Extended Gauge Theories in Euclidean Space with Higher Spin Fields

The extended Yang-Mills gauge theory in Euclidean space is a renormalizable (by power counting) gauge theory describing a local interacting theory of scalar, vector, and tensor gauge fields (with maximum spin 2). In this article we study the quantum aspect

arXiv:hep-th/9909117v3 7 Sep 2000

The extended Yang-Mills gauge theory in Euclidean space is a renormalizable (by power counting) gauge theory describing a local interacting theory of scalar, vector, and tensor gauge fields (with maximum spin 2). In this article we study the quantum aspect

ExtendedGaugeTheoriesinEuclideanSpace

withHigherSpinFields

E.Gabrielli

DepartamentodeF´ sicaTe´orica,C-XI,UniversidadAut´onomadeMadrid

E-28049Madrid,Spain

InstitutodeF´ sicaTe´orica,C-XVI,UniversidadAut´onomadeMadrid

1E-28049Madrid,Spain

Abstract

TheextendedYang-MillsgaugetheoryinEuclideanspaceisarenormalizable(bypowercounting)gaugetheorydescribingalocalinteractingtheoryofscalar,vector,andtensorgauge elds(withmaximumspin2).InthisarticlewestudythequantumaspectsandvariousgeneralizationsofthismodelinEuclideanspace.Inparticularthequantizationofthepuregaugemodelinacommonclassofcovariantgaugesisperformed.WegeneralizethepuregaugesectorbyincludingmatterfermionsintheadjointrepresentationofthegaugegroupandanalyzeitsN=1andN=2supersymmetricextensions.Weshowthatthemaximumhalf-integerspincontainedinthesefermion eldsindimension4is3/2.Moreoverwedevelopanextensionofthistheorysoastoincludeinternalgaugesymmetriesandthecouplingtobosonicmatter elds.Thespontaneoussymmetrybreakingoftheextendedgaugesymmetryisalsoanalyzed.

The extended Yang-Mills gauge theory in Euclidean space is a renormalizable (by power counting) gauge theory describing a local interacting theory of scalar, vector, and tensor gauge fields (with maximum spin 2). In this article we study the quantum aspect

1Introduction

Theinteractingtheoriesofhigherspin elds[1]haveattractedagreatattentionmostlybecauseoftherelevantroleplayedbythespin-2 eldgravitonwhich,itiswellknown,shouldcoupletoanykindofparticle.Thereisnodoubtabouttheexistenceofhigherspincompositeparticlesinnature,aclassicalexampleisgivenbytheobservedhadronicresonances.However,uptonow,noelementaryparticlewithspinhigherthan1hasbeenobserved(amongthesethegraviton).

Themaintheoreticalproblemswhicha ecttheconstructionofaconsistentinter-actingtheoryofelementaryhigherspin elds[2]inMinkowskispacecanbebrie ysummarizedasfollows:onemustrequirethecancellationofallnegative-normstates,acancellationwhichisperformedbymeansofhigher-spingaugeinvariances.Theselocalinvariances,though,imposetoomanyrestrictions(ontheinteractingterms)whichcannotbesatis edinmanycircumstances.Theserestrictionscanbecircum-ventedbyrelaxingsomebasicrequirementsofquantum eldtheory.Indeedageneralclassofconsistentinteractinghigher-spingaugetheoriesindimension4,3,and2existsanddescribesin nitelymany eldscontainingallthespins[3].Intheformulationofthesetheories,though,inordertoimplementthegaugesymmetries,necessarytoeliminateallthenegativenormstates,in nitelymanyauxiliary eldsmustbeintro-duced.Thismechanisminduceshigherderivativesintheinteractiontermsandtheseinturngiverisetonon-locality[1].Thesetheoriesareofinterest,however,sincetheyalsoestablishaconnectionwithstringmodels,eventhoughinthelatteralltheelementaryhigherspinexcitationsbeyondthegravitonaremassive[4].

UptonowinMinkowskispace,theonlyconsistentlocalinteracting eldtheoriesofmasslessspinhigherthan1/2aretheusualabelianandnon-abelianYang-Mills(YM)gaugetheoriesforthespin-1[5],thegravityforthespin-2[6],andthesupergravitytheories[7]forboththespin-2andspin-3/2.Inadditionifonerequiresthesetheoriestobealsorenormalizablethentheabovelistwouldfurthershortensinceitdoesnotcontainthegravitationalinteractions.Atpresentstringtheories,wheregravityisconsistentlycoupledtomatterandgauge eldsofanyspin[4],arelargelybelievedtoplayacentralroleinthesolutionofalltheseproblems.

Inthispaperweanalyzeagaugetheoryofhigherspininteractionsina atEu-clideanspace-time.Inthisspace,infact,weshallseethatitispossibletoconstructageneralclassofrenormalizablehigherspingaugetheories.Thistheorywas rstintroducedanumberofyearsagoin[8]whereanextensiontotheabeliangaugetheorywithscalar,vector,andtensorgauge eldswasproposedinEuclideanspace.Thismodelisdescribed(ina4–dimensional atspace-time)byanon-abelianU(4)gaugetheoryoftheYM’stypewheretheconnection eldtakesvaluesintheusual

3

The extended Yang-Mills gauge theory in Euclidean space is a renormalizable (by power counting) gauge theory describing a local interacting theory of scalar, vector, and tensor gauge fields (with maximum spin 2). In this article we study the quantum aspect

Cli ordalgebraofspinors.Herethescalar,vector,andtensor eldscanbeidenti edasthecomponentsofthegaugeconnectionalongtheCli urdimensionsthecontentofthemaximumspinofthegaugemultipletisaspin-2andthefullinteractinglagrangianisrenormalizablebypowercounting.Thismodel,inthefullbasisofCli ordalgebra,containsthreespin-2 elds:twoofthemareinastan-dardrepresentationoftherotationgroupandaredescribedbyasymmetrictracelesstensorofranktwo,whereastheremainingoneiscontainedinoneoftheirreduciblerepresentationsofatensorofrank-3withtwoantisymmetricindices[8]-[10].

Aninterestingaspectofthismodelisthatthegaugetransformationsmix,inaconsistentway,di erentirreduciblerepresentationsofthespace-timerotationgroup.Herethegaugespin-2 eldsdonothavetheusualcouplingtotheenergymomen-tumtensorwhiletheydocoupletothelowestspinparticlesinaconsistentway.2Moreoverthefreegaugetransformationsofthestandardspin-2 eldscoincidewiththeusualspin-2gaugetransformationsoftheFierz-Pauli eld[11].

AcontroversialquestionistheanalyticalcontinuationofthistheorytotheMinkowskispace.Wenextrecallsomeproblemsrelatedtothisissuethatarestillopen.SincetheelementsofthegaugegrouparenotinvariantundertheLorentztransformations,thequestionwhetherornotthemodelin[8]isforbiddenbytheColeman–Mandulano–gotheorem[12]mightarise.AspointedoutinRef.[10],themodelin[8]circum-ventsthehypothesisof[12].Themainreasonforthisisthatthetheoremin[12]appliesonlytotheglobalsymmetriesoftheSmatrixanddoesnotdealwiththelocalsymmetriesoftheLagrangian(seeRef.[13]foradetaileddiscussiononthisis-sue).Moreover,sincethepresenttheoryisoftheYM’stype,weshouldexpectthecon nementphenomenontoarise.Ifso,thenthephysicalspectrumwillbedescribedbygaugeinvariantoperators,suchasforexamplethehadronstatesorglueballsinQCD,andthissymmetrywillnotbemanifestintheSmatrixofthephysicalstates.Howeverwestressthat,inthismodel,oneofthemainstatementsoftheColeman–Mandulatheorem,whichistheanalyticalbehavioroftheSmatrix,canbedirectlycheckedinperturbationtheory.Inparticularonecanverify(bymeansoftheanalogywiththegluonscatterings)that,inthepureYang–Millssector,thetree–levelgauge–invariantamplitudessatisfyalltheanalyticalrequirements[14].

ThistheoryiswellformulatedinanEuclideanspacewherethegaugegroupiscompactand,beingaYM’stypegaugetheory,itshouldsatisfyalsotheOsterwalder-Schraderaxioms[15].Thereforeitispossibletoquantizeitbymeansofthestandardpathintegralmethodappliedtogaugetheories.However,whenthistheoryisformulated

The extended Yang-Mills gauge theory in Euclidean space is a renormalizable (by power counting) gauge theory describing a local interacting theory of scalar, vector, and tensor gauge fields (with maximum spin 2). In this article we study the quantum aspect

directlyintoaMinkowskispace(wherethegaugegroupisnon–compact),problemswithunitarityoftheSmatrixmightarisebecauseofunwantedghoststates[8],[10].Neverthelesswestressthattheappearanceofaninde nitemetricsintheHilbertspace,duethenon–compactnessofthesymmetrygroup,isnotalwaysanobstacleforbuildingaconsistenttheory[16]–[19].ApioneeringstudyinthisdirectionwasstartedbyLeeandWickinRef.[16].Moreoverintheliteraturevariousnon–compactsigma-modelswithinde nitemetricsalsoappearinsomeextendedsupergravitieswhenthereductiontofourdimensionisconsidered[17].ThegeneralconclusionforthesemodelsisthataunitarySmatrixinthephysicalsubspacecanbeobtainedfromapseudo-unitaryS-matrixinthefullHilbertspace[17],[18].

Inthepresentmodeltheanalysisofunitarityisamorecomplicatedissuethaninthenon–compactsigma–models,mainlybecausethesymmetryislocalanditisnotaninternalone.AcarefulanalysisoftheunitarityoftheSmatrix(inMinkowskispace)isstillmissingforthistheoryanditwouldbeworthinvestigatinghowtheunphys-icalghostsectorcoulddecouplefromthephysicalamplitudes.Theclari cationofthisproblemcouldbehelpfulforunderstandingtherelationbetweenunitarityandrenormalizabilityofthespin-2 eldinteractionsinMinkowskispace.HoweverinthepresentpaperwedonottackletheissueofunitarityandrestrictourselvestotheEuclideanspacewherethegaugegroupiscompactandthetheoryisconsistent.RecentlyinRef.[10]aninterestingproposaltoincludefermionsinthemodelin[8]hasbeengiven.WhereasinRef.[10]onlythesub-groupSO(4)isconsidered,inthisarticleweshallseethatitisstraightforwardtoextendthesefermioncouplingstothelargergroupU(4)whichwas rstconsideredinRef.[8].Moreoverinthisworkwegeneralizethemodelin[8]toitsN=1andN=2Euclideansupersymmetricextensions.Thepaperisorganizedasfollows.Insection[2]webrie yrecallthemodelpro-posedinRef.[8]andanalyzethefreeparticlespectrum.Insection[3]wequantizethismodelinacovariantgaugeandgivetheexpressionfortheghostlagrangian.Insection[4],byfollowingtheapproachofRef.[10],wegeneralizethemodelin[8]byincludingfermionsintheadjointrepresentationofthegaugegroupandanalyzeitssupersymmetricextensions.Insection[5]weextendthemodelin[8]soastoin-cludethestandardinternalgaugesymmetries.Theexpressionfortheuni edgaugelagrangian,whichincludestheinternalSU(N)gaugegroup,isgiven,togetherwiththecorrespondingin nitesimalgaugetransformations,inappendix.Insection[6]westudythecouplingoftheextendedgauge eldswithbosonicmatter eldsandmakesomeremarksonpossiblecouplingswithordinarymatterandgauge elds.Insection

[7],thespontaneoussymmetrybreakingoftheextendedgaugesymmetryisanalyzed.Finallythelastsectionisdevotedtoourconclusions.

5

The extended Yang-Mills gauge theory in Euclidean space is a renormalizable (by power counting) gauge theory describing a local interacting theory of scalar, vector, and tensor gauge fields (with maximum spin 2). In this article we study the quantum aspect

2PureGaugeAction

InthemodelproposedinRef.[8]theusual

abelian

gaugetransformationshavebeenextendedtonon-abelianoneswhichmix eldsofdi erentintegerspin.Asacon-sequencetheelementsofthegaugegrouptransformnon-triviallyundercoordinaterotations.Inadditionthelagrangian,whichisinvariantundertheseextendedgaugetransformations,describesalocalinteractinggaugetheoryofhigherspin elds.Thismodelhastheattractivefeaturethatitisrenormalizablebypowercounting.(Duetothegaugeinvariance,webelievethatthemodelisalsofullyrenormalizable,howeverwedonottacklethisissueinthepresentarticle.)Moreoveranotherinterestingchar-acteristicisthatthemaximumvalueofthespincontents(S)ofthegaugemultipletis xedbythespace-timedimensiond;inparticularford=4wehaveS=2.

Beforepresentingouranalysiswebrie yrecallthemodelproposedinRef.[8].One ij(x)inEuclideanfourdimensionalspace, rstconsidersaspinorial-vectorial eldAµwhereiandjareindiceswhichbelongtotheDiracspinorialspace(i,j=1,...,4).Inparticularthis eldisde nedtotransformundertheEuclideancoordinaterotation3

νx→x′µ=Λµνxν,withΛαµΛαν=δµ(1)

(2)asfollows

Notethatin(2)S(Λ)istheusualspinorialrepresentationoftherotationgroupO(4)whichisgivenby

S(Λ)=exp( (x)µ→A ′(x′)=S(Λ)A ν(x)S 1(Λ)Λν.Aµµ

√

WeusetheconventiontosumupthesameindicesandtheEuclideanmetricisgivenbyµδµν=δν=diag(1,1,1,1).4Inthecaseofin nitesimaltransformationswehaveΛµν=δµν+ωµν+O(ω2),withωµν= ωνµTheexactrelationshipbetweenΛandωcanbefound,forexample,inRef.[20].3

6

The extended Yang-Mills gauge theory in Euclidean space is a renormalizable (by power counting) gauge theory describing a local interacting theory of scalar, vector, and tensor gauge fields (with maximum spin 2). In this article we study the quantum aspect

under

O(4)rotations(thereadercaneasilycheckthispropertybymeansofEqs.(2)¯µ,Tµν(andanalogouslyand(3)).Thisimpliesthatthe eldsA

4

4

3ASδµνφ+Sµν+Vµν+Vµν,¯+δµνφVµν+1A(δµαBβ δµβBα)

Vµν)areantisymmetrictensorsinthe(1,0)and(0,1)representations,

respectively.(Notethatwiththenotation(x,y)werefertotheusualSU(2)×SU(2)complexspinorialrepresentationoftherotationgroupO(4)[21].)Thetensor eldsSADµαβandDµαβbelongtothe(3/2,1)and(1,3/2)representationsrespectively;they

(S,A)(S,A)areantisymmetricintheα,βindicesandaretraceless.MoreoverVαβandDµαβsatisfythefollowingself-dualityconditions[10]

(S,A)VµνS,A=±1

αβγδDµγδ,(S,A)

2(6)

wherethesigns(+)and( )referto(S)and(A)respectively.5

(S,A)(S,A)Wenowseethat,byusingtheself-dualityconditions(6),theDµαβandVαβtensor

eldshave8and3degreesoffreedom,respectively.Notethat,ifthe eldsare¯µνandD(S,A).Onmassive[22],aspin-2 eldiscontainedineachofthetensorsSµν,Sµαβthecontrary,ifthespin-2 eldsaremassless,accordingtotheWeinbergtheorem[22]onlytheleft-handedandright-handedpolarizationsareconsistentlydescribedbytheSDµαβVµν elds)andA+Dµαβ,whichappearinEq.(5),arereplacedbyonlyoneirreducible eldrepresentation.(S,A)

7

The extended Yang-Mills gauge theory in Euclidean space is a renormalizable (by power counting) gauge theory describing a local interacting theory of scalar, vector, and tensor gauge fields (with maximum spin 2). In this article we study the quantum aspect

SADµαβandDµαβ elds,respectively.Therefore,ifparityisconserved,onemayconclude

SAthatamasslessspin-2canbedescribedbythereducible eldDµαβ=Dµαβ+Dµαβ.

µtoaReturningtothemodelin[8],

the

succeeding

step

is

to

promotethe eldA µtransformsunderalocalgaugetransformationgaugeconnectionbyrequiringthatA

U(x)asfollows

G(x)=U(x)A µ(x)U 1(x)+1Aµ

√

16 µν(x)= µA ν νA µ+ g[A µ,A ν],F µνF µν],Tr[F(11)

whereintheaboveexpressionthecommutatorandthetracearetakenontheCli ordalgebra.Byusingthecomponent eldsgivenin(4),thelagrangianin(11)takesthefollowingform

LE=L0+gL1+g2L2,(12)

8

The extended Yang-Mills gauge theory in Euclidean space is a renormalizable (by power counting) gauge theory describing a local interacting theory of scalar, vector, and tensor gauge fields (with maximum spin 2). In this article we study the quantum aspect

where

L0=1

¯αβCγβδ( γT¯α αT¯γδ)2[TαβCγβδ( γTαδ αTγδ)+Tδ

¯µ[Tαβ( αT¯µβ µT¯αβ)+T¯αβ( µTαβ αTµβ)]+A√√+[

¯αγCµβA¯¯¯µαβγ TµαA¯µT¯βγCβαγ].2[2TαβTγµ TαβTµγAC(15)

Thelagrangianin(11)isinvariantunderthefollowinglocalin nitesimaltransforma-tions

1δAµ=

g

1¯µν), µ¯ +2(¯ νTµν νT

√¯µ+2 Cµνα ¯ νAαδTµν= √

¯µα αν),2T

δCµαν= g1 µ¯ ν+2( ¯Tµν+2( αTµν νTµα)+√

(16)2( βνCµαβ βαCµνβ).

9

The extended Yang-Mills gauge theory in Euclidean space is a renormalizable (by power counting) gauge theory describing a local interacting theory of scalar, vector, and tensor gauge fields (with maximum spin 2). In this article we study the quantum aspect

ClearlytheAµisafree eldsinceitcorrespondstotheU(1)gaugeconnectionofU(4)andtheinteractingtheoryisdescribedbytheSU(4)gaugegroup.Notethatthesmallestsub-algebraofSU(4)isgivenbytheσµνgeneratorswhichbelongtothealgebraofSO(4).Thesmallestgaugelagrangian,whichisinvariantundertheSO(4)gaugetransformations,isgivenbythetermsin(13-15)containingonlytheCµαβ eld.ThelatterwasconsideredinRef.[10].

Oneinterestingaspectofthismodelisthatthelagrangian(12)canbewrittenintermsoftheO(4)irreduciblerepresentationsbyinsertingthe eldsdecomposition

(5)intheexpressions(13)-(15)(see[8]).Thenthecorrespondingin nitesimalgaugetransformationsfortheirreducible eldsareobtainedbymeansofthestandardde-compositionmethod,aspreviouslyshowninEq.(16).Forexample,thecorrespondingin nitesimalgaugetransformationsforδφ,δSµν,δVµν(S,A)aregivenby

δφ=δT1

µµ,δSµν=δµνδTαα,

δV14

µν(S,A)=2δTαβ αβµν ,(17)

withtheobviousgeneralizationfortheother elds.TheexpressionforthelagrangiancontainingonlytheCµαβ eld,intermsoftheirreduciblerepresentation,canbefoundinRef.[10].

Wehereanalyzethephysicaldegreesoffreedomandthefreeparticlespectrumofthemodelintroducedin[8].InordertodosowerestrictouranalysisbyonlyconsideringthefreelagrangianL0whichisinvariantundertheabeliansector6ofthegaugetransformations.We rstconsider7thefreelagrangianL0(T)inEq.(13)whichcontainsonlythetensor eldTµν.FromEq.(16)weseethatitispossibletomakeagaugetransformationwhichensures

µSµν=0andφ=0.(18)

Thisresult

the eldS isjusti edinthefollowingmanner.Thefreegaugetransformationsfor

µν=Sµν+1/4δµνφaregivenbyδS µν= µ ν+ µ ν.Then,bymeansofthesegaugetransformations,the rstconstraintin(18)canbeimposed.Thisconstraint,whentheon-shellmasslessequationsforSµνandφareused,isinvariantunderanewgaugetransformation.Thisnewgaugedegreeoffreedomallowsusto

The extended Yang-Mills gauge theory in Euclidean space is a renormalizable (by power counting) gauge theory describing a local interacting theory of scalar, vector, and tensor gauge fields (with maximum spin 2). In this article we study the quantum aspect

eliminatethecomponentoftheon-shellmasslessscalar eldφ.(Notethatthegauge µνandthegauge xingin(18)arethesameonesen-transformationsforthe eldS

counteredinthespin-2 eldoftheFierz-Paulilagrangian.)Byadoptingthegauge xing(18)allthegaugedegreesoffreedomareusedandnonewconstraintonTµνcanbeimposed.Inparticularnoconstraintmaybeintroducedfortheantisymmetric

ASAtensor eldTµν=Vµν+Vµν.

(S,A)Nowifthe eldsVµν,inthe(1,0)and(0,1)representation elds,weremassivethen

theywoulddescribetwospin-1 elds.Howeversincetheyaremassless elds,accord-ingtotheWeinbergtheorem[22],onlyonemasslessspin-1 eldcanbeassociatedtothe(1,0)⊕(0,1)representation,wheretheright-andleft-handedpolarizationarecontainedinthe(1,0)and(0,1)representation,respectively.Herethetworesidualdegreesoffreedomarecontainedinthelongitudinalcomponents.Thesedegreesareassociatedtothetransversepolarizationsofthevectorial eldLµ≡ νTµν.Indeedthesepolarizationscannotbegaugedoutinthismodel.TheonlycomponentsofthereducibletensorTµνwhichcanbegaugedoutcorrespondtothevector µTµν.Asaconsequencethephysicalpolarizationsofthe eldTµν,inthemomentum

T(S)(A)spacek,arethe“transverse”ones µα(k)and T(k)(where(S)and(A)refertoµαthesymmetricandantisymmetrictensorinµandα,respectively)and“longitudinal”

ones Lµα(k)whichsatisfythefollowingconditions

(A)kµ T(k)=0,µαT(S)kµ µα(k)=0,kµ Lµα(k)=0,(19)

αLnotethatintheaboverelations Lµαhasnotade nitesymmetryinµ,αandk µα(k)=

T(S)(A)0.FromEq.(19)wehave µα(k)and T(k)eachcontainonlytwoindependentµαLpolarizations.Onthecontraryin µαtherearefourindependentpolarizations,each

ofwhichcorrespondtoonemasslessspin-1andtwomasslessspin-0.Asaresulttheon-shellTµν elddescribesthefollowingspectrum:onemasslessspin-2,twomasslessspin-1,andtwomasslessspin-0.Thereforeintotalwecount3×2+2×1=8degreesoffreedomfortheon-shellTµν eld;thisresultisinagreementwiththenaivecountingbasedonthegaugedegreesoffreedom.

WenextanalyzetheparticlespectrumdescribedbythefreelagrangianL0(C)inEq.(13)whichcontainsonlythe eldCµαβ.Wecanusethegaugedegreesoffreedom

Cµαβ→Cµαβ+ µ αβ

inordertosetthefollowingtransversalityconstraintsonthe eldsDµαβ

µDµαβ=0.

11(S,A)(S,A)(20)(21)

The extended Yang-Mills gauge theory in Euclidean space is a renormalizable (by power counting) gauge theory describing a local interacting theory of scalar, vector, and tensor gauge fields (with maximum spin 2). In this article we study the quantum aspect

TheresultsinRef.[22]enableustoseethatinthemasslesscasetheDµαβwhichsat-is esEq.(21)describestwodegreesoffreedom.Thesedegreesoffreedomcorrespondtoaright-handedspin-2andaright-handedspin-1 eld.Analogouslywecanseethat(A)theDµαβdescribesthecorrespondingleft-handedones.Therefore,sinceinthismodel

(S)(A)parityisconserved,thephysicalpolarizationsofthereducible eldDµαβ+Dµαβwill

describeamasslessspin-2andspin-1 eld.

Asaresultofthegauge- xingin(21)therearenogaugedegreesoffreedomavailablewhichwouldenableustoeliminateothercomponentsinthevector eldsBµ¯µ.Thusthe eldBµcontains4independentpolarizations.ThetwotransverseandB

one(respecttothethree-momentum)correspondtoamasslessspin-onepolarizationswhereasthelongitudinalonesareassociatedwithmasslessspin-0 elds.Thespectrum¯µisobtainedanalogously.Asaresulttheon-shellCµαβ elddescribestheforB

followingmasslessspectrum:onespin-2,threespin-1,andfourspin-0 elds;sointotalwecountrespectively4×2+4×1=12degreesoffreedom,inagreementwiththenaivecountingbasedonthegaugedegreesoffreedom.

Itisclearthattheon-shellparticlecontentsofthismodelisgaugeinvariantandonecanreachthesameconclusionsonthespectrumbyusingdi erentchoicesforthegauge xing.Finallywenotethatthespin-0particles(orlongitudinalphotons)whichappearinthespectrumarestrictlyconnectedtothefactthatsomelongitudinalcomponentsofthetensororvector eldscannotbegaugedout.(S)

3CovariantQuantization

Whenthemodelin[8]isquantizedintheEuclideanspacethenegativenormstatesareabsentsincethespace-timemetricistheδµνandthegaugegroupiscompact.Moreover,duetothecompactnessofthegaugegroup,thetheorycanbequantizedbymeansofthestandardpathintegralmethod.Clearlythefactthatthetheoryiswellde nedinEuclideanspaceitisnotenoughtoguaranteeitsanalyticalcontinuationtotheMinkowskione.EveniftheOsterwalder-Schrader(OS)axiomsaresatis ed,andinparticularthepropertyofre ectionpositivity[15]isveri ed,onecannotuseherethereconstructiontheorems8ofRef.[15].Indeed,intheproofofthesetheorems,thegaugegroupisnotchangedbytheanalyticalcontinuationtotheMinkowskispace.Onthecontraryinthemodelin[8],inordertomaintaintheLorentzcovariance,wemustrotatethegaugegroupU(4)tothenon-compactoneU(2,2)whenthe

The extended Yang-Mills gauge theory in Euclidean space is a renormalizable (by power counting) gauge theory describing a local interacting theory of scalar, vector, and tensor gauge fields (with maximum spin 2). In this article we study the quantum aspect

analyticalcontinuationtoMinkowskispaceisperformed.However,aswediscussedintheintroduction,thepresenceofextranegative–normstates(inducedbythenon–compactgroups)isnotalwaysanobstacleforbuildingaconsistenttheory[16]–[19].Inparticular,forthismodel,itshouldbeinterestingtoseeifaLorentzinvariantHilbertsubspace,wherethetheoryisunitaryandtheunphysicalstatesdecouplefromthephysicalamplitudes,exists.HoweverinpresentpaperwedonottackletheissueofunitarityinMinkowskispace.

Nowweanalyzethecovariantquantizationofthismodelinthemostcommonclassofcovariantgauges.InEuclideanspacethepathintegralrepresentationofthegeneratingfunctionaloftheGreenfunctionsW[J]canbeformallywrittenas

W[J]= µDηDA Dη exp µA)dxLE+LGF+ LGH Tr(J4 µ ,(22)whereLEisthefulllagrangiangiveninEq.(12)andLGFandLGHcorrespondtothegauge- xingandtheghostlagrangian,respectively.Inthelasttermthetraceis µ,whichcanbedecomposedasA µinEq.(4),istakenontheCli ordalgebraandJ µ.thesourceforthegauge eldA

Inthepresentstudyweconsiderthegeneralclassofcovariantgaugeswhosegauge– xinglagrangianisgivenby

LGF=1

TµνandCµαβaregivenby

P(A¯ν)=¯µA

P(TµαTνβ)=

P(Tνβ)1k21 , k2

1

2 ,=P(TµαTνβ),P(CµαγCνβδ)=δµν (1 ξ)kµkν

The extended Yang-Mills gauge theory in Euclidean space is a renormalizable (by power counting) gauge theory describing a local interacting theory of scalar, vector, and tensor gauge fields (with maximum spin 2). In this article we study the quantum aspect

representationsonlyinthet’Hooft-Feynmangaugeξ=1.Howeverforpracticalcalculations,suchasthescatteringamplitudes,itismoreconvenienttoworkwiththepropagatorsinthebasisofthereducible eldsTµν,

Tµν η √µ ¯µνTµν η¯ ηνAµ η¯ √ µν ¯µ η¯νT µη¯+η¯νA µην+

+2 ηανCµαβ µηβνµν√Tµα ηµνα µναη¯νC µη¯α

Theghostmultiplet,whichappearsinEqs.(26),iscomposedbythefollowing elds:acomplexscalarη¯,twocomplexvectorsηµ,η¯µ,andacomplexantisymmetrictensorηαβ,allofwhichareGrassmanvariables.Notethatvectorialghost eldsalwaysappearwhengaugespin-2 eldsarepresent,aclassicalexampleisthequantumgravity.Itisworthnotingthat,althoughthephysicalspectrumshouldbedescribedintermsof eldswhichbelongtotheO(4)irreduciblerepresentations,therenormal-izationpropertiesofthelagrangian(12)canbedirectlyanalyzedbymeansofthe¯T,O(4)reducible eldbasis(A, T µη¯αµν .(26)

The extended Yang-Mills gauge theory in Euclidean space is a renormalizable (by power counting) gauge theory describing a local interacting theory of scalar, vector, and tensor gauge fields (with maximum spin 2). In this article we study the quantum aspect

in4dimensionsandtheabovementionedproblemofγ5doesnotexist.Moreover,eventhoughthefermionsmatter eldsarecoupledtothegaugeconnection,thelatticeregularizationdoesnotspoiltheSU(4)gaugesymmetry.Indeed,aswewillshowlateron,thegeneratorcorrespondingtothegaugetransformationcontainingtheγ5matrixisnotconnectedtothe“standard”chiraltransformationsandsotheWilsonterm,whichisnecessarytosolvethedoublingproblem,doesrespectthegaugesymmetry.Clearly,whenfermionsareaddedtothetheory,theWilsontermbreaksthe(global)“standard”chiralsymmetry.

4SupersymmetricExtension

µtofermionmatterInthissectionweconsiderthecouplingsofthegauge eldA

eldswhichareintheadjointrepresentationofthegaugegroup.InRef.[10]thesecouplingshavebeenproposed,andthesmallestgaugesub-groupSO(4)studied.WegeneralizethisapproachbyconsideringthelargergroupSU(4).MoreoverwederivetheN=1andN=2supersymmetricextensionsofthepuregaugeaction(12).

Inordertointroducethesefermioncouplingswefollowthemethoddevelopedin ijwheretheRef.[10].Wede neinEuclideanspacethefollowingfermionmultipletΨkupindicesi,jandthedownindexkaretheusualDiracindices.Beforegivingthe ijsomede nitionsareinorder.InadditiontocoordinatetransformationrulesofΨkthespinorialrepresentationoftheO(4)rotationgroup,namelyS(Λ),weintroduce¯(Λ).anotherindependentspinorialrepresentationofthisgroupthatwewillcallS¯(Λ),intermsofanewCli ordalgebrabasisThematrixS

¯i=1¯5,γ¯µ,γ¯5, γ¯µγ¯5,σ¯µν,ΓIΓ¯,γ(27)

(where1IΓ¯istheunitymatrix)isassumedtocommutewithS(Λ)andtohavethe¯ibasisissamerepresentationasS(Λ)(seeEq.(3)).NotethateachelementoftheΓassumedtocommutewithanyotherelementoftheCli ordΓibasis. ijinNowwecande nethefollowingcoordinatetransformationpropertiesofΨkEuclideanspace,theseare

x→x′µ=Λµνxν,

ij(x)→Ψ ′ij(x′)=Ψkk

orinamorecompactnotation bj ¯(Λ)Skm ab(x)S 1(Λ)(S(Λ))iaΨm

(28) x)→Ψ ′(x′)=S¯(Λ)S(Λ)Ψ( x)S 1(Λ)Ψ(

15 (29)

The extended Yang-Mills gauge theory in Euclidean space is a renormalizable (by power counting) gauge theory describing a local interacting theory of scalar, vector, and tensor gauge fields (with maximum spin 2). In this article we study the quantum aspect

¯ andanalogouslyfortheadjoint eldΨ

′¯¯¯′ 1 ¯ 1(Λ),Ψ(x)→Ψ(x)=S(Λ)Ψ(x)S(Λ)S(30)

¯(Λ)matricesactsontheupanddownDiracwherethemultiplicationofS(Λ)andS

indices,respectively.NotethattheS(Λ)matrixinEq.(29)isthesame

matrixap-pearinginthecoordinatetransformationruleofthegaugepotentialinEq.(2).We onthesamebasisΓioftheA µ eldasfollowsnowdecomposethe eldΨ

µµµν(σµν)jkjkjk jk. (γγ)+λΨ=λ(γ)+λ(γ)+λi5µ5µi5iii2jk(31)

Inthesequelthenotationforthespinorialdownindex“i”inthecomponent elds,appearingin(31),willbesuppressed.Asaconsequenceofthecoordinatetransfor-mations(29),thecomponent eldsλ,λµ(oranalogouslyλ5µ),andλµνtransforminthefollowingmanner

¯(Λ)λ,λ→λ′=S

¯(Λ)λα,λµ→λ′µ=ΛµαS

1λµν→λ′µν=

√

√

2µνσ¯ijξj+1

Thespinorialindices“i,j”appearingin(33)havebeentemporaryreintroducedtoavoidconfu-sionswiththenotation,andthesameindicesareintendedtobesummedup.9

16

The extended Yang-Mills gauge theory in Euclidean space is a renormalizable (by power counting) gauge theory describing a local interacting theory of scalar, vector, and tensor gauge fields (with maximum spin 2). In this article we study the quantum aspect

The eldsψ,ψ5,ξ,(resp.ψµ,ψµ5,ξµ,ψµν)describespin-1/2(respectivelyspin-3/2) elds.BymeansofEq.(32)itisstraightforwardtoprovethatthedecompositions

(33)areO(4)irreducible.

Inordertocouplethe eldΨ tothegaugeconnectionA µweneedtorequirethat,underthegaugetransformationsU(x),the eldΨ, anditsadjointΨ,¯ transformasfollows

Ψ ij

k(x)→Ψ Gij

k(x)=(U(x))iaΨ abk(x)U 1(x)

Ψ¯ ij(x)→Ψ¯ Gij(x)=(U(x))iaΨ¯ ab(x) bjkkk)(35)

where,asusual, U 1(x bj

derivativeD thesumoverrepeatedindicesisunderstood.Asaresultthecovariant

µactingonΨ isgivenby

D µΨ = µΨ g Ψ ,A µ ,(36)

wherethecommutatoristakenontheΓiCli ordalgebrabasisandgisthesamecouplingappearinginthelagrangian(12).

WenowanalyzetheN=1andN=2supersymmetricextensionsoftheaction(12),by rstrecallingtheknowntechnicalsolutionsforconstructingsupersymmetricthe-oriesinEuclideanspace.

Inthefour–spinorformalism,N=1supersymmetry(SUSY)requiresthe

Majoranafermions.Howeverthe¯existenceof

t(whereCis

thechargeconjugationmatrix,ψ¯Majoranarealityconditionψ=Cψ

istheadjointofψ,andthesu xtstandsfortrans-

pose)isinconsistentinEuclideanspace[21],[25].10Thisisapparentlyatechnical,butnotfundamentaldi cultytoimplementsupersymmetryinEuclideanspace.11IntheN=1supersymmetricYM(SYM)theoryinEuclideanspace,bymeansofaspecialde nitionforEuclideanMajoranaspinors[26],[27],supersymmetrycanberestoredatthepriceofgivingupthehermiticityoftheaction.Then,byconstruction,

The extended Yang-Mills gauge theory in Euclidean space is a renormalizable (by power counting) gauge theory describing a local interacting theory of scalar, vector, and tensor gauge fields (with maximum spin 2). In this article we study the quantum aspect

theexpectationvalues(Schwingerfunctions),generatedbytheEuclideansupersym-metricmodel,aretheanalyticalcontinuationsofthecorrespondingGreenfunctionsinMinkowskispace[26].Indeed,asobservedinRef.[26],inEuclideanspacetherelevantnotionisnothermiticity,butratherOsterwalder-Schraderre ectionposi-tivity[15]whichguaranteestheabovementionedanalyticalcontinuation.MoreovertheEuclideanN=1SUSYtransformationsarenothermitean.Howeverthisisnotaproblem,sincetheSUSYtransformationsarejustaformaldeviceinordertoobtainSUSYWardidentities[26].

Inanotherapproach,proposedbyZumino[28],thehermiticityoftheactionisretainedatthepriceofgivinguptheexplicitconnectionbetweenrelativisticandEuclidean eldtheory.InparticularintheZuminomodel[28]thenumberoffermionicdegreesoffreedomisdoubled(withrespecttoN=1SYM)inordertogetanhermiteanaction,butadditionalbosonic(scalar)matter eldsshouldbeaddedtotheN=1SYMactioninordertorestorethebalancebetweenfermionicandbosonicdegreesoffreedom.AsaconsequencetheZuminomodelisanN=2SYMtheoryandthereforetheanalyticalcontinuationwiththerelativisticN=1SYMtheoryislost.

WestartouranalysiswiththeN=1SUSYextensionofthelagrangian(12).Paral-lelingthetechniquedeveloped

addition in rstreferenceof[26],the rststepistointroduce,inΞ¯tothecomplex eldΨinEq.(31),acompletelyindependent(complex) eld whichtransformsastheadjointofΨ inEq.(30).TheexpressionofΞ¯ incomponentsisgivenby

jk

Ξ¯ jk

i=¯λi(γ5)jk+λ¯µi(γµ)jk+¯λµ5i (γ5γµ)jk+¯λµν(σµν)

i2.(37)

Thenextstepconsistsinimposing

component eldsofΞ¯thefollowingMajorana–likeconditionsonthe andΨ, namely

λ≡C¯λt,λµ≡Cλ¯tµ,λ5µ≡Cλ¯t5µ,λµν≡Cλ¯tµν(38)

wheretstandsfortranspose,C(thechargeconjugationmatrix)isde nedasC C=1andC 1= C,andthestandardspinorial

fermion eldsλaor¯multiplicationbetweenCandthe

λaisunderstood.

consistentlysatis edsince¯Notethattheaboverelations(38)canbenow

λa

forthe elds¯λ=λ aγ0.Inotherwords,Eqs.(38)arejustde nitions

a[26].Then,inordertoimplementtheanalyticalcontinuationto

Minkowskispace,oneshouldrequirethatλaareEuclideanMajoranaspinors,whoseformalde nitioncanbefoundinthe rstreferenceof[26].12

The extended Yang-Mills gauge theory in Euclidean space is a renormalizable (by power counting) gauge theory describing a local interacting theory of scalar, vector, and tensor gauge fields (with maximum spin 2). In this article we study the quantum aspect

FinallytheO(4)–gauge–invariantlagrangianLFforthefermionsectorisgivenby

LF=1

LIF2 µ¯αµα¯¯=2Aλαγµλ5 2Tλγµλ5α+ √µα¯λγµλα T

2Cµαβ ¯ µλ+λ¯αγ µλ+λ¯5αγλγµµαµ µλα5¯αβγ µλ+λ√µαβ ,

Inderiving

Eq.(40),thefollowingMajoranarelationsfor

anticommuting eldsλahavebeenused

¯aγµλb= λ¯bγµλa,λ¯aγµγ5λb=λ¯bγµγ5λa,λ¯aλb=λ¯bλaλ(41)

¯a≡λtC)areun-whereinEqs.(40)and(41),thede nitions(38)(orequivalentlyλaderstood.Then,duetothefactthatthelagrangianinEq.(40)doesnotdependon¯ theadjoint eldsλ a(oranalogouslyλa),thehermiticityislost[26].

Forcompleteness,wereportthein nitesimalgaugetransformationsforthecom-ponentfermion elds

δGλ=2(λα ¯α λα5 α), √δGλα=2 λ ¯α+ √δGλ5α=2λ α+√δGλαβ= 2λβα β ¯αγµλβ+λ¯5αγµλ5β+2λ¯αδγµλδλβ ,(40)2λβα¯ β, ,

NotethatinEq.(40)wehaveeliminatedfromthenotationthe“bar”overtheγ¯µmatricessinceinthefollowingweworkonlywiththecomponent eldsalongtheΓibasis.132 Dµ

19

The extended Yang-Mills gauge theory in Euclidean space is a renormalizable (by power counting) gauge theory describing a local interacting theory of scalar, vector, and tensor gauge fields (with maximum spin 2). In this article we study the quantum aspect

Notethatin(43)theLEandL

FaretheonesgiveninEq.(12)andEq.(39),re-µspectively,andtheauxiliary(bosonic) eldsD,andDµν(withµD,Dµν,being,respectively,ascalar,vector,andantisymmetrictensorunderO(4))shouldbeaddedtothelagrangianinordertoclosetheo -shellSUSYalgebra.WerecallthatthelagrangianinEq.(43)isnothermitean,butitsatis estheOsterwalder–Schraderre ectionpositivity[26].

Finally,the(o -shell)N=1SUSYtransformationswhichleaveinvariantthela-grangian(43)(uptoatotalderivative)aregivenby

δSAµ=ω¯γµλ,

δS

δSTµα=ω¯γµλα,σµνFµνα+γ5Dαω,

δSλ5α=

δS2 ¯µν+γ5σµνFDαω,δSλαβ= 2 D=ω¯γ5γµDµλ,δSDα=ω¯γ5γµDµλα,

¯µνα= µFFµνα= µTνα 2g √αTν (µ ν),Tνα (µ ν),

Tνβ 2CµβδCναδ2gTµαTνβ+

Thecovariantderivativesarede nedasDµλa= µλa+g µλa,wheretheexpressionsfor µλaare

µλ=2λα

2 (µ ν).(45)Tµα+√

2λβ5Cµβα¯µ λαA

δ µλ5α=2λ√ µλαβ= √ 2λβαTµ

βTµβ

Tµβ+2λαCµδβ (α β).

20 , ,(46)

The extended Yang-Mills gauge theory in Euclidean space is a renormalizable (by power counting) gauge theory describing a local interacting theory of scalar, vector, and tensor gauge fields (with maximum spin 2). In this article we study the quantum aspect

Thelagrangian(40)isnotwrittenintermsoftheO(4)irreduciblecomponents,butthiscanbeeasilyobtainedbytakingintoaccountthedecompositions(5),(33),andtherelations(41).Weobservethatnon–trivialcouplingsbetweentheirreduciblerepresentationofthegaugeandfermion eldscanariseintheinteractinglagrangianLIF.Aftersomestraightforwardalgebraicmanipulations,intermsofthefermionirreduciblerepresentations(33),thefreeLagrangianL0Fis,uptototalderivatives,

L01

F=2ψ¯αγµ µψα+12ψ ¯µψ1µ+

2ψ¯5γµ µψ5 √2ψ¯µνγα αψµν 2ξ¯ν( νξ+ µψµν),(47)

where,inderivingtheexpression(47),therelations(41)wereused,andthefollowingde nitionshold

λ≡C¯λt,ψµ≡Cψ¯tµ,ψ5µ≡Cψ¯t5µ,ψµν≡Cψ¯tµν,ξµ≡Cξ¯tµ.(48)ThecorrespondinggaugeorSUSYtransformationsfortheO(4)irreduciblerepresen-tations(seether.h.s.ofEq.(33)14)cannowbesimplyobtainedbyprojectingthereducibletransformationsδλµ,δλµν,...etc.,de nedinEqs.(42)or(44),intothefollowingones

δψ=1

2γµ(δλµ),δψ1

5=2γµ δλ5µ ,

δψµ=δλµ 1

2γµ(δψ),δψµ1

5=δλµ5 2γµ(δψ5),

δξ= σµν(δξ)

2 ,

δψµν=δλµν µ

4[γ(δξν) γν(δξµ)].(49)

WenowanalyzetheN=2SUSYextensionofthemodelin[8]byrequiring,asintheZuminomodel[28],thehermiticityoftheactioninEuclideanspace.In

anhermiteanaction,onehasto ordertoobtain

anditsadjoint

one,namelyΨ.¯introduceacomplexfermionic eldΨ

TheΨ andΨ¯ have,respectively,theO(4)coordinatetransformation

propertiesas andΨ¯de nedin(29)and(30).ForourconvenienceweformallydecomposeΨ

asinEq.(37)andEq.(31),respectively.Themaindi erencewiththeprevious

本文来源：https://www.bwwdw.com/article/yvm4.html