Scale Invariance

Scale Invariance Wilson 1969 wet ,fo) PHYSICALREVIEW VoLuMe179,NUMBERS 25MARCH1969. Non-Lagrangian Models ofCurrent Algebra* e Kexnern G.Wesoxt ‘ Laboratory ofNuclear Studies, Cornell University, Ithaca, NewYork14850 a (Received 25November 1968) ° Analternative isproposed tospecif Lagrangian models ofcurent algebra. Inthisalternative thereare aoenn fyntSper poiotiinepntLin,76)vo ote,itaszanedthatsaleinvarianceisabrokensymmetryoftonginteractions, x8Propo aTest, ee ntti ofequaltinecommuticors issuned:Operatorproductsathaetdtances haveexpansions involving localfedsmultiplying singular functions. Ttisassumed thatthe © . dominanteldsace{heSUGSEO)currents‘andtheSU(3)X5U(3)multiplet containing thepionfeld. .ip aseuned thatthepionRedwaleslkeaGeldofdimension4,where4isunspecifiedwithiatherange ‘ ‘ Meee etceeaSeacquence ofrenormalization. Thesehypotheses implyseveralqualitative 's a predictions: ThesecondWeinberg sumruledoesnotholdforthedifference oftheK*‘andaxial-X* propa-a i ators, evenforexact SU(2)XSU (2);electromagnetic corrections require onesubtraction proportional to e a Brereton OeGeld;+Svandra»2yaeallowedbycurentalgebra.Octetdominanceofnonleptonic.’. speakpresscanbeunderstood,andanewfomofsuperconvergence elationisdeducedasaconsequence‘ i 1generalization oftheBjorken litisproposed s 4 IL.INTRODUCTION “operator-product expansions” forproductsoftwo(or(5°OPPHEREaresnumberofproblemsinstrongintrina)lotanhsanelocalfields(neat\the samepointForexample, ”ie actionswhichinvolvetheshort-distance behaviorO"eCa"COnStHict,Expansions for‘productssuchas:rsoftheSU(3)XSU() currentsbutwhichcannotbe£050)orHG)H0)i-10) whenyandsarenearx.5 %solvedbyGell-Mann’s currentalgebra!alone.These penyn'roryicontainfunctionswhicharesingular M "problems includetheconvergence ordivergence ofwheny=oryisonthelightconethroughx.‘These’ .Wbecr sumnules®divergences inradiative corcec- Srounsions giveamoredetailed pictureoffeshoitionstostronginteractions, thenatureoftheBjorken knoncebehaviorofproductsthanonegetsifoneonlyie oneace MavebeenproposedbpKiowaouutimecommutator, Theseecpansionsogiyhandletheseproblems, suchasthealgebraofficlds,*natedindetailedstudiesofrenormalization inpertur- & éthequarkmodelorthe#modelThesemodelsgivebationtheory.’Theimportance ofscaleinvariance for=ifconflictinganswerstosomeoftheproblemsmentioned. theanalysisofshort-distance behaviorisapparentit 52Onethereforemustconsiderwhatfurtheralternatives eaeaoeaae eninteractioniy to thesemodelsexist,andhencetogetanideaofthengie.\mensiot ated outbandoaneedliHrangeofanswerspossibletotheproblemslisted. socalinvarianceConeeoughtof2aoe of >‘Thispaperpresentsaframework inwhichonecanaeecartateiieee uoath*ofafeatures discussSomealternatives tospecificLagrangian models. SPeciltocertain strictlyTagrangina theories,However,The presentframework doesnotinvolveLagrangians: jaye: theThirringmodel”showsthatscaleEa‘Therearenocanonicalfieldsintheformalism, andthecanonical persistinatheoryretcarte, Fi ~ Hse ¢canonical commutators havebeendestroyedbyEsoeniatproductsatihsane.aia,feforexample8renormalization effects.(TheThiecing,modelinvolvesFe atdunmethodsofanaycngabort3cPn%Selinnsspace, oetnedinesin i.distancebehavior, twohypotheses areproposed. The‘ecmicoupling.) Whilescaleinvariance persists,-4ifirstisthatthesironginteractions becomescale-in- scalinglawsforticularfieldschangeasthecouplingaati ctdist Tt _‘Sonstant changes. Thiswillbeassumedtoholdfor}oea toe proposedbyKas-Stronginteractionsalso,sothatthescalinglawsfor ibrokensymmetry inthesamesenseaschiralSU(3)strongly interacting fieldswillbeassumed todifferXSU(3). Theotherhypothesis isthatthereexist(because ofrenormalization effects)fromfreefields.‘ ). ‘Thehypotheses ofthispaperleavemuchtobedeter-<~Supported inpartbytheOfficeofNavalResearch mined;nevertheless, whencombined inasimpleway*|Teautoe thankstheAicedBSianFoundationfocsupport, withcurrentalgebra,onecanmakeanumberofqualita- SaesoePhys.Rev.125,1067(1962);Physics1,63tivepredictions. Theapplications considered inthis S 78weiaberg, Phys,Rev, Letters 18,507 (1967);8.Glashow,—~——— H.Senter Sad&Weinberg, abid.19,139(1967); T.Das, "J,Valatin, Proc.Roy.Soe.(London) A222,93(1954); A222s-Visfathug and'5.Okubo,thal48,764,(1567). 228(1954);A325,535(1954);A226,254(1954);W.Zimmermann, TDBioko M8467260),'RaoveChncato10,971938),K,Neijima,Phys.Rev.IIT, 8 sebdespe Beebeatbn, Re.LesPECREHagOeUPHECGSBanaeo MBcm aneS nmeC ee ae1678 Ds 1968), andreferences cited Phys.(Kyoto) 1,377(1952); Umea hi’7,8(992 therein PEERJohnson,Nuoveaeene hae 4791499 1 >»* { 1500 KENNETH G.WILSON 179 | paperinclude thevalidity ofWeinberg sumrules,di-morefieldsclosetothesamepoint,(‘These statementsVergencesinradiativecorrectionstostronginteractions,willbedemonstratedinaseparatepaper.)Onecan ro)2-93andx*—+2ydecay,nonleptonic weakinter-computetheequal-time commutator of1andB,given +actions, andtheBjorken limit. theoperator-product expansion forA(«)B(y) (see‘Thispaperisonlyasummary oftheideasinvolved below);onecanalsocompute equal-time commutators“ jnthetwohypotheses andasurvey oftheirapplica- ofanytimederivatives of4andB.tions,Manydetails,someofconsiderable complexity, ‘Thereareseveralreasonsforusingoperator-product .3]havebeenomitted. ‘expansions inplaceofequal-time commutators.to des-“i,Operator-product expansions areintroduced inSec.cribetheshort-distance behavior ofafieldtheory. One ¥#"}TE.Scaleinvariance appliedtooperator-product ex-reasonjsthatequal-time commutators caninvolvein-atpansions isexplained inSec.III.Thescale-invariant finiteconstants, whereastheexpansion cocfficients_ ‘bpartofstronginteractions isdiscussed inSec.IV.The“C,(2—y) cannot.Forexample,theSchwinger termin3effectofmasstermsintheLagrangian (themassterms thecommutator oftwoquarkcurrents contains adi- 3.fretobetreated asinteraction Lagrangians inthesense vergent constant." Incontrast, thefunctions Ca(s~y) ofperturbation theory) isconsidered inSec.V.Aspeci- mustbedistributions inthefour-vector xysincethe «ficsetofmassterms isproposed inSec.VJ.Theappli- operators 4(x)andB(y)are,andadistribution cannot 5.cations areanalyzed inSec.VII.Section VIIIcontains contain infinite constants.” Arelated result isthe ~“efinalcomments. following: TheBjorken limit,formulated intermsof48) equal-time commutators, predicts thatFourier trans- 4 IL,OPERATOR-PRODUCT EXPANSIONS Forms ofamplitudes, such2s(a|TA (z)B(0)|8), willbe#X] . have asapower series inthetransform variable go‘Anequal-timecommutator oftwolocalfieldsA(=)whengoislarge,qbeingheldfixed?Withthe‘noregandB(s) isexpected tobeofthegeneralform generaloperator-product expansionitisfoundthat& [A0x),B(oy)]= LeDalya), (2.1)fractionalpowersotgearealsopossible;theyorca9 ‘”” the Thirring model inone spaceand one time dimen- 4%wheretheO,(2)areasetoflocalfieldsatx[including sign,1*Otheradvantagesoftheoperator-product ex-ff]/theunitoperator7whichforthepurposesofthispaper’pansionsare:Theyaremanifestly covariant, theyexistaSisalocalfieldQx(2)=J;itislocalbecauseitcommutes forTproductsofoperators, andonecangiveasimplewith theother local felds forspacelike separation]. discussion ofsymmetry-breaking effectstoallorders. ‘Thefunctions D,(&-=9)are&functionsorderivatives“Onecanrelatetheoperator-productexpansionof& fe) offunctions dereisthinary4@B0),10 theequal-timecommutator of4andB.BAThegeneralization popeed berithatanotinary SypporeCs—9)bebavesas[ (ey)?eco)700# productA(x)B(Q)hasanexpansion whenthefour-somepowerp.(IfAandBarenotLorentzscalars,there vector yisneara,oftheform would alsobeapolynomial inthecomponents ofx--y.) GFA(2)BQ)= EnCale7)Oal2) (2.2)Theiecomesinbecauseintermediate stateshaveonlyRypositive energies andonly intermediate states ofvery Herethefunctions C,(«—y) depend onafour-vector, largeenergies (largerthanthefixedinitial-andfinal- ynotathree-vector. Insteadofbeing5functions, theystateenergies) contribute tothelight-cone singularity. %involve powers ofx~y. They canhave singularities onThese high-energy states areexponentially damped if thelight cone oftheform [(z—y)*—ie(zo—e)F?,Ponegivesx»anegativeimaginarypart.So(*—3)* beinganyrealnumber(itneed’notbeaninteger).becomes(ze-yomfe)*—(a—y)% ‘Thecommutator 45 Theyalsoinvolve logarithms of(zy). Thecomplete [A(z),B()] forynearx(butno!equaltimes)has %expansion ingeneral involves aninGnite number ofanexpansion Jocal fields O,(x) buttoanyfinite order in2—y only a3 finitenumber offieldscontribute. Theexpansion is [AG),BO)]=LsEnle—y)On(2), (2.3) ‘validintheweaksense:Onemustsandwich theproduct 9)ks xAG)BG) between fixedfinalandinitialstates(alandWMe*®Zale~3) is *18).Theexpansion isthenvalidforysufficiently close Ey(t)=E[(—2*pie)?—(—s*-ies)?)] (2.4) fh tox‘‘Theseoperator-product expansionsexistforthefreendBisaconstant.Thesecondtermcomesfromtheascalarandspinorfieldtheories andforrenormalized ProductBOy)A(z): ItmakesZe(z)vanishforspacelike —¥interacting felds.to allorders inperturbation theory, #Onecanconvert Fa(s)intoasumofBfunctions in) =Tneverycasetheyarevalidforanyelementary or10Fanynonzerobutsmallze,Oneusesthedefinitionof2 compositelocalfields:A(z)andB(y)canbeelementary yey. catclationiimi ; scalarorspinorfieldsorlocalcurrentsortheStressenecalgaetinissimilartothatofJ.Schringer,Phys.Rey.energytensororanylocalWickproductinafree-field “TheWightman axiomsareassumed. *theory.Similarexpansions existforTproducts orcom-_,,”»ThisismconsequenceoftheanalysisofSoc.YTpineknowngh e)Inutatoreatsmalldistances orproductsofthreeorBOSGsaye®nmematelsseeXJohnson,NuovoCimento} FS 2 e : ce)ss179 NON-LAGRANGIAN MODELS OFCURRENT ALGEBRA 1501 7 ba3function ingfinite-mass vector-meson theories'#) bytheexactya andbrokensymmetries ofthetheory.Themostcrucialba fH(a)pl2)=(0), (25)Ofthesesymmetries isbrokenscaleinvariance." The .ics . freescalarandspinorfieldtheories withzeromassare‘ exactly scale-invariant. Mass terms andrenormalizable: a f[v5%(2)'Jo(z)= —Vp(0),ete., (2.6)inl;tions!by:hesymmetry buttheghostoff%, s * scale invariance stillgoverns thebehavior ofthesineular * “functions.!* Exact scaleinvariance means thatthefieldwhere p(z)isadifferentiable function ofz,andJz“theory isinvariant toaone-parameter group oftrans- jae=Sade,Onewrites| formations U(s).ThelocalfieldsO,(z)transformasy %; eUNG)Oa(2)U(6)=54°0 aCs2).1) va[rene[o00r+sVolO)+:Hele)OT)”5eceeseldtheoriestheconstantd(x)isthedimensionP ofthefieldOn(c), thatis,On(x) hasdimension m#)Te makes sense toexpand p(z)inaTaylor series because jnmass units. Forexample, afreespinor field¥(x)is ¢Eq(z)vanishes unless|z|<|z0l. Onenowseesthatthetransformed tos¥*f(sx), thepowerofsbeingdeter-*function p(s) isequivalent toasumof8functions: mined sothat thecanonical commutation rules are 2 _ sq) foee invariant. Afreespinor fieldhasdimension m**,again% Bele)=Fle1Fuca) VO)+(28)HEANCfthecanonicalcommutation rulesTtisi-*where portant forthedimension ofythatthereisnodimen-e sional constant inthecanonical commutation rule. Onex Fols)=fEaleot), (2.9)canalwayschangethedimension ofybymultiplyinga . itbyapower ofamass mbutthisputs adimensional a . constant intothecommutation rule.Multiplying ybyBe Fuad=—fBena,etc., (2.10)aconstantdoesnotchangeitstransformation proper-Fig + tiestoU(s). bsba. ..Imanexactlyscale-invariant theorythebehaviorof fo)Jc»Thedependence ofthefunction Fels)ete»on#1shefunction Cyle~9) idetermined excepforacon-x? determined bydimensional analysis tobe staatbyscaleinvariance. Performing ascaletransfor-My Paleo)furs*™*, (2.11)mationonBq,(2.2),onebas SS Pi(ee)=fies”,ete., (2.12)ste4404(sx)B(sy)=DnCale—y)sOn(52). (3.2)iwherefoandf,areconstants (proportional toZ).Expanding theleft-handside, |Z Actually f,vanishes because ofrotational symmetry, ashen e—sy)On(2)= EinCalt—y)s%04(s3) He” butthere willbetensor quantities fay,etc., which do*EnCalse—sy)On(sa)= EnCalsy)980.Os . not vanish, = ‘Theequal-time commutator isobtained byletting Ifthefields O,(x) arelinearly independent (this can bw0.hecoefiicientof6%(x-~y)Zinthecommutator alwaysbearranged),onemusthave isFo(0),whichis, ©gayegtedateenfg. 0,ifp<is Culse~sy)=5* Cale—y). 3-4) fe,ifpois ‘Thisequation saysthatC,(r—y) mustbehomogeneousan oforder —da—dg+d(n) inx—y. TheLorentz trans-@,ifpris formation properties ofCq(s—3) thendetermine the * ; Sena) Fk 4__behavior ofCu(s—y) completely exceptforoneormore Similarly, thecoefficient ofVi¥j5%(x—y)I isnonzero it imp>2.5andinfiniteifp>2.5,ThisanalysisisbasedonConstantsCone,ifCu(e—>)mustbeasealit]Tn.pareoe W’partiular definitionoftheequal-timecommutator cularthestrengthofthe light-conesingulasinris de:yp0"A (thelimitforzo0oftheunequal-time commutator). termined _bythedimension dytda—d(w),wesonlydetdeSdn)andbmarps, Teenaynocppltootherdefnicons desler thattenul onlydrde2dw)andbecomesmoresings>oll +operator-product expansions havegreaterflexibility inTnafteefieldtheorythefeldsO,(s)thatoccurinytheformofthecoefficient Ca(x—y) thandotheequal- oe 5 ae ener time commutatorsunlessonepermitsinfinitecoefficients °PeTstor-product expansionsarethefreefelditself,its intheequal-time commutators. "Whe short-distance behavior ofvector-meson theories iscom-Ss plicated bythelongitudinal partofthevector-meson propagator. ‘ MI.SCALE INVARIANCE “tSealemearianee inree-Seld theoriesisdiscussed(aspartof ‘Thenature ofthesingularities ofthefunctions "scpnforml group)byJWess,NuowyCimento18,1986(190). Cale—y)isdeteriined inknownfieldtheories(exclud- SeeSex.vrnuentanSheory 1502 KENNETH G.WILSON 179 derivatives (ofanyorder),andallpossiblelocalWickmetryinanexactlyscale-invariant theory.‘Thenthere fe) products.Forafreescalarfield(x),examplesoffieldswillbefieldsA(x)satisfying Outs)are [AG@),Q}94G) co) : 5(8),VaV.VadCe), 28%2):, VordX@)VAG)VB):, €-Foysomeconstantg.ThisequationisinvarianttoscaleTtisusuallydesirable tousealinearly independent settransformations onlyifQisinvariant:ofthesefields, sothatonewould exclude V,V#@(x) and 7 - 3.8):4(a)V2Veé(x): andotherswhicharefixedbythefree- UHQUE)=O. G8)fieldequation, ThesetoffieldsOn(x)forafree-scalar- IfQisthespaceintegral ofacurrent jo(x),thenje(+) aa fieldtheory canbeordered by‘dimension, starting with musttransform as. |theunitoperator (dimension 2e70)andthescalar field . a(6)itself(dimension 1).Therearetwofieldsofdimen- UNG)JU=#5ebs2). G9)sion2[V,o(2) and:¢%(x):]. Themumber offields Consider theunequal-time commutator ofj,(y)withrultipliesrapidlyasthedimensionincreases, butnever-A(x)which,forsneary,hasanoperator-product €x-theless there areonlyafinite number oflinearly inde- pansionpenident localfieldsofdimension Dorless,foranyfinite [AGjO)IEL= Kale9)0a(2). (3.10)bound D.Thisistruealsoofthefree-spinor-field theory. 6. . .Theorderingbydimension isausefulconceptbecause SinceSrisaconserved current,Ku.(s—) isconserved.thefunctionsCa(a-—3)becomelesssingularasthedi-Thismeansthattheintegrals “mension d(n)increases. Co(z—y) isthemostsinscular .Junction(ififdoesnotvanishidentically), andonlya haefKolo) ‘finite number oftheCy(z—y) canbeatallsingular. A. Onecanconstruct explicit operator-product expan- | ;sionsinafree-field theory.Anillustration willsuffice. F€independent oftime.Hencetheycannotdiverge 5,Consider theproduct :4%(2)::4%):.Itcanbewritten for9-40,Atequaltimes(x=) onehas byWil’ theorem: TAG)AOYIH Leks"-VNO)EST. G1) B:$%(2):14%):=2[D(e—y)FI4-4D(e—-y):4(@)6(): wheretheSchwinger terms(ST)involvederivatives ofBS -+:4%(2)6%y):, (3.5)8functionswhosecoefficientsmaybedivergent.Tobe“» fe)whereDisoneofthefree‘eld singular functions. 1tconsistent withFa(3.7),onemusthavescalesas(x—y)-%, ExpandtheWickproducts ina GA(@)=Zinbsnl) (3.12) “Taylor series in y—x, for 4 lor seriesiny=,forexample, NotethatB,canbeanonzeroconstantonlyifKuis)© 4()66):= 8%): HO~2)er6G) POL)++,(6)sealesas4,whichmeansOyhasthesamedimension ‘ewas A(z). *. andlikewisefor:¢*(2)#*):.Onethenhasanexpansion similaranalysisofthetranslationgeneratorPy offeeredformexceptwatweaoperators showsthatP,hasdimension1andthelocalstress: involved isnot linearly independent [forexample, energy tensor hasdimension 4.wae)¥.Vg(a): isincluded), Thefinalstepistoreduce“ *™ my . theoperators toalinearly independent set,which is formally straightforward butinpractice rather com-TV. HADRON SKELETON THEORY . plicated. Theresulting expansion hasalltheproperties ‘Eiddtheories withexactscaleinvariance arepot*discussed here. physically interesting, since‘theycannot havefinite- &Tnanexactly scale-invariant theorythesingularities _massparticles. Butonecanhypothesize thatthereexistsofthefunctionsC.(2—y)aredetermined bypuresym-~ascale-invariant theorywhichbecomesthetheoryoftymetryconsiderations (scaleinvariance andLorentzin-stronginteractions whenoneaddsmasstermstotheFFvariance), exceptforconstants. Ifthereareinternal Lagrangian. This’leads totheideaofbroken scalein- .symmetries, someoftheseconstants willbezero.‘Thevariance proposed byKastrup andMack.* Thisideascale-invariance requirements override anyothercon-willnowbeexplained indetail.Inthissection the‘scale- * siderations; forexample, onecannot demand thattheinvariant theoryunderlying stronginteractions willbe *equal-time commutators ofalllocalfieldsbefinite.Ifdiscussed, andinSecs.VandVItheeffectofmassterms .A(z)andB(y)arefieldsofhighdimension, then,theirwillbeconsidered. .‘commutator willcontain terrifyingly singular functions, Itisassumed thatthestrong interactions containTeading toanequal-time commutator withmanyderivar somearbitrary fundamental parameters justasthemasstivesof8functions andmanydivergent constants, andcharge oftheelectron arefundamental parametersThecurrent commutators areaspecial case,where inelectrodynamics. However, thegreater complication equal-timecommutatorsmustexist(apartfromSchwin-ofstronginteractionsmeansthattheparameters.of fo)gerterms).LetQbethegenerator ofaninternal sym- stronginteractions arenotphysical masses‘andcoupling weES bgfe)a19 NON-LAGRANGIAN MODELSOFCURRENT ALGEBRA 1503J constants; theyshowupexplicitly onlyintheshort-eGistancebehaviorafstronginteractionsTmplicily,they Jomf[xp6ol2)—2040(2)] Kaa)<jdetermine allofstrong interactions, buttocalculate :{physical masses andcoupling constants onehastosolve ;2thestronginteractions, whighisnotpossibleatpresent. GivethatOoistractless,symmetric andconserved,[8Inphysies thesebarameters haveparticularvalues,but_"ese#enerators arealsocomehietl1Soy’thetheoryofstronginteractions sassumedtobeself..“TheskeletontheoryipresumedtohaveoperatotKitheoryneinteractionsassumedtobeselpoductexpansionslikeBq.(2.2).Theéoeficents 1PPanital!theparaseters arezero itwilbeassuried thatCal2)a7determinedexceptforconstantsbysesle ae |theparameters arezeto,ikwil beassumed that.evariance andothersymmetries. Noproposal willbe)fealpastielsymmetries becorme-exact. Thetheese Peeparameterssetequaltozerowillbecall made‘here fordetermining theseconstants.However,it fjtlLreeparameterssetequalto2es0wilbeaedtbeyibeassumedthattheseconstantsarealluniqueand:iFreeeiualphysical masses cqunltomensions. Then dimensional analysis oftheope. ero,whichisexactly scale-invariant, andexactly in-tF-Praduct expansion showsthatthedimension of :HG,TenantwsPGPOTe”SU)XSUC), ndbaryonlteldAG)theanestsealeinvaiance quan ) Number, Thequatemodelsuggests thereshouldalsobe‘umnumberda.Therearealsooperator-product expan-an“qaialbacyonnumber” butthiswillnotbecon.‘i005forproducts ofthreelocalfieldsAx(s)As(9)4xG),. ISdered hereforthesakeofexpediency. ManyoftheOFevenmore.Inthiscase,theexpansion functionscomplications oftheGnite-mass theoryshouldbeabsent C+; #~2)candependinanarbitrarily complicated#fromthezerornass theory: Allfnite-smass thresholds W8Y00theratio(s—9)*/(—2)* without violating anySegone,replacedby@continuum startingfrommass!8Va¥lance, NoprocedurewillbeofferedfordeterminingK,zero,IfthereareReggefamilie ofparticles, thewhole *Hedependence onsuchdimensionless arguments . '%,familyistelescoped intothezero-mass point.Atwo-Itwillbeassumed thatthefieldsQn(2)ofthebasis. pointfunction G(e),which couldbeextremely compli. 2ordered bydimension. Foranygivendimension % itdinteGaitemacetheory,isnowasunplepowerthevewilbooneormoremultipletsoffiesbeled ) orebecause ofsealeinvariance, bytheirLorentz representation, baryon number, SU(3) tAHR“Theskeletontheoryhasasetoflocalfields.ThissetSUG)representation, andPandCproperties. OF‘bgdividesintotwolinearspaces:thespaceoflocalBosePettieularimportancearethefieldsoflowdimension, ° fieldsandthespaceoflocalFermifields.EachsetcanSincethesefieldshavethemastsingular coeflicients in{bedefineasthestoffeldswhichcommutewiththeoperator-product expansions,Asaresult,theywillde- ) HeSuxS0U3) caretforspaclice separation. Afri salts eradiativecorrections, conver-)Jfnearcombination ofsuchfieldsisalsoalocaleld,sogence.Weinbergsummls,ete1ppractice.Behe.) 8Sthatthesesetsdetinelinearspaces,Eachspacewillbe elds.ofdimension 4orlessthatareimportant.ThereEgssumed tohaveacountablelinearlyindependent basis.6severalfieldsthatmusthavedimension 4o:lessThetwobaseswillbelumpedtogetheranddenotedbyFitst,therearetheSUG)XSU() currentsandthe : »(Osea Howthebascostae islnrguyae.EenuretTheewlbeassumedtosatisfyGell- :%—Sitrary, andtherewillbemanyequallyvalidchoices Mann's curzent algebraintheskeleton theoryaswell ¥%pofbasis.However,itconvenienttodefinesubsetsof|$°thephysicaltheory.Sothecutsentshavedimension. <j,®fieldswhichbelongtoparticularirreducible representa- +Thestressenergytensorhasdimension4.Thefems Jftionsofthesymmetries andchoosethebasiseldsfrom{80beanSU{3)XSUG) multiplet including thepionr thesesubsets Inpractice, itiseonvenient toletapar-eld,withadimension lessthan4,andgreaterthanor- ticularOn(a)beanindividualGeldandnotamultiplet; 8!*fabdimensional TC- fe ee ug attial conservation ofaxial-vector current:reitmightbeocomponent of&vectoreeexample. (PCAC)workwhenSU(3)XSU(3) isbroken;thiswille ergytensordyoftheskeletonisatracelesssymmetric Be&xPlinedlater.Thismultipletwillbeassumed6 . a=~YionoftheLorentzgroup.Thisisthecaseforthequark©.3)tepresentation ofSU(3)xSUC)Allthese :*Mhodel.TeimakessalenvarianceandLoreateinvariance PropertiesaretrueoftheSU(3)%SU(3) «modelit 5+tomatic, giventranslational invariance, becausetheWichthepion-feld multiplet hasdimension 1,and sFenn Seale uansformations andLorenta trans. {quark modelinwhichthepionGeldhasdimension a formations becorae —-"For each fed O4(s) there isone arbitrary nocmlizaionTe) ftwlthoneietobeasinaasian : D=f20y0(2), MN)scheRSchdettantheiesionoftheft 8 — : ialElda thebana wieguanine sme for" ©SincePCPisautomatic inaloeateldthaoy,teamandwitMSUROaSeeven OeOEEbeignored 1Thiswasproposed byGell-Mann (Ref.1) | 1504 KENNETH G.WILSON 479 4 3.Herenoassumptionwillbemadeaboutthedimen-“field,super-renormalizable interactions(whichrequire. Te)sion&beyondtherestriction1A<4. “Gubtractions onlyinlowordersofperturbation shears) +‘Arethereotherfieldsbesidestheonesmentioned Fenormalizable interactions, andnonrenormalizable in|,aeeeeseepetion4orlee?Ttwillbecomeevident{ernctions, Fachinteraction ‘corresponds 0Tocal :> abetthisavitalquestion.TheauthorbasnowaytoLagrangiandensityscalartoLorentztransformations - answeritconclusively. Asanadhocapproach,itwillandinvariantto‘exactinternalsymmetries.Clearlyany beassumed, tostartwith,thattherearenootherfieldsLagrangian densitymustbeaneoncombination oftheeraiorless,Anextrafield[anSUG)XSUG) bussoflocalfields(Qs(z)),60thatitislogicaltouse Sialetslateldwillbeproposed laterbecauseitithesubsetoffelfromthebasitwhichorescalarsand |neededasamass{ermintheLagrangian. exacttointernal symmetries asthespecificinteractions. i= caeaeManodels[theSUG)XSUC) ©modelorOnethenhasa.basis{2(e))ofpossibeinteractions.z thefree-quatkinodel]thebasissetoffields(Q.(a))culledfromthecompletebasis(Qx(s))Tewasshownby3 eeeaaatedfromWickproductsandderivatives Umezaweaefal.*thatarenormalizable interactionmust: @ftheclementary fields,Thus,inthequarkmodel,thehavedimension 4orless.TLithasdimensionJesthan_|5 HeeotSincnaion4orlessandzerobaryonnumberare4,itisasuper-renormalizable interaction (e.g.the+$*—= seeoeciesYasor:U,day:or7VaVa):,whereinteractionofscalarfield).Free‘eldmassterms(for FSseteoopiandSU()matrixandyisthequark scalarorspinorfields)havedimension Jessthan = foldThee fields allhave dimension 3or4Astudy Aslongasinteractions onafree-feldtheoryare- “oftheThitting model_hasconvinced theauthorthattreatedinperturbation theory,onefindsthatoperator-3 thadimensions offieldsinfreefeld modelshaveliteproductexpansions atsmalldistances [ikeEa,(2.2))= ‘ornobearingonthedimensionsoffieldsinstronginter-existinthepresenceofinteraction, butthatthedifferent = aaa theThivring modelthereiasingledimen- typesofinteraction haveprofoundly different feels = . acapling constant andthetheoryisscale-ontheexpansion coeficients Cu(e).IEone-hasonlin,= invariantforallvaluesof4.Inthefree-fieldlimitofthe_masstermsandsuper-cenormalizable interactions..then‘Thirringmode)thespinorfieldybasdimension }asthedomipantterminCala),forssmall,isthesheeexpected fromthecanonical commutation rulesinone term l.creeedimension, However, thedimension variescon-coupling constant aresmallerhy-a_pawer of4.1fonetlomously with)andapproaches =asapproaches hasazenormalizableinteraction,theinteractionsci=rae e)2m.This isaconsequence ofrenormalization;afterre-2s3Smulusisate)5 normalization thefielddoesnotsatisfycanonical com-“skeleton term,Finally,anonrenenmalizable interaction aGmutationrulesandhencedoesnothavetohavedimen-treatedinlowestordergeneratesatconnocesinalatHonk, owever, theThirring modelhaschargeandby-apower of:thantheskeleton term.Foreverytypeseriargeconservation andtheequal-time commuta- ofinteraction ormassterm,thefunctions C.(x--3) cantongofthecharges withthespinorfieldforcethecor-bewritten asapowerseriesinthecoupling constant orresponding currentstohavedimension 1,whichtheydo.mass(onlytofrstorderfornonrenormalizable interThais thecameasthedimension oftheproductPy,¥,actions).Thenth-order termintheseriesscalesasnlyinthefreeield limit.Sointhinking aboutthe2"¢-40relative totheskeleton. term,wherediistheGimensions ofoperators inthestronginteractions wedimension oftheinteraction. Therecanalsobelogs- _ donotassumethatthereareelementary fieldssatisfying rithmsof(s)*/m!,wheremisthefree-fieldmass. 4 =cenonical commutation rulesanddonotassume that Similar conclusions applytointeractions onthe= creoeticr localfieldshavethesamedimensions asYAdronskeleton. Onedefines thebasisofpossible in-- products ofelementary fields.Ttisalsonotassumed Aeractions asthesubsetoffields(0,(2))fromthebasisprottdsexistwiththesamedimension astheproduct/{O,)whichareLorentzscalars,eventoPandC,andGftwopionfeldsortheproduct oftwocurrents. (Thy/haveJ=¥e=0. (Oneexcludes theunitoperator andfatteraxsumption affectsthetheoryofnouleptonic weak_allderivatives offieldsfromthislist,sincethesearenotinternetione/*) ‘Thisleavesonewithenormous Spki-meaningful interactions.) Therearethreetypesofin-Biltyinchoosingdimensions offej, ‘ealteractions, depending ontheirdimensions: Generalized(/soghsoass.termshavedimensionlessthan4,renormalizable V,MASSTERMS:GENERALITIES /7¥",interactions ifany,havedimension4exactly,andnon- 5 i"fa" genormalizableinteractionshavedimensiongreaterthan sonehantheBroblemofinteractions atSA.A.generalperturbationformulacan,beset ueeecory.ICaheskeletondescribeanyoftheseinteractions,Toavoidinnumerable theorywereafree-fieldtheory,therewouldbefour eet ere_woule complications ofperturbation theorytoallordersone ThediferenceindimensionaffectsthebehaviorofthepionUaeenpagetyJ:Eaters,PAD.thesComeselpartyteeta octet thtian. snc, cotage hel andsuperenSMEaleNansPhyo.Rev.125,1061(1962);Ref,20.renormaliableinteractionsin,stronglyinteractingtheorybe: BS‘There isalcotheunitoperator. ‘Suse’onedocsnotwritelocaleldsa8products ofelementaryHeThe results stated below canbegleaned fromRef.12. fields. r %ee, -nn Es re)le479NON-LAGRANGIAN MODELSOFCURRENTALGEBRA 1505fis. . BESwritesonlyafirst-order formula givingthechangeintheequal-time commutators associated withthesym-quire Ueanylocal(Heisenberg) fieldOa(x)whenanycoupling metry.”However,thenumberofsubtractions neededoy WB.constantischanged.Thatis,if(A)arethesetofinEq,(6.1)isunchangedbythepresenceofinteraction,Tocal 342couplingconstantsassociated‘withtheinteractions sothatonehasarenormalizable interactioninthecon-tows E,(o},oneobtainsaformulaforA04(x)/A%s, Theusualventionalsense.Nonrenormalizable intisackions Paty (uncenormalized) formulais duceexpansionfunctionsmoresingularbyapowerof vexzythantheskeleton terms andhence forceoneto ofthe ines 20.(8) makeextrasubtractions duetotheinteraction inEq.ous3 Ref Cove)2soreey (8) hattheyasenonrenarmazble inthecontone 4 ne ventionalsense. .|tions, REwhere[rw:meanstheretardedcommutator (yo<xo)._Iwillnowbeassumedthattheinteractions onthemby *.,Thisformulahastobecorrectedbothfornonadiabatic hadzonskeletontheoryareallgeneralizedmasstermsmust stacts(whenphystalparticlemassesvarywithA)andThismeansthatscalefavariance,aswellssSUG)than forulaviole singulatities at2=y.‘Thenonadiabatic SUG),is&brokensymmetry. Thisis2pofhocas-sigh Seeeayvorcrly nccourtedforandwillnotbecon-sumptionmotivatedinpartby’thesuccessofbrokenis(or [E3,Siderdhere.Theultravioletsingularities canbeanalyzedSU(3)X.SU(Q) andinparebecatse hardtousethe 2singtheoperator-product expansion forthecommuta- ideasofthispaperifrenormalizable of‘unrenormalizableyare SREtor[Oa(x),L:(9))] andthesingulartermscanthenbeinteractions arepermitted(thisisnottrueiftherenor-Sore {removedbysubtraction.Onecanthenshow"thatoper-malizableornonrenormalizabie rsetOtpresentOn) L.Gtocproduct expansionscontinuetoholdinthepres-butsmall;thuselectrodynamic andweakcarestionstoTrent REeeetheperturbation andobtainformulasforderiva-stronginteractions donotcreatedifficulties), |effects ativesofexpansionfunctionssuchasdC,(s)/a%.These__Inorderthatalltermsinagivenexpansionfunction ay Lefaveruingwhlnotbequotedhere.TheseformulascanCx(~3)havethesamedimensions, onemustassignthe HEbeusedtoshowthattheexpansionfunctionshaveteachcouplingconstantA;thedimension-m*#, where‘eter BEpowerseriesinalnteractions. Onefindssinthefree-disthedimension of2s,OoA!‘generalized masstermsi Seeldcasethatthenthordertermscalesae24?havecouplingconstantswhichcarrydimensions;hence,It &felativetotheskeletonterm,wheredsisthedimension_itispossibleforphysicalmassestobegeneratedbyanyne BAeoftheinteraction, Theremayalsobelogarithms ofgeneralized massterm.nthe BSxy.Onecanavoidlogarithms ofphysical massesnition fob dependinaverycomplied wayonthecou- VI.HADRONMASSTERMSgular [2%plingconstants\)byintroducing anarbitrarysubtrac- Considernowthepossiblegeneralized masstermsinytype kemaceoatorelogarithmshavetheformstronginteractions. Giventhelistofldsofdimension y)can fapnLe—2)%/0r),notlate9)/n,intheexlellessthan4assumedcaitheonlypossibilitiesare‘antor Bioteeubaaeateritisnolongerpossibletotwofieldsfromthepion-feld multiplet. Theseareosinter- SEchoosethesubtractionsforEa,(5.1),sothatthevacwum andgqwhereoyistheI=¥=0,SU(S)-actet sealerulesasBeexpectationvalue(210.612) vanishes, field,andooistheSU(3)-singlet scalarfield,Assumeisthe oeoiesence eeoe Tiass.Ssti vithd4SA.xive-cOtees: thatthereisanSU(3)XSU(3) singletfield,(2),also>loga-rg(2oRa1A TS asebyspowerofwithdimension lessthan4,Theneedforthiswillbe (thanthe skeletontemTnthiscase,thedivergencss seenshortly.Thentheinteraction Tagrangian canbeinthe By,ina(61)ateprimarily duetotheskeene007 rieniad Bz°-,Also,theequal-timecommutators ofsymmetrygenera 1)dora)-+darsl)+ be(tos.acetosomeextentwanfectedbethepusenceofB4(2)~dala)tres)thts).(6) Gand f(enetalizedmasstes[moreprecisely,theequal-timeOnewouldlikeanorderof-magnitudeestimatefor oeand ":VGommutator ofaeureent(3)withalocalfieldA(x)eachterminBx.ThetermNera)istheSU(3)-beeaking arenot __ischanged onlybyfldsofdimensionless thanAitself].teem;thisfsknown,tocauseenergyseparations withincent AySogeneralized masstzrnsatethelogicalchoiceofja:=multipletof450MeV(theN*decuplet)orless.ASalized ;,teractionwhenonesenotsassanmetey“oftheskeletonameanenergy,sayNss~300MeV.Thisisarelatively’‘eable YB"toheahmben symmetryolthetheorywithinteraction.” smallenergy,sinceitisnowknownthatoneshould‘anon £5."Therenormalizable interactions producecorrections measureenergiesrelativetomyofmaybeMsandnoterthan %toexpansion functionswhicharelogarithmically morem+.OnecangetanestimateontheRezotermbutby&uptoESsingularthantheskeletonterms.Hence,ifaninter-moreindirectargument, Oneknowsthatintheinitnerable SE,actionisnotinvariant toasymmetry, itcandestroy . . ers one7"“GTRparagraphsuramarizesaverycomplexanalysis Peraateeel OFSUC. 4_MIREPESRARYSayotedbybcGelcafannandF.GaLos,—™TheSa)gCOygemmettybreakingtermsaCe88,pro- ffvigete95ADDGOSH.Teicanon toesbaegeent posedbyM.GallMann,PhysRew12,1067(962),EasG2),1super aly thatrsetoytespuranandaafSLTaecnotheReFetes20,984CS), cory be reaissuch,lariwoulestoytheeemASandMGMan,RJOaks,andBsRene,Physie178 mentaty Drejoer seenfaNsessential 2195(i968). 1506 KENNETH G.WILSON 179 ofexactSU(2)XSU(2) symmetry thepionmassmust. ‘VII.APPLICATIONS I bezero.”SotheSU(2)XSU(2)-breaking termmust . fo)raisethepionmassfrom0to140MeV;inotherwords, A,Weinberg SumRules theSU(2)XSU(2)-breaking termissmaller thaneven ‘Toillustrate theapplications oftheoperatorproducts theSUG)oeaking term.TodiscussSU(2)XSU(2) expansions, considerfirstthequestionofconvergence¢ SeekingonemutexpresbothgoandopintermsofanofWeinbergsumrules.*Let4ya(x)beanarbitrarylinear aU (2)-violating operatoroy[belonging tothecombination ofvectorandaxial-vector mesonpropaga; (2,2)representation] andSU(2)XSU(2)-conserving tors,andGy(p) beitsFourier transform. Thefirstfieldoxo: Weinberg sumruleholds forGyo() ifA(x) islessa=(Devt(V3)o.0, (6.2)_singularthanx-¢asx~0."Thesecondsumruleholds* ifitislesssingularthanx-*,Mor sey,onecaroomWider (Von 63)felheSond’sumvaleeerthegytmotheSinceopandogarepartofasingleirreducible repre *»%rterminSyo(s)islesssingular thana~*.SooneneedssentationofSU()XSU(), thisdecomposition ofotheshort-distance behavioroftheproductTVy«(+) andosbySU(2)XSU(2) representation isunique. NowXV-a(0) andTAya(a)4,3(0) forxnear0;aandBareonehas SU(3) indices. Thebehavior isneeded toorder7. SinceVyeandAyehavedimension 3,onewillneedto- Dasort-Ases=L— (V/4)dot (V/4)Ae expand interms ofoperators O,ofdimension 4orless.+HLQ/DAeHOV2)AcIoie. (6.4) Since onewants thevacuum expectation value ofthe Ifmgwere22r0,thecoefficient ofoywouldbezero,7Product oneneedsonlyoperators OnwithanonzeroWithmeonby140MeV,thecoeficient ofvyshouldbeYAewum expectation value.‘Theonly.possiblities oeroughlyzero,thatis, I,6»,#1,andw.Termsinvolving wwillalwaysbeless ’ ” wysingular than terms involving J,sothatwwillbe *deYrs. (6.5)ignored.Letooandosbethevacuumexpectation ‘‘TheerrtumsouttobeofordermewhichevenVANES(Ale(x)|) and(M(2)}9). Thelinearcome *compared to.(300MeV)"issmall.‘Thismeansthebination Ay-(s)hasanexpansion SU(3)-symmetric termAgerepresents anenergy of, Dyo(2}XHls)+Haye(a)ootHawla}on (7-1) perhaps, 200MeV. This iswhy another mass term is . needed tochange thep,nucleon, andotherheavier forxsmall,withunknownfunctionsH,H,andHs. particles frommasszero(intheskeleton theory) toThequantity Sp(z)isavacuum expectation valuetheirobserved masses. Thisrequires energies oforder of@linearcombination ofTproducts. ‘Thelinearcom-1BeV,sothatAwshouldbeoforder1BeV. bination ofTproducts canbechosen tobelong toaDGiztheinteractionLagrangian,onecanderivetheparticularirreduciblerepresentation ofSo formula, ItfollowsfromEq,(5.1)(eveninre-¢currents themselves belongto(8,1,8),)so.normalized form),thatifj,(x)atconserved current theproductoftwocurrentscanbelongto(1,1),(8,1)intheskeletontheoryand£ycontainsonlygeneralized (1,8),(10,1)@(1,10)@(46,1)(1,10),(27,1)(1,27),: ‘massterms,then or(8,8).Whenonetakesthevacuwin expectation value, . ,thedecupletrepresentation disappears. Thepropagator Veju(2)= iLO), £1()], (6.6) combinations which correspond totheother representa-whereQ(xs)isthechi jatedwithj.,InthetionsarelistedinTableI,Todetermine thesingularity hereOG)isthechargeassocisted WithjpIntheoFA(x),Hi(a),andH(s)foreachlinearcombination, aseofthestrangeness-conserving isovector axial-vectorcurrentA,,oneobtains onecandoaspurion analysis. IfA(z)belongstothe‘samerepresentation ofSU(3)XSU@) asao,then(x) VA(2)=[—Ist VIAT)A2), (6.7) willhaveaskeleton term,Ifnot,onemustfindouthow here6isthepi manypowersofthesymmetry-breaking parameters Xs where @isthepionfield. | and2gareneeded forH,(2) nottobezero, Thisisde-‘Onecannowseewhythepionfieldmusthavedimen- termined by ryysiswiththe.sionAleesthan4.Thereasonisthatthepionfieldmust.SEINEBYSPne Thespurion. sentingthesymmetry-breaking interaction. Thespurion beinthesame SU(3)XSU(3) representation asthe 5, Dates).Comin thSUC)XSUG) breakingtermsiniy,duetoEq,(6.6).beoresto33IOG: ).CombiningonesPurioneeeButsheLaganiian contains only”generalized macy07%,Whichalsobelongto(3,3)@@,3),onecanproduce ‘.©om allrepresentations except(27,1)®(1,27):Toproduce termsinordernottojeopardize thecurrentcoramuta~ i 5 ‘i iy theJatterrequiresthreespurionsatleast.Theskeleton tion relations. The lower bound A?1isanelementary Aconsequence oftheKillén-Lehmann representation. termin7,(2)wouldhavebehavedas«-***(6forwocurrents,Aforthedimensionoftheofield).Withone Tywonsn6,Jonata spurion,onehasonepowerofNeoFAs,andcorrespond- 2y.NambaandG,Jona-Lasinio, Phys.Rev.122,345(1961);————— Fee see eeDachen,Curent ThiswouldmeanA(2)beinglessingularthange/standAlgebras (W.A.Benjamin, Tne,New York, 1968). nls. ‘« fea5 re)¥119 NON-LAGRANGIAN MODELSOFCURRENT ALGEBRA 1307 =. ‘Tanz I. Combinations ofvector andaxial-vecton, em ivergeace inRadiative CorrectionsSeduncotinine ase Divas inadaOatiEEntatfons,andmasimumsingulaityoff,ZandHae ‘Asasecondexample,considerthequestionofdi- SGU) SUDMSUD Hr, VerBences intheradiativecorrections tostronginter- ere z ; = ion: are deseri ‘an effective interactionRoe ee) 2)22a—2_—_‘Fetlons; thesearedescribedbyaneffectiveinteracti . a wD 88) ay) 242% 0-2 -erangian s NO (18) ap 226-2 ef 75.(2)j(0)D"(a) y : et a) a) 6 2: ba 2:SGA)200K") ~(etD) wherej,(2)isthehadronelectromagnetic currentand: sp GANARE) $304D) D(x)isthephotonpropagator. Thedivergences in ) Hy; —"Storand: icepomagtae, Ostheaxialpomagatnccounenart integralat=O;theyoccuriftheTproductisas:BG, roger,tecStbelchaieonIGGeaCercreac,singularasx-*,Oneisconcerned onlywiththeisospin-14otBaryoncurren and-2partsof£/(0).Sooneneedsonlyfieldsinthe:ingly afactora(apartfromlogarithms). Henceexpansionofthe7productwhichcass,opin,Ye.4-Hy(z)scalesas~*apartfromlogarithms, exceptforSimaaoeee‘Soonewrites:yethetercombinationofpropagatorsassociatedwith“mension4oFless.90oneWal " the (27,1)(1,27)representation. Thesameistrueof ju(2)j0)= TinCane(#)0n(0) ,7.3)E—_Hi(e.ThecompleteresultsareshowninTableT.One Tjple)jAO)= ZeCom(s)O(0) os2seesfromTableIthatthesecond Weinberg sumrulewiththesumover0,(0) sorestricted. Finally, theHG_canholdonlyforthe(27,1)(1,27)combination. How-integralsof‘Cam(x)D(x) overxwillvanishbecauseofaeever,aspointedoutbyWeinberg,’ itshouldbeagoodLorentzinvariance unless0,(0)isascalarfield.SooneapproximationtogototheexactSU(2)XSU(2) limitsrestrictedtotheJ=1,J,=0scalarfieldo3inthepion- ig(QueVAX).Thentheoperator-product expansionoffieldmultiplet.Sotheonlypossible.divergenttermin fe)SX Eq.(7.1)mustobeySU(2)XSU(2). Butthefieldsco(0) hastheform iandosbelongtothe(1,1)and(2,2)representations ofHF—SU(2)XSU(2), sothattheycannotappearintheex- y arpansion ofthepond,linearcombination. Sotheoig- offcote(@))os0 wa)& inalsecond Weinberg sumruleconverges intheexact *isSU(2)XSU() limit.ThisisnottrueofanyoftheThebehaviorofC,-(x)isfoundbyaspurionanalysisto %—SU(3)XSU(3) generalizations [exceptthe(27,1)bex,exceptinthecaseofexactSU(2)XSU(2) sym- SF @(1,27) term, ifitconverges without theSU(2) metry, when itisidentically zero. This means that, for ASU(2)limit]AlltheSU(3)XSU(3) sumrulescon-finitepionmass,Cyo(x)hasafactorm,*.Thestandard .{verge intheexactSU(3)XSU(3) limit;howgoodanprocedure forobtaining afiniteeffective Lagrangian is% approximation thiswould betophysics, Idonotknew. tosubtract theoffending termandaddanarbitrary~ Tf4is3orlarger, then even thefirst Weinberg sum constant times o3: ruledoesnotconverge,exceptforthe(27,1)@(1,27) bdcombination. This whole analysis ismodified ifthere .yee tudeafclmenson €orleseangingtote£406[CTHCIO—ColsdedQVOr C+fo, (8,8), (8,1)(1,8),oF(27,1)@(1,27)representations of . as) “SU(3)XSU(3). Such fields would destroy thesecond. ‘Weinbergsumrulefortherelevantlinearcombinations where/isanarbitraryconstant.Thea4(0)termisthe Isc ofpropagators, whether ornotSU(3)XSU(3) isexact. tadpole term proposed_by Coleman andGlashow.* If ‘Tw isemai theassumption: of§,TyGlahow ands.thereafeotherisospin-1 orisospin-2 scalarfieldswith Wary, BhpHeites202(ieapare, dinenson orls,theytoocancnusedivergenes inforthedifereneeoftheKeandKa*propagator whichon:radiativecorrections.Hei tethagtn'o htepaperstovecepeated(oanthat STH VS.Mathue,and'S,Okubo,ibd.19,470(196: ©a9attaDecay vaedieeerieofSCG)beatingafeatfordifenesy_TWOspeciic problemsinvolvingelectromagnetic cor-wPvegiorimeson propagators, butholdinsecond-order SUC) rections willbeconsidered here:the»—»3decayand fe) Fe eeaetorfears,Sg)WeT—?2ydecay.Firstsconsiderthe>Sxdecay: xu‘breakingintheLagrangianandviolationofSUG)Experimentally theamplitudefor7—»3xiswellfitted SCG inthevacuum expectation values oftheo fields, whichmeans one can hace volslon oftheWeinberg sum rule even inSecond ovder ofS043) beeaking. "MS, Coleman and§,T.Glashov, Phys. Rev.13,D671(1961) 1508 KENNETH G.WILSON 179 bytheformula®® netic Lagrangian, which isnecessary ifitistoaccount . Fi yg)forthe7+3xdecay. e)Ala ettat pad[1—m,-ta(Se~S)], (7.6) " . where D.2° yyProblem=Lala)90 7.) . meSomLaln)—a00F, (7) sryedecayx°~+2yhasbeenshownbyVeltman and Syemet+im?, (7.8) Sutherland tobeforbidden bycurrent algebra and a=0.240.015, (7.9) G8ugeinvariance, Theinvariant amplitude forn®decaycan bewritten % and9(n) andg(x") arethe»andxfour-momenta. It , 7 cnerne . wasshown bySutherland® that theamplitude Ashould TlPk)=eranh°T(A), (10) F vanish when anypionfour-momentum goest0280. where pisoneofthephoton momenta andhisthe«* 1fthea*four-monientumgoestozero,then9(9)—9() omentum, Thephotonmomentawillbekeptonthe mustequalq(z~), andSoism.*.Inthiscase,Eq.(7.6) sacsshell[p=(—p)*=0']. Theargument ofVeltman givesanamplitude lessthan0.14.ingoodagreement andSutherland predicts that7(0)willbezero.BelwithSutherland’s prediction. Butwhenthex°four-andJackiw®haverecentlypointedoutthatthereexistsmomentum goesto2er0,SebecomesM,?andthey3x4specificcalculation of7(0)inthe¢model,andthat=amplitude fsaboutSd, (0)doesnotvanish.Thecalculation isanoldcaleu-—&Sutherland’s calculation assumed theunsubtracted —jation ofSteinberger. > form(7.2)fortheeffective electrodynamic Lagrangian. Tyeresolution ofthiscontradiction isinabreak- 4 Withthesubtracted form(7.8)thecalculation isdifgyyn‘of theusualarguments thatoneusesiawritingHeeee enaera)theFeldngWardidenties. Namely,onesusedtotakingamatrix + inEq.(7.5)butnottothetermfox(0). a:Janne PSUGB)andrewritingit.GFcommutes withthechargedaxial-veetor currentsbut2ementsuchas(AT3p(2)¥¢j-O)|B) andrewritingitynotwiththeneutralaxial-vector current.[Therelevant v,*(A|T°j,(2) je(y)|B)+8(vo—yo)(4 ILin(2),5o(0)]1 B)-éSU(3)XSU(3) commutators aregivenbyGell-Mann.1] . .Sh .‘Asaresult, onestillpredicts thatthen 3xamplitude Butinthecaseofthex*~»2yamplitude, which involves ¢vanishes forzerox*momentum butnotforzerox#thematrix element (9)73,(2)j,)V"A4(@)|9),extea momentum,‘Thisagreeswithexperiment. WhentermsarisewhenV"isbroughtoutsidethematrixele-—& g(=")—»0,they—»3xamplitudewillbeproportional ™ent.Theextratermsinvolvethesingularityata=y Atothesubtraction constant f.Since fisanunknown =Oandinvolve complicated integrals. Theauthor has i parameter, itcanbechosen tofitthe»—»3r decay otevaluated these integrals eveninthee-model ex- ap amplitude. ‘There remains aproblem in7—>3xdecay, @™Ple, Jetalone strong interactions; however, onecan =namely, itsrclation totheKand=massdifferences guethattheydonotobviously vanishineithercase.obtained byfirsttakingtwopionstozetomomentum _Abriefaccount oftheanalysis willnowbegiven. . inthe>3amplitude, whichreducesittothe9-x‘Thex°—2yamplitude canbewrittenelectromagnetic mixing amplitude, then relating the ‘ 5 mixing amplitude viaSU(3) totheKandxmass dif- tem ipnyaf [oimeeive iferences.” ‘Thisanalysis doesnotworkverywell,butTm(Ps#)=(B—m!)(Pam Jf ere itinvolves more dubious approximations than the . simple current-algebra calculation. X(Q|Tj,(s)j(OVA(2)]9), (7-11) Tthasbeen proposed that thetadpole contributions toelectromagnetic corrections beobtained dynamic- where F,isthePCAC constant, : ally—in terms, say,oftheA»Regge-pole contribution t tothevirtualforwardCompton amplitude.™ Itremains VA(x)=Fems79(2), (7.42) :tobeseenwhethersuchatermhasadifferentSU(2)4gj. is isiisthe2?field,and4,istheneutralaxial-vector field. XSU(2) behavior fromtheunsubtracted electromag- Thequantity 7(0)canbeextracted fromtheformula The _yalue foroistaken from M.Gormley, E,,Hyman, rae ealW.Lees.Nash,J.Peoples,C.Schule,andS.Stein,Phys.Rev. _VaPVp'Tyolk)|panneCorea(0).(7.13) ‘Letters21,402(1968). .MD. Sutherland, Phys. Letters23,384(1966) 18M,Veltman,Proc.Roy.Sas.(London)A301,107(1967); 318. K.Howe and A.Ki,Zimmerman, Nuovo Cimento 43A, D.G.Sutherland, Noel. Phys. B2,433 (1967). For related work 1168 (196), R.Ramachandran, fhid. WA, 609 (96H); R.WE. onthewpe aystein, aceKR.Perrin, ‘Phys. Rev. 170, 1367 (1968); Grats, O'Raifenrtaigh, and&,Pakcava,ibd.48h,890(1967);$C,BrownandG.1.West,isd:174,1777(1968);R,Amowitt, Y.T. Chiu,J.Schechter," andY.Ueda,Phys.Rev.161,1612M.H.Friedman, andP.Nath,Northeastern University (4967); D."G.Sotherland, Nucl. Phys. B2,433 (1967). Fora(anpublished). Circleeaeolor deysee].S.Belland D.G. T'SBellandR.Jake,NuovoGimeno (obspubli) Siherland2d.Bt,315(1968). 1withfothankDr.Jackiwforveryhelpfuldiscussionsonthe %SceS.{.AdlerandR.L,Dashen(Ref.31),p.137. ‘x*—+2yproblem.SeealsoS.L.Adler,Phys.Rev.177,2426 8 Okubo, Phys. Rev. Letters 18,256 (I96F); D.J.Groce (1960);hisconclusionsareequivalenttoours" ro)andH.Pagels,Phys.Rew.172,1381(3968) 9,Steinberger, Phys.Rev.76,1180(1949). Bo . nn “ . ra Ke);.179 NON-LAGRANGIAN MODELS OFCURRENT ALGEBRA 1509ntaSo wherecenit emffatalTeiOvrAst). teffveteestndsall, 0.20 adba (7.14)sndIyandIsareanalogous.Thesintegralcanbewrit- ndE Inthee-model calculation inlowest-order perturbation tenintermsofsurface integrals onthesurfaces foay theory, thismatrixelementinvolvesthecurrentsofa=e"and29=xote”.Consider,forexample,thesur-freecharged Diracfield faceintegrals at29xo-te’’: Theyare _9)fi H@)=Va: Gs)ff[Bleaso€)—Bleasee'sel)» =” A(e)= diVer): (1169) et (any be ‘Thesecurrents commute forequaltimes(exceptfor‘Thexintegral mustbebrokenintotwoparts.Inonea c-number Schwinger termswhichcannotaffectthepart,xisrestricted tobefarawayfromtheorigin,‘s three-current matrixelement), justasthecurrentsjcompared toe”.Withthisrestriction onecanlet¢”>0AL andAofstronginteractions do.Also,thefree-field insidetheintegral without making mucherrorand 1current j,(x) isconserved and«1,(x)ispartially con- theintegrand reduces toanequal-time commutatorieserved.Furthermore, T(0)isdefinedbyaconvergent whichvanishes.Thisisnolongertrueif+isoforderk integral; infact,ifonerotatesthex»androintegrals ¢’orsmaller,forthenPy,(t,2) hasanontrivial de-ne. iatotheimaginary axis(thusgoingovertoaEuclidean pendenceonthevariablezaswellaszoe“(thisisA %,metric), theintegral for7(0)isabsolutely: convergent. evidentinthecaseoffree-feld curcents), andonecan* je (Thiscanbeverified explicitlyforthefree-fieldcurrents, nolongerreplace¢”byzero.Actualevaluation ofthe iandisaconsequence ofthehypotheses ofthispaper for integral isnow nontrivial. ). thecurrents ofstronginteractions.) Consider theeffect ‘Thedifficulty causedbyxbeingoforder¢”isavoided 7 bof theconventional rulesforinterchanging aTproduct ifonerequires that¢bemuchlarger than€”;then+ sfe) 7paagradient.OnecansimplybringV*outsidetheisalwaysmuchlargerthan¢”.Inthisevent,thesurface ral productbecauseallequal-time commutators aretermsatze=xocte”approximately cancel.Likewise, if. 4zero.Define ©>¢,thesurfacetermsatso-b¢’approximately cancel.FA ie Fale)(0|TIp(0)jA0)A -2)|9), (7-17)TOBePreciselit—0)lim(e”40)i=! (7. x2 andconsider thefollowing identity: ~Henle 0)Hin(e’ ODF=0 (2) an forfixed nonzero «.(Ifthecurrents j,andA,hada , a80Vo"Fprelty2) =VetUtaralwelt) ]—(VertVe) nonzeroequal-time commutator, f,wouldhavereduceda XLetatelpesTEVSL vagtenet Vet)Fyep |toanequal-time commutator term.)Soonecanmakeia wunVetPrag. (718)Z1,vanishbyletting¢—»0bethelastlimitthatone . takes. Similarly, Z2vanishes if¢’—>0 last, andJs ra‘Thefirstthreetermsaretotalderivatives; thelasttwovanishes if«’—+0last.Unfortunately, thesethreere-* termsvanish according totheconventional analysis quirements areincompatible. Ifonelets<”—»0 last,i because oftheconservation ofj,.Theintegrals oftheeliminating J,thenJ;and/;neednotbezero,Inthis ») izfirstthreetermsarezero;hence7(0)shouldvanish. case,J;reducestotheexpression (7.21)withxunre- Thistypeofargument treats cavalierly theintrinsic stricted (Sothat1,nowdepends onlyon¢”).Only J singutaities ofPreis2) whentwoormoreoperators small:xand(ofordere”areimportantintheintegral: areevaluated at’the samepoint. Tosetupamore Ifeither xor2islargecompared to¢”,thetermat2 J& careful teeatment, thefrststepistodefinetheintegral a=syte"cancels thetermatsy~.to-¢", Forxand2d ‘$—_(7.14)suchthatthesingular pointsdonotoccurintheoforder¢”,Fyre(x,e) scalesas(¢'becauseofscalet israngeofintegration. ‘Thisiseasytodo:Onecalculates invariance atshortdistances.’ Withthisscalinglawtheintegralexcludingtheregions[x0]<«,[zo]<e,andforFyre,theintegral(7.21)isindependent of¢. 3)& —|e—zo] <e”, forsmall but finite «¢,and <”.Then “acthiseac 1thatPore)i. inetakesthelimit«00,¢°-0, ande’—>0. ThesingletanCOS"Poresp60hedeteattheseals %integral,beingabsolutelyconvergent,isindependent ¥%,theoperator-product expansionforTju(2)44()andthe x |. oftheorrlerthatonetakestheselimits. techniofSeecaseFale)includesapropagatorSPC).°* te . Onenowcalculates theintegralforfinite¢,¢,and¢”When=islargeand#isnears,onecanangen Sra)by.iy usingtheidentity (7.18). Because thesingular pointsof5702);thisapproximation isessential inrelating (7:21)to3p»8+Rona)areexehidedfromtherangeofintegration, Retaymeitat's'on goaeet*AM#7BeeBOA he theidentity holds without question, andthelasttwo: ©The (¢")* scaling lawforFye(x,2) canbeseenexplicitly26 termsarezero.Oneisleftwith {phefreeeldexample, Forstrongater eon ceaTOmEME hebE), (LAS) EE, ieaeons Sektage andcanbe :. i 1510 KENNETH G,WILSON 419 ."Theconclusion ofthisanalysis isthat, ifonetakes condition thelimits«0,e—+0,andthen¢”—>0last,Jsis denen fe) zeroandJ;andTyapproachconstants, butthereisno fWe(s,A)=e". (7.24)reason tosuppose thatJ;andJyarezero:Onthecon- : s trary,inthecaseof{ree-field currents, (0)isnotzero,Thiscondition arisesifthesamecutofffunction occurs: sothatoneofJ;orIsmustbenonzero.!# insemileptonic weak decays. Thecutoff massMfwill For7(0)nottobezero,itiscrucial thatFye(z,2) beassumed tobelargecompared totypical strong-scalesas(¢’)-*whenxand2areoforder¢”.IfFuve(22) interaction masses, ie.,MO>1GeV. :‘werelesssingularthanthisforxand20,thenJ,Because1V"(x,M) hasashortrangeinpositionspace, .andTswouldvanish inthelimite”—¥0 because ofthetheintegral in(7.23)involves onlysmall«,andinthissmallrangeofintegration (x,2~e"). Inthiscase,7(0) regiononecanwriteanoperator-product expansion for wouldpavetobezero.TheauthorsuspectsthatT'j,m(a)j-s(0). Thiswillhavetheform . Pyrslsyz) willbelesssingular than¢"~?inthealgebra - ea):(Qa wy 5 +oifeds."ThentheSutherland argument wouldapply Tjge()j.dO=Le ConMOOU0), 128)tox°~+77decayinthealgebraoffields.Thiswouldwhich,substituted inEq.(7.23),givesmake thealgebra offields inbadcontradiction with , .theobserved?» yydecayrate,whichagreeswellwith ©w(0)=AV2G cossindHuCulAA)00(0), (7.26) } theSteinberger formula. with‘Thereisnoproofthattheoryofstronginteractions c=[CoMWeltnM). (2a) satisfying scaleinvariance atshort distances willgive s $ thenumerical °—>vyratecorrectly,butatleastit - ¥ willnotbeinerrorby:afactorofat,(mg?intheampli-.Becauseofexactsymmetryrequirements thefields tudeandm,4inthedecayrate). (04(0)thatcontribute toEq.(7.26)mustbescalarorpseudoscalar fieldswithAP=1, AZ=—4, andA=} or}. : : E.Fonleptonic WeakInteractions TheleadingterminCy™(a), foranygivenmand ba‘TheeffectiveLagrangian forweakinteractions cansmall2,scalesasx7,wherea(n)dependsonthedi- 4beanalyzed qualitatively, usingthehypotheses ofthismension ofOa(2x) andthenumber ofsymmetry-break- paper,OctetdominanceinnonleptoniedecayscanbeingspurionsneededtomakeCyye"(x)nonzero.Ifno fe)Understood without introducing neutral currents andspurions areneeded, then €some new“four-dimensional” superconvergence re- aln=de-6. *Jationsareproposed.ViolationofCPisignored. (n)=da—6 (7.28)‘. ConsidertheAY=1componentoftheeffectiveweak-Bydimensionalanalysis,Cn(3)behavesas 2 interactionHamiltonian. Itwillbeassumed tohave CMe Mme, (7.29) 3 : eform : Infree-fieldmodelsconealgebratherearean* L1y(0)=3VIG cosesind|Tj,w(a)j,s(WG, infinitenumberofSU()XSU(3) multiplets ofscalar ()—126cousseTinwle)iesWM), eyNeeudosealar fieldswhichcancontribute tothe(7.23)expansion (7.26)for£1r(0).ButifMislarge,thecoefi- =cients C,(M) willbeverydifferent fordifferent multi- where GistheFermi constant, @istheCabibbo angle, _pletsbecause different multiplets usually havedifferentGuwisthenonstrange weakcurrentwithAQ=—1, andvaluesfora(n).Soonlyoneorafewmultiplets will FrcistheA=1,AQ=1strangeness-changing current.dominateinEq.(7.26);apartfromsymmetry-breaking ThefunctionW(x,Af) mightbea1¥-mesonpropagator considerations, themultiplet oflowestdimension will :(apart from ascale factor), oritmightbeamorecom-havethelargestcoefficient.The¢-¢multipletisthe plicated function, Mistheeffective cutofi mass of multipletoflowestdimension,byassumption.Therele- Woz),ie,We(x,34)fallsoffexponentially whenvantfieldsforEq.(7.26)aretheAY=1oand¢fields; =Mjslargeandspacelike; itisassumed thatdimen- however, thesefieldsarebothdivergences (ofthe sionally H»(ayM)isoftheformM‘H"(xM)."Thefune- _strangeness-changing vectorandaxial-vector currents),tionW(z,3f) isassumed tosatisfy thenormalization 80theywillnotcontribute toanydecay process.Hence onemust gobeyond thelistoffields proposed‘ ~ . nek _inSec. IV. cei ambigulyJnJaminedwihdlzentSitingpro-Tnthefree-quarkmodel,thenextscalarandprcudo- Sforestas) already painted outbyBellandJackiw (Ref.41)andscalar multiplets: withappropriate quantum numbers AdlerRel41). . havedimension 6(Wickproducts oftwocurrents). ‘Field algebralsts claim that thevacuum expectation value “v*oftwocurrents Gal3),(0)[0) isesssingular atsOinthe SeeSL.AdlerandR,F.Dashen. (Ref,31),p.133;C. Seer aeeseSeenieTsSITG90N,Rev17521800969), nn”SGPespnntaeMw ——————————————— ce isre)179 NON-LAGRANGIAN MODELS OFCURRENT ALGEBRA ist EF—phereareseveralirreducible multiplets withthesame|B)areone-baryon states(forexample, |A)mightbe7.24) dimension-—in fact,allmultiplets contained intheanucleonand(8)a%state);jaw(0)isanonstrange . product oftwocurrents. Instrong interactions, thecurrent andj5#(0)astrangeness-changing current. Ttis [eiauthorconsiderssuchdegeneracyindimensionunlikely:assumedthattheproductofcurrentsatthesamepoint seus:IfRenormalization effectsshouldbedifferentforeachexists.Itiscalculatedbyputtinginacompletesetof willBs different irreducible SU(3)XSU(3) multiplet. This intermediate states; inpractice onlyafewsingle-‘ong- #meansthatonecanhopethatasingleSU(3)XSU(3). particleorresonance statesarekept.DefineM4s(9)EmultipletwilldominateEq.(7.26).Theexperimental tobethesuroverthoseintermediatestateswithmo- nace,G.—factofoctetdominanceindicatesthatthedominantmentumqrelativetotheinitialstate;thenonecom- this, HX multiplet shouldcontain onlySU(G)octetsandsinglets; putesfirstM.1n(q), andonehas afor fsthismeansthatthedarian multiplet mustbeeeHan (8,1)(18)ora(3,3)@3,3)multiplet. Theauthor -123) frknowsofnowaytodistinguishthesetwopossibilities, MasfMasta). as” a butwillguessthatthedoininant multiplet is(8,1)1,8). ‘Thismutliplet willbedenotedo’-¢',Its_Tocomparewiththeoperator-product analysis,some126) Hfdimension willbecalled4’.Neglecting othermultiplets furtherassumptions willbemade.AssumethatA’is.%inEq.(7.26), anddenoting theAY=1 I,=—} fields eitherequalto6orslightly smaller. Thismeans thatBy os’andgy",theellective weakAY=1 Lagrangian isfortheo’—#’ multiplet theconstant a(n)isalmostzero., e . _ , ‘Assumethatforallotherscalar-pseudoscalar multipletsaldGF(0)=4V2Gcos0sinda*Los'(0)—#4'O)], (7-30)except6-4),a(n)isappreciably abovezero.Thenonelds &whereaisanunknownconstantdepending onlyoncanwriteanexpansion fortheordinaryproductres FEstronginteractions. Thecombination ay'44’resultsJen(o)is"(0):7-4 gefomasumingpurechiral‘weakcurents(equalvectorjan(x)js*(0)=Bi(x)om(0)-+Ba(2)}m(0) fF and axial-vector Cabibbo angles). Le../(0)— be!and 4Formula(7.30)isnotexpectedtobeexact,butthe _Tele!()—$0'(0)], (7.52). therrorwilldependinversely assomepoweroftheweakwheremis4or5.Thefunctions Bi(z)andB1(s)scaleH ©cutoffM.TheratioofK+toKes.lifetimesshouldde-#8*°*forsmallx(sceSec.VICA),B(x)as2.a 4pendontheweakcutoffMf,sincetheK*decaywillbeThe#-multiplet mustbeconsidered because theidetermined byamultiplet containing A=}fieldswhichbatyonstates|)and|B)donothavethesameenergy.willappear asoneofthecorrections toEq.(7.30). TheerrorinEq.(7.32) goestozeroasx—»0. Asate-‘.28) keEquation (7.30)predictsoctetdominance inallstrange- sult,onecanwritethefollowing momentum-spacef, _ness-changing weak processes. Itmustbemade clear formula: »)that octet dominance isnotaprediction ofthehypothe- ‘s ses ofthispaper. Theabove analysis simply allows one “sag Bilgeas— an toincorporatetheexperimentally ‘observedoctetdomi.fiveCg)Brla)aaa~Balaban ‘erfinanceintothetheorywithoutintroducingneutralcur- ~B(Q(oan'~da0')]=0, (7.33) rentsintothecurrent-current yrangian. a ihe i"ThefactorM™*"inEa.entreansthatnonlep-wereBug,etearetheFourtransformsofBis),uti F tonicdecays canbeenhanced orsuppressed compared“? * =Alen(O)1 1.38)ue F——tosemileptonic processes,depending onthevalueof4”. can=(Alon(0){B), etc. (7.34)ct However, recentcalculations" usingalocalcurrent- Because theintegral converges, onecanhopethatonlya i current weakLagrangian suggest thatneither enhance- moderate values ofqarethemostimportant values“ > mentnorsuppression occurs. Therelation ofthese andcutifoffatanottoolargeGass.Thenonehas‘he :calculations totheoperator-product analysiswillnowte 2 bediscussed.5* wen ,tae aanMatonofRef.50involve(forbaryon Ma0lq)=BrQeas)oaotBr2Qna)oas i ° lecays) eliminating thepion bycurrent algebra, “* (gman) (a8—ba a |:afterwhichonehasamatriselementoftheformwhere FBrGeadloas'—ta0), 35) ae = (A|jew(0)js(0)[B) (denoted Mun), where JA)and meLa Baltaud= [Bidee.138 sed“V. S.Mathur and P.Oiesen, Phys. Rev. Letters 20,1527 *(1968).Prespecitemechanismoftheseauthorsformakingtheou,py¢ \do-WiganoffbcuapendentdeowakecoNowBri(gn“)andBelgoatbebavegofoae verost¥Hara, Progr Theoret, Phys. (Kyoto)37,;¥-1une,oWingtothebehaviorx?ofBy(x)andB(x). oe) rig.saJSepghte, PhosReytie1sorAGEForverylargeguns,Br'Qous)behaves2guns*, c (1946);187,(317(960), Biawas,&-Kumar,andR.Sasena, butfornottoolargegous,Br'(quax) isapproximately 1 ie ioeieeters 268G80);8Nanasae©.Preparesdueto(s)"-*beingotexcept.atverysmall. tye. SstothankProfessorJ.Schechterforraisingthis_Whatoneneedsforcalculating weak-interactionaqestion matrixelementsarethematrixelementsos’andgaa’. . 1512 KENNETH G,WH.SON vg H Soonemustsomchowcliminatethematrixcementsthen&fractionalpowerdominatesR,(q)unless4is| fo) casaddasineq(35)OnecaydothisbegoingnegoFMalina | {otheexact SU(3)X SU(3)limit. ThematrixelementscanandGandonotvanishinthislimitbutthefunctions VII,FINALCOMMENTS* Bi)andBe(2)do[becausejjanda4belongto_Whatisproposedhereisanewlanguagefordescrib»orthogonal SU(3)XSU@) multiplets]. IfB,'(Gux) iingtheshort-distance behavior offieldsinstronginter-approsimately 1,thenEq,(7.35)givesthedesired actions. Onetalksaboutoperator-product expansionsjratrixelements. Tf4’issomewhat different from6,theforproducts ofoperators nearthesamepoint,instead .‘matrix Gement on’woulddifferfromthematrix cle-ofequal-time commutators. Onediscusses thedimen- .ments needed fortheweak-decay calculations onlybysionofanoperator jnstead ofhowitisformed from ‘.auniversal factordepending ongnaandM,"but notproducts ofcanonical fields.Analyses ofdivergences in £thestates AandB.Evenifonecannot eliminate theradiative corrections, etc.,arecarried outinposition = capmatrix elements, onecanderive sumruleswhere spacerather thanmomentum space. Furthermore, oneas,ec.,allvanish,namely, bytakingthelinearcom-hasqualitative rulesforthestrength ofSU(3)XSU(3)- . bination of2tonucleonmatrixelements whichprojects symmetry-violating corrections atshortdistances. To 3 outtheA/=4 partofthecurrent-current product.Calltheextentthatonecananalyzeproblemsatshortdiss thislinearcombination Mzw3(g); thenwepredict tancesusingonlytheSU(3)XSU(3) currents andthe *o-multiplet, thehypotheses ofthispaper havethe ¥ ewig)! elegance ofsimplicity, onceoneisusedtothelanguage. .[remo° (7.37)Seninthenonleptonicweakinteractions,whereaxnewmultipleto’-¢"isintroduced,onecaneasilyobtain : ‘Onemightcallthisfour-dimensional superconvergence arapportbetweentheoryandtheexperimentally ob-relation. Theauthorisnotprepared todiscusstheservedoctetdominance. Theresultsofthehypotheses ia validity inpractice ofeitherEq,(7.35)or(7.37). areallqualitative, butwiththemonecanresolve some &ofthequalitative difficulties withprevious current 8¥.Bjorken Limit algebra calculations of7»3zandx°~»2ydecay, a[Asafinalapplication, considertheBjorkenlimit? ‘Thereareformidable obstaclestobeovercomebefore a‘Theproblemistodeterminethebehaviorofanampli-thehypothesesofthispapercanbemadequantitative. § tude Thisisbestseenbyreturningtothex°—»2problem. (oe)Tocalculate thex°—2yrate,oneneeds toknow the i Tes()=(alfef?TA(x)B(0)|8), (7-38)vertexfunction(@[7j,(x)j-(0)4 «(2)|2)whenxand=i * aresmall. Oneneeds thismatrix element forthehadron. y whengo1withqfixed.Withtheoperator-product Skeleton theory. However, ifoneknowsthismatrix ;expansion, thereisnoneedtokeepqfixed;onevibes clementforsmallxands,oneknowsitforall=and#. tohaveqlarge.Thentheintegral isdominated bytheit’thehadronskeleton theoryduetoscaleinvariance.Behavior ofthe7product forsmallx;ifCa(x)areTtisbardtoimagine thatonecouldhaveacompleteexpansion functions forthe7product, then formula forthisvertexfunction without havingacom- *plete solution ofthebardon skeleton theory. Thepros- # yectsforobtainingsuchasolutionseemdimatpresent. TaldBwolOuafects), 9)” ” ‘escinewten cece, OMENS ca . scussionswithProfessorK.Johnsonwerecrucial $ Tas(Q™LXn Ru(g)ax}On(0)|6). (7.40) foranearlystageofthiswork.Inthegeneral field- ¥ Bydimensional analysis, Re(q)contains askeleton theoretic work,Ihavebeenhelped byconversations 3‘termscaling as withProfessor W.Zimmermann, Professor K.Syman-Ra(qyrgtsteoneinr—e zik,Professor D.Boulware, andmanyothers.Inap- »._,plications tostronginteractions, 1haveusedconver- . forlargeo.TherearealsofinitemasscorrectionswhichsationwithProfessorS.Glashow,ProfessorJ.Bjorken,* art)wilbecpomofLorentzinvariance, eachtermProfessorW.Weisberger, ProfessorJ.Schechter, Dr. inRq(q)willbeapower ofg?,notnecessarily anintegral apower,multiplying apolynomial ing.Theremayalsop-Sutherland, DrBrown,Dr.G.West,andDr.R.Petogarithis ofg2-ToobtaintheBjorkenlimitwysigeJackiw.TalkswithProfessor K.Gottiied andotherscally,onecanletcecewithqhelidBeawhichseatyaCornellwerehelpfulthroughoutthiswork.The thag!isreplacedbygo?teeeeeetemandSutborhasalsobenefitedfromtalkswithProfessor g*isreplacedbyge?toafirstapproximation, andJosephson, ProfessorJ.Langer,ProfessorB.Widom, ‘onegets anexpansion interms ofgo?, butonemay get 1dProf M.Fisher ab tical ph igue.getsanexpanton feo tebe:tid PofenorM.Fisheraboutcritephenamenaio A 'A-classicalstatisticalmechanics."* fe)tional powers areseldom ofimportance whenAandBoTao ay,Gatze,D.Hamblen,R.Hecht, . aretwo currents. They will beimportant ifAandBeg,Sos1,Kadanol,W.Gitze,DyTan Tees >weented gulpan0.00)iearcarene,TAestheMiaaga83bas mePAPO me y 7 Wilsom :paperreedNov1968omoperator product expansions 4 Fe) 1.Introduction. Attimeofwriting current algebra wasalltherage,withits various equal time commutators. This paper will expand the realm of same by assuming that scale invariance holds somehow athigh energy (small distances), and that there exist operator product expansions. Paper will show why this is reasonable. Idea istonot always sit right on top of sharp singularities, Paper will doboth theory add applications, 2.Operator Product Expansions. Weshould beaware that ingeneral equal-time commtators contain not just one term but derivatives ofdeltas and possibly divergent constant coefficients ofsome ofthese deltas, asubject usually referred to as"schwinger terms". Why not, says Witton, just consider the product of two fields without any commtator orequal time. Perhaps for x-y small in sense of(x-y)* youcanshowthatA(x)B(y) =sum0,(my)0,,(x)where0,is somehow acomplete setofoperators. Ifyou assume that thefunctions C,have acertain simple power form, then you will find the desirable property that (A,B) vanishes outside light cone, And ther asyou approach the equal time comtator ligit you can see how the delta functions arise and how infinite fo)constants mightalsoarise,clearlydescribed ompage179.Soclearly, ifcorrect, operator product expansions allow clearer vision ofwhat isgoing down, Obviously the thing todoisstudy expansions inmodel theories to see what they look like. 3.Scdle Invariance. Suppose inadditiom tooperator product expansion you also have exact scale invariance (ie, all operators transform imusual way ,each operator having adimension d(m). Then you get the important result (3.4) which says that you can utilize this scale transformation to access the lightcone slingular region oftheCO,(zy)functions. Youlearnthattheoperators 0,(x) oflowest dimension will then bemost important, ones ofhigher dimension less so, ifyou are interested inthe lightcone limit. Inany situation you can always group all your operators according todimension; for any finite dimension there are only afinite number of operators under that dimension, sonow you have a way toorganize your operator expansion, anotion ofconvergence. The operator oflowest dimension andthus themost important oneisalways }0,(x)=»identity. If, inaddition toscale invariance, you have some other symmetry generated say by charge Q,you get interesting results shown on page 1502. 8 Wee beBes |Ceres Chicas*| aseh Wy.2=spent,QenC2)70 anhdl=2,Obaurargerghacka. eea \ :YocamB'seppuhua?RHE2(D+iW) SkgOLO) =?odypomgeSLOnsksconhulerdesJurelZoAldial, teduwallea,, SotnDoknageSie)aSO)+2Vg+Ba,Vis+ bepilaorarstentss Jee B=VRE); 7 =\Kem[bostecFetees J=Raso+Fars DcurabenJoannyfobGG)Deskopentbeverslve| ©Dru,fratchgorAdremeneSS,sepeedy IdchYaro, donFERJoe.gmmyantich fe) Jackiw P.T. 1972 wal q aeRend 1.) Results ofsome MIT~SLAC scattering experiments have aroused interest inatheory ofdynamical quantities thatare independent ofdimerisional parameters. Roman Jackiw When symmetries arepresent thesolu-. parity isospin. Chirality becomes antionofalmostanyphysicalproblemis’exactsymmetry onlywhenthemassof|[SHE isimplified, becausewecangetatthethepioniszero.Inthatcase,itcorre-|[<== eerProperties! ofasystemwithoutcom-spondstothepossibilityofchangingthe|BRWNNy=== ===( Pletelycalvingalltheequationsthatparityofastatewithoutachangeof|HAMINMIWHSBE=SHmelff describe the system. Inhigh-energy mass, byemittingamasslespion ROTIaREE= Seer physics,symmetry principles havebeen HereIshalldescribeanapproximate |i1HinesFeriNe studiedwithaneyetocircumventing space-timesymmetry,scaleordilaca.|[iiteccenry HUT twoobstacles totheunderstanding oftioninvariance, thathasrecently be- anemmeRelementaryparticles:lackofprecisecomethefocusofmuchattentionbe. Seeeeea ie fe)knowledge aboutinterparticle forcescauseofresiltsfrominelasticelectron iSSH h4(withtheexceptionofelectromagne- nucleonscatteringexperiments done|Bagg)casaae Netism)andthemathematical intractabil- attheStanfordLinearAccelerator in|A@DeS aaa eeityofanyrealisticmodelsthatproposecollaboration withagroupfromthe|BOgVayeqpees SomaiE toexplainobservedparticlephenomena. Massachusetts InstituteofTechnology. |[Seaamamrae auladAan ‘Theusefulsymmetries, andtheiras-Weshallseethatthesymmetry isexact — omnissociatedconservation laws,arooftwoonlyiftherearemasslessparticlespres-|thopartcietrajectoryshownaboveKinds:“space-time” and’“internal.” ent,sothatitmustbebrokenforstrong|angonthecoverofinalancefetenatInspace-time symmetries, kinematic interactions. Ifthesymmetry wereex-|anonrelativistic particlemovingintheProperties suchasenergyandmomen- act,itwouldbepossible torescaleor|potentialtumareconserved. Forinternal sym-“dilate” thespace-time coordinates for vin==metries, properties such ascharge orasystem, without changing thephysical (0=~3p baryon number areconserved. High- content. Anunderstandingofthework- Timeisplottedontheverticalaxis. energytheorists havefounditprofitable ingsofthisapproximate invariance TheHamiltonian governing thismo- toconsiderapproximate symmetries asprinciple promisestohelpunravelthe|tion(foraunitmassparticle)tewellasexactsymmetries, ‘These ap-intricacies ofphysical systems atvery 4proximate symmetries arenotcomplete- high energies, such asthose available =i oelyvalidinNature, buttheirphysical atSLACandattheNational Accelefa- |whichcorresponds tothecasewithn=2predictions aremoderately wellsatis- torLaboratory, wherewecanhopethat|discussed byRomanJackiwonpage24.fied.Sofar,onlyinternalapproximate thesymmetrybecomesexact,because|theequationsofmotionoftisparesymmetries havebeensuccessfully ex-inthatkinematical domainitshould|iceainvariantunderinewanna ploited. Examples areWerner Heisen- bepossible toignore masses. This tion berg’s isotopic spin (SU(2)), which ispoint ofview has been advocated fora tpt conservedexactly onlyifweignore elec- longtimebyH.A.Kastrup andJulius raphSomagnetism, andchirality(SU(2)xWess.? popgs (2),atransformation, developed by which leads toanewconservation lawMurray Gell-Mann, FezaGarsey, Mau- Natureofdilatation symmetry 4tice Lévy, Yoichiro Nambu, Julian Initsmost intuitive form, dilatation ot Schwinger and Steven Weinberg, that symmetry isrelated todimensional ware extends theconcept ofisospin byin-analysis. Consider any“dynamical 0-8 +metroducing both positive and negative quantity, forexample aphase shift fortere neeetee articlescattering, compated inawa —TheEattors Roman Jackiw isastalfmember atthe tumtheory. Itwilldepend ontheap. Center forTheoretical Physics attheMassa- propriate kinematical variables, suchas chuselts Institute ofTechnology. energy andmomentum, aswellasonthe pastesTooaanwreys Callan, Coleman &Jackiw 1969 — “ - 4 SN }waisoFmvstes: $9,42-73(1970) A.NEWIMPROVED ENERGY-MOMENTUM TENSOR B | gravitational field,Therefore, itisdesirable fortheenergy-momentum tensor tobe i ‘ renormalizable; thatistosay,foritsmatrix elements tobecut-off independer itt 1 * (Ofcourse,thematrixelements atzeromomentum transfer arefinite,sincethese ANewImproved Energy-Momentum Tensor ‘aresimply connected tothetotalenergy andmomentum. Likewise, thefirst : : derivatives atzeroarefinite, sincetheseareconnected totheLorentz generators. 1CurrisG.CALLAN, JR: ‘Butweknownothingaprioriabouthigherderivatives). The InstitueforAdvancedStudy,Princeton,NevJersey08540 urprincipalresultisthatforanyrenormalizable fieldtheory,itispossi. indanenergy-momentum tensor whose matrix elements ar However, this ‘StpNEY COLEMAN, ‘energy-momentum tensorisnotalwaystheconventional one.Forexample, forLymanLaboratory ofPhystes, Harvard U.,Cambridge, Mastachusetts 02138 thesimplest renormalizable quantum fieldtheory, described bytheLagrangian ‘AND f=tagHp—dheto*—dept, ay Roman Jacerwt theconventional tensor, Laboratory forNuclearScienceandPhysiesDepartment, . ©Ta=8498.9—Bau soma (1.2) ‘MassachusettsInsureofTechnology, itA Cone Neha doesnothavefinitematrix elements eventolowest orderinA,However, the modified tensor, Received December 2,196, * ’ . ur=Tor—40.2,—Sw 9 maar (1.3) \Weshowthatthematrixelementsoftheconventional symmetricenergy-momentum hasfinitematrixelementstoallordersinA.Notethatthistensordefing¢‘the same. :tensorarecut-offdependent inrenormalized perturbation theoryformostrenormalizable four-momentum andLorentz generators astheconventional tensor.Geldtheories. However, weargue thet,foranyrenormalizable fieldtheory, itispossible InSection 2weprove thatthematrix elements ofthetensor (1.3) arefinite toall {0constructanewenergy-momentum tensor,suchthatthenewtensordefinesthesame ordersinA.InSection3weextendthisresulttoanarbitrary renormalizable four-momentum andLorentz generators attheconventional tensor, and, further, has ores . “|eeeeaemesisicnereendesofeneroalinel perturbation theory,(*Finite” |_interaction ofmassivescalar,spinor,andvectorfield(Wweexcludemasslessfieldsto -peneindependent ofthecut-off inthelimitof largecut-oR) Weexpicly construc this -—-'-_avoid considerations ofinfrared divergences). Wehavewritten Section 2asclearly tensorinthemostgeneralease |48wecan,giving thedetails ofeverystepintheproof. Section 3ismuchmore“Thenewtensor isanimprovement overtheoldforanother reason: thecurrents telegraphic instyle, withmany details suppressed.asiociated withscaletransformations andconfotmal transformations haveverysimple Our proofisnotrigorous.** Thereason isthis;inordertodisentangleefficiently ‘expressionsintermsof.thenewtensor,ratherthanthevery.complicated onesthey rs7 0 " ! ee eeol ‘overlapping divergences, weusetherenormalization procedure ofBogoliubov, ‘Wealsoshowhowtoaltergeneralrelativityinsuchawaythatthenewtensorbecomes }Parasiuk, andHepp(BPH)[1].Thevalidity ofthisprocedure hasbeenrigorouslythesourceofthegravitational fe,anddemonstrate thatthe newgravitation theory proved onlyforaspecial cut-off, onesostrong thatanypolynamial interactionane inhiswaymeetsalheexperimental teststhatavebeenapplied togenerat |(notjustarenormalizable one)ismadefinite,Unfortunately, ourproofalsousesrelativity. certain consequences (Ward identities) oftheconservation of@,,, 1,Inrropuction 6,=0. anf Inanyloca!fieldtheory, theenergy-momentum tensor isanimportant object; “Throughout thispaper,“finite” and“cut-off independent” mean“possessing afinitelimitknowledge ofitsmatrixelements isneededtodescribe scattering inaweakexternal fasthecut-offgoestoinfinity,ineveryorderofrenormalized perturbation theory.” . : *We use ametric such that thesignature ofthemetic tensor is(+—~—). “Thisworksupported inpartbyAirForceOfficeofScientific Research Contracts AF49 +Noteyaddedinproof:Indeed, ourarguments haverecently beencriticized byK.Symanzik638-1330 andAFOSR 70-1866 andU.S. Atomic Energy Commission Contract AT(30-1) 2098. |(private communications). InAppendix 4,wediscuss thealterations needed inthedefinition"AlfeedP,SloanFoundation RescarchFellow, 1ofthe energy-momentum tensor ifthese criticisms arevalid. 42 | re) Anoverallcomment:This paper written in1969 isabout several different subjects. Onfirst reading Iknew nothing sbout any ofthese subjects and had trouble following the paper. Now Iknow alittle about some ofthe subjects. They are: a)theory ofscale ordilatation invariance. (section 5) b)theory ofconformal invariance (section 5) ¢)good Noether understanding, Belinfante symmetric tensor etc. (section 5) d)derivation andmeaning of"Ward Identities" with Oy,inside theVEVTOP. e)renormalization theory. f)gravitational theory. Atthis point Iknow pretty much about the first four topics. Gravitation isof course the motivation for the paper, why you want renormalizeable stress tensor soyou can dogravity. Myreal weakness is, asusual, renormalization. 8.26.78 re)Callan,Coleman,Jackiw:ANewImprovedEnergyMomentumTensor.(1281970) Comment: this paper isvery peripheral towhat Iamnow doing. Ijust wanted tosee something about scale transformations endthe conserved cutrett. However, jnline with myphiloschvy that itishealthy toread papers that areaccepted as"famous" Ihave read alittle ofthis thing anyway, although not presently relevent. 1.Introduction. Forasimvle g*scaler field theory they write downtheusual canonical etree tensor T,,, andinvent their owntensor @,, which isequivalent inthesense that ityields thesameP"andMY Poincare groupy currents. However, this new tensor isnicer inthat its divergence is"soft" inseome sense. This tensor ,,is,like T,,, conserved. 2.Continue: Theobject ofinterest is(2.8a), ie,theVEVTOP ofthetensor 0, which isofcourse just afunction ofthe fields. Why they are interested in matrix elements ofthis tensor Idonot know, but they want toprove that ifyou take this tensor inplace ofthe usual one, then itispossible toarrive et matrixelementswhicharecutoff-indesendent. Ie,insomesensetheobjectO., fo)is renormalizeable. 3.General case. Here itisclaimed that themost general renormalizable Lagrangian can only have mesons, fermions, and vector bosons. Moreover, there has tobe asymmetry tothat the vectors couple toconserved currents. This allows you to replace N°withk?inthevector propagators, thereby retaining renormalizeability. Forthis generel theory, the0,tensor isdafined asin(3.12) .They goonto vrove insimiler fashion tothe special case treated above that the matrix elements ofO,,areWfinite. «Samvle Calculetion. The above proofs deal with renormalizability toall erders, and are based oncounterterms and BHP theorems etc etc. Here they just want toshow how things work tolowest order perturbation theory. They return tothe simple guLagrangian andthey will compute thematrix elementx of@.,with twoexternal varticles only. This involves doing some feynman graphs. Aninterestin gtrick I had not peeveiously heard ofisthat ofhaving ficttious "regular" mesons fields addedtothelagrangian, justawaytoimplement thecutoff,Thematrixelement 6 ofinterest isM,andthey goontoshow that asyouremove thecutoffs, Muy is finite. Again, Ihave very poor motivation here because Idont know why you want a“renormalizeable" streetensor inthefirstplace. g 5.Scie and Conformal Invariance. Now weare gettin gcloser tomycureent interest. Suppose you have aLagrangian and you make ascale transformation. Infancy generel matrix notation this isgiven by(5.1), butitmeans that you stretch each field according toits canonical dimenSion. The claim, which seems almost obvious, isthat ifyou have only dimensionless coupling constants in your lagrangian, this scale transformation will beasymmetry. Asasymmetry, there will ofcourse beaNoether conserved current J,. Theinteresting fact isthatthiscurrent J,isprecisely thegiven by Jy=XYOywhere 0, isthe tensor they have been describing aZl along. Probably Iwould dowell to gustworkthisoutonmyom? . Asidelight ofthis section isadefinition ofthe conformal group. Itis infact SO(4,2) and contains theusual Poincare group with its 10generators, nlus there are5newgenerators oneofwhich isthescale generator. Theother four are conformal generators ofsome wort. &Gravity.HereIthinkmyquestionisanswered,RecallthattheEinstein 7] equations tell youthat thespin-2 gravity field ismore orless 7.,,, thecenonical stress tensor. Actually, this isthesource, andR,.,theRicci tensor canbe taken asthegravity field (orh,,), similar tothefield ¢forameson. Now, Ithink the point isthis: ifyou consider graviton emission between two states, youaregoing toendupwith R,,andthen @T,, inyour VEVTOP, Ithink they cleim that ifthey choose instead todrive gravity with the0,tensor, then gravity iscloser tobeing renormalizeable. Ie,thefield 0,apoears inyour matrix element because itisroughly the graviton field. Sogravity istheir main motivation for having astress tensor that has finite matrix elements, ie, which is "renormeilzaeable. " Needless tosay Ihave not followed mich oftheir discussion. But atleast their motivation is clear. *) aae ake aR _2BAl 34 Be Baw beth adeo —$e Sn== FQaraget QU =“peta 3-4(ea ere =3aSo=TP—SYost“nosaanunad eunacaee - we mts ae ate eee se Pe ET SIPS TTB T RTRSY) T(t gOTESEMOTE fo20SSan =ah=FREREEXEPPORP STE Nae: BueVike, Gabequan WS(AdnangteGH. eo SA(eee =Ske Wo=ES Re) WaaVMS ol Se Youetten. SkQalte a.QkGa TT oe oeNeBae ree TS RE Wesek poeeadenalanaDeck ESQR 224s Ns. ee OPSysSend=92bh=284=8)<2R=BE Sea Sede SERRE Sa . 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Obviously thereisexactly onewell-defined meaning for__ —_—..a time_derivative_on thistimelattice, themeaning Ihavewritten downabove. -...--—But_now_you.can takeoneoperator totheright andtheother totheleft, toget: :=Aw) vseTatamental —dapastee| aa)2 wee Ee - eee -----__..-At_this point, wedowhatwealways te:theresult isthenthis: 00 ~ =Llu)4Gengi(Steal)=Nag)4)opSEQ) fo)pe 4nd AQ) add =aCaN Olegar See -——Soit,doesanpearthettine“derivative movesthroughthepathintegrals The__ general statement ofwhichresult isgivenbys9 a OS0¢sO Kel G@lo> =9.\tagtfamy] glySo 7p sa .. _ oa wo La edGO ae Ee we we A RAID OO ©) terewithontyone.field,vedonthevetoworryabouttimoordering operator. The —_- -spacetime_derivative, goesrightthrough thepathintegral, justasyeuwould we ee —2AT (2kRG) 2GaolTCaanaa)jad woe _ Re po pa git) Cattle telgetes “leJalal 8te).eas aeOne Ges = --9 fe ee = eCEN aoe iene ee AaNSQs}onaffhan Sa ao_)Sm. teeCe oe n° Ra,4tobeuasgalsoenoahbSee VT (3h.4)ey=Laat) GAO) ).S aaReoAD)IS=YasBAO Ra)LP _Conclustons exceptfortheusual‘dlactinction betweenthedgdpversionandthedq. _version ofthepathintegral, you_handle thederivative ofafield injustthe ___ _____same wayasyouhandle thefield.itsefl, Notdce. that_there isaninteresting _______non-trivial result here, however: namely, you_cannot. trivially move_operation.. .—— eee justond(x),becausethenyou. _.| sa12tosedeltaofthetaterms.Te,0|a - — nS Saab =Shanwoage aSoeaoage—— OnBeBARNS someQiCaRN FAT USIONS. =A GAOSON TT Oa. 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AnRAWWeradaoorcmeSavwheeWen=O.Quaee ©assyatamstaiw aywnstuoomenor eens,SupposeyorYarameWendeamask YramdWIp=D2wotemseund. Dhancyan cng qoaluind ond define, Bowne) =LOTHAS-4-)187+Bronluallyy yorwllmasdtobooLae,gons®Jadeaaaremsook GrinteBerm) Hy).DrewourMaskeDanmar pate Tea}pOo en)=Te) 2.Boe¥)AITAeKhoaSEY abst Mery vn)=LOUT(aH a-d)I9>- Dhara,framy,cumadsackbeakTROPOEBreyKOH, Osavidgha"Warkdaoosby"qdNcegeneralfos3maakwt cuaaecomarwed, These"Wat heron merely payaagi Awe &daadineDeng TOPmemmopekupjue) cauad byernurcte hie, 8 “OWlehyBergudanGEDWardJduattn UnWeWey. 0$aY=TFAandwaeLvsMakMeo+TJre= Syoiyts ateiy.eens GPM =eA 3LV=were kt AppsOostopesschon nosdoh EwenEigsnel =eatyar frampaUasomeFIancarBK=HseOh BOER] =ey. Wuch ibosay 6Peal-wend =eK > Py=-wTeK Wrgolsacke audLoviect He-PLR =ers Bp =Lene RKYET frre Ke] =O Ohne rearAomiudad OroOre:Mac( : ASpray 4fsBogwungous ,VO)=hengegouges, PrineDinatearoadbeshack,Wt We) SeyUwJona “\Arsepumals 7 “OWlehyBergudanGEDWardJduattn UnWeWey. 0$aY=TFAandwaeLvsMakMeo+TJre= Syoiyts ateiy.eens GPM =eA 3LV=were kt AppsOostopesschon nosdoh EwenEigsnel =eatyar frampaUasomeFIancarBK=HseOh BOER] =ey. Wuch ibosay 6Peal-wend =eK > Py=-wTeK Wrgolsacke audLoviect He-PLR =ers Bp =Lene RKYET frre Ke] =O Ohne rearAomiudad OroOre:Mac( : ASpray 4fsBogwungous ,VO)=hengegouges, PrineDinatearoadbeshack,Wt We) SeyUwJona “\Arsepumals 7 5&®Misl®=LRAN =[Werte AG=ilaeee,Ancol =UNTO [heyAula]+«Lanta Atay} 0) Maybe Deradh arate dspadsrolyApadmstAah.Qu (hier behVendy, © Tro, Atil*=0 [0qange above Qrae Auth omocluss Wek : SATG AGoAney-. ARDY =© Qromnar Hero andard—comwmuctardn foun. OYor, odabot Bo,Agloyy™=antsTageava)+UaNalA}EE 3 we =~Sbe-a)\KY). TBads) BOUT FreyBones --WGm) _—(Zsa) LITHKE) RO)I9> Qarwermonuwn SpaceDrewillbe: ™, &) PS)RITA Bete)=Ze&hyfeak,--Pm) . Wer.Wy,Juldeonersaef)roathsonssrcspt2g SLT"(Beyer) TUeslep =-ESen) +3-x)|20DONG) 6 RCS Pg=nbCRE #6OCG,Bt) =1Sei)¢SeCaek) ~iAT(hypp)=Spgrk)—SeQ) | GhiaPecleaurrfunieyabeardosleundardWark,bikBdockbund ubakdodoaloakow WekYrou,Wordfraodeto. Waead4waydoragiacct YawillApvecuteDaceqrusneSte! 8 . 6 Coleman &Jackiw 1971 fo) fo) ANNALSOFPiYstcs:67,552-598(1971) DILATATION GENERATORS $53 1 nature, scaleinvariance issurelynotanexactsymmetry, itisimportant forrealistic ' ."applications tohaveaclearunderstanding oftheprecise mechanism ofsymmetryslat violation. ' WhyDilatation Generators DoNotGenerate Dilatations* Thepurposeofthepresentworkistoattempttogainsuchanunderstanding by : Stoney CoLeman, examining, inexplicit perturbation-theory calculations, thescaling behaviour of ! severalfour-dimensional fieldtheories withonlydimensionless couplings.’ In q LymanLaboratory, Harvard University, Cambridge, Mass02138 thesetheories, itappears thatonlythemasstermsinthe‘Lagrangian areresponsible forthebreaking ofscale invariance, andonewould expect thatthere would ANDremainmanytracesofexactscaleinvariance. qRoman Jacxiw? Atfirstglance, thisexpectation seemstobefulfilled: Usingformal canonical 1 reasoning, onecanconstruct adilatation current, whosedivergence isproportional | Loboratory forNuclear Science andPhysics Department, tothemass terms intheLagrangian, andtheintegralofwhosefourthcomponent ‘MassachusettsInstitue ofTechrology, transforms thefieldsappropriately under equal-time commutation. From theseCanibridge, Massachusetts 02139 statements aboutoperators onecan,byconventional methods, deriveWard Received December 28,1970 identities, andfromtheseWard identities, onecan,inturn,derive theorems about theasymptotic behaviour ofGreen’s functions. Unfortunately, thesetheorems arefalseinperturbation theory. Forexample, ‘WeshowthattheWardidentities associated withbrokensealeinvariance contain, foneofthemstatesthattheinverse propagator foraspinless meson is,athighanomalies inrenormalized perturbation theory.Inloworders,theseanomalies eanbe energies, proportional top%,while,inperturbation theory, itisproportional top?atsorbed itoaredefinition ofthese incon oe eanacetycing timesapowerseriesinlogp2,Thismeansthattheequations fromwhichthese eneeeeconstorobecietherthancanonialHelés,egurrens.These falsetheoremsarederivedmustthemselves befalse;incurrentparlance,theWardIlsreeuabIanedtofrnontrivialorderinperturbation theorybyexplicitFeynman identitiesassociatedwithbrokenscaleinvariancemustcontainanomalies falculations(whichgiveusinformation atallmomentum transfers), andinhigher Thetechnical reasonfortheoccurrence oftheseanomalies isnotdifficult_ta ‘ordersbythemethodofCallanandSymanzik (whichgivesinformation onlydonut understand: Themanipulation ofcanonical commutators required toproveWard‘momentumjransfer)Thetwoapproachesareconsistentwithintheiecommondomain identitiesisjustifiedonlyifweintroduceafGutoMHowever,thisdoesusnogood ; He Teaalcntecealanstonnaton andofthedesvaton ofWardiene, unlessthecutofTischoseninsuchawaythatthecutofftheorystillobeystheWard reehetappendswederivetheCalan-Symansik equationsfarGreen'sfonetons identities?Forsuchfamiliarcasesasquantumelectrodynamics orthesigma ofcurrents,andshowthatnoredefinition ofscaledimension isnecessary forthese model,forexample, thiscondition presents nodifficulties; itisstraightforward to‘objects, although theother anomalies remain, introduce acutoffinsuchawaythattherelevantequationsofthetheory(gauge invarianceinonecageandPCACandcurrentalgebraintheother)remaintrue. Forscaleinvariance, though, thesituation ishopeless; anycutoff procedure I.INTRODUCTION necessarily involves alargemass,andalargemassnecessarily breaksscalein- . yianceinalargeway.Thisargument doesnot,ofcourse, showthattheoccurrence ‘Thereisarecurrentideainparticlephysicsthat,insomesense,athighenergies, |TMA#RGEINATangeway.” Particlemassesshouldbeunimportant. Inrecentyears,thisideahesledtothe "Anearlierversionofthisworkappearedinpreprintform,butwasnotsubmittedforpublica: conceptofscaletransformations (dilatations) asapproximate symmetries ofnature, tion[5].AlsotheresultspresentedherewerereportedattheSymposium onDeSitterandCon- withpossibleapplications tothestudyofhigh-energy scattering [1],low-energy |formatGroups,UniversityofColorado,Uoulder,Coloradoure1910);andattheCasternphenomenology [2,3],andshort-distance behaviour infieldtheory[4].Sincein Treoretical PhysicsConference, StevensInstituteofTechnology, Hoboken,NewJersey,(October 120)Casaations sinitothoseofSetonIthavebendoeindspendey byBe,Bernt, P nie Energy Commis aad Sitin (6), $F24620-70-c-0030. ecutralpiondecay[7]. ‘552 } ~~, vpsdobishat, Tidvhoiuclion, Tusk saddhinQaaaaItispossibletoderivecertain _ . Wardidentities forscale invariance (brockn scale invariance!), called the_ AtraceidentityinthispapersHowever,itturnsoutthatthisidentityisme) ______ felseinseveral ways:first, youcancorrect themsotheyagreewitha _. lowest pert-theory calo byallowing fields toacquire anomdlous dimensions But - tohigher orders youhavetodoevenmore,thissuggests thattherearemore_| _-.. -.than_just masstermsinthedivergence ofthediletation current. Theirresults _ __ confirm earlier conjecture ofWilson based onhis study ofThirring model (4-fermion __... im42), andtheirresults arealsconfirmed bytheCallan Symanzki equations. __ . This papercameout_as preprint imMay70,wassubmeitted inmodifiedforminDec70.Theirbusiwork isunderstandeble, buttheir interpretation seems_ "Weak,Theydonotreallyunderstand exactly whatthepoopis.Buttheirlong appendices onSeale andWardareverynicee ___ ee _ Outlines | ce — |. _—. —I+_Introduotion. __ - ee —————. 11sSeale Invariance andBreaking Basics weeee ee gg 1: basics: D,,thedimensiomd, 0,2+TheWardidentities=0. ra)- 3.falsetheorem: showasymptotic contradiction basedonWard's =_me} - ______.. _4+Post-mortem: theythinkmabybe Bothanomdimensions andanomdivergences __ --—__1H;_Anomolous Dimensions - oe eee _______ dsGeneral: anomoly A=difference inGammawithandwithout cutoff fields My _ 24 4theory: toIowestorder) thereisnoanomolyatall.(fieldtheorycalc) _______3, Mukana:toorderg?anonolycanbeabsorbed bymodified fielddimensions, |. feDirectcheckthataappearsincomrelee| ___WV.PurtherAnomolies eee ee _ ~ts tngtthereipstintcontradiction atorderX’evmallowing at.=. 2+Comparison toCallan-Symanzik equations. coe.VsComelugionse 0 8 ceee _._ Appendix AtScale Invariance Theory, including Conformal stuff. _ _.Appendix B:General Derivation ofWard identiy foranysymmetry, ----.. Appendix¢:somefynman graphcalos __ . Be . Appendix D:Callan Symequations foyourrents. ne OycucaSeite ahcaggama6BaueedSaaleaveoonMokDasOpesatagaateaaah.=(08/2)AgVK-wes SinMarsachaScatelrsabiig pada. OK,Oaascktin paisuns beak a.WandVad WSR GaGa =excorQuonGeAgMalss “=eGDFRA).Geysah nn OR, eGBo ~QijpetkDese2akunigaBak,Aganng. Spemecca a+--BQ8)_Baty(Baa guna nakSpee Et =Peg[ecg aa : —.. .Sted ont ee ee Wa: akalbaquid ys)daowas.aad. Fake. ~~TILAnemoloua: OumenaiosaOca ihpnoh0Naaanan chao~~ 98Swann ha—(BS).—Te,Dasasonehamoatdocanagiht TORY. a —Rllgemodsh. Oesonal.Addo*aspatie Jild"Teasaglasar ge .Aig,}Sldecossdad baGocduadoudsiachuda. oddLaideweSR20-aGun)tse OKie,gahoOsoosWersaliany neared. lense - <<NeoJesadeadephe: ONTCSEGRNSSonwrdal(2.8). ——&, =MWopeasnbey ee ooMeee tinMg U5 peMe QkSEUeDemoancombolen 82 oe ae ene a mn ee ee and bende 8Ga)neeGa)balendbengly a DGaWiasS= Pay =akg sede feoDaneaSamomalNeomen Oo ___QikDaainden) = Wa. - Treat.Wiedeondebuckoonslok ODSSestaan! 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Ghoagtwr\ouath Arties Onany,Ora,onernaalies aa SSsores nn Appendix A: Basics ofScale and Conformal Trensformations. ie] PartI:Dilatations. IhaveworkedthesedetailsoutonmyownunderDilatations inGeneral Field Theory binder. The idea isthat the Poincare Group has 10 generators called PY andMY .These are assocaited with translations androtations andboost transformations onfunctions ofspacetime. Similarly, you can condider scale . trensformations and invent anew generator D,Ihave worked out its relation to the other generators, ie, commutation relations. Ifyour lagrangien has scale invariance (anyregular andmassless field theory with dimensionless couplings Ithink, you can just test tosee), then ofcourse there will be@conserved cureent. Ihave shown inmyNoether notes how this current comes out being that shown in(A.9a). Ifyou like, disaBiktle diagonal matrix which conbins the correct canonical dimensions forthefields, oritisjust d,aspart ofthesum. This isthe canonical dilatation current. Itisconserved inany scale invariant theory. Now, you canrewrite this diletation current interms oftheBelinfente symmetric tensor which weknow is"equivalent" tothecanonical stress tensor. Dropping a term which ha: noeffect onthe generated charge Dorthe conservation ofthe Co) dilationcurrent,weredefinethecurrenttobe(A.15).Ie,BforBelinfante.Now, ifyouforthemoment assume that the"virial" VW"canbewritten asin(A.16), then you canredefine your dilatation current once more toget (A.21). Ie,here 9isstill enother strees tensor beyond theBelin@nte one, andisgiven by(A.18). So, the divergence ofthis finally established dilation current isthe trace ofthis new stress tensor. Obviously ifyou have scale invariance this stress tensor must be traceless. But suppose you donot have scale invariance, ie,itisbroken saybysome term(s) inthelagrengian. Then D=D(t) isnolonger aconserved chanrge and yougetmodified commutators asin(A.24). Interestingly, though, [D(t), (x)] is not affected. This pert ofApendix Aends with aproof that themost general boson lagragian that isscale invariant andsatisfies theassumption (A.16) isa+theory, aswe might imagine. Moreover, the new symmetric tensor relevant tothis theory isjust the new tensor this paper istalking about all along. Ie, their new tensor ofthe paper maybeidentified with themodeified ciletation-current tensor ofthis appendix, atleast forthe#+theory. 6 Part 2:Conformal Trensformations. This ismyfirst encounter with thie conformal group. Ihave nofeel atthemoment forwhat thefour conformal transformations fo) shownin(A.30)reallydo.Theirgenerators arecalledkK".Soifyoucombinetheregular Poincare group withDofdilatations andthefourKY,theconformal generators, yougetthe15parameter "Conformal Group" which isEquivalent to S0(4,2). Obviously there isaLiealgebra, this isgiven in(A.31) forthénew commutators. Andthen youfigure outhowthefields transform under theK's, answer given in(A.3h). Ie,thething appearing onthe right side ofA.34 Iwould probably call#4similar toiydifferential operator. Next, wewant tofind conditions for alagrangian tobeconformally invariant, je,invariant just under theconformal generators (ahd Poincare ofcourse). The result isthis: inorder tobeconformally invariant, your theory must bed scale invariant anditmustbepossible towrite A.16. Well, alltheir theories have this A.16 property, soyou can say that infact they are dealing with theories offull conformal invariance, not just dilatation invariance. Next, inA.38 the conserved (inconformal invariant theory) "canonical conformal currents" are quoted, ie, Noether theorém. Itisnoted that scale implies conformal asshown in(a.36). Then you goonand drop terms and soontomodify this conformal current until itissimple. The result isA.40 orA.4l which Twill write fo) downhere: KaC0 1%OY K=O usw p=xoe ne dwdabahé $ BNv wr=eer xe WED aUnQewunnr quvek vorlk CY=DomudshoreJonsa. Theappendix goes ontodothecommutators ofKwhen itisK(t) inporesense of conformal breaking, and does some other things Idont care about now. This ismyonly reference onthis stuff, agood appandix, Iamglad they decided toadd these self-contained appendices. 6 OWarmdvrar Y= comm dil.eon ie) =XO 4Jon. Drspsxdotem aadraw veATwhe becswee os StWont Woofoskasenant OuchAL3408InsameOMfun TexaPALaOU . Oulu, nplygsi po oFS Yee a8" ew=or,adg? Why »XKaaa3'soeqtoxs?Agop—cioys| ++of|3%e"-3"a". . xSP Key yoniaad amdSMasaperadlly sigpmundeCes),(He). 6nfs ayy. .Xa =Kaa symYyensurth,boldogallan. Qua: ysy=yor. *Ghoahrs, BMasllsyrwmb. Natt dep N °)BPX =genni gQath q— gop)aN =938 wy)_oh3 98s 03")4"(asS)+97"(dag)stage”) a=Wo eyo8sghagar vay oS ~~ ppd3aa9K\"= gS3385=”=%rag=o. ~g¥Sayoxs” ->ros” +PMSand5OF *ong033TN » “-3dpWeots —YrsoS as30gyeweVy=~:=dws)dga Wot,saansidinmea ‘pslaaswe=(For "4. =o, Ore, ‘DpdAdsXsn° remmt+Gai0X9” ff +)claam=o. 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