A Brief Course in Spontaneous Symmetry Breaking II. Modern Times The BEH Mechanism

ABriefII.CourseModerninTimes:SpontaneousTheBEHSymmetryMechanismBreaking

1

Fran¸coisEnglert2

ServicedePhysiqueTh´eorique

Universit´eLibredeBruxelles,CampusPlaine,C.P.225BoulevardduTriomphe,B-1050Bruxelles,Belgium

Abstract

Thetheoryofsymmetrybreakinginpresenceofgaugefieldsispre-sented,followingthehistoricaltrack.Particularemphasisisplacedupontheunderlyingconcepts.

I.Introduction

Itwasknowninthefirsthalfofthetwentiethcenturythat,attheatomiclevelandatlargerdistancescales,allphenomenaappeartobegovernedbythelawsofclassicalgeneralrelativityandofquantumelectrodynamics.Gravitationalandelectromagneticforcesarelongrangeandhencecanbeperceiveddirectlywithoutthemediationofhighlysophisticatedtechnicaldevices.Thedevelopmentoflargescalephysics,initiatedbytheGalileaninertialprinciple,issurelytributarytothiscircumstance.Itthentookaboutthreecenturiestoachieveasuccessfuldescriptionoflongrangeeffects.Thediscoveryofsubatomicstructuresandoftheconcomitantweakandstronginteractionshortrangeforcesraisedthequestionofhowtocopewithshortrangeforcesinquantumfieldtheory.TheFermitheoryofweakinter-actions,formulatedintermsofafourFermipoint-likecurrent-currentinter-action,waspredictiveinlowestorderperturbationtheoryandsuccessfullyconfrontedmanyexperimentaldata.However,itwasclearlyinconsistentinhigherorderbecauseofuncontrollablequantumdivergencesathighen-ergies.Inorderwords,incontradistinctionwithquantumelectrodynamics,theFermitheoryisnotrenormalizable.Thisdifficultycouldnotbesolvedbysmoothingthepoint-likeinteractionbyamassive,andthereforeshortrange,chargedvectorparticleexchange(theso-calledW+andW−mesons);theo-rieswithfundamentalmassivechargedvectormesonsarenotrenormalizableeither.Intheearlynineteensixties,thereseemedtobeinsuperableobstaclesforformulatingatheorywithshortrangeforcesmediatedbymassivevectors.Thesolutionofthelatterproblemcamefromthetheoryproposedin1964byBroutandEnglert[1]andbyHiggs[2,3].TheBrout-Englert-Higgs(BEH)theoryisbasedonamechanism,inspiredfromthespontaneoussymmetrybreakingofacontinuoussymmetry,discussedintheprevioustalkbyRobertBrout,adaptedtogaugetheoriesandinparticulartononabeliangaugetheories.Themechanismunifieslongrangeandshortrangeforcesmediatedbyvectormesons,byderivingthevectormesonsmassesfromafundamentaltheorycontainingonlymasslessvectorfields.Itledtoasolutionoftheweakinteractionpuzzleandopenedthewaytomodernperspectivesonunifiedlawsofnature.

Beforeturningtoanexpos´eoftheBEHmechanism,weshallinsectionII

1

review,inthecontextofquantumfieldtheory,theanalysisgivenbyRobertBroutofthespontaneousbreakingofacontinuoussymmetry.SectionIIIex-plainstheBEHmechanism.WepresentthequantumfieldtheoryapproachofBroutandEnglertwhereinthebreakingmechanismforbothabelianandnonabeliangaugegroupsisinducedbyscalarbosons.Wealsopresenttheirapproachinthecaseofdynamicalsymmetrybreakingfromfermionconden-sate.WethenturntotheequationofmotionapproachofHiggs.Finallyweexplaintherenormalizationissue.InsectionIV,webrieflyreviewthewell-knownapplicationsoftheBEHmechanismwithparticularemphasisonconceptsrelevanttothequestforunification.SomecommentsonthissubjectaremadeinsectionV.

II.SpontaneousBreakingofaGlobalSymmetry

SpontaneousbreakingofaLiegroupsymmetrywasdiscussedbyRobertBroutin“ThePaleoliticAge”.Ireviewhereitsessentialfeaturesinthequantumfieldtheorycontext.

Recallthatspontaneousbreakdownofacontinuoussymmetryincondensedmatterphysicsimpliesadegeneracyofthegroundstate,andasaconse-quence,inabsenceoflongrangeforces,collectivemodesappearwhoseener-giesgotozerowhenthewavelengthgoestoinfinity.ThiswasexemplifiedinparticularbyspinwavesinaHeisenbergferromagnet.There,thebrokensymmetryistherotationinvariance.

SpontaneoussymmetrybreakingwasintroducedinrelativisticquantumfieldtheorybyNambuinanalogytotheBCStheoryofsuperconductivity.TheproblemstudiedbyNambu[4]andNambuandJona-Lasinio[5]isthespon-taneousbreakingofchiralsymmetryinducedbyafermioncondensate1.The

¯󰀓=chiralphasegroupexp(iγ5α)isbrokenbythefermioncondensate󰀒ψψ0

andthemasslessmodeisidentifiedwiththepion.Thelattergetsitstinymass(onthehadronscale)fromasmallexplicitbreakingofthesymmetry,justasasmallexternalmagneticfieldimpartsasmallgapinthespinwavespectrum.Thisinterpretationofthepionmassconstitutedabreakthroughinourunderstandingofstronginteractionphysics.Generalfeaturesofsponta-neoussymmetrybreakdowninrelativisticquantumfieldtheorywerefurther

formalizedbyGoldstone[6].Here,symmetryisbrokenbynonvanishingvac-uumexpectationvaluesofscalarfields.Themethodisdesignedtoexhibittheappearanceofamasslessmodeoutofthedegeneratevacuumanddoesnotreallydependonthesignificanceofthescalarfields.Thelattercouldbeelementaryorrepresentcollectivevariablesofmorefundamentalfields,aswouldbethecaseintheoriginalNambumodel.Compositenessaffectsdetailsofthemodelconsidered,suchasthebehaviorathighmomentumtransfer,butnottheexistenceofthemasslessexcitationsencodedinthedegeneracyofthevacuum.

LetusfirstillustratetheoccurrenceofthismasslessNambu-Goldstone(NG)bosoninasimplemodelofacomplexscalarfieldwithU(1)symmetry[6].TheLagrangiandensity,

L=∂µφ∗∂µφ−V(φ∗φ)withV(φ∗φ)=−µ2φ∗φ+λ(φ∗φ)2,λ>0,(1)isinvariantundertheU(1)groupφ→eiαφ.TheU(1)symmetryiscalledglobalbecausethegroupparameterαisconstantinspace-time.Itisbrokenbyavacuumexpectationvalueoftheφ-fieldgiven,attheclassicallevel,√bytheminimumofV(φ∗φ).Writingφ=(φ1+iφ2)/

Aroundtheunbrokenvacuumthefieldφ1hasnegativemassandacquiresapositivemassaroundthebrokenvacuumwherethefieldφ2ismassless.ThelatteristheNGbosonofbrokenU(1)symmetry.Themassivescalardescribesthefluctuationsoftheorderparameter󰀒φ1󰀓.ItsmassistheanalogoftheinverselongitudinalsusceptibilityoftheHeisenbergferromagnetdiscussedbyRobertBroutwhilethevanishingoftheNGbosonmasscorrespondstothevanishingofitsinversetransversesusceptibility.Thescalarbosonφ1isalwayspresentinspontaneousbreakdownofasymmetry.InthecontextoftheBEHmechanismanalyzedinthefollowingsection,itwasintroducedbyBroutandmyself,andbyHiggs.WeshalllabelittheBEHboson2(Fig.1).Intheclassicallimit,theoriginofthemasslessNGbosonφ2isclearlyillus-tratedintheFig.1.Thevacuumcharacterizedbytheorderparameter󰀒φ1󰀓isrotatedintoanequivalentvacuumbythefieldφ2atzerospacemomentum.

→

Suchrotationcostsnoenergyandthusthefieldφ2atspacemomentaq=0hasq0=0ontheequationsofmotion,andhencezeromass.

Thiscanbeformalizedandgeneralizedbynotingthat󰀁theconservedNoethercurrentJµ=φ1∂µφ2−φ2∂µφ1givesachargeQ=J0d3x.Theoperatorexp(iαQ)rotatesthevacuumbyanangle󰀁α.Intheclassicallimit,thischargeis,aroundthechosenvacuum,Q=󰀒φ1󰀓∂0φ2d3xandinvolvesonlyφ2atzeromomentum.Ingeneral,󰀒[Q,φ2]󰀓=i󰀒φ1󰀓isnonzerointhechosenvacuum.Thisimpliesthatthepropagator∂µ󰀒TJµ(x)φ2(x′)󰀓cannotvanishatzerofour-momentumqbecauseitsintegraloverspace-timeisprecisely󰀒[Q,φ2]󰀓.ExpressingthepropagatorintermsofFeynmandiagramsweseethattheφ2-propagatormusthaveapoleatq2=0.Thefieldφ2isthemasslessNGboson.

Theproofisimmediatelyextendedtothespontaneousbreakingofasemi-simpleLiegroupglobalsymmetry.LetφAbescalarfieldsspanningarep-resentationoftheLiegroupGgeneratedbythe(antihermitian)matricesTaAB.IfthedynamicsisgovernedbyaG-invariantactionandifthepo-tentialhasminimafornonvanishingφA,s,symmetryisbrokenandthe

a

vacuumisdegenerateunderG-rotations.TheconservedchargesareQ=󰀁

∂µφBTaBAφAd3x.Asintheabeliancaseabove,thepropagatorsofthefieldsφBsuchthat󰀒[Qa,φB]󰀓=TaBA󰀒φA󰀓=0haveaNGpoleatq2=0.

III.TheBEHMechanism

-Fromglobaltolocalsymmetry

TheglobalU(1)symmetryinEq.(1)canbeextendedtoalocalU(1)in-varianceφ(x)→eiα(x)φ(x)byintroducingavectorfieldAµ(x)transformingaccordingtoAµ(x)→Aµ(x)+(1/e)∂µα(x).ThecorrespondingLagrangiandensityis

1

L=Dµφ∗Dµφ−V(φ∗φ)−

where

aABBaaabcbc

(Dµφ)A=∂µφA−eAaφ,Fµν=∂µAaAµAν.(4)µTν−∂νAµ−ef

Here,φAbelongstotherepresentationofGgeneratedbyTaABandthe

potentialVisinvariantunderG.

ThesuccessofquantumelectrodynamicsbasedonlocalU(1)symmetry,andofclassicalgeneralrelativitybasedonalocalgeneralizationofPoincarein-variance,providesampleevidencefortherelevanceoflocalsymmetryforthedescriptionofnaturallaws.Oneexpectsthatlocalsymmetryhasafundamentalsignificancerootedincausalityandintheexistenceofexactconservationlawsatafundamentallevel,ofwhichchargeconservationap-pearsastheprototype.Asanexampleofthestrengthoflocalsymmetrywecitethefactthatconservationlawsresultingfromaglobalsymmetryaloneareviolatedinpresenceofblackholes.

Thelocalsymmetry,orgaugeinvariance,ofYang-Millstheory,abelianornonabelian,apparentlyreliesonthemasslesscharacterofthegaugefieldsAµ,henceonthelongrangecharacteroftheforcestheytransmit,astheadditionofamasstermforAµintheLagrangianEq.(2)or(3)destroysgaugeinvariance.Butshortrangeforces,suchastheweakinteractionforces,seemtobeasfundamentalastheelectromagneticonesdespitetheapparent

5

4

aFµνFaµν,

(3)

absenceofexactconservationlaws.Toreachabasicdescriptionofsuchforcesoneistemptedtolinktheviolationofconservationtoamassofthegaugefieldswhichwouldarisefromspontaneoussymmetrybreaking.Howevertheproblemofspontaneousbrokensymmetryisdifferentforglobalandforlocalsymmetry.

Tounderstandthedifference,letusbreakthesymmetriesexplicitly.TotheLagrangianEq.(1)weaddtheterm

φh∗+φ∗h,

(5)

whereh,h∗areconstantinspacetime.Letustakehreal.ThepresenceofthefieldhbreaksexplicitlytheglobalU(1)symmetryandthefieldφ1alwaysdevelopsanexpectationvalue.Whenh→0,thesymmetryoftheactionisrestoredbut,whenthesymmetryisbrokenbyaminimumofV(φφ∗)at|φ|=0,westillhave󰀒φ1ofthedegeneratevacuainperfect󰀓=0.analogyThetinywithh-fieldtheinfinitesimalsimplypicksmagneticuponefieldwhichorientsthemagnetizationofaferromagnet.Asinstatisticalmechanics,spontaneousbrokenglobalsymmetrycanberecoveredinthelimitofvanishingexternalsymmetrybreaking.Thedegeneracyofthevacuumcanbeputintoevidencebychangingthephaseofh;inthisway,wecanreachinthelimith→0anyU(1)rotatedvacuum.

Whenthesymmetryisextendedfromglobaltolocal,onecanstillbreakthesymmetrybyanexternal“magnetic”field.Howeverinthelimitofvanishingmagneticfieldtheexpectationvalueofanygaugedependentlocaloperatorwilltendtozerobecause,incontradistinctiontoglobalsymmetry,itcostnoenergyinthelimittochangetherelativeorientationofneighboring“spins”;thereisthennoorderedconfigurationingroupspacewhichcanbeprotectedfromdisorderingfluctuations.Asaconsequence,thevacuumisgenericallynondegenerateandpointsinnoparticulardirectioningroupspaceastheexternalfieldgoestozero.Localgaugesymmetrycannotbespontaneouslybroken3andthevacuumisgaugeinvariant4.Recallingthattheexplicit

presenceofagaugevectormassbreaksgaugeinvariance,wearethusfacedwithadilemma.Howcangaugefieldsacquiremasswithoutbreakingthelocalsymmetry?-Solvingthedilemma

Inperturbationtheory,gaugeinvariantquantitiesareevaluatedbychoosingaparticulargauge.OneimposesthegaugeconditionbyaddingtotheactionagaugefixingtermandonesumsoversubsetsofgraphssatisfyingtheWardIdentities5.

ConsidertheYang-MillstheorydefinedbytheLagrangianEq.(3).LetuschooseagaugewhichpreservesLorentzinvarianceandaresidualglobalGsymmetry.ThiscanbeachievedbyaddingtotheLagrangianagaugefixing

aν

term(2η)−1∂µAµa∂νA.Thegaugeparameterηisarbitraryandhasnoobservableconsequences.

A(a)

(b)

(c)

BqFig. 2

Theglobalsymmetrycannowbespontaneouslybroken,forsuitablepoten-

tialV,bynonzeroexpectationvalues󰀒φA󰀓ofBEHfields.InFig.2wehaverepresentedfluctuationsofthisparameterinthespatialq-directionandinaninternalspacedirectionorthogonaltothedirectionA.TheorthogonaldirectiondepictedinthefigurehasbeenlabeledB.Fig.2apicturesthespontaneouslybrokenvacuumofthegaugefixedLagrangian.Fig.2band2crepresentfluctuationsoffinitewavelengthλ.

Clearlyasλ→∞thesefluctuationscanonlyinduceglobalrotationsintheinternalspace.Inabsenceofgaugefields,suchfluctuationswouldgiverise,asinspontaneouslybrokenglobalcontinuoussymmetries,tomasslessNGmode.Inagaugetheory,fluctuationsof󰀒φA󰀓arejustlocalrotationsintheinternalspaceandhenceareunobservablegaugefluctuations.HencetheNGbosonsinduceonlygaugetransformationsanditsexcitationsdisappearfromthephysicalspectrum.

ThedegreesoffreedomoftheNGfieldswerepresentintheoriginalgaugeinvariantactionandcannotdisappear.ButwhatmakeslocalinternalspacerotationsunobservableinagaugetheoryispreciselythefactthattheycanbeabsorbedthroughgaugetransformationsbytheYang-Millsfields.TheabsorptionofthelongrangeNGfieldsrendersmassivethosegaugefieldstowhichtheyarecoupled,andtransferstothemthemissingdegreesoffreedomwhichbecomestheirthirdpolarization.

Weshallseeinthenextsectionshowtheseconsiderationsarerealizedinquantumfieldtheory,givingrisetoanapparentbreakdownofsymmetry:despitetheabsenceofspontaneouslocalsymmetrybreaking,gaugeinvariantvectormasseswillbegeneratedinacosetG/H,leavinglongrangeforcesonlyinasubgroupHofG.

-Thequantumfieldtheoryapproach[1]α)BreakingbyBEHbosons

LetusfirstexaminetheabeliancaseasrealizedbythecomplexscalarfieldφexemplifiedinEq.(2).

Inthecovariantgauges,thefreepropagatorofthefieldAµis

0Dµν

=

gµν−qµqν/q2

8

q2

,(6)

whereηisthegaugeparameter.Itcanbeputequaltozero,asintheLandaugaugeusedinreference[1],butweleaveitarbitraryheretoillustrateexplicitlytheroleoftheNG-boson.

Gauge field

Complex scalar field

Fig. 3

Inabsenceofsymmetrybreaking,thelowestordercontributiontotheself-energy,arisingfromthecovariantderivativetermsinEq.(2),isgivenbytheone-loopdiagramsofFig.3.Theself-energy(suitablyregularized)takestheformofapolarizationtensor

Πµν=(gµνq2−qµqν)Π(q2),

(7)

wherethescalarpolarisationΠ(q2)isregularatq2=0,leadingtothegaugefieldpropagator

D=

gµν−qµqν/q2

µνq2.(8)ThepolarizationtensorinEq.(7)istransverseandhencedoesnotaffectthegaugeparameterη.Thetransversalityofthepolarizationtensorreflectsthegaugeinvarianceofthetheory6and,asweshallseebelow,theregularityofthepolarizationscalarsignalstheabsenceofsymmetrybreaking.ThisguaranteesthattheAµ-fieldremainsmassless.

Symmetrybreakingaddstadpolediagramstothepreviousones.Toseethiswrite

φ=12

(φ1+iφ2)󰀒φ1󰀓=0.(9)

TheBEHfieldisφ1andtheNGfieldφ2.TheadditionaldiagramsaredepictedinFig.4.

BEH tadpoleNG propagator

Fig. 4

Inthiscase,thepolarisationscalarΠ(q2)inEq.(7)acquiresapole

Π(q2

)=

e2󰀒φ1󰀓2

qµqν/q2

q2−µ2

+η

Π(q)=

ab2

e2󰀒φ∗B󰀓T∗aBCTbCA󰀒φA󰀓

q2−µa2

+η

qµqν/q2

q2

=

λ=1

3󰀃

(λ)(λ)

eµ.eν,q2=µa2,

(17)

(λ)

wheretheeµarethethreepolarizationvectorswhichareorthonormalintherestframeoftheparticle.

Inthisway,theNGbosonsgeneratemassivepropagatorsforthosegaugefieldstowhichtheyarecoupled.Longrangeforcesonlysurviveinthesub-groupHofGwhichleavesinvariantthenonvanishingexpectationvalues󰀒φA󰀓.

Notethat(asintheabeliancase)thescalarpotentialVdoesnotenterthecomputationofthegaugefieldpropagator.ThisisbecausethetrilineartermarisingfromthecovariantderivativesintheLagrangianEq.(3),whichyieldsthesecondgraphofFig.5,canonlycouplethetadpolestootherscalarfields

11

throughgrouprotationsandhencecouplethemonlytotheNGbosons.ThesearetheeigenvectorswithzeroeigenvalueofthescalarmassmatrixgivenbythequadratictermintheexpansionofthepotentialVarounditsminimum.Hencethemassmatrixdecouplesfromthetadpoleatthetreelevelconsideredabove.AnexplicitexampleofthisfeaturewillbegivenfortheLagrangianEq.(32).β)Dynamicalsymmetrybreaking

Thesymmetrybreakinggivingmasstogaugevectorbosonsmayarisefromthefermioncondensatebreakingchiralsymmetry.ThisisillustratedbythefollowingchiralinvariantLagrangian

1¯¯L=LF−eψγψV−eψγγψA−FµνFµν(A).(18)VµµAµ5µ0

4

HereFµν(V)andFµν(A)areabelianfieldstrengthforU(1)×U(1)symmetry.Chiralanomaliesareeventuallycanceledbyaddingintherequiredadditionalfermions.

TheWardidentityforthechiralcurrent

qµΓµ5(p+q/2,p−q/2)=S−1(p+q/2)γ5+γ5S−1(p−q/2),

(19)

showsthatifthefermionself-energyγµpµΣ2(p2)−Σ1(p2)acquiresanonvanishingΣ1(p2)term,thusadynamicalmassmatΣ1(m2)=m(takingforsimplicityΣ2(m2)=1),theaxialvertexΓµ5developsapoleatq2=0.Inleadingorderinq,weget

qµ

Γµ5→2mγ5

Thevalidityoftheapproximation,andinfactofthedynamicalapproach,restsonthehighmomentumbehaviorofthefermionselfenergy,butthisproblemwillnotbediscussedhere.

7

12

Γµ5γν5axiovector propagatorfermion propagator

Fig.6

Thisexampleillustratesthefactthatthetransversalityofthepolarization

tensorusedinthequantumfieldtheoreticapproachtomassgenerationisaconsequenceofaWardidentity.ThisistruewhethervectormassesarisethroughfundamentalfundamentalBEHbosonsorthroughfermionconden-sate.Thegenerationofgaugeinvariantmassesisthereforenotcontingentuponthe“treeapproximation”usedtogetthepropagatorsEqs.(11)and(16).Itisaconsequenceofthe1/q2singularityinthevacuumpolarisationscalarsEqs.(10),(13)or(22)whichcomesfromNGbosoncontribution.-Theequationofmotionapproach[2,3]

Shortlyaftertheaboveanalysiswaspresented,Higgswrotetwopapers.Inthefirstone[2],heshowedthattheproofoftheGoldstonetheorem[6,7],whichstatesthat,inrelativisticquantumfieldtheory,spontaneoussymmetrybreakingofacontinuousglobalsymmetryimplieszeromassNGbosons,failsinthecaseofgaugefieldtheory.Inthesecondpaper[3],hederivedtheBEHtheoryintermsoftheclassicalequationsofmotion,whichheformulatedfortheabeliancase.

FromtheactionEq.(2),takingasinEq.(9),theexpectationvalueoftheBEHbosontobe󰀒φ1󰀓,andexpandingtheorder

NGfieldφ2tofirstorder,onegetstheclassicalequationsofmotiontothat∂µ∂F{∂µφ2µν=−e󰀒eφ󰀒φ1󰀓Aµ}=0󰀒φ,

(23)ν1󰀓{∂µφ2−e1󰀓Aµ}.

(24)

Defining

B1

µ=Aµ−

Inthisformulation,weseeclearlyhowtheGoldstonebosonisabsorbedintoaredefinedmassivevectorfieldwhichhasnolongerexplicitgaugeinvariance.Thesamephenomenoninthequantumfieldtheoryapproachisrelatedtotheunobservabilityofthe1/q2polementionedinthediscussionofEq.(15);thiswillbemadeexplicitinthenextsection.

Theequationofmotionapproachisclassicalincharacterbut,aspointedoutbyHiggs[3],theformulationoftheBEHmechanisminthequantumfieldtheorytermsofreference[1]indicatesitsvalidityinthequantumregime.WenowshowhowthelatterformulationsignalstherenormalizabilityoftheBEHtheory.

-Therenormalizationissue

ThemassivevectorpropagatorEq.(16)differsfromaconventionalfreemas-sivepropagatorintworespects.Firstthepresenceoftheunobservablelongi-tudinaltermreflectsthearbitrarinessofthegaugeparameterη.SecondtheNGpoleatq2=0inthetransverseprojectorgµν−qµqν/q2isunconventional.ItssignificanceismadeclearbyexpressingthepropagatoroftheAµfieldinEq.(16)as(puttingηtozero)

aDµν

≡

gµν−qµqν/q2

q2−µa2

+

1

q2

.(27)

ThefirsttermintherighthandsideofEq.(27)istheconventionalmassivevectorpropagator.Itmaybeviewedasthe(non-abeliangeneralizationofthe)freepropagatoroftheBµfielddefinedinEq.(25)whilethesecondtermisapuregaugepropagatorduetotheNGboson([1/e󰀒φ1󰀓]∂µφ2inEq.(25))whichconvertstheAµfieldintothismassivevectorfieldBµ.

ThepropagatorEq.(16)whichappearedinthefieldtheoreticapproachcon-tainsthus,inthecovariantgauges,thetransverseprojectorgµν−qµqν/q2inthenumeratorofthemassivegaugefieldAaµpropagator.Thisisinsharpcontradistinctiontothenumeratorgµν−qµqν/µa2characteristicofthecon-ventionalmassivevectorfieldBµpropagator.Itisthetransversalityoftheselfenergyincovariantgauges,whichledinthe“treeapproximation”tothetransverseprojectorinEq.(16).Asalreadymentioned,thetransversalityisaconsequenceofaWardidentityandthereforedoesnotdependonthetreeapproximation.Thisfactisalreadysuggestedfromthedynamicalexample

14

presentedabovebutwasproveninmoregeneraltermsinasubsequentpub-lication8[9].Theimportanceofthisfactisthatthetransversalityoftheself-energyincovariantgaugesdeterminesthepowercountingofirreduciblediagrams.ItisthenstraightforwardtoverifythattheBEHquantumfieldtheoryformulationisrenormalizablebypowercounting.

OnthisbasiswesuggestedthattheBEHtheoryconstitutesindeedacon-sistentrenormalizablefieldtheory[9].Toprovethisstatement,onemustverifythatthetheoryisunitary,afactwhichisnotapparentinthe“renor-malizable”covariantgaugesbecauseofthe1/q2poleintheprojector,butwouldbemanifestinthe“unitarygauge”definedinthefreetheorybytheBµpropagator.Intheunitarygaugehowever,renormalizationfrompowercountingisnotmanifest.Theequivalence,atthefreelevel,betweentheAµandBµfreepropagators,whichisonlytrueinagaugeinvarianttheorywheretheirdifferenceistheunobservableNGpropagatorappearinginEq.(27),istheclueoftheconsistencyoftheBEHtheory.Afullproofthatthetheoryisrenormalizableandunitarywasachievedby’tHooftandVeltman[10].

IV.Consequences

ThemostdramaticapplicationoftheBEHmechanismistheelectroweaktheory,amplyconfirmedbyexperiment.Considerableworkhasbeendone,usingtheBEHmechanism,toformulateGrandUnifiedtheoriesofnongrav-itationalinteractions.WeshallsummarizeherethesewellknownideasandthenevoketheconstructionofregularmonopolesandfluxlinesusingBEHbosons,becausetheyraisepotentiallyimportantconceptualissues.Weshallalsomentionbrieflytheattemptstoincludegravityintheunificationquest,inthesocalledM-theoryapproach,andfocusesinthiscontextonaninter-estinggeometricalinterpretationoftheBEHmechanism.-Theelectroweaktheory[11]

Intheelectroweaktheory,thegaugegroupistakentobeSU(2)×U(1)

a′′

withcorrespondinggeneratorsandcouplingconstantsgAaµTandgBµY.

TheSU(2)actsonleft-handedfermionsonly.TheelectromagneticchargeoperatorisQ=T3+Y′andtheelectricchargeeisusuallyexpressedintermsofthemixingangleθasg=e/sinθ,g′=e/cosθ.TheBEHbosons(φ+,φ0)areinadoubletofSU(2)andtheirU(1)chargeisY′=1/2.BreakingoccursinsuchawaythatQgeneratesanunbrokensubgroup,coupledtowhichis√themasslessphotonfield.Thusthevacuumischaracterizedby󰀒φ󰀓=1/

4

whosediagonalizationyieldstheeigenvalues

2MW+

=

v2

4

g,

2

2

MZ

=

v2

2G)−1.

Althoughtheelectroweaktheoryhasbeenamplyverifiedbyexperiment,theexistenceoftheBEHbosonhas,asyet,notbeenconfirmed.ItshouldbenotedthatthephysicsoftheBEHbosonismoresensitivetodynamicalassumptionsthanthemassivevectorsW±andZ,beitagenuineelementaryfieldoramanifestationofacompositeduetoamoreelaboratemechanism.Henceobservationofitsmassandwidthisofparticularinterestforfurtherunderstandingofthemechanismatwork.-Grandunificationschemes

Thediscoverythatconfinementcouldbeexplainedbythestrongcouplinglimitofquantumchromodynamicsbasedonthe“color”gaugegroupSU(3)ledtotentativeGrandUnificationschemeswhereelectroweakandstronginteractioncouldbeunifiedinasimplegaugegroupGcontainingSU(2)×U(1)×SU(3)[12].BreakingoccursthroughvacuumexpectationvaluesofBEHfieldsandunificationcanberealizedathighenergiesbecausewhilethe

16

renormalizationgroupmakesthesmallgaugecouplingofU(1)increaseloga-rithmicallywiththeenergyscale,theconverseistruefortheasymptoticallyfreenonabeliangaugegroups.

-Monopoles,fluxtubesandelectromagneticduality

Inelectromagnetism,monopolescanbeincludedattheexpenseofintroduc-ingaDiracstring[13].Thelattercreatesasingularpotentialalongthestringterminatingatthemonopole.Forinstancetodescribeapoint-likemonopolelocatedat󰀴r=0,onecantaketheline-singularpotential

A

󰀴=gz3z3BA3ry2Bry2x1unobservabilitygauging outx1Fig. 8AnexampleistheSO(3)monopole,representedinFig.8,arisingfromthepotential

g

Aai=,eg=4π.(31)

r2

BreakingthesymmetrytoU(1)byaBEHfieldbelongingtotheadjointgroupSO(3)onecanremovethepointsingularitytogetthetopologicallystable’tHooft-Polyakovregularmonopole[14].

ThisprocedurecanbeextendedtoLiegroupsGofhigherrank[15].ForageneralLiegroupG,thepossibilityofgaugingouttheDiracstringdependsontheglobalpropertiesofG.Namely,themappingofasmallcirclesurroundingtheDiracstringontoGmustbeacurvecontinuouslydeformabletozero.ClosedcurvesinGarecharacterizedbyZwhereZisthesubgroupofthe

˜ofGsuchthatG=G˜/Z.GaugingoutcenteroftheuniversalcoveringG

onlyoccursforthecurvecorrespondingtotheunitelementofZ.Thisistheoriginfortheunconventionalfactorof2(4π=2.2π)inEq.(31)asSO(3)=SU(2)/Z2.

Theconstructionofregularmonopoleshasinterestingconceptualimplica-tions.

ThemixingbetweenspaceandisospaceindicesinEq.(31)meansthattheregularmonopoleisinvariantunderthediagonalsubgroupofSO(3)space×SO(3)isospace.Thisimpliesthataboundstateofascalarofisospin1/2withthemonopoleisaspace-timefermion.Inthisway,fermionscanbemadeoutofbosons[16].

18

OnecandefineregularmonopolesinalimitinwhichtheBEH-potentialvanishes.ThesearetheBPSmonopoles.Theyadmitasupersymmetricextensionsinwhichthereareindicationsthatelectromagneticdualitycanberealizedatafundamentallevel,namelythattheinterchangeofelectricandmagneticchargecouldberealizedbyequivalentbutdistinctactions.TheBEH-mechanism,whenGsymmetryiscompletelybroken,isarelativisticanalogofsuperconductivity.Thelattermaybeviewedasacondensationofelectriccharges.Magneticfluxisthenchanneledintoquantizedfluxtubes.Inconfinement,itistheelectricfluxwhichischanneledintoquantizedtubes.Thereforeelectric-magneticdualitysuggeststhat,atsomefundamentallevel,confinementisacondensationofmagneticmonopolesandconstitutesthemagneticdualoftheBEHmechanism[17].

-AgeometricalinterpretationoftheBEHmechanism

TheBEHmechanismoperateswithinthecontextofgaugetheories.DespitethefactthatgrandunificationschemesreachscalescomparabletothePlanckscale,therewas,apriori,noindicationthatYang-Millsfieldsofferanyinsightintoquantumgravity.Theonlyapproachtoquantumgravitywhichhadsomesuccess,inparticularinthecontextofaquantuminterpretationoftheblackholesentropies,arethesuperstringtheoryapproachesandthepossiblemergingofthefiveperturbativeapproaches(TypeIIA,IIB,TypeIandthetwoheteroticstrings)intoanelusiveM-theorywhoseclassicallimitwouldbe11-dimensionalsupergravity.OfparticularinterestinthatcontextisthediscoveryofDp-branesalongwhichtheendsofopenstringscanmove[18].Thisled,forthefirsttime,toaninterpretationoftheareaentropyofsomeblackholesintermsofacountingofquantumstates.HereweshallexplainhowDp-branesyieldageometricalinterpretationoftheBEHmechanism.WhenNBPSDp-branescoincide,theyadmitmasslessexcitationsfromtheN2zerolengthorientedstringswithbothendattachedontheNcoincidentbranes.ThereareN2masslessvectorsandadditionalN2masslessscalarsforeachdimensiontransversetothebranes.TheopenstringsectorhaslocalU(N)invariance.Atrest,BPSDp-branescanseparatefromeachotherinthetransversedimensionsatnocostofenergy.ClearlythiscanbreakthesymmetrygroupfromU(N)uptoU(1)Nwhenallthebranesareatdistinctlocationinthetransversespace,becausestringsjoiningtwodifferentbranes

19

havefinitelengthandhencenowdescribefinitemassexcitations.Theonlyremainingmasslessexcitationsarethenduetothezerolengthstringswithbothendsonthesamebrane.

Fig. 9

Dp-branesThissymmetrybreakingmechanismcanbeunderstoodasaBEHmechanismfromtheactiondescribinglowenergyexcitationsofNDp-branes.Thisactionisthereductiontop+1dimensionsof10-dimensionalsupersymmetricYang-MillswithU(N)gaugefields[19,20].TheLagrangianisL=−

1

2

DµAiDµAi−

1

ToidentifythelatterweconsiderthescalarpotentialinEq.(32),namelyV=Tr1

4,Aj]|m󰀓.

(34)

i,j󰀃

Aj]|n󰀓󰀒n|[Ai;m,n

󰀒m|[Ai,Wewrite

Herethediagonalelements󰀒m|Aj{xj|n󰀓=xjmδmn+yj

mn.

(35)

yjmmn(=−[yjnm]∗)defined(N2}aretheBEHexpectationivaluesandtheymn=(ymn)1+i(ymn)−2,Nm)hermitian>n,andscalardisthefieldsnumber(ymn)a(a=1,2)

wherejjj

oftransverse

spacedimensions.Themassmatrixforthefields(yi

mn)ais

∂2V

andnotvacuumexpectationvalues.Thenondiagonalquantumdegreesoffreedom󰀴ymn⊥(󰀴xm−󰀴xn)haveapositivepotentialenergyproportionaltothedistancesquaredbetweentheD0-branesmandn.HencetheygetlockedintheirgroundstatewhentheD0-branesarelargelyseparatedfromeachother.Inthisway,theD0-braneAi={ximn}matricescommuteatlargedistancescaleanddefinegeometricaldegreesoffreedom.Howeverthesematricesdonotcommuteatshortdistanceswherethepotentialenergiesof

i

theymngotozero.Thissuggeststhatthespace-timegeometryexhibitsnoncommutativityatsmalldistances,afeaturewhichmaywellturnouttobeanessentialelementofquantumgravity.

V.Remarks

Physics,asweknowit,isanattempttointerprettheapparentdiversityofnaturalphenomenaintermsofgenerallaws.Byessencethen,itincitesonetowardsaquestforunifyingdiversephysicallaws.

OriginallytheBEHmechanismwasconceivedtounifythetheoreticalde-scriptionoflongrangeandshortrangeforces.Thesuccessoftheelectroweaktheorymadethemechanismacandidateforfurtherunification.Granduni-ficationschemes,wherethescaleofunificationispushedclosetothescaleofquantumgravityeffects,raisedthepossibilitythatunificationmightalsohavetoincludegravity.Thistrendtowardsthequestforunificationreceivedafurtherimpulsefromthedevelopmentsofstringtheoryandfromitscon-nectionwitheleven-dimensionalsupergravity.ThelatterwasthenviewedasaclassicallimitofahypotheticalM-theoryintowhichallperturbativestringtheorieswouldmerge.Inthatcontext,thegeometrizationoftheBEHmechanismissuggestiveoftheexistenceofanunderlyingnoncommutativegeometry.

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