform of the Diracequationis:

[u (p — eA)]2~i= (w2 — m2)~ (2.29)

and

[o- (p—eA)12 p_eA)2_eu.B=~(r.p)(p.r)+~[L2_n2+nu. ~] (2.30)

where

L2—n2+no.i~=(uM)2+(uM). (2.31)

The bispinor solutionsaretheneasily found to be:

~ R~(r)fl~[ (w+mu/2 1M 2wj I

t/JkJnm[ w—m)°2 I (2.32)±i(

RkM(r)[2~~~mj2w

where

324 P. Rossi.Exactresults in the theoryofnon-Abelian magneticmonopo/es

R~M(r)=\/k/rJM±!!S(kr).

Again, as in the scalarcase,thereis a completeanalogywith the correspondingsolutionsfor a freeparticle. The difference amountsto the introduction of an anomalousangularmomentumeigenvalue1’(j’) definedby the relation:

(l’+~)2=(l+~)2-n, l�n(2.33)

~~~l)2~+I)2fl2 j+~�n.

A peculiarcharacteristicof the spin ~caseis the possibility that j + ~= n, correspondingto M = 0, andto formal solutionsof the equationsof motion that are not regularin the origin. This is a “fall on thecenter”phenomenon,wherethe absenceof mechanicalangularmomentumallows for the particle to gothroughthe sharpsingularity of the potentialat the origin.

The pathologyof the quantummechanicalresult is bestunderstoodin termsof non-Hermicityof theHamiltonian operatoron the correspondingsubspaceof functions [221.In the samephysical situationthe classicalparticle shows the unexpectedphenomenonof an abrupt changein direction of its freemotion when it comesto the origin wherethe monopole is located.

An amusingproperty of electricchargesin pointlike monopole fields is the possibility of explicitlywriting down scatteringsolutions (i.e. solutionsasymptoticallybehavinglike a planewave plus onlyoutgoing (or ingoing)sphericalwaves).

In the scalarcase[14,171, if the particleentersfrom z = —~ onefinds that the z-componentof its totalangularmomentumis —n.

Rememberingthe familiar expansionof e1~onemay show that the sumsof partial waves

= e~”~ (2/ + 1) e~e1~2ji,(kr) Y~1,~(0,~) (2.34a)

~out) = ~ (2/ + 1) e~”2j~.(kr)Y~

1~(6,~) (2.34b)

correspondto the outgoing andingoing scatteringsolutions.In order to build up the spinor scatteringsolutionswe must first definethe “helicity” eigenstates1141

(eigenstatesof the conservedoperator.~ . (p — eA))

(p — eA)i,/i~,,,= ~ (2.35)

where

~ ~ j�n +4 (2.36)

t/1kn~1/2,nm= [(w + rn)’12 ±(°— rn) 2] e”’~~ , = n. (2.37)

2w 2w Virjn~ r

We thenfind the scatteringsolutions(correspondingto independent(±)helicity states)[781:

P. Rossi, Exactresults in the theoryof non-Abelian magneticmonopoles 325

~‘kn(in) = ~ \/2j + 1 e”~ e”°e~M/2~~±,i2 (2.38a)

jjnI—!/2

hI4n~(out)= ~ \/2j + 1 e~M12lfr~~±,,

2. (2.38b)j = Jn~—!/2

Thesescatteringsolutionsturn out to be especiallyusefulin the approximatediscussionof the scatteringamplitudesfor non-pointlikemonopoles,as the onesthat emergein non-Abeliangaugetheories.

3. Monopolesin non-Abelian gaugetheories

It has beenfound that non-Abeliangaugetheorieswith spontaneouslybrokensymmetrymay admitclassical static finite energysolutionsto the field equationsthat are to be interpretedas (extended)magneticmonopoles[42,67]. The appearance(andstability) of thesesolutionsis relatedto the presenceof non-trivial topologicalpropertiesin the mappingfrom the groupmanifold to the sphericalsurfaceatinfinity wherethe boundaryconditionsare defined[6,24,59, 65,85]. Due to the non-Abelianstructureof the theory, thereare gaugesin which no Dirac string appearsin the solution, that turns out to beregular everywhere.It is howeverthe topological natureof the magneticcharge, as we shall see,thatinsuresthe correctchargequantizationcondition[29,30].

For the sake of definiteness,let us considerthe simplest theory admitting magneticmonopoles:an0(3) gaugetheory coupledto a triplet of Higgs fields in the adjoint representation,with a potentialtermallowing for a non-zerovacuumexpectationvalue for the norm of the Higgs field [37]:

..~f=__~F~F~+~Da~pa _~A2e2(P~2P~1— v2)2 (3.1)

where

~ = — ~ + e�A~A~ (3.2a)

D~.= + e�~1!~cA~.. (3.2b)

Some propertiesof this system are obscured by the way the coupling constantsand scales areintroduced.Let us then operatethe rescaling

b—*vk, A~—*vA,.~, x~-3x,jev (3.3)

andwrite down the staticenergydensity (3~= 0, A0 = 0)

= gv�(A)= gvf ~{~BaBa+ ~ ~ + ~Az(~k~2~r— 1)2} (3.4)

where g = 4ir/e is the unit of magneticchargeand �(A) turns out to be 0(1) for all values of A[10,16,~31.By partial integrationit is possibleto write � in the form:

= ...JdSx~(Ba~~a)+JdSx{1[Ba D~r]2-l-’A2(cb’~k’~— 1)2}. (3.5)

326 P. Rossi, Exact results in the theoryof non-Abelianmagneticmonopoles

The gaugeinvariant quantity

n = J91(B) (3.6)

is called “topological charge” andis the integralof a total divergence.Its contribution is then non-zero(for finite energy configurations)if and only if there is a non-zero vacuum value for ~ and some

non-trivial topological structurein the behaviorof the fields at infinity. It is possible to show that n isbound to be an integernumberand that it measuresthe flux of the magnetic field generatedby theclassicalgaugefield configuration. It is alsopossibleto show that n is relatedto anothergaugeinvariantproperty of the system. i.e. the “zero set” of the Higgs fields tJY

4 [31;indeed the following relationholds:

= f dS ~IJk ~ ~ (3.7)j 8ir

and in all gaugeswhereno Dirac strings are present

~J d~SI~IJk ~a& ~

3~b ik’k (3.8)

is equalto the numberof zerosof k’~enclosedby the surfaceS0. We may then relatethe locationandstrengthof the magneticchargesassociatedwith a given field configurationto the location andstrengthof the zerosof ~O

A local, gauge invariant characterizationof the field configurations is also possible by the intro-duction of the ‘t Hooft tensor[421

= F~,r — ~abc ~jja D~P”D~g-~a~(A~I”)— a~(A~I~) — b~j~ t9çh a~I’ (3.9)

whose spacecomponentsplay the role of the divergencelessmagneticinduction field for this model:indeed its divergencerelatesto the density of topological chargeand is zero everywhereexcept fordelta-functionsingularitiesat the zeroesof the Higgs field.

An important consequenceof the expressionfor e, eq. (3.5), is that, due to the positive-definitenatureof the last two terms [15,26]:

~f�ngv (3.10)

and the equalitywill requireA = 0.However we have not yet shown that finite energy solutions do indeed exist. The first known

monopolesolution, correspondingto the casen = 1, was found by ‘t Hooft [42] and independentlybyPolyakov [67] by consideringthe situationwhen the field configurationsenjoy the property of beingsphericallysymmetric.

A property of the Yang—Mills fields whoseconsequencesare far-reachingis that there are ways to

P. Rossi,Exactresults in the theoryof non-Abelianmagneticmonopoles 327

realize physical spacesymmetriesthat involve internal spaceindices. As for our specific example,explicitly spherically symmetric configurations, among others, are those that may be put in theform [951

= EiabXb 1 + ç~2(r)~(0 — 1i~a)~~ ~aAi(r) (3.lla)

~pa = IaAo(T). (3.lhb)

(Obviously, all the configurationsobtained from thesethrough a non-sphericallysymmetric gaugerotationarephysically sphericallysymmetric,too, even if the symmetryis not apparent.)

Moreover, if we furtherrestrict the Ansatz to

= �iabXb 1 — K(r) (3.12a)

= ~a~I~(r) (3.12b)

we find that the static field equationsreduceto:

1)K— P2K= 0 (3.13a)

~ -~~ 2

U ~I’ h. U~f’ ~ ‘~‘

2~ ~‘ ~ —

dr rdr r 2

It hasbeenshown that this systemadmitsof a solutionwhich is regulareverywhereandsuch that

K—~1+O(r2), ~-=r, r—~0 (3.14a)

K—~O(e’), P—~1 + 0(e~7r), r—~. (3.14b)

Substitutionof the Ansatzeq. (2.12) leadsto

= ~ijk xkIr (3.15)

explicitly showingthat we arein the presenceof a monopolewith chargeg. It hasbeenproven that thissolution is stableand that thereareno other staticsphericallysymmetricsolutions:in particular thereare no generalizationsto n> 1 [92]. In connectionwith this last remark we would like to stressthatsince n> 1 would correspondto more thanone magneticallychargedparticle, when A � 0 a repulsiveforce would act amongthem, such that in particular no stable staticsolutionswould be possible.Theonly residual possibility is that of unstable maxima of the energy correspondingto many chargessuperimposedat the samelocation;howeveraswe havealreadypointedout the solutioncould not havesphericalsymmetry.One would expect axially symmetric unstablesolutions to exist with n> 1 andenergy:

= n�1(A)+O(A

2). (3.16)

328 P. Rossi,Exactresults in the theory of non -A belian magnetic monopoles

An interestingextensionof the single monopolesolution comesfrom consideringthe possibility of anon-zerostatic sphericallysymmetricCoulombicfield:

A~= Xa V(r) (3.l7a)

V(r)~V~+e~/r, r—~. (3.l7b)

The appropriatelymodified equationshavebeenshown to possesssolutions, which are calleddyons,andcorrespondto (extended)particlescarryingboth a magneticchargeg andan electricchargee0 [51].The massof thesestatesis consistentlylargerthan that of the correspondingpuremonopolestates.

A further extensioncomesfrom consideringgauge groupslarger than SU(2) and more complexpatternsof symmetry breakdown.This subject is obviously relevant to the problem of the physicalexistenceof magnetic monopoles.The appearanceof monopole solutions for the groups that areexpectedto describegrand unified theoriesof the physical interactionsmay be a suggestionof theirpresentexistenceor, at least, of their appearance(and possibly relevance)in some phaseof thecosmologicalevolution. We shall not belaboron this point. We want only to recall that sphericallysymmetricmonopoleshave beenfound in SU(N) gauge theorieswith different patternsof symmetrybreaking,and that dyonsare alwaysassociatedwith them [9, 11, 13, 28, 39, 93, 94].

More generally,a very interesting link hasbeen found betweenthe symmetry propertiesof themonopolemultiplets andthe symmetryof the residualunbrokengaugegroup H. It is alwayspossibletodefine a dual groupH”, enjoyingthe property(H~= H, which classifiesthe monopolemultiplets andexplains the patternsof their quantumnumbers [33,38]. (In particular, for the case of SU(2), theresidualunbrokensymmetryis just the e.m.U(l), andthe dualgroupis another— magnetic— U(1).) Thisphenomenonsuggeststhe relevanceof the study of “fundamental” theoriesof monopoles,wheretheycould appearas constituentparticles,whereasthe original “gauge” particlesshouldappearas solitons.Little progresshoweverhasbeenmadein this direction.

The canonicalquantizationprocedurein the presenceof a classical,finite energy field configuration

requiresknowledgeof the time-dependentfluctuationsaroundthe classicalsolution.In order to analyze the problem, let us supplementthe SU(2) Lagrangian,eq. (3.1), with terms

coupling “small” scalarand fermion fields to the original gauge-Higgssystem:

(3.1$)The corresponding small fluctuation equations are:

1—DCD~+ ~ GS’P’~T”+ ~ = 0 (3.l9a)

Iiy~DC+ rn + GFP’~T’~It/J= 0. (3.l9h)

Once we have replaced the fields A~and P” with the known spherically symmetric classical solution wefind that a total conservedangularmomentumfor the fluctuations can be definedand its expressionturns out to be:

J=rAp+T+S (3.20)

where S, the intrinsic spin, is zero for the scalars and equal to ~ for the fermions.

P. Rossi,Exact results in the theoryof non-Abelian magneticmonopoles 329

The classical expressionfor the angularmomentumis now supplementedwith an “anomalous”contributioncoming from the isospin: thisphenomenon,generalizingwhat we havealreadyseenin theAbeliancase,is a peculiarfeatureof the gaugemonopolesystems[41,46,48,581.

Explicit solutionsfor the fluctuationequationsmaybeobtainedonly whenoneconsidersappropriateapproximations.In particular the choice that the operatorT~~ commuteswith the approximateHamiltonian (a property which is not sharedby the exactproblem in general)allows the statesto beclassifiedby meansof their eigenvaluefor T ~. Since T ~ is a chargeoperatorfor the fluctuations, thismeansthat we aresuppressingall charge-exchangeprocesses.The eigenfunctions,both in thescalarandin the fermion case,will thenhavea factorizedisospin dependence:

= U~~(j, rn, w), ~ = U~~(j, rn, w) (3.21)

where

~ T2~~=t(t+1)~~ (3.22)

and U is the isospin rotation matrix such that:

U~(rAp+T)U=rA(p—eADT3)+i~TS (3.23a)

U’(T. I~)U= T3 (3.23b)

whereAD is the Dirac potential, eq. (2.2). Not surprisingly this gauge(correspondingto the standardunitary gauge)is also called Abelian gauge,becausethe 3rd componentof the gaugefield is now themasslessphotonfield, a purely Dirac pointlike monopolefield, as it follows from eq. (3.15).

Let us now considerseparatelythe scalarcase[17}.The eigenvalueequationin the monopole-dyonfield can be put into theform

{4 (r p)(p r) + -~ [(J — KT)2 — (1— K)2(T~

= {[w + VT — ~2 — G5 ~It(T Il) — H5~P

2}~c. (3.24)

The pointlike limit is once more exactly solvable.It amountsto taking the asymptotic valuesfor theclassicalfields:

K=0, 1=1, V= V,~+e~/r. (3.25)

Eq. (3.7) turns into:

p)(p . r)+ ±~(J2— T’ ~2)]~ = [(w + V~T~~+ T• )2 - ~ - G~T• — Hs]ço (3.26)

andthe radial equationfor theeigenstatesof J2 and Tr is:

[_~sr+4[l(l+1)_n2_e~n2]_2(w+ V~n)~n+,.t2+G5n+H~—(w+Voon)2]~ 0. (3.27)

330 P. Rossi, Exactresults in the theoryof non-Abelian magnetic monopoles

Eq. (3.10) has the general structureof a Coulombic problem, with anomalousangularmomentuml’(l’ + 1) = 1(1 + 1)— n2. Theeigenfunctionsmaybeexpressedin termsof hypergeometricfunctionsandthediscreteenergyeigenvaluesare:

w ~2+H~+nG~ —nV~ 1 + n2e~/(V(l+ 1~2— n2 — n2e~+ flr + 1)2 -

where ~r = 0, 1,2,... is associatedto a radial degreeof freedom.Notice that we are assuminge~~ Iandthat V

112+ H~+nG~—nV,, plays the role of the effectivemassfor the elementaryexcitationsof the

scalarfluctuationfield.The pure monopole casecorrespondsto V(r) = 0; it is straightforwardto see that the equation

reducesto the alreadydiscussedAbelian equation,with no boundstates.Let us now considerthe constraintson the quantumnumbers:T = J . 1 implies that n ~ I. We

then find, for any given (flr, 1, m), 2t+ 1 eigenstateswhen t ~ 1, but only 2j + I eigenstateswhen / ~ t.

Thesemultiplicities correspondto thoseof a systemhavingan internal spin T, which is consistentwiththe interpretationof the angularmomentumformula, eq. (3.3), as conversionfrom isospin to spin.

We want to commentbriefly on the inclusionof corecontributionsandcharge-exchangetermsin theHamiltonian. While core contributionsin generalwill just affect the shapeof the wave functionsin away that may be analyzedperturbativelyfor eachchargesectorseparately,the term

‘~K(r ~- ~ ‘[T.J--(T.,~)2] (3.29)

r

will alsoinducecharge-exchangereactions.In the distorted-wave Born approximation the charge-exchangeamplitude is obtained by the

formula:

—~-~ ~ {—t~ . J— (T• ~)2I}~~)) (3.30)

involving the previously determinedscattering solutions. The angular integral may be explicitlyevaluatedwith the aid of the operators:

X.= T.J_(T•i~)2±i(TA,2).J (3.31)

enjoyingthe property

[T.i~,X±j=±X± (3.32)

so that theymay be qualified as chargeraising andlowering operators.They are normalizedthrough

(3.33)

It is apparentthat in this approximationthe only non-zerocharge-exchangeamplitudesinvolve theexchangeof a single unit of electriccharge.

P. Rossi,Exactresults in the theoryof non-Abelianmagneticmonopoles 331

Let usnow turn to the fermioniccase[78].The eigenvalueequationhas the Dirac form:

(3.34)

and thespinorversion of its second-ordercounterpartis:

VT~~]2 [ 1K(~ T)]+(UA ~)(TA r~±(~. ~)(T~ ~)K_ 1_ GF~2(T~

= — ie[(u A i~)(TA i~)~-~+(u~i~)(T

— iGF[(u A i~)(TA i~)~+(u• ~)(T ~)~](‘~). (3.35)

In the pointlike limit howevereq. (3.35) reducesto:

{4 (r . p)(p ‘ r) + 4 [(u . M)2 + (o. . M) — e~(T.~)2 ±ie~o~T•

= {[~+V~T~+~-~T ~]_(m + GFT ~ (3.36)

where

M= rAp+ T—(T.i~)i~. (3.37)

The angularoperatorscan be given the form:

(~y. M~)2+ (o . M~) (3.38)

M~=M±ie~(T~ (3.39)

and i/i may be chosenin such a way that T ~ is diagonalizedand the samehappensfor o M~,replacedby its eigenvalue—1 ±VM2_ e4

0n2. Once more we haveto deal with a Coulombicproblem,

andthe energyeigenvaluesfor the boundstatesare:

m + nGF—nV,,. (3.40)

Vi + n2eo4/(\/M2_e~n2+nr)2

a straightforwardgeneralization_ofthe problem of a Dirac particle in a Coulomb field, once werememberthat M= V(j+~)2—n2. The effectivemassis now m + nGF— nV~.

The conservedoperatorof the relativistic Coulombproblemis alsogeneralizedto our system[781:

1 yo(.~~M+ 1)75(H—YOGFT• ,~— m)—i(GFT’ i~+m)e~Zt~. (3.41)

332 P. Rossi.Exact resultsin the theoryof non-Abelianmagneticmonopoles

Let us recall that

1T = J• ~— £‘ i~/2~~ J• ~l+ 1 ~/2~~j +4. (3.42)

When n <j + 4 then flr admitsthe value0. Thereare two seriesof solutions,correspondingto the twopossibilitiesfor M~,and the detaileddiscussionof degeneraciesis againconsistentwith the spin-from-isospininterpretation.

Again, we may think of the effect of core correctionsand of the charge-exchangeterm in theHamiltonian:

x’ (i A T)K(r)/r. (3.43)

In analogywith the scalarcasewe define:

Y~ (TA i~)(uA ~)±i(TA i~ a’ (3.44)

IT~,Y-~]±Y+ (3.45a)

1~o~~. Y±~= ~Y. (3.45b)

= ~Il + u• i~1T2 T i~(T. j~± 1)1 (3.46)

andevaluatehelicity-conservingandhelicity-flip amplitudes.The operatorsdefinedin eq. (3.44) arealsohelpful in performingthe separationof variablesfor the

exactproblem of fluctuationsin the fields of extendedmonopolesanddyons.Let us considerthe spinorversion of the first-orderequationwhere

x~=~+ix. ~ =~+i~. (3.47)

Deletingthe explicit massterm for simplicity we obtain from eq. (3.34):

[‘(a’. ~)r . p +1(u• ~)(u . M)+ ~u. (r A T)±iG ~(r)T = 1w + V(r) T ~(3.48)

Let us expandthe spinorsin the form:

x~= ~ R~)~~+)-R~Q~±)+R~Q~)- R~~Q~) (3.49)

where

= ~U[~~(Qj~m ~m)(±)x(Q~1~~1~flJ-n,n )1. (3.50)

We thenget the following set of coupledradial equations:

P. Rossi, Exactresultsin the theoryof non-Abelianmagneticmonopoles 333

~ i)— n(n + 1)R~~’~= iwR~+ inVR~ (3.51a)

[~-+L~nG~]R _~‘~tR~—~Vt(t+h)— n(n — 1)R~lH= iwR—in%7R~+’~ (3.Shb)

supplementedwith the conventions:

R11~= (~)R~1 (3.52a)

R°~= R°, R~:)= (~)R~±1,R~°±3

1~~= (~)R~. (3.52b)

In principle, the systemof linear first-orderordinarydifferential equations(3.51) allows the discussionof the mostgeneralproblemfor fermionswith arbitraryisospinin the field of extendedmonopolesanddyons. In practice,analysisof eqs.(3.51) is exceedinglyhardbut for somespecialcases.

The system of equations(3.51) turns out to be especially appropriatefor the discussionof thefermioniczeromodes(w = 0) in the puremonopolefield (V = 0). We shalllater discusstherelevanceofthesemodesto the problemof quantizationof the classicalsolution.

Notice that in the specialcasewe areinterestedin the r.h.s.of eqs.(3.51) is set equalto zero,andthetwo systemsof equationsdo then decouplewith respectto x~andx. It is easyto show that thereareno normalizablezeromodesof thex~type, andthe systemthen reducesto:

[~-+~_ ~ Vt(t+ 1)— n(n + i)R~1~= 0 (3.53a)

[~4~+~+ ~ 1)— n(n — i) R~t1~= 0 (3.53b)

with the convention

R~”~=

andthe specialequation:

~ h)R~+1=0. (3.53c)

(We havesuppressedthe— index of x.)It is possibleto diagonalizethissystemaroundthe origin andat infinity.At the origin the behaviorsof the independenteigenfunctionsare labelled by the eigenvaluesof

o(r Ap), running from ±(j+~+t)— 1 to ±(j+~—t)— 1 when j+~>t, and to ±(t+~—j)—1 whent � j + ~.The two different parities decouple(correspondingto the (±)sign), and the eigenvaluesaregroupedaccordingto:

(j+~+t)— 1, —(j+~+t—1)—I, (j+~+t—2)— 1,... (3.54a)

334 P. Rossi, Exact results in the theoryof non -Abelian magneticmonopoles

and

—~+~+ t)— 1, ~+~+ t— 1)— 1, —~+4+ t— 2)—i,... (3.54b)

with a multiplicity given by the smallestbetween(2t+ 1) and (2j+ 1).In order to count the numberof regulareigenfunctionswe must count the numberof positive (or

zero)powersin eachmultiplet:

when j + ~> t and t half-oddt + ~ positive t + ~ negative in both groups

and t integert positive t+ 1 negative in onegroup (+)t+ 1 positive t negative in the other (—)

when j + ~ ~ t and t half-oddj positive 1+ 1 negative in onegroup(+)j+ 1 positive j negative in the other (—)

and t integerj + ~ positive j + ~negative in both groups.

Let us now proceedto the diagonalizationat infinity:

whenj+~>t

R~,R~—A e”’ + B e’1’, R~±)—r’~M,M>i (3.55)

when jjj< t

M= 0 when n =j+~ so thatR~~”2~andhigher values of n arezero

when t half-odd

R”~,~ -~-A e’~+ B e”’, R~1~sH— e’’2~’ (3.56)

when t integer

~ R~ A e~”’+ B e”~. R1’~I/2~- — e -0~2)~ (3.57a)

~ M�l whenj�~. (3.57b)

Collecting all results we find that:when (I + ~)> t there are (2t + 1) functions and (21 + I) regular conditionswhile when (I + ~)~ t and I

half-odd there are 2j + I functions

j + 1 regular conditions at infinity

and one parity choice (—) gives

j + 1 regularconditionsat the origin

P. Rossi,Exact results in the theoryof non-Abelian magnetictnonopoles 335

when (j + ~)~ t and t integerthereare2j + 1 functions

j + ~regular conditionsat the origin

while the other parity choice (+) gives

j + ~ regularconditionsat infinity.

Solutionsmay existwhenthe spaceof singularsolutionsat the origin (at infinity) hasa dimensionalitysmallerthanthe spaceof regularsolutionsat infinity (at the origin): in this casea linear combinationofregular solutionson oneside existssuch that it spans the subspace not spanned by the continuation onthat sideof the singularsolutionson the oppositeside.

The total numberof zeromodesis now easilyobtained:

N= 1)= {t(t+1): ~ (3.58)

This result, based on a direct analysis of the equations,matcheswith the more formal and rigorousresult by Callias, obtained by means of an index theorem appropriateto the problem [23].

The solutions for I = ~ and I = 1 may be worked out explicitly [49]. When t = ~ there is only onesolution, corresponding to j = 0, M = 0:

(3.59)

R~L~=~exp{-J dr’ [~G~P+ 1 (3.60)

When t = 1 therearetwo independentsolutions,correspondingto j =

[~-+-~+ G~P]R~i1—~V2R~±1= 0 (3.61a)

0. (3.6th)

Eqs. (3.61) are solvedby:

R~ = ~ (r2V2R~±~) (3.62a)

V~R~±)= 4 u(r) exp{—f dr’ F(r’)}, F = ~[G~ — K’/K] (3.62b)

where u(r) solves

336 P. Rossi,Exact results in the theoryof non-Abelian magneticmonopoles

— a” + (F2 + F’ + 2K2/r2)u = 0. (3.63)

The relevance of the fermion zero modes comes from their significance when the problem ofcorrectly quantizing the monopole states is dealt with. We shall not enter the generalproblem ofsemiclassicalquantizationof solitons.Let us just rememberthat oneusually associatesa single quantumstate to eachclassicalsolution supplementedwith all its bosoniczero modesthat reflect the standarddegeneraciesof the quantum state (translation invariance, Lorentz propertiesand so on) and thesymmetriesbrokenby the classicalsolution that are restoredin the quantumstate.The fermioniczeromodeshowevercannotin generalbe naively interpretedas the result of the breakdownof a standardsymmetry.The only circumstancein which this possibilityemergesclearly is whenthe way thefermionsare introducedrendersthe system explicitly supersymmetric[79]. In that case,the appearanceof thezero modeshints at the supermultipletdegeneracy,that is superficially brokenby the classicalsolution,andsuggeststhat the modesthemselvesare to be usedas anticommuting(Majorana)chargesin order tobuild up the supersymmetricmultiplet of soliton statesthe sameway as in the standardapproachthewhole supermultiplet of fundamentalexcitationsis built up by applying the anticommuting chargeoperatorsto any oneof them.This proceduremaybe naturally generalizedto the situationswherethereis no apparentsupersymmetry:notice that the zero modesare electrically neutral, so they may beproperly usedas Majoranacharges.The resulting physical multiplets will then be realizationsof thealgebrathat the anticommutationrelationsfix on the quantumoperatorsassociatedwith the zeromodesin the eigenfunction expansion of the Fermi wavefunction in external field, as expected on the ground ofa more standard approach to the problem of quantization [49].

As for the degeneracyof the soliton multiplet, it is a standardresult from supersymmetrythat, in thepresenceof N independent charges (N fermion zero modes) the multiplicity of a supermultipletreconstructedstartingfrom a sphericallysymmetric(J = 0) state is 2”.

In particularwhen I = ~, N = 1 and the fundamentalstate is a couple of spin 0 (fermion number±~)[49]states,while when t = 1, N = 2 and the fundamental multiplet of monopoles contains four states; acoupleof spin~states(forming aDiracspinor),ascalaranda pseudoscalar.By theway, the t = 1 case maybe supersymmetrized and the resulting soliton spectrum is then expected to be itself supersymmetric.Afour particle multiplet like the one wehave nowdescribed is consistent with the so-called scalar multiplet ofsupersymmetry with two anticommutingcharges.

4. The Prasad—Sommerfieldlimit

As we pointed out in section 3, an inequality may be found for the energy of a monopoleconfiguration, and saturationof the inequality is possibleonly if the parameterA is set equalto zero.Sincethe massof theHiggs field is proportionalto the parameterA, this choiceamountsto introducinga new masslessparticlein the model,whoseLagrangianis now simply:

,~= __F~+~DaD,~ba (4.1)

supplementedwith the conditionthat thereis still a non-zerovacuumexpectationvalue v for the normof the Higgs field. This Lagrangianwould not in general survive quantization:however the model,known as the Prasad—Sommerfield(PS) limit [76], may havea quantizedversion when we considerits

P. Rossi,Exact results in the theoryof non-Abelian magneticmonopoles 337

supersymmetricextensions.In that casethe masslessnessof the Higgs field is intrinsically linked tosupersymmetryandrequiresno justification.

Most of the fascination of the PS limit comes from the remarkableproperty that explicit, exactsolutionsare known for the static field equationsand manyother propertiesof the monopole systemmay be exactlydetermined.This solvability in turn is relatedto thepossibilityof replacingthestandardsecondorder field equationswith a set of simpler first order equations,known as the Bogomolnyequations[15]:

B~±D~P~= 0 (4.2)

(the ±sign correspondsto monopolesandantimonopoles,respectively).Recallingthat the energymaybe put into the form:

� = nI + J~ [B~±D.rJ2~ (4.3)

the origin of Bogomolnyequationsbecomesapparent.There is howevera deeperway to look at the origin of theseequations,that will prove to be very

appropriatein order to get furtherexact resultsfor this problem [55,57, 80]. Let us indeedconsiderapureYang—Mills Euclideantheory:

~YM = F~PF~-F~F~,+ ~(D~A~— a0A~)2. (4.4)

If we considerthe staticcaseo~= 0 and relabel A0 cIi, the resultingstaticLagrangianis:

~F~JF~J+ ~DgY~D4~. (4.5)

This in turn is exactly the static version of the PS energydensity. We then obtain the followingcorrespondence:a solution of the PS model is associatedto each static solution of the free YMEuclideanequations.Thesolutionsof the YM Euclideanequationshavebeenwidely studied.They areknown as instantonsand theysolve the first orderequation:

= 0. (4.6)

Notice that thestaticcounterpartof eq. (4.6) is exactlyeq. (4.2). We may thenexpectthe PS solutionstobe related to static instantonconfigurations. Actually, when we considerthe casewhen an infinitenumberof instantons(in the ‘t Hooft Ansatz [43,47]) are equallyspacedalong a straight line (to beinterpretedas the time axis from now on):

= �iabôb In p + 0at9o ln p (4.7a)

A~~t9alflP (4.7b)

p(r, t) = ~ r2 + (t— t~)2~ t~= 2irn (4.8)

338 P. Rossi,Exactresults in the theoryof non-Abelianmagneticmonopoles

(21T/(ev) is the distancebetweentwo instantons,and (ev) is their scale)we find that the sum may beperformedexplicitly andwe get [80]:

lsinhr(4.9)2 r coshr—cost

The resultinggaugefields are periodic in time, but by performing an appropriatetime-periodicgaugetransformation [801:

u = exp(—iT. ~0) (4.10)

~ sin I sinh r \0=tan (4.11)\coshr cos t — I j

we find that the gaugerotatedfields are

A~’= EiabXb(~ sin~r) (4.12a)

A~’~a(cothr) (4.l2b)

which implies that all gaugeinvariant quantitiesare not simply time-periodic,but also static.Then thefield configurationwe havefound turns out to be a solution for the PS model.It describesa magneticmonopolewhosechargeis g and whose mass is ~f’= (4ir/e)v as expected.

This solutioncould havebeenfound simply by solving the radial Bogomolnyequations:

dK/dr=KcP (4.13a)

d12/dr= (K2— i)/r2 (4.13h)

andfinding

K=r/sinhr, ~I~=l/r—cothr (4.14)

but we wantedto stressthe interpretationof the monopoleas “instanton string~’.It is amazing that the dyon solution, not corresponding to a solution of the Bogomolny equation, may

also be explicitly written [761:

A~’= �iab~4(l— K)/r (4.l5a)

A8 = 1a~Psrnhy (4.lsh)

= ~a~1’c05h y (4.lSc)

whereK and 1 arethe sameasin eq. (4.14) and y is a free parameterin the classicaltheory.Let usnow considerthe irrotationalgaugeinvariant e.m.field:

(4.16)

P. Rossi,Exactresults in the theoryof non-Abelianmagneticmonopoles 339

This field is the “e.m. field in the matter” counterpart(D, H) of ‘t Hooft tensor,eq. (3.9) (E, B). Its

valueon the dyon solution is1-K~ _QEU 1-K2

~jj — ~ijkXk r2 — i— x r2 (4.17)

wherewe havedefinedthe magneticandelectricchargesof the dyon:

QM = 4i-rle g, QE = —(4ir/e)sinhy. (4.18)

Let us stressthe striking identity of theelectricand magneticdyon form factors,eq. (4.17).The massof the dyon is readily evaluated:

MD = (4ir/e)coshy (v cosh y) (4.19)

andtheobservationthat the vacuumexpectationvalueof theHiggs field, obtainablefrom eq. (4.15c), isnow

(P) = V cosh y (4.20)

leadsto the dyon massformula:

M=(k)VQ~+Q~. (4.21)

But noticethat eq. (4.21) may bereinterpretedthefollowing way: it expressesthe valueof the massforall the particlesof the model (massivevectormesons,photons,Higgs bosons,monopolesanddyons)ifQM and QE arethe respectivevalues of the magneticandelectric chargefor eachparticle. By “mass”we areobviouslymeaningherethe classicalvalueof theintegral of the energydensity.

Thepropertiesof the dyonandthe universalmassformulaeq. (4.21) show an apparentduality at theclassicallevel in this model:the exchanges

B~-*E and g�*e (4.22)

leavethedyons andthe massformula unchanged.The six-dimensional formulation of the supersymmetric version of the model, where the electric and

magneticchargeareunderstoodas fifth andsixth componentof momentumandthereis a 5—6 rotationinvariance,shedssomelight on this phenomenonandexplainsthe massformula [96].Howeverthe fullquantuminterpretationis still unclear.

5. The small fluctuations and the construction of propagators

The relationshipbetweenthe PS monopolemodel and the staticEuclideanpure YM fields can beeasily extendedto the small oscillations problem [80]. The explicit form of the YM static smalloscillationsequationsin the backgroundgauge~ = 0 is:

D~D1a~+ [A0, [A0,ar]] + 2[F,~,,a~1= —fl(fl)a~. (5.1)

340 P. Rossi, Exactresults in the theoryof non-Abel/anmagneticmonopoles

and the monopole fluctuations equations are obtained by the replacements:

A~~’M)~_*~a(PS) , a~—~~a(n) (5.2)

2(n)~w(n). (5.3)

Therefore,extendinga known resultof instantontheory [19],it is possibleto show that a solution to thescalaroscillation problem:

D~D~p+ [A,),[A0,~1]= —w2~ (5.4)

generatesin a straightforwardway eigenfunctionsfor the coupled vector-Higgsequations,eq. (5.1)correspondingto the sameeigenvalue.Moreover if we introduceminimally coupledfermions into themodel,the solutionsto the Dirac equation

[—iy0y~D~+ A0~/i= wçli (5.5)

arealsoobtainedfrom the scalarsolution to eq. (5.4) andoncemorecorrespondto the sameeigenvaluefor the frequency. Four vector eigenfunctions are obtained from each scalar solution:

= iii,~,D~co (5.6a)

(5.6b)

Correspondingly, four independent components for the spinor eigenfunction are obtained through therelations:

(5.7)

x = ~ x = [—iy,y,D,+ A01~ ~5)~ (5.8)

For all non-zeroeigenfrequenciesthis constructionamountsto a completesolutionof the problem,oncethe scalarequationis completelysolved.Theseresultswill be implicitly used in the constructionof thepropagatorsfor scalar,vector andfermion particles in the monopolefield.

The propagatorin the YM backgroundfield obeysthe equation:

[D2 8~.+ 2F~IGX(x. y) = 8~i ~4t(x— v) (5.9)

admitting the formal solution:

(5.10)(n)

Whenthe YM fields arestatic the propagatoris time-translationinvariant andits Fourier transformmaybe defined: it obeysan equationwhose w = 0 limit is:

P. Rossi,Exact results in the theoryof non-Abelian magneticmonopoles 341

{0~[D~D1+A0A0]+2F~j}O~(x,y,0)=0, 0°~(x—y). (5.11)

Eq. (5.11) in turn implies that

(n)j \ (n)+j \

G~(x,y, 0) = d(x0— Yo) G~(x,y, x0 — ~ = a~.‘~ (5.12)

is the staticpropagatorfor the isovector(isoscalar)fields in the monopolebackgroundfield.It is now possible,using knownresultsin the theory of propagatorsin multiinstantonfields [19],to

constructexplicitly these static propagatorswheneverthe monopole solution may be expressedininstantonform.

As a specialcase,the propagatorsin the n = 1 spherically symmetric monopolefield are explicitlyconstructed[801,thanksto the “instantonstring” representationfor the monopole.

Let us notice indeed that, oncewe know the propagatorfor the scalarfield (for arbitrary isospin),definedby the equation:

—D2zi(x,y)= 0~4~(x—y) (5.13)

the fermion (massless)propagatorturns out to be:

S(x,y)= y .DLI(x,y) ~~+LI(x,y)y ~D12~~ (5.14)

and the gauge spin 1 propagator in the ~ gauge is:

G~(x,y) = —0~,DA[zI ]2DA + (1 — ~) D,~[LI]2D~,— (0,~A0.,,. — 0~,c&,A + �~~A

5)DA[LI]2D~. (5.15)

Theseresultsarefairly general[19].If we restrict ourselvesto instantonconfigurationsexpressiblein the ‘t Hooft Ansatz,eqs. (4.7), we

may write down the isospin~scalarpropagatorin the form:

LI(x, y)= p(xyh/24~~~1,)2p(y)~2= 4~(X~Y))2 (5.16)

where

F(x, ~ = ~ r~~~(x_zstr~(.y—z~Y’ (5.17)

The isospin 1 scalarpropagatoris then:

A — Wab Cab~.Iab 2 2m 2

41T (x—y) 4ir p(x)p(y)

where

342 P. Rossi,Exactresults in the theoryof non-Abe/ian magneticmonopoles

Wab = ~Tr TaM(X, y) r~M(y,x) (5.19)

Cab = ~ ~rsa(X) Crstu~tub(Y) (5.20)rstu

and

(x—zr),L (x—z5)~ (5.21)~rsa(X) = Th~~’a(x — z~)

2(x — z5)

2

while

Crstu — OrU&v — 0rv0sU 1— (z~-z

5)2 (ZrZs)2~0f5U~T~ 1 2 (5.22)(z~— z~)

where

~ = &~,, g~= [~ 2]8 1 — (5.23)r~s(Zr — z

5) St— (z, — z~)2

Amazinglyenough,all sumsmay be explicitly performedin the static limit that leadsto the monopolesolution, and the appropriate gauge rotations restore the explicit time-translation invariance, so that thestaticpropagatorsare obtained[1,80]. Unfortunatelythe isospin 1 propagatorshavevery cumbersome,albeit completely analytic, expressions.

We just wish to present,as explicit results,thefunction involved in the isospin 1/2scalarcase:

1/2

M(x, y,~t) (si1~h1IxIy”

2(sir1h1IYly~

2J dw exp(wT~x~’) exp(w~yt’) (5.24)

—1/2

andthe static scalarisospin~propagator:

— 1 (sinhIxI)-!/2 (sinh lYli~’2~j t~t(~) 4ir \ lxi \ 1YI /

x { 1+ 1.x~y [(IYI+IxlIY]+lxl lYIHxl’~ / I2lxi l~i— 2x . ~ — 2 + cosh 2 ) exp~—Y ~ xj) - i]

1—T~xr~y[(lyjjjy~(Iy~J)I (5.25)~2IxliyI-2x ‘y \lYXI

showingthe correctasymptoticbehaviorandthe free particle limit:

___________ 1ii~~a!(x, ..v) — ~. (5.26)

P. Rossi, Exact results in the theoryof non -Abel/an magneticmonopoles 343

6. Bosonicand fermionic zero modes

As we havealreadypointed out, knowledgeof the zero-energyfluctuationsarounda classicalstaticfield configuration (small deformations) is especially relevant to the quantization problem, sincenormalizablezero modesare associatedto symmetriesbrokenby the classicalsolutionandallow, whenproperlytakeninto account,for the quantumrestorationof the symmetriesthemselves.

The discussionof the zero modesin the PS limit is dramaticallyeasedby the observationthat wedon’t have to resort to the secondorder equationsdiscussedin the previoussection for the non-zeromodes.It is apparentfrom the structureof the energyfunctional that the zero energyfluctuationsshould again satisfy the Bogomolny equations.To be more precise,since we consider infinitesimalfluctuations,we expectto find solutionsto a linearizedversion of the equationsin the backgroundfielddescribedby the knownsolution.

Let usconsiderthe expansion:

= A~(PS)+3A~ (6.la)

cpa = cpa(p~)+4~cpa (6.lb)

The equation0B~= 3D~Patakesthe form:

E,jk[t9jOAk + �a?~CA~(PS)0A~]+ �~I~(PS)0A~= a0b’~+ �~A?(PS)0~’. (6.2)

The general solution to this equation in the case n = 1 has been constructed[5], up to gaugetransformations:

0A~+(D1Ayt (6.3a)

or + ~ ~. (6.3b)

Sincethe linear operatorappearingin eq. (6.2) commuteswith a total angularmomentum

J=rAp+T+S (6.4)

whereS = 1 for the gaugefields andS = 0 for the Higgsfield, andsinceit anticommuteswith the parityoperator,the modesmaybe classifiedaccordingto the eigenvaluesof J2, J

2 and parity, and a partialwave decompositionis possible.We shall not enterthe details of the derivation, but just quote theresults.

The solutionsin the “natural” parity sector(—1)’ may assumethe generalform:

0A~= sinh r (~iaa — Oja ~)(sin~ i”) (6.5a)

= —sinhr�0~~11~a~(r A~ (6.5b)~sinhr /

whereA is a solution of the Helmholtzequation:

344 P. Rossi, Exact results in the theoryofnon-Abel/anmagneticmonopoles

V2A = A. (6.6)

Solutionsto eq. (6.6) maybe summarizedin thepartial wave expansion:

A = ~ a,,~I,(r)Yim(O, tp)+ bim K,(r) Yim(O, (p) (6.7)

The “unnatural” paritysector(_1).~±1admitstwo distinct classesof solutions.Oneclassmay assumetheform:

0A~= cosh r + �aibab( ~‘ii r A) + (cosh r — 1)fZ�iim~i9m (sin’~i r A) (6.8a)

r A’~+coshr(3a—Ia-~--~( r A~ (6.8b)ar \sinh r J ar/ \slflh r /

whereagainA is obtainedby eq. (6.7). The secondclass,however,hasthe form:

0A~= �aibôb(~ r V) + (1 — coshT)~aEzini~iôm(sin~i r V) (6.9a)

pa = Ia -I-(rcoth r V)+ (ôa — ~a ~)(sin’i ~1~~’) (6.9b)

where V obeysLaplaceequation:

V2V=o (6.10)

whosegeneralsolution is:

~ ~ ~+ b~mr’ Yim(O, (p). (6.11)

Let us observethat the solutions dependingon A could have beenfound [2] by going back to the“instanton” gauge,wherethe monopolesolutionhastheform (alternativeto theonewe havepresentedin section 4):

A~= ~iab3b In a- + lOia (6. l2a)

A~=—öalng (6.12b)

a-=sinhr/r, V2u=a- (6.13)

andobservingthat a moregeneralsolution is obtainedby replacingsinh r/r with sinh r/r — A.Howeverthis approachdoesnot accountfor theclassof zeromodefluctuationsdependingon V, that

cannotbe gaugedinto the restrictedinstantonAnsatz.The existenceof three distinct series of solutions to the zero mode equationsas well as their

P. Rossi,Exact results in the theoryof non-Abel/an magneticmonopoles 345

asymptotic behaviorat infinity is to be expectedon physical grounds: it relatesto the existenceofmassivechargedandmasslessneutralexcitationsin the spectrumof the theory.

Let usnow considerthe (massless)fermionzeromodes,in the casewhenthe fermionsareminimallycoupled.This coupling choicehasa deepersignificance:it correspondsto the supersymmetrizationofthe model. The fermionic wavefunction will obey a Dirac equation in the form:

[a (p_~A T)+ a(~AT) K(r)—f3~T. = 0. (6.14)

Actually, dueto the absenceof theright handside,the equationmaybe turnedinto a spinorequation:

[~(~~A T)+ a’(.!A T)K(r)+ icPT~ = 0. (6.15)

Whenthe fermionicfield hasisospin1, a relationakin to the onewe havefoundfor the non-zeromodesconnectsthe spinor solution for eq. (6.2),when thebackgroundgaugeconditionis imposedto the gaugefluctuations:

(D~a~)a=0. (6.16)

It is possibleto show [20] that two zero-energyvector-Higgssolutionsare associatedto each spinorsolutionby meansof:

_ia-~a~’)rr(~~~z:) and —io~a~2~=~ ~) (6.17)

whereUa andd~are the spinorcomponents.We may then rederivethe solutionsfor the zero-energyfluctuationproblem by solving the associate

fermion problem [61]. This rederivation turns out to be especially interesting for the low angularmomentum(1 = 0, 1) modes,correspondingto the fermionicequationfor j = ~.Among theseare theonly renormalizablezero modesthat arepresentin the n = 1 monopolesystem.

The j = ~fermion0-modeequationsare:

(~~+1+~)R )—1~R~±)=0 (6.18a)

(6.18b)\dr r T/ r

It is easy to show that a complete set of solutions is:

1— 1 11 \ I cothrR(÷)a

1 . i——cothrp+a2i— -sinh r \r / \ rsinh r(6.19a)

0 / 1 1\ /cothr 1R(+) = 1(~h2r~)+ a2~~r2 + rsinh2 r

346 P. Rossi,Exact results in the theoryof non-Abel/anmagneticmonopoles

R~i)= f3i sin’Ii r (1 — r cothr) + /32(— ~ (6. 19b)

R ~ = s1(51flh2 r — cothr) + I32sinh2 r

It is to be remarked that only one of these solutions turns out to be normalizable, and precisely

R~)=(~_cothr) - 1 , R~÷)=- ~ (6.20)r sinh r sinh r r

It gives origin to the two independent (spin up and down) fermionic zero mode solutions that areexpected to be present on general grounds in this model, as we have shownin section3.

Wewant to point out that these zero modes could have been obtained by a more fundamentalconsideration [79]: since the coupling is now supersymmetric these modes have to be the supersym-metric zeromodesof the model, andindeed:

= [R ~~0ai — 1a~i)+ R~±)~a1jIa-~. (6.21)

It is now straightforward to obtain the vector-Higgs normalizable zero modes in the form:

a~(A)=~ 7~aMAR~i)1= F~. (6.22)

It is especiallyinterestingthat thesemodes,that areautomaticallyin the backgroundgauge,eq. (6.16),can also be put into the form:

a~(0)= (DMA )~ (6.23a)

a~(l)=~,M~(D~AY~ (6.23b)

where

= a ~(r) (6.24)

is a solution of the scalar zero mode equation

DMDMAa=0. (6.25)

This would be true also for the second(singular)parity (+) solution with

A” = ~a coth r/r (6.26)

and is a special case of the property that holds for all non-zero modes.

P. Rossi,Exact results in the theoryof non-Abe/ianmagneticmonopoles 347

Finally we want to commenton the meaningof the normalizablezeromodes.Let usobservethat

a”~(A)= F~M= — D4~LAA + ÔAA~ (6.27)

so that a gaugetransformation

a(A)+DMA~ (6.28)

will leadto the (gauge-equivalent)form:

a(A) = aAA~. (6.29)

Eq. (6.29) is particularly appropriate in order to stress the significance of the vector-Higgs zeromodesasreflecting the translation(andgauge)invarianceof the original equationsof motion.

7. Multimonopole solutions

As we haveseen,the single monopole,sphericallysymmetricsolutionfor the PS model is explicitlyknown. When we turn our attention to the possibility of further solutions characterizedby highertopologicalcharge,wefind out to be in a ratherdifferentsituationfrom the generalcase(A � 0).

The long-rangerepulsivepotentialbetweenmonopoleshas now been completely screenedby theequalstrength-oppositesign scalarmasslesspotential,and it is possibleto show that the semiclassicalforce betweenmonopolesis zero,while the monopole—antimonopoleforce is nowdoubled[56].In theabsenceof a repulsiveforce, nothing preventsin principle the existenceof multimonopolesolutions.Theseconfigurationsare expectedto satisfy the Bogomolny equationswith a different topology, andthis would leadto the massformula

= n~ (7.1)

consistentwith the absenceof interactions.Moreover it has beenshown by meansof index theoremsthat any solution to the equationswould

admit a numberof infinitesimal deformations(4n — 1) consistentwith the existenceof noninteractingmultimonopolestateswith a U(l)M local gaugefreedom,as expected[90].

The main problemin the searchfor thesesolutionsis the alreadyquotedtheoremthat preventsthemto enjoy a sphericalsymmetry,evenin the casewhenall the monopolechargesare superimposedin asingle location. It is alsopossibleto show that, for any monopoleconfigurationwith separatelocations,evenwhen the locationslie on a line, not evenaxial symmetryturns out to be possible[44,77]. Thesepreconditionsforce the attentionon the circumstancewhen the monopole chargesare superimposedandthe hope is for axially symmetricsolutions.

For the purposeof parametrizingaxially symmetricfield configurationsassociatedwith n units ofmagneticchargewe introducea set of orthonormalvectors:

~~n)= (cos fl(p, sin fl(p, 0) (7.2a)

348 P. Rossi,Exact results in the theoryof non-Abe/ian magneticmonopoles

u~°=(0,0,1) (7.2h)

= (sin n~,—cosnip, 0). (7.2c)

In this section r = Vx2+ y2 is the axial radius.The vectorseq. (7.2) enjoy the properties:

= ‘! (7.3)

= �~u~°°. (7.4)

We now expandthe fields accordingto:

A~= u~’~u~’°W~(r,z) (7.5a)cpa = U~”~(p”. (7.5b)

If we now require parity reflection invariance we find that:

W~=W~=W~=W~=W~=(p3=() (7.6)

andby defining x’ = r, x2 = Z

W~= V~ (7.7a)

W4 = —~1/r, W~= (n — ‘q2)/r (7.7b)

we find that

B~= ~ — ~ �MMV~u)u~~~ (7.8a)

= (8~(p~— ~ ~ ~ (7.8b)

and Bogomolny equationsreduceto [571

— �~$VMn$)= a~(pa— �~~V~çv,3 (7.9a)

= ~ (7.9b)

This is a set of five equationsfor six unknownfunctionswith a residualAbeliangaugeinvariance:

(p~—~ (p,~.= (p~cosx + �a$(,O$sinx (7.lOa)

~)a CO5X+�a~flssinx (7.lOb)

VM—~V,~=VM + aMx. (7.l0c)

We are looking for regular solutions to this set of equations.

P. Rossi,Exact resultsin the theoryof non-Abe/ian magneticmonopoles 349

The most general asymptotic behaviorwith properboundaryconditionsat infinity is best describedinterms of the gauge invariant quantities

h=Vco~p,.and ~//77~c~~ (7.11)

obeying the equations

�M~ôM~b=a~,h (7.12)

outsidethe exponentiallydampedcorewhere the monopolesare located.The generalsolution of eq.(7.12)is an arbitrarydistributionof pointlike monopoles.All othergaugeinvariant field components

AM = VM + �a~(paô~92~and E = �~$fl~(~~ (7.13)

areexponentiallydampedat largedistances[77].The veryexistenceof consistentregularexpansionsaroundtheorigin speaksin favor of the existence

of n-monopolesolutions.Moreover, the structureof the zeroesof the Higgs field, which are not separablealong the axis of

symmetry,but only in the planeorthogonalto it, is a furtherhint of the impossibility of finding axiallysymmetricsolutionswith finite separationamongthe locations.

It is possibleto analyzenumericallythe problem, by meansof variational techniques.Actually, thefirst evidenceof existencefor multimonopolesolutionscamefrom this kind of analysis[4,77]. It is alsopossibleto give anexplicit analyticconstructionof the multimonopolesolutionswhenall the monopolechargesaresuperimposed[34,35, 36,54, 63, 71,73, 74, 87, 88]. This requirestheuseof two resultsthathavebeenoriginally derivedin the studyof the self-dualYang—Mills fields:Yang’sR gauge[98]andtheAtiyah—Ward Ansätze[8, 27,69, 751. Onehasto employ the complexifiedversion of the theory:

x1+ix2 - xi—ix2 x3—ix4 - x3+ix4‘~ ‘ ‘~ V~ ‘ q V~ ‘ q V~ (7.14)

andYang’sR gaugecorrespondsto:

‘-~-~ o~ /~-~ -~

= f 24 A0 = ( 2~ ~ (7.15)~Pu ~ui ‘0\~ 2çb/ \ 2~

where A,. = (a-’~/2i)A~, u = p, q and the subscriptdenotesderivation.The self-duality equationstakethe form:

(a~a~+ aqa4) In 4~+ (app 3~5+ t

9q~34,5)= 0 (7.16a)

~~(~)+a4(~)=o, ~ (7.16b)

350 P. Rossi.Exactresults in the theory of non-Abel/anmagneticmonopoles

Let us noticethat the self-dualfields can be cast into the form:

Ap~~R10pR. AqR~aqR (7.l7a)

A~=R~’19~R. Aq=R

1t94R (7.17h)

where:

/ 1 01V~ ~

R=l~ ~, R=( ‘~\. (7.18)l~~P_\/~I I 0 -i---\~/~ ‘17

Gaugetransformationsare the transformations:

R-~V(,5,q)RL, R-*V1(p,q)RL (7.19)

where L is an arbitrarycomplexmatrix function and

A,. —* L’A,.L + L 13,.L (7.20)

while

JR1(~ ~ ~(p,q)JV(p,q). (7.21)

In order to find explicitly real fields one hasto find matrices V. V such that Vi V is a positivedefiniteHermitianmatrix, andchooseL to be a square root of the matrix:

LL~= (R~V~V’R~’ (7.22)

The Atiyah—Ward Ansätzearecharacterizedby the introductionof atransitionmatrix as a functionof acomplexvariable ~[861:

G~(w1,w2, fl = (~I1~(w1,w2,~)) (7.23)

whereV2w1 = (4—pr) and \/2w2= —(q+j7~~). R gaugesolutionsof the self-duality equations(6.19)are obtained by [271

nxn nxn nxn— “k—i (n) — i_i \n “k—i—i —(n) — i_i \n+l .H’k_i±l— Hn—lXn—

1 , P — ~ ‘j n—lXn--1 , P — ‘, 1) n—lXn—1 -k—i k—i k—i

P. Rossi,Exact results in the theoryof non -Abelianmagneticmonopoles 351

where H’,i?,+,,, is the (Toeplitz) determinant of the j x j matrix whosekth row and lth column is givenby LIk_1+m and

LI, = ~— ~ 12~(w~,W2, ~‘) ~ 4~. (7.25)

Let usobservethat the functionsLI, where—n s 1 ~ n satisfythe following equations:

3pLIj’34LIi±i, 3qLIiôpLIi+i (7.26)

andthereforetheyarealsosolutionsof the Helmholtzequation:

(t9p3p+3qt94)LIi0. (7.27)

In order to obtain monopolesolutionswe have to insure that the gaugefields ate (at least in somegauge)static, real andnonsingular.Staticity is insuredby the choice [871:

W2, fl = e’°’~~2F(w), Li) = Wi — ~ (7.28)

Realitymaybeachievedby choosing[74]

e’° + (_1)n e’”

2P~(w) (7.29)

whereP~(w)is a polynomial of order n in w with real coefficients.This choice, together with the

observationthat the following relationholds [70,75]:

h2= 1—V2lnH~i7 (7.30)

insuresus that whenwe considerthe nth Ansatzwe aredealingwith an n monopolesolution.It is found that all singularitiesdisappearwhen one assumesthat the function 11~’°(wi,w

2) cor-respondingto the n monopolesolution is generatedby the following “splitting rule” [74]:

flt’°(w

1,w2) = L1~”1~(x

3+ ~i1r) + ‘1~”’~(x3— ~i1T) (7.31)

startingfrom the knownsingle monopolesolution:

u1~1~(wi,w

2) = e~~~+~02sinh w (7.32)

This choice corresponds to:

= (n — 1)! ~ 11 (w — Zk) (7.33)

352 P. Rossi.Exactresults in the theoryof non-Abe/ianmagneticmonopoles

where

/n+l ~Zk = i7r~ k~, k = I n. (7.34)

We thenfind:

1 ~—‘ 1+t t/2

LI~”=e4e ~(_lY~J dtet~(2cos.1t) (1-17) 11(r\/1 — t2). (7.35)

Regularityat the origin is insuredby construction.Absenceof singularitiesand the correctbehavioratinfinity are proven by showing that these solutions are also a special case of a different class of(generally complex and singular) solutions of the self-duality equations.These new solutions areobtained[75] by solving the recursionequations,eqs. (7.26), without resortingto a transitionmatrix butrather defining

= (—1)’ e~(x~±ix2)~(l~ a3~A~ (7.36)

where

A1 = p -b--A,±1= p -~-~A,÷1 (7.37)

andeqs. (7.37)are solvedby

A1 = 13k(Pk)Il/2-iCOk) (7.38)

where1-k = \/r2 + (x3 — q,. )2 1, are the modified Besselfunctionsof imaginary argumentand /

3k, q,< arearbitraryc-numberparameters.One can then find explicitly [741

/ — — ‘.i Ok / — L ‘.1 Pk

— ~ ~ i x3 gk Pk~ e — (x3 gk Pk e~,—e ~~pk ~ I— I I—

k=1 ‘ v2p /

2Pk \ v2p 2Pk

By dropping the exponentiallydampedcorrectionsO(e2°”)one can evaluate exactly the relevantdeterminants.The norm of the Higgs field turns out to be:

h = 1— ~ ~+ O(e2~k) (7.40)

k = Pkwhile the streamfunction is found by solving eq. (6.12):

(7.41)

k=1 Pk

andthe correctasymptoticbehaviorsare shown.

P. Rossi, Exact results in the theoryofnon-Abel/anmagneticmonopoles 353

The constructionis nowcompletedby the observationthat, by choosing

k_ak=(kl),(flk)~~ qk—zk

we areled to the following non trivial identities

LI,(ak,zk)=LI,, —n<l<n (7.43a)

/i±fl(ak,Zk) = LI±~— (x1 ±ix2)~[(21T)”’(n — 1)!] exp(ix4~x3) (7.43b)

suchthat the gaugepotentialsobtainedfrom A, are (complex)gaugeequivalent(andindeedequal)tothose obtained from LI,. The representationin terms of the LI, allows us to find explicitly thetransformationturning the fields into manifestlyreal solutions[72,751:

c~=~, v= ( 0 y1(V2p)~~ (7.44)

\—y(V2p)” 0 /

where:

y2(2pp)fl (~2 + ,5~)~ = y2[(21T)~’(n— 1)!]2 = 1 (7.45)

such that y is a real constantas required.It is theneasilyfound that

/ (n) —in~ (n)~

LL~—’ ~ ye (746)— ‘~~ye”Q~—yK°°I

where

= (V2 p)flp~~~e~,

K°°=(V2p)fh3*~e~4 (747)Q(n) =

T’~~’~ ~*(n)

andK, K, Q arereal functionsof r andz = x3, obeyingthe constraint

~n) ~ + Q~Q~= — 1/y~. (7.48)

The choice:

fV~exp(_~) o \ (V~exp(~) ~J~exp(-~)L=1 P~ Iinço\ 1 /in(p\ ~ L —‘ 1 / ~fl(p (7.49)

~ ~_~exp~,,—~--)/ \ 0

leadsto the (anti)Hermitianfields:

354 P. Rossi,Exactresults in the theoryof non-Abel/anmagneticmonopoles

/-~0\= ( ~ ~ 1= (A~, Aq = ~ = (A4)

t (7.50)\p~ —~J \~-~J

q5 2~ ~‘ 2~

where

(“1t(n) —z

~ J(fl) , p(n)= yJ~(fl)e (7.51)

are the real potentials,that turn out to be also staticand (p-independent(axisymmetric).An immediatecomparisonwith the axisymmetricparametrizationwe haveintroducedis now possibleand leadsto:

— 19zP’ —

(7.52a)

a (752br 4’’ r4V

V2=—çc1. (7.52c)

Eqs. (7.52c) moreovercorrespondto the gaugechoice leading to solutionsof the Ernst equationofgeneralrelativity

Re eV2�— (V�)2= 0 (7.53)

with the identification

E ~‘~‘~+ip”’. (7.54)

The superimposed n monopole solutions are thus found to be axisymmetric and mirror symmetric.The gauge invariant field componentscan then also be obtainedfrom a superpotentialformulation

I~7.771:

r=1nH~<? (7.55)

h2 l—V2r (7.56a)

E2+ ~2 = n2— r2(3r0. ~3r+ a2a2)r (7.56b)

h~/i—3~r (7.56c)

A — r3,.h +E L-~)

A proof of existence of multimonopole solutions has also been given for (widely) separated locationsand with no assumptions on the symmetry [84].

P. Ross4Exact results in the theoryof non-Abe/ianmagneticmonopoles 355

In the more general case of separatedmonopoles no completely explicit construction can beexhibited.An implicit constructionis howeverpossible,leadingto staticmonopolesolutionspossessingn units of magneticchargeand(4n— 1) degreesof freedom,as expected[28].We outline the procedure,observingfirst that the constructionis still basedon the Atiyah—Ward Ansätzeandamountsto a set ofprescriptionson the transitionmatrix G~”~(wi,w2,

Staticity is ensuredby a generalizationof eq. (7.28):

w2, ~)= e”~’~’°2f(w,~)

The reality condition can be shownto force the following form of thefunction f:

— e~+x+ (—1)~C~X

2P,,(w,~)

wherex is a polynomial of degreen — 1 in w~

x = x5(~)(wfl5

with coefficientsx5(~)that are regularfor l~l< 1 + �. Pfl(w, ~)in turn is a polynomial of degreen in w

~

with ~s(~)~’ being apolynomial of degree2s in ~satisfying

=

such that thecondition

P~(w, ~)= (—1)”~[Pfl(w, ~)]*

is satisfied.The regularitycondition requiresin thefirst placethat f be an entire function of w. By defining the

roots of P~:

Pfl(w,~)=JJ(w—ws(~))

we needto choosex so that

eO(~0C)= ±1 for cv = w5(~), 1 � s~ n

where

356 P. Rossi. Exact results in the theory of non-A belian magnetic monopoles

O=2w+X+X5.

Therefore,given the integers(+) or half odd integers(—)n5, 0 is uniquely determined:

O(w, ~)= 2iri ~ flr ~ (Wr w~)

= _fl5 if Wr(~)

Continuity with the axially symmetricsolutionsrequires

n5=~(n+I)—s, l~s~l

and the + (—) sign as n is odd (even) respectively. A naive counting of the number of free realparametersleadsto the expected4n — 1 degrees of freedom. In the n = 2 case121, 89] one is led to aone-parameterfamily of solutions (up to translationsand rotations)characterizedby

P 2 1222W +~1Tf

~L

Further study of this solutionwould requireexplicit numericalanalysis.The most recentdevelopmentsin the study of multimonopole solutionsare related to two main

topics: the representation of multimonopoles within the ADHM formalism originally employedformultiinstantons [7] and the extension of the SU(2) results to more generalgaugegroups.

As for the first problem, it has been in principle solved by Nahm [621by showing that eachconfiguration(for arbitrarychargeand gaugegroup) is characterizedby asolutionof a certainordinarynon-linear differential equation,which has chancesto be completely integrable, and in the axiallysymmetric case reduces to the integrableTodalattice equations.

A further ordinary linear differentialequationmustbe solvedin order to constructthe potential.In the axially symmetric case, extension to arbitrary groupshas beenconsideredin detail [12]and an

algebraic constructionmethod of the self-dual solutionshasbeen provided,which makesuseof theexistenceof Bäcklundtransformationsin the associatedtwo-dimensional(curvedspace)problemandofthe inversescatteringmethod.More generally,the problemhasbeenconsideredfrom an abstractpointof view by means of group theoretical arguments and index theorems leadingto a parametercountingfor the solutions [911.For each group and choice of symmetry breaking there is a set of fundamentalmonopoles with minimal topological charges and no internaldegreesof freedom.When the unbrokengauge group is non-Abelian all solutions with higher topological charges belong to familiesparametrized by position and group orientation parameters. It is thereforearguedthat thesesolutionsshould be interpretedas (superimposed)multimonopole solutions correspondingto a set of non-interactingfundamentalmonopoleswith the given topologicalcharge.

P. Rossi, Exact results in the theoryof non -Abe/ian magneticmonopoles 357

8. Supersymmetricmonopoles

We want to analyzein detail the interestingpropertiesthat arisewhenthesupersymmetricextensionsof the PS model whoseeffectswe havealreadymet aretakeninto full account.

The simplestpossibility correspondsto N = 2 supersymmetry[31].The Lagrangianis:

= ~ + ~ D,.cpa+ ~D,.P’~D,.P’2 — ~ pb]2 + i~/i(y,.D,.+ ‘1~+ iy5P)i/i (8.1)

where a pseudoscalar field pa and a Dirac spinor (two Majorana charges)havebeenintroducedwithminimal couplings.The infinitesimalsupersymmetrytransformationsare:

= — y,.D,.i(~” + iy5P”)c (8.2a)

8A~= �y~Ii” (8.2b)

= (8.2c),~5pa= ei~’~6~. (8.2d)

There is a chiral symmetry between scalarand pseudoscalarfield. The symmetry is spontaneouslybroken througha non-zerovacuumexpectationvaluefor the scalarfield.

The generalbosonicsolution is obtainedas a trivial generalizationof the PS solution:

A~= ~“cpsinhy, A~ �iabXb K—i (8.3a)

cpa = ~acpcoshy cos~ pa = ~“cpcosh y sin6 (8.3b)

where y,i5 are free parametersin the classicalsolution. It is possible, by iterating the infinitesimaltransformationseqs. (8.2), to obtain the completesupersymmetricsolution.

Let usconsiderfor instancethe puremonopolesolution y = 8 = 0. The complete solution is then:

= u~o~B~’(PS)u, A~(PS) (8.4a)

= u~o1B~(PS)u, cp’~= (JY~(PS) (8.4b)= (—iu,B~(PS) u, 0) (8.4c)

whereu is an (anticommuting)constantspinorwith two independentvaluesand becauseof anticom-mutation no more than two powers of u may appear. It is easyto check that the full set of staticequationsfor commutingand anticommutingvariablesaresolvedby eqs. (8.4).

The interpretationof the quantumspectrumis now straightforwardand it matcheswith what wewould haveexpectedsimply on the basis of the appearanceof two fermionzeromodes:

= (—icr, B~(PS) u, 0), j = (8.5)

obviouslygeneratedin this caseby the supersymmetryinvariance,eq. (8.2a). Therearefour quantumstatesfor the monopole:the two componentsof a spin ~ fermion, a scalar and a pseudoscalarstate.

358 P. Rossi,Exactresults in the theoryof non-Abelianmagneticmonopoles

These statesform a supersymmetrymultiplet, and their common massis ~ There is a perfectlyanalogousmultiplet for the dyons, with mass:

M=(P)\/Q~4+Q~. (8.6)

The massformula eq. (8.6) generalizesto all the particlesin the model, as we havealreadydiscussedinsection4, andits appearanceis especiallyrelevant in thesupersymmetricmodel. Indeed,its origin maybe relatedto thepresenceof centralchargesin the model,andthis fact ensuresus thatthe massformulawill survive full quantization and hold for the physical spectrum of the theory once the charges arereplaced with their renormalizedcounterparts.One may explicitly check this statementup to one loopcorrections by analyzing the small oscillationsspectrum[311.The pseudoscalareigenvalueequationis:

_D,.D,.pa = w~. (8.7)

For any pseudoscalarsolutionfour vector-scalarsolutionsare obtainedthat satisfythe equation:

D,.D,.a~+ 2e”F~a~,.= —w2a~. (8.8)

Eq. (5.7) relatesthe vectorsolutionsto the pseudoscalarone. As we haveexplainedin section6, fourcomponentsolutionsfor the spinorequation:

—iy,.D,.~i”= wi/i” (8.9)

are also obtained. Since the ghost equation is the same as the pseudoscalarone but carries twoindependent components, for each eigenfrequencythereare:

4 vector— 2 ghost= 2 gluon eigenfunctions1 scalar + 1 pseudoscalareigenfunction4 fermion eigenfunctions

andkeepingall contributionswith the propersign:

M = M(ps) + ~[o~ ~0B 2 ~ ~0gh _4 ~ WF] = M(ps). (8.10)

There is only another possible supersymmetricextension,correspondingto N = 4 anticommutingcharges [18]:

~ i$’B~’Iy5)i/(�ab~~ (8.11)

with 3 scalars,3 pseudoscalarsand2 Dirac spinors,enjoying a global 0(6) = U(4) symmetry.The mostcompactdescriptionof this theory comesfrom usinga ten-dimensionalnotation:

= — ~(FABF~)+ ~i(A,FADAA) (8.12)

where the scalarsand pseudoscalarsare the 5th to 10th componentof the Y.M. field and A is a

P. Rossi, Exactresults in the theoryofnon-Abel/an magneticmonopoles 359

MajoranaandWeyl spinor.The infinitesimalsupersymmetrytransformationsare:

ÔA = (8.13a)

6AA = ~iEFAA— ~iAI’A�. (8.13b)

Most of the previousargumentscan berepeated,including vacuumsymmetrybreaking,the existenceofclassicalsolutionsthat aretrivial generalizationsof the PS solutionsandthe possibilityof building up afully supersymmetricsolution,now dependingon no more thanfour powersof the (Grassmann)spinora [811:

AA = AA(PS)+ ~á(iFA CF~)a+ ~ ~ (8.14a)

A IBcF~)+l[2~cD(iF~0EF(~)1 (8.14b)

Again the massformula, eq. (8.6), holds, andit is expectedto survivequantization.Buta striking featureof this modelemergeswhen we considerthe spectrumof the monopoles:there

is one massivevectorstate,five scalarsanda coupleof Dirac fermionstates.Noticethat thisspectrumisexactly the same as the spectrumof fundamentalcharged(massive)excitationsof the theory [64].Moreover the interactionsamongthe monopoles,betweenmonopolesand antimonopolesandbetweenmonopolesandunchargedmasslessparticlesare, at leastsemiclassically,the sameas for thefundamen-tal chargedmesonsandantimesons[561,anda U(1) local gaugefreedomis enjoyedby the monopolesaswell as by the mesons[90].Furthermore,since thereis only one N = 4 supersymmetricrenormaliz-able Lagrangianhaving such propertiesfor the multiplets and the dynamics,the low energyeffectiveLagrangian (and the low energy S matrix) for the monopole sector is obtainedfrom the originalLagrangiansimply by the substitutione—* g, and theexactpolesof the S matrix in the monopolesectoraregeneratedby the samerule.

All thesephenomenastrongly hint at a property of self-duality for the N = 4 supersymmetricLagrangian,that is, one might expectthat the monopole Lagrangiandescribingthe monopolesas itsfundamentalsectorcould be interpretedas an exactLagrangian,andthe chargedmesonswould thenappearas its solitons,with dyonsand unchargedstatesidentified (after interchangeof the electric andmagneticphoton)[60,641.

Howeverit is not obviousat all that this picturecould really survivequantization.When a /3 function is introducedfor the electric charge, and the Dirac quantizationcondition is

extendedto the runningcouplingconstants[251:

ae(t)ag(t) 1 (8.15)

whereae= e2/4ir and ag = g2I41Tthen from the definitions:

/3(ae)daeldt ,6(ag)z’dagldt (8.16)

the condition

360 P. Ross), Exact results in the theory of non -Abe/ian magnetic monopoles

f3(ag)= —a~/3(1/ag) (8.17)

follows immediately.Self-duality is preservedonly if /3 hasthe sameform as /3, that is if the equation:

/3(ag)= —a~f3(1Iag) (8.18)

is satisfied.The generalsolutionto eq. (9.18) is

i/3(ag)= ~(ag)_~(~-) (8.19)

where~ is an arbitraryfunction of its argument.This would in generallead to the existenceof a secondorder phasetransitionat the fixed point ae = = I with a light mesonphaseand a light monopolephase.Amazinglyenough,a featureof this modelseemsto point to a ratherdifferentpossibility. Indeedit has been shown up to 3-loop perturbative calculation that the /3 function vanishes identically[40,50,681. If weadmit that this phenomenonmight persistto all ordersandthat /3 could be identicallyzero,we find that self-duality of the model immediatelyfollows [81].Actually from eq. (8.17) we easilyextractthe informationthat /3 = 0, too. Then the S matrices for the two sectorshave,up to an e *-* gtransformation,the same low energybehavior,the same high energybehavior (dictatedin the scalingregion by /3 = 0, /3 = 0), and the samepoles and thresholds.Rememberingthat the supersymmetrypropertiesmustbe the samein the two sectors,we are led to an identificationbetweenthe two matrices.More directly, if morenaively, /3 = 0 implies the absence of countertermsin the effectiveLagrangianforthe monopoles. The vanishing of /3 would then lead to a proof of self-duality.

Acknowledgements

I wish to thank Profs. A. Di Giacomo, R. Jackiw and C. Rebbi for their constantattention andencouragement.Thanks are also due to the Center for Theoretical Physicsof the MassachusettsInstitute of Technology,wheremost of the original researchwork was completed.

References

[11SI.. Adler, Phys. Rev. DIS (1978)411.[21SI.. Adler. Phys. Rev. D19 (1979)2997.

[31SI.. Adler. Phys. Rev. D20 (1979) 1386.[41SI.. Adler and T. Piran,in: Proc.XX Intern. Conf. on High Energy Physics,AlP Conf. Proc. 68. Pt. 2 (1980) p. ~[5] R. Akhoury. J.H. Yun and A.S. Goldhaber.Phys. Rev. D21 (1980)454.[6] J. Arafune,P.G.O. FreundandC. Goebel.J. Math. Phys. 16 (1975)433.

[7] M.F. Atiyah, N.J. Hitchin. V.G. Drinfield andYu.I. Man/n.Phys. Lett. 65A (1978) 185.[8] M.F. Atiyah and R.S.ward, Comm.Math. Phys. 55 (1977) 117.[9] F.A. Bais, Phys.Rev. D18 (1978) 1206.

1101 F.A. Bali and J.R.Primack, Phys.Rev. D13 (1976) 819.[II) F.A. Bali and J.R.Primack.NucI. Phys. B123 (1977) 253. -[12) F.A. Bats and R. Sasaki.On thecomplete integrability of the staticaxially symmetricselfdual gaugefield equationsfor an arbitrary group

(1981)preprint.[131F.A. Bais and H. Weldon. Phys.Rev. Lett. 41(1978)60!.

P. Rossi,Exact results in the theoryof non-Abel/an magneticmonopoles 361

[14]PP.Banderet,Helv. Phys.Acta 19 (1946)503.

[151E.B. Bogomolny,Soy. J. NucI. Phys.24 (1976)449.[161E.B. BogomolnyandM.S. Marinov, Soy. J. Nuci. Phys.23 (1976)355.[171D.G. Boulware,L.S. Brown,RN. Cahn,S.D. Ellis andC. Lee,Phys. Rev.D14 (1976) 2708.[18] L. Brink, J.H. SchwarzandJ. Scherk,Nuci. Phys.B121 (1977)77.119] L.S. Brown,RD. Carlitz,D.B. Creamerand C. Lee, Phys.Rev. 017 (1978) 1583.[20] L.S.Brown,RD. CarlitzandC. Lee,Phys. Rev.D16 (1977)417.[21] S.A. Brown,M.K. PrasadandP. Rossi,Phys. Rev.D24 (1981)2217.[22] Ci. Callias,Phys.Rev.D16 (1977)3068.[23] C.J.Callias,Comm.Math. Phys.62 (1978) 213.[24]S. Coleman,in: New Phenomenain SubnuclearPhysics,ed.A. Zichichi (New York, Plenum, 1975) P. 297.[25] 5. Coleman,unpublished.[26] S. Coleman,S. Parke,A. Neveuand CM. Sommerfield,Phys. Rev.D15 (1977) 554.[27] E.F. Corrigan,D.B. Fairlie, R.Y. YatesandP. Goddard,Comm. Math. Phys. 58 (1978) 223.[28] E.F. CorriganandP. Goodard,Comm.Math. Phys.80 (1981)575.[29] E.F. Corrigan andD. Olive,NucI. Phys.BilO (1976)237.[30] E.F. Corrigan, D. Olive, D.B.Fairlie andJ. Nuyts, NucI. Phys.B106 (1976)475.[31] A. D’Adda, R. Horsley and P. Di Vecchia, Phys. Lett.76B (1978)298.[32] PAM. Dirac, Proc.R. Soc. A133 (1931)60.[33] F. EnglertandP. windey,Phys.Rev. D14 (1976) 2728.[34] P. Forgacs,Z. Horvathand L. Palla, Phys.Lett. 99B (1981) 232.[35] P. Forgacs,Z. Horvath and L. Palla, Phys. Rev.D23 (1981) 1876.[36] P. Forgacs,Z. HowathandL. Palla, Nonlinearsuperpositionof monopoles,ReportKFKI-1981-23,Hung. Acad.Sci. Budapest(1981).[37] H. Georgi andS.L. Glashow,Phys. Rev.D6 (1972)2977.[38] P. Goddard,J. Nuyts andD. Olive,NucI. Phys.B125 (1977) 1.[39] AS. GoidhaberandD. Wilkinson, NucI. Phys.B114 (1976)317.[40] M. Grisaru,M. RoçekandW. Siegel, Phys.Rev. Lett.45 (1980) 1063.[41] P. Hasenfratzand G. ‘t Hooft, Phys.Rev. Lett. 36 (1976) 1119.[42] G. ‘t Hooft,NucI. Phys.B79 (1974)276.[43] G. ‘t Hooft, unpublished.[44] B. Houston and L. O’Raifeartaigh,Proc.Seminaron GroupTheoreticalMethodsin Physics,Moscow 1979.[45]R. Jackiw,Ann. Phys.(N.Y.) 129 (1980) 183.[46] R. JackiwandN. Manton,Ann. Phys. (N.Y.) 127 (1980)257.[47] R. Jackiw,C. Nohl and C. Rebbi, Phys.Rev. D15 (1977) 1642.[48] R. Jackiw andC. Rebbi, Phys. Rev.Lett. 36 (1976) 1116.[49] R. Jackiw andC. Rebbi, Phys. Rev.D13 (1976)3398.[50] D.R.T. Jones,Phys.Lett. 72B (1977) 199.[51] B. Julia and A. Zee,Phys. Rev.Dil (1975)2227.[52] Y. Kazama,C.N. Yangand AS. Goldhaber,Phys.Rev. D15 (1977)2287.[53] T.W. Kirkman andC.K. Zachos,Phys.Rev. 024 (1981)999.[54] S.C.Lee,Phys. Rev.D24 (1981)2200, 2209.[55] MA. Lobe, Phys.Lett. 70B (1977)325.[56] N.S. Manton,Nucl. Phys. B126 (1977) 525.[57] N.S. Manton,Nuci. Phys. B135 (1978)319.[58] N.S. Manton,Ann. Phys. (N.Y.) 132 (1981) 108.[59] MI. Monastyrskyand AM. Perelomov,JETPLett. 21(1975)43.[60] C. Montonenand D. Olive, Phys. Lett. 72B (1977)213.[61] E. Mottola, Phys. Lett. 79B (1978) 242.[62] W. Nahm,All selfdualMultimonopolesfor Arbitrary GaugeGroups,CERN preprint (1981).[63] KS. Narain,Phys. Rev. D24 (1981)2211.[64] H.Osborn,Phys.Lett. 83B (1979)321.[65]A. Patrascioiu,Phys.Rev. D12 (1975)523.(66] H. Poincaré,C.R. Acad. Sci. Paris 123 (1896) 530.[67] A.M. Polyakov,Z8ETF Pisma20 (1974) 430; JETPLett. 20 (1974) 194.[68] E.C. PoggioandH.N. Pendleton,Phys.Lett. 72B (1977) 200.[69] M.K. Prasad,Phys. Rev.017 (1978)3243.[70] M.K. Prasad,Physica10 (1980) 167.[71] M.K. Prasad,Comm.Math. Phys.80 (1981) 137.[72] M.K. Prasadand P. Rossi,Phys. Rev.023 (1981) 1795.

362 P. Rossi,Exact results in the theor-vof non-Abel/an,nagneticInonopo/es

[73] M.K. Prasadand P. Rossi. Phys.Rev. Lett. 46 (198!) 806.[74] M.K. Prasadand P. Rossi.Phys. Rev. D24 (198!) 2182.

1751 M.K. Prasad.A. Sinha andLI. Chau Wang, Phys. Rev. D23 (198!)232!.1761 M.K. Prasadand C.M. Sommerfield.Phys. Rev. Lett. 35(1975)760.[77] C. Rebbiand P. Rossi.Phys. Rev.022 (1980) 200).[78] P. Rossi, Nucl. Phys. B127 (1977) 518.

[79] P. Rossi, Phys.Lett. 71B (1977) 145.18)!] P. Rossi, NucI. Phys. B149 (1979) 17)).[81] P. Rossi, Phys.Lett. 99B (198!) 229.[82] i.E. Schonfeld.J. Math. Phys.21(198)))2528.[83] J. Tamm. Z. Phys. 71(1931)14!.[84] C.H. Taubes,Comm. Math. Phys. 8)) (198!) 343.[85]Yu.S. Tyupkin, V.A. FateevandA.S. Schwartz.JETP Lett. 21(1975)41.[86] R.S.Ward, Phys.Lett. 61A (1977) 81.[87] R.S. Ward.Comm.Math. Phys. 79 (1981) 317.[88] R.S.Ward, Comm.Math. Phys. 80(1981)563.[89] R.S.Ward, Two Yang—Mills—Higgs MonopolesClose Together.Preprint (198!).[90] E. Weinberg.Phys.Rev. D2)) (1979) 936.[91] F. Weinberg,NucI. Phys. B!67 (1980)5)8).[92] El. Weinbergand A.H. Guth. Phys. Rev. D14 (1976) 166)).[93] D. Wilkinson and F.A. Ba/s. Phys.Rev. D19 (1979) 24!)).[94] D. Wilkinson and A.S. Goldhaber,Phys. Rev. D16 (1977) 122!.[95] E. Witten, Phys. Rev.Lett. 38 (1977) 121.[96] E. Witten and D. Olive, Phys. Lett. 78B (1978) 97.[97]T.T. Wu and C.N. Yang. NucI. Phys. B107 (1976)365.[98] C.N. Yang, Phys. Rev. Lett. 38 (1977) 1377.