B.W.Lee Papers

Benjamin W. Lee papers B.W. Lee Transformation Properties of Proper Vertices Lee & Zinn -Justin Spontaneously Broken Gauge Symmetries I - Preliminaries Lee & Zinn -Justin SBGS II - Perturbation Theory & Renorm Lee & Zinn -Justin SBGS II I - Equivalence Lee & Zinn -Justin SBGS IV - General Gauge Formulation Fradkin & Tyutin A.A .Slavnov Phil Lucht notes 197 9 Bl. Lee (1a73) i Votune468,number2 musics LETTERS 17epomber 1973 vate :0/ Usuall] :TRANSFORMATION PROPERTIES OFPROPER VERTICES Fy { INGAUGETHEORIES 1: ifGis : BW.LEE* tie: CERN,Gene,Switzernd a:andInstituteforTheoreticlPhysics",StateUniversityofNewYork,StonyBrook,USA ¢ if GisReceived2July1973 na “TheWard-Takahshi identitiesfor(singleparticlereducible) properverticesatederivedfornon-Abelian gauge Feynn tneories;therenormalization ofsuchtheories,previouslydiscussedintermeoftheWard-Takehashi identitiesfor dedued Greenfunctions, cannowbeconsiderably simplified Seal] -where| Indiscussingtherenormalizability andunitarity l=,4(ASHHB), wee oftheS-matrixin(spontaneously broken)gauge BORO OTCe rr1theories,theWard-Takahashi (WT)identities satisfied where69,=Wa(%q)isthespace-time dependent para a byGreen functions playacentral r6le,Therenormal- meter ofacompactLiegroupG.Aisoftheform M,gl0)] ization procedure isusually stated interms ofproper aJI], 94 i ;(ie,single-particle irreducible) vertices,sothattheAga|Z)"3,549), if=EO), theopWTidentitieswrittenintermsofpropervertices, =0 otherwise, cae ‘would enormously simplify thediscussion ofthere- ° malizea|normalization procedure. Itisthepurpose ofthis where£4=£a5apisthegaugecoupling matrixt?, We ilynotetoprovideaderivation ofsuchidentities. Inpre- have oeewke viousdiscussions(1,2]oftherenormalizability of 5MyBO tA Gant EP)thesetheories, onehastocontend witht?theWE Heth *ND—alGEy®)*ND :identitiesforGreenfunctions,andaconsiderable =f, [F{i =fog(theytAD ale amountofworkwasnecessarytoextractinformation Foss(Tete *ND i‘onproper vertices therefrom. wherefay,isthestructure constant ofthelocalgauge Weshalladopttheconvention previously used[3] group. Theinvariance oftheLagrangian underthis Thequa thatallfields[gauge fieldsbg(x)andscalarfields saugetransformation maybeformulated as Hare ¥_(&)]} arerepresented by@,where istands forallat- ae source Jtributesofthefields.ThusforthegaugefieldB%(x)(at+9)sel=0. oy Theaistands forthegroup index aytheLorentz. index pand athespacetime variablex,collectively. Summation and Weshallalwaysusearealbasisfor¢;,90thattherep- Wid}= integration overrepeated indices shallbeunderstood, _esentation matrix oftheLiealgebra (¢*]=1iseal isthege ‘Thus,forexample, antisymmetric. fonctionon4 Jnquantizing suchaLagrangian, wemustchoose : =a4 ‘agauge.Weconsider agaugecondition linearin¢: TA}=7| afte Doeenoseye” +DY,C)v,0) FIgh=Fo,=0 « @ isthegei Theinfinitesimal localgaugetransformation of 1°Thiswaseapeciallytrueinthefirstctedreferencein(1) Joteruse The infinitesimali 1?Itthegioup Gisadirect product ofmsimple soups, X=¥ 1@Gy©@~@Gythereacoimgeneralmgrvse gi‘couplings £1521«++.neWithin thesamefactor group XplAIS *supported inpatbytheNSFgrantGP-32998X. Gj,forcourse6”Bp e Permanent address. Anca 214. i | eign —’ Sse ROR MER an” SRRGAGENRMRRIEL MeBiBlinRCRA SE aa eres = 1973 Volume46B,number2 PHYSICS LETTERS 17September 1973 Usually, onechooses F,tobe ice,Xy[A] arethepropagators whenthefields@are ‘ constrained tohave thevacuum expectation values AFate ein Nowweareinaposition towritedowntheWT * identity forC[A]. First observe that if};isagaugefield, 118] mfgyt5] ve Meal;aglexeczupagia +!2) 0)VA 2ad O sanere ifisscala eld where a,&aregaugefsingparam.4,¢4B=4,+2X41]& eters,andcfaresomeconstants. Inthisgauge,the 18) iFeynman rules forconstructing Greenfunctionsare =exo(zut Sexpazyy deducedfrom theeffective actionSug(4] AZT5pex(ZU) Syl]=LlO]-4(F19]?+7M,gl], 3) eal)=LOFELD?+7Malley ©RecallingthatFinan,wewet0,(4)at where@andcarefictitiousanticommuting fields sma 5 whichgeneratetheso-calleddeWitt-Faddeev.Popov-F,[A]=m4[at+f(4+i,[Alall ghostloops[5,6]andMyyisgivenby fi Wg paras 3 =F(A: , ty sixD]e fiMegld]=F,(+89) - xangaer5]0.‘Theoperator Misingeneral notHermitian sothatthe Nowdefine [4]byhost line isorientable.. > fe) Inthefollowing weshallbedeating withunrenor. PLA]=Tl) ~FF,[4] 1} malized fieldsandcoupling constants, anddimension. Thenfromeq.(8)follows Weallyregularized[7]quantities,Ashasbeenshown 6rd] 16 1selsewhere[3],thegeneratingfunctionalWJ}of 0G[vw+4(4+.Ebel +ae Greenfunctions satisfiestheWTidentityt? a toy t 18 so) 6) afl s (io(<f8]salegt ay)![Eo]#1=0.wemustnowexamin ouge @ plbya[ayh5] ' ThequantityM-!(5/i5J]WU]istheGreenfunction 7184BL"By. forthefictitiousfieldcinthepresenceoftheexternal 5 5sourceJ. =iXgl)SeMgt[4+xsl an to} Thefunctional Z[J]definedby * rep WU=exp(zis)) safes [avir} real isthegenerating functional oftheconnected Green functions. TheLegendre transform [4]ofZU]: *Upondefiningt* se : MA]=2U1-J4, A=8218, J,=50(AY64, eat 3(3GpnlAleos[asixgs] isthegeneratingfunctionalofpropervertices[8].ForsothatGga[Alistheghostpropagatorwhethe (11 laterusewedefineX,,[A]by fields@areconstrtined tohavethevacuumexpecta: Me- tionvaluesA,wecansolveeq.(11)for¥f[A 7% Hilal=64/84, o eOD ettAlLou Xi{ABT [AI/5A,84,=6, 1°tereonemustdistinguish betweenanoperatorequation ”iAIST =by suchas(4x9)"=1494249?andtheequationresuling fe) 19Anequivatent diagrammatic derivation ofea(4)igiven fomitsaetiononaesetainFunctionofx,suchas(L20)*L inret 2). =I 25 Volume 460,number 2 PHYSICS LETTERS 17September 1973 Volume bo fields!$.Furthermoreitcanbeshownthatthedi- O)wriar-ig x,iG,412G2A.62)serpentparteoftheenormatzation constantsare kinsensitive totheparameter ineq.(2)andthere- Noweq.(10)maybewritten as foretherenormalized S-matrix isindependent ofthe 5 gaugest7,Thedetailshavebeenworkedoutand LilAl gzPyl4]=0 (13) willbereported elsewhere, ‘ 1thank S,Coleman, G.‘tHooft,M.Veltmanand whereJ,Zinn-Justin formuch help they have given me. LlAl= At+4,+7714] aa) ‘+5‘Thediscossonsinrefs.(1]weretailoredtothegroup an‘SU()andothercompletely reducible compact Liegroups:114} = inwhich theproduct ofwoiereducible re[A]=Fybgi{Al. 19 (Gesgroupsinwhichthep 7Gal)=Foal] a9 resentations RandR’containsathirdR"atmostonce). 16The renormalizabiity inthemot general case, end theNotethatG~1[A]isthegeneratingfunctionalof ‘gaugeinlepeadence oftherenormalized Smatsxhaveproper vertices with twoghost lines, sothat alsobeenworked out,priortome,byG.‘tHooft and tageM.Veltmanbymeansoftheirdiagrammatic analysis G523G;210) (private communication) andbyJ.Zinn-Justin (unpub- lished manuscript) onthebasisof6.(4. istheinverseghostpropsgator: 47Thaveaotfilivestgated thecaeInwhich Fisof tet mension twoandnotlinearinfeds,PGEay w haty=(0 re fat64; References °isthepropervertexoftwoghostsatcandandthe Sincefieldatete. [1]BLW.Leeand3.ZinnSustn, Phys.Rev.DS(1972)3121, explaine .i 3137,3155. : ‘Thesystem,eqs.(12)(15),isourmainresult. 9)OuchandM.Veltman,NucleasPhys mechani Eq,(13)defines non-linear gaugetransformation ae Homan, NuclearPhys.B50(1972) withtherich taves7,4)invariant,Itisgeneralization of [3]BIW,LeeandJ.Zinn-Justin, Phys.Rev.D7(1973)1049. Hamilteeq.(1)andreducestoitinthetreeapproximation. [4]G.‘tHooft,NuclearPhys.833(1971)173. HoH(nthebassofes.(12}-(15), theconsiderations [6]BS.deWi,Phys.Rev.1620967)1195,123, inSsofeas. tne ‘ddeevandV.Popov,Phys.Lett,258(1967)29. 7 oftef,(1]canbegeneralized readilytoanycompact 14)&«(HooftandM.Veltman,NuclearPhys.B44(1972) hasbeen Liegroups andarbitrary representations ofscalar 189. sumrule [8]G.Jona-Lasinio, Nuovo Cim.34(1964) 1790. anexpla main dif which is. with the planatios particula Lagrangi o-model, Thea ‘can acco XSU(3) ii)theSt 2"Apers baryon PA deta breaks handle 216 AEN, Peecerrc hE oe, itiiseiatabMehEaseseEEa Ben Lee's letter Le) Thepurposeofthisletterisveryclearandsimple:converttheWTidentities from the old ones interms ofW[J] (which thus involve the Greens functions )to new ones interms ofGAMMA[A] involving the proper vertices. Obviously the proofs ofrenormalizeability are going toinvolve the proper vertices sonice tohave Wards in terms of these vertices. This stuff is of course contained in Lee's review, but Ilike the idea ofhaving the original letter just topin dates. 6 8 274 Yotume601,number 1 PIIYSICS LETTERS 22December 1975 Voluine xJ arepol: 13 thermoPe GAUGEDEPENDENCE OFRENORMALIZATION GROUPPARAMETERS alltern igINGHOST-FREE NON-ABELIAN GAUGETHEORIES fete we satistie J.FRENKEL functic| ES InstitutodeFisica,UniversidadedeSooPaulo,Brasiia pearsa Ee.stants, te Received 30September 1975 integra & view of 3 Wediscuss thegauge dependence oftherenormulization group parameters inadss ofghost-fiee non-abeian gauge onyPst theories.Weshow,usingthedimensional regularization withthe“minimal”renormalization procedure,thatthese mensic % srameters are gauge independent. 2parametersaregaugeindependent. etsothat: “Therenormalization groupequations{1]playan oeyeRysih ofu(timportantroleinthestudyoftheasymptoticbehav. 8°"M2318 io} Weno zjourofrenormalizedfieldtheories,especiallyinview_where¢=4~nandzisanarbitrarymass,whichsets which “4ofthefactthatnon-abelian gauge theories areasymp- _thescaleofthetheory. Furthermore, itisnecessary to ofx2 * totically free[2].Aninteresting problem isthegauge _rescale thegaugeparameter n@suchthatthegaugefix: neede: ~ independence oftherenormalization group param- ingtermin(1)remains invariant. Thisrequires: resid B; eters,orlackthereof. Thiswasalready discussed by@9«7-1/2 ® peinde“ numberofauthors {3}fortheclassofcovariant wees Mae +1)7] ¢ Lorentz gauges. Wewilldetermine the(dimensionless) counterterm indep: “4 Here, using then-dimensional regularization Zz,using the“minimal” regularization scheme [8], We 3 scheme [4],,Wewould liketodiscuss thisquestion in which inthiscasecontains onlyinverse powers of¢inthat, “4thecontext ofnon-covariant “axial” gauges, which do anexpansion about ¢=0. | mst} fi notIead totheappearance ofFaddeev-Popov ghosts ‘Thepropagator forthe.vector-boson fields,fors© atedi[5].Forsimplicity, wewillrestrictourselves toapure =0: and( Yang-Mills theory[6],sinceallessentialfeaturesare 1 nDyt "Pp,PyPv> thee»3 alreadypresenthere.ThistheoryisdescribedbytheDEMpn)=oti ot ve depen](unsenormalized) Lagrangian: P P (py .Feasor| a 0=109,492—2,492+gyabeaobace)? leads,bypowercountingarguments,toarenosmalized presei y LOSSQWASEDAE+BPAPASS theory.Remarkthat,sinceD&®isafunctionofzeroth ofZ3 22) “degreeinn,ithastheimportant property ofbeingin- As| ;1 ‘a ia y—35(ngage)? variantunderthescajfign~>sn,wheresisanarbitrary show2p MwAn (#0) parameter: ticle where/*%arethetotallyantisymmetric structurecon 2p,n)=DX. fiest 1stantsofthegaugegroupGandn?isanyfixedfour. Pawo") =Dab(P.s")- © . : vector (ngn9 #0).Inorder toobtain finite Green Using thisrelation,wewillshowthat2isindepen 3, gfunctionswerescalethefields42? dentofthe gauge parameter n,whichimplies,aswe* oc=Z1? 4a ‘willsee,thegauge independence oftherenormaliza ontE aia “Fon groupparameters. Theproofisinductive. grou “Inthedimensional regularization scheme itisconven- ‘Consider aFeynman diagram, which ismade finite, - ient torescaleg®suchthattherenormalized coupling orderbyorderinaloopexpansion,upto/loops,by («3 constant isdimensionless. Asaconsequence ofthe ‘counterterms which areindependent ofn.Thisistriv- Slavnov-Taylor identities [7],which hold intheab- iallytrueinthetree-approximaton (7=0),where Z3 whe 3 sence ofghosts, onefindsZ=Z3,sothatwecan =1,Toorder (I+1)thediagram willexhibit poles (up= write todegree/+1),withthepropertythattheresidues = 4 = ene ei ©) Volume608,umber1 PHYSICS LETTERS 22December 1975 = arepolynomial intheexternalmomenta [4,9].Fur- tanthermore,using(6)andbytheinductionhypothesis, pat 5a. Pic allterms,including theresidues, mustbeinvariant un- alles |Bllgoing” -derthescalingn+sn.Notethatthisproperty mustbe ono ‘owe ©) ia‘satisfied independently byrational andlogarithmic yd | &functions.Wenowobserve(rom(3))thatyonlyap-7"Hu987virosr Ei pearsasapowerofu*/?,multiplying couplingcon- ‘orto astants.WhentheLaurentexpansion ofeachFeynman Althoughthedimensionless parameters 8,6=54it,/n?, ayintegralismade,onlylogarithms of»willappear.In‘ycouldaprioridependonnviafunctions ofn2/u?, Bxviewofthefirstproperty,theresiduescoulddepend -_wewillnowshowthattheyare,infact,independent a‘onp,viatheselogarithmi¢ functions, onlythrough di- ofthegaugeparameter n.Toseethis,letusapplythe be mmensionless ratios ofn2/42. However, these terms can- operator7,0/87,oneq.(8).Usingrelation(7)wear- Senotsatisfythesecondrequirement, mentioned above, tiveat; oa sothatweconclude that theresidues areindependent a ‘‘ofu(Voradiscussionofthispointseealsoref.(10]).[rodeha(ze5,)aN(teae" ar ‘WenowremarkthatZisadimensionless quantity, * * . ~ Ra,whichcandependontonlyviadimensionless ratios =0. (10) ae‘ofn2/y2.Asitspolesaredefinedtobeprecisely those aneeded tosubstract thepolesassociated withtheabove Wehaveimposed nonewconditions onI,sothat aresidues,whichareindependent ofy,Z3mustalsobethisequationmustbesatisfiedidentically. Thus,the aejrindependent, Itfollowsthen,thatalsotoorder( coefficients vanishandweobtaintherelations: beois 1),wecanremovethedivergenceswithcounterterms 4,agian.=0, ra independent ofn,whichconcludes theproof. taOB/9rtq=O» (i SaWecanreachthesameconclusions byobserving ng.36/9ng=0, nb) anathat,sincethetheoryisrenormatizable, theresidues ‘oo esrusthavethesameformasthecounterterms gener- - igatedintheLagrangian(1)bytherescalings(2),(3) aOya=O, (19, Eatand(4).Separating outexplicitlythedependence onwhichimplythegaugeindependence oftheparam- ie theexternal momenta, weseethattheresidues could eters, 8.andysincetheyaremomentum independent ed dependonmonlythroughfunctions ofn?/u2.FortheFrom(9)weseethat(1Ic)istriviallysatisfiedasthe oereasonsdiscussedabove,suchfunctionscannotbe counterterm Zisgaugeindependent. Eqs.(11s)and _present,whenceweconcludethegaugeindependence _(11b)canalsobederivedbynotingthat,fromeqs batofZ3 (3),G@)and(9),wehave(inthelimite->0) ve Asaconsequence ofthisproperty, wecaneasily pe b= a2 BFshowwiththehelpof(5)thatanyN-particle (one-par. ag r widticleirreducible) Greenfunction™(pz,g.n,u)satis: Usingtheserelationsweseethat(1.12)and(11)fol ae fiestherelation: lowfrom(1c). q#8 3‘Asanillustration,weobtain,withoutmakingany Ayng5e-PMPs8soH)=0» (7)assumption aboutn,,inthelowestorderofperturba- Mae a theory 1 Ee‘Ontheotherhand,P™satisfiestherenormalization . og” BSSgroupequation[1} zaidSpt ayet Pee 3fane® m4" 4= (udrogers,3--™)P(pj.gsn,4)=0 (8)whereCisthevalueofthequadraticCasimiroperator fe. intheregular representation ofGandisgivenby b* where 8,8, and7aregivenby: SanC=Saatoar:Relation(13)showsthat2isinde- a ©,pendent ofthegaugeparameter andyields, using YS (9)and(12),thesamegaugeinvariant resultforthe6 is 75 bye wet | PAS z | *Volume 608, nubmer 1 mysics LETTERS 22December 1975|Votun i function,astheoneobtainedincovariantLorentz References ‘ ofgaugtis[1] | ~ Finally, wewould liketoremarkthat,despitethe11]CG.Callan,Phys.Rev.DS(1972)1670. | are gaugeinvariance oftheresult(13),itisnotpossible to K.Symanzik, Comm.Math,Phys.23(1972)49. ¥choosethegaugen?=0,whichwouldgreatlysimplify ee9me)0zeeYsRew.DB.(1973)3633;| “Sthecalculations(seeq.(5)).Infact,anexplicitcaleu-13)pat.JenesNuclPhys.BMS(1974)$31. : BsationperformedinthisgaugeyieldsforZ3adifferent W.E.CaswellndF.Wiezek,Phys,Lett,B49(1974)291| Es(and,hence,incorrect)result.Thereasonforthisdif- H,Kluberg-SternandJ.B.Zuber,SaclayPreprintD.Ph.T/| “8 ferenceisduetothesingularities n~?whichappear 14-56, |whencomputingtheFeynmanintegrals[11].These [4G.'tHooftandM.Veltman,NuctPhys,B44(1972)189, Be singularities, whenmultiplied bythefactorn?,which ts]FonPaddsev andVAN,Popov,Phys,LettB25(1967) : appears inthe propagators, yieldwelldefined results ESS.Fradkin andLV.Tyutin, Phys,Rev.D2(1970) 2841 3 which,ingeneral,donotvanish.Duetothisfact,in [6]CN.YangandR.Mills,Phys.Rev.96(1954)191. =ordertoobtainconsistentresults,itis(unfortunately) [7]1.0.Taylor,Nucl.Phys.B33(1971)436. Intro EE necessary tousethefullexpression (5)forthevector 10AkBren,TearandMathPye100097298,bosonpropagator, withn?#0. 19]G."tHooftandM.Veltman, Diagrammar, Cernzeport K= J 739(1973). tude {10} 5.C. Collins, Nucl. Phys. B80 (1974) 341. nece a [11] W.Kainz,W.KemmerandM.Sehweda,Nucl.Phys.B79 data %(974)484, wi 2 expe « In] ; phas| secti = relet ce inessc| “4 applie evide| : that inth ™ thisy a| a 1.Ay | |sh4 = w a fishe J eros: = bya a] = “y — R = sc ag _ ;oo i 16 ~ pe ee .3; a 2s . @) _—_sFronleel's letter ofDeo1975 Interesting .and perhaps relevant toour simple anzatz. Heclaims that, when you work inthe dim reg framework with that mass scale parameter MU, you can show that intheaxial gauge theZ,andthelittle renorm group functions like BETA areindependent completely ofthevector ETAorn,.Thus youcannot have Z(ourgamma). The Z's ere ofcourse the residues ofpoles atn=4; asanexample hecomputes what isIthink ourBorn loop and shows that theresidue (ieZ)isinfact independent ofn. Ishould check this calculation. Youonly eed theleading term. Ishould also goread about AFnow that Iamwith the dim reg stuff. Another conment mateinpassing isthatyoucannot teken”=0because something goes singular andyou get thewrong answer. This sounds similar tosomething wedid * wrong once. Asabonus, Iamreinforced about renorm when heclaims that our form for the propagator makes the theory, bypower counting argument, renormalizeable. Iagree. eo) Thisisagoodpapertostuflyinmoredetail.Seeifyoucanduplicate itsbasicresults. 6 Lee«Zinn~Tistin T yy. . 7 = +oe th ioe SORE s FIRST ANDSECOND FACTORIZATION IN... 3121 yeeed|ae oeLixssy,Nambu, Phys. Rev.D4,1199(1971);1.T.Drum- .PraPiag’Praag) es Fi7e mond, Nucl. Pays. B35, 269 (1971);F.Csikor,ITP- Faageai—Fay i .iSelapest Report No.292,1971(unpublished). Pre Pratngog 4HS,tasanexample,weconsiderthecase!=3,may1=2, oenang" Ftalng19). :BSmy.g2, m0otherwise, Thenwehave cc) ‘ Se Shoe ey west . (Prctatiengoi"Pa-tayy) .fe +aPLAPLRPDTPAPER ana .Toy a :PS.+mBq.@4),C,thepolynomial inthecrossedvariables, peteyay5Thypilpalmyed ‘ Fi/3_waswriteninafactorized form.Ifweinverttheorder ®EaPwPay °he ingofvectorsinbothLgiageeea,andRF",theresult yalmytae.pra)”es" . tmwillremainvalid,Le.,wehave (Pagina PP). es) ye “sienna, ‘Theamplitude ofEq.Q0)follows fromthefactortzedBeOrbayopeg EE 4) formofEa.24").Fornotational convenience, wehave .where : also made thechange my--m,-1- > t bes Oe - . PHYSICAL REVIEW D VOLUME 5,NUMBER 12 15JUNE1972 i Spontaneously BrokenGaugeSymmetries. [T.]Preliminaries 2 Hi BenjaminW.Lee g4 National Accelerator Laboratory, P.0,Box500,Batavia, [Uinois60510 3}andtastituteforTheoretical Physics,StateUniversity ofNewYorkatStonyBrook,StonyBrook,NewYork11790 5 J and 3 ae JeanZinn-Justint 4Af nstitute forTheoretical Physics, StateUniversity ofNewYorkatStonyBrook, StonyBrook, NewYork11790 4 (Received10March1972) ‘ 2ea ‘Thisisthefirstofaseries ofpapers addressed totherenormalizabllity question ofspon— 3 L taneously broken gauge theories. Wegive abriet outline ofthe motivation forsuch aninvos- 2 ‘ tigation anddesecibe themanner inwhich therenormalizabiity ofsuchtheories willbe proved ; Ry Inthe soquel, Putbriefly, wewillshow thetinanappropriate gauge, ultraviolet divergences an ofaspontaneously broken gauge theory areremoved completely bythegauge-tnvartant coun- ’ Fr terterme intheLagrangian whichwouldmaketheGroen's functions ofthocorresponding un- ). 2 broken gaugethoory finite, thatthe$matrix computed inthisgaugeisunitary, ahdthatthe :H $matrix isindepondont ofthegauge chosen, Inthispaper, therenormalizability question ofis theunbrokengaugetheoryisconsidered, WedorivetheWard-Talahasht Identitiesofthe ; theory. Wodiscuss soveral ways ofregulating divergent Feyaman integrals ofthe theoryid ‘withoutdestroying gaugeinvariance. Infrareddivorgonces areavoidedhythedeviceofinter~ tnediate renormalization, wherein wechoose assubtraction points some points whore exter i nalmomenta areBuclidean. Thissuffices toestablish thattheBogoliubov-Parasiuk-Hepp re- ‘ oa normalization will give renormalized Green's functions which satisfy theWard-Takahasti > Idontities. ‘The existence offinite, renormalized Green's fuictions satisfying theWard-Take- hashl idostities provides uswith themeane ofproving therenormalizability ofthespontane~ . tusly broken symmetry case. TheWard-Takahasht identities wore previously derived forthe . gauge bosons bySlavnov.. Woprosent horeanewdarivation..‘The discussions onrogulstiza~ : tion methods andintormediate renormalization procedure and therenormalization conditions forsintter fields, webeliove, arenew contribations ofthepresent pape. _ELINTRODUCTION _ whosemassesaregenerated byspontaneous break-down ofgauge invariance ofthesecond kind,*and ‘This isthe first ofasertesofpaperswhichwillofconstructing afinitetheoryofweakinterac deal with therenormalizabilityofspontaneously tions*’~*promptacloserexamination ofthequan- broken gauge symmetries. Theintriguing possi- tization andrenormalization questions oftheories bilities ofunifying electromagnetic andweak'inter~ ofthis genre. actions interms ofYang-Mills gauge bosons,“ Inthesequel ofthis series, wewish toexamine 4 Lee &Zinn-Justin I, Preliminaries written March 1972 1,Introduction. Motivation isunification ofweakandEMinteractions. NomentionofQCD. fe) There willbe(19Discussion ofHiggs andstability (2)Theorem: renormalizeability of unbroken gauge theory implies renorm ofbroken theory. Actually, this first paper will not getinto thebroken theories. (3)IntheRenorm gauge, unitarity will beun-obvious; nevertheless, unitarity willbeverified (4)theRenorm andUnitary gauges willbeshown equivalent. This isthe program for the entire series ofpapers. The aim ofthis particular paper istoderive the Ward Identities and then construct Greens functions that satisfy them. These should berenormed and finite greens functions, sothey will have todiscuss how they are made finite, where they are subtracted, etc. The BPHZ will be used. Intermediate renorm means that they will choose asspacelike renorm point some point other than where the IRproblem lies. Nobig deal. Ward Identity derivation credited to Slavnov. . 2.Quantization. Theusual stuff. TheFPdeterminant iswritten asexp[Trflog...)] andthen Trin(..) appears aspart oftheLagrangian. Itisnoted that this extra term canbe interepreted asghosts intheusual way, andIcould ifIwanted figure outwhythey are fermion-like. Itisshownhowgreensfunctionsareobtainedbyfunctional derivatives on fe)2[3], which bythewayisthething that gives connected graphs. 3.Ward-Takahashi Identities--I. Ihave made detailed notes onderivations ofquations inthis section, Iderived everything except, unhappily, Iwasunable toexecute thefinal step toget the key result ,that will have towait because Isimply don't know how to doitright now. Here ishowtings went: W,isthegenfunc without theghosts: theFP ghost detispulled outasin(3.1), anew idea tome. Now, obviously making agroup variation onintegration variable ofanintegration cannot change the number that is theintegral, soRHS of(3.4) iszero. Onthe other hand, your variator reaches in and"varies" parts oftheaction which arenotgauge-invariant. Recall that, although Fy, does rotate asacolor vector, A,does not, sothegauge-fixing term (arising from the delta) andthesource term aregauge-violating asshown in(3.3). Thus, ineffect, the gauge-fixing variation must cancel against the source-term variation, and thus you have Eq.(344) which isWITforW,. Butofcourse youwantWIIforthefull.W.Putting the ghosts back inisaccomplished bythe horrendous identity (3.5) which was derived in appendix A.Thus, you get (3.8), where Histhe greens function osoperator Dd. This resultismanipulated finallyinto(3.13)whichistheresulthewilluse.Thisisan 6 amazingly complicated, functional-derivative condition onthefull W[J] forapure Yang-Mills theory quantized correctly. Igetaresult close to(3.13), but Ihave asign difference, plusIcantdumpttheoutside Ddoperator. ‘wadok -2- This (3.13) isthefamily ofWTI's fortheGreens Function genfunc. The claim isthat fe)thesimplarresultforthe1PIfunctions ismuchmorecomplicated! Slavnoviscreditedwith first derivation ofthis. they are called Taylor-Slavnov identities. ksWerd-Tekehashi Identities-II. Basically, theideahereistodefine therenorm parts ofthetheory (they are: ghost-prop, gauge-prop, orrespective self-energies, the triple-glue vertex, theglue/ghost/ghost vertex, andthe4-glue vertex. )Intheir discussion, onlythe1PIversions ofthe4-glue and3-glue enter. Called [",The ghost/glue triplevertexiscalled4andisalso1PI. Inthemiddle ofthis section wehave theghost self-energy Dyson, (4.14). Idont see aglue-Dyson orother Dyson equations. Themajor Ward identities are(4.14) [which relates glueprop,ghostprop, and3-glue andghost-glue vertex[] andalso (4.17) which relates the4~glue to3-glue andthe glueprop. Strangely, theghosts dont seem toenter this oneatall. Whyarewedoing allthis? REcall howinQEDyouusetheWard identity tosimply avoid worrying about theelectron self-energy. Iassume theFancy Wards will beutilixed inasimilar way inNAGT. Notice ofcourse that all functions sofar defined are “unrenormalized". Later C7) inezine hewillrewrite theseequations fortherenormed objects, andwillstaterenorm conditions, defining various X's and soon,This isthe part Iamwating for!!! 5.Regularization. Igather that the dimreg method was not yet popular, perhaps tHV had notcomeoutyet.Sohereisthegistofthissection: Firstg theyconvince. themsevles that they can render finite any graph involving only one loop. This was done inapaper ofHooft.earlier. Themethod isthis: justaddsome"regulator fields" withmasses mj” (spinor and scaler fields may both benecessary). Then adjust the couplings ofthese regular fields, and their masses, tosatisfy certain linear conditions such as(5.3) thwough (5.5). Then you discover that, when you include Feynman graphs which include these regulatro fields, you have cancelled all divergences atthe one-loop level. Ipresume that you can interpret these “regulator fields" asterms: added tothe original lagrangian, afinite number ofterms, sothis isOKtodo. But, itseems that this “regulator fields method" will fail for higher-loop diagrams. Somore stuff is needed. That "more stuff" istoadd fancy derivative YMcoupling terms tothe Lagrangian asshown in(5.7); cutoff appears inthese terms. The effect ofthe addition ofsuch terms is:1)regulatesthegluepropagatordowntopowerceinsteadofkK"?sothismakes fe)higher-loop graphs tend toconverge fast since lots ofpropagators.; 2)butyou also have added all kinds ofcouplings, like &-glue couplings! Nevertheless, itisclaimed that theresult ofthis propagagstor modification is: only one-loop graphs aredivergent!! -3- Thus, bysochanging theglue propagator (byadding those derivative couplings), youhave fo)putalldivergence probléms intotheone-loop sector.Buttheseone-loop graphscanberendered finite bythe regulator field method! Thus, you have solved the whole thing. Moreover, amajor point, the idea isthet the counterterms you have added tothe Lagrangian todothis regulation were gauge invariant! Eg, you know that Fisatrue color vector, éFisnot, but DFis, and only DFtype stuff appears in(5.7). Thus, these counterterms have noeffect onthe Ward Takshashi Identities !!!! Touhave regulated without hurting gaugeinvariance, andthisisthemainpoint.Ofcoursethedimregdoes this too inamuch fancier way. You can goontoadd higher propagator derivative couplings toimprove the W convergence ofyour propagator, but this never makes the one-Loopers finite. Appendix Dshows alittle how you power count graphs with this propagator modification| and you draw your own concludions, fine. 6.Renorm Conditions andIRDivergences. Suddenly, without warning, alltheGreens function: theyareusingaretobeunderstood as"renormalized". Thus,firstworking withof-0as. theirrenorm point, theychoose J(Kk*)=1 atk°.0. Obviously 4fwehadanunrenormed theory youcouldnotaribtrarijy setthis,ifwouldbesomedivegent thingdepending onthe ©) cutort. aut inthefunctional method, you can Just let everything berenormalized. Recall that the multiplicative renormalization isaccomplished byadding counterterms tothe Lagrangian. So,Jatatk°20istherenorm condition onthe,glue-prop (orself-energy). The ghost residee isalso settounity. Thetriple-plue vertex isproportional tothe bare vertex since other tensors involve more p's and are thus smaller! The renorm condition onthetriple-glue isthat atthezero point itisGtimes bare tensor. What about theghost-glue vertex? Itstarts outwith itsownconstant G',but then theWard sets this tosame Gasintriple glue. Andwhatabout the4-glue? Ithastwotensor forms withconstants FandF'.But thentheother Wardidentity putsF-0”andF!=0.Soforallthevertices, there is only one renormalized charge G. Now theywant toshift thewhole thingtoanewrenorm pointsothingsareoff- shell attherenorm point. Itisnoted that holding things offshell tends toremove IR divergences. Thisiseasytoseewithalittle Ploopgraphdk/(k7k'2) which onlygoes IRdivergent ifq=0. Sothemotivation to"renorn" offshell isclear. Howisthis imimplemented? For theglue andghost prop, its easy. You just choose Jeland Z_=1 atfe)w?o-a®,Trivial.Theverticesarenownotsotrivial.Thetriple-giue nowhasSaree tensor forms, the bare vertex plus two more. Constants are called G,H,J. And the -he ghost-glue vertexnowhasG'plussixmoreconstantstoworryabout.Thenthefirst ce]Ward identity relates all these constants. But, asyou can see, only Gand G'are divergent constants because those other tensor forms have higher powers ofmomenta inthem. Soyou can read equation (6.9) to also say: (Clay =Goiy,,, Butwait, these constants arenotdivergent. OK,I think they have left something out: ie, they never defined divergent Z's, but Ithink Inow see how to do ite Bythe way, notice that they say nothing about the 4-glue vertex inthis off-shell renorm case. Probably ithas amillion tensor forms. One ofnylittle problems isto use group; theory totry and find these tensor forms. Conclusion: good tosee some real live renorm points being selected. 7.Renorm ofthematter fields. Uptonowwehave had pure Yang Mills fields only. In this section wethrow inavector ofscalars (massive) belonging totheadjoing rep. Then the theory has more vertices which are given names. Then arenorm condition ismade for thenewvertices. Ie,youhave abunch ofconstants multiplying tensor forms, asusual. Now, theWard identity including thenewfields connects thenewvertices only withthetriple-glue, whichyourecallhadaGconstant. Whenyoustuffyournew 6renorm conditions into theWard, yougetcertain results like OaGetc. Again, everything can be related back to G. Ofcourse thenewfields couple with thesame bare coupling go,sothis result does not surprise me. Finelly, 1&2gobackandredoallthisstuff using -a”asrenorm point because thefirst timethrough theyusedq”-u thescalar mass, andthishastheusual IRproblem. Then suddently the paper ends. [oneofthemainpointsisthis:towritedownyourRennpmConditions, yousimply pickarenorm point andyouwrite down allrelevant tensor forms andyougive each of them aconstant multiplier. Then yourelated thevarious constants asmuch asyoucan using thevarious Ward identities. when youdodo,everything boilds down tojust one constant, theGonthetripleglue,therenormed charges\ Appendices: Ihave elready done AandDinthemidst oftheabove. Appendices BandC deal with theidea ofproper vertices andWards interms ofthem. Iwill keep inmind that these appendices are available, butnow isnot the time formetoread them. 8 . BD 23) . ~ Sone - =taBRAS Bo aFBR) LYOES adstegphaaaahdspaaarneh. > — a 0 r =) =spacermoluyy Statute Me GR) DT (©stvomnes \e, MaGas)=-3wn.a3G-9)lee (Way=StyVrs) ond Bay >KOSe) avon Onan?KeCPlkT- QWeALR . on --@lbs-y &Le] ONS2I Que,dung)beaJ, *4 ach=ayea]ok|8] oanyoke [EWESE-A',| ax(y)= ee BW,KOOWas) ThsaLees]TZee)=aks fe enesio nt aRLEOERS] eels-yEPROT we CUOEPA Fol +o Fo[ sy &GSRpe ee Asem du- — ~t- ° 2,OF[SRaoT” wee © * A - Gokakmed. Exgorswk a =Gay+SBRAM AKSAteteAKG 3 “sp )\¢AcaypF) 304,%) &Rw)56am), 3ks) : > py Oy g. a nROSH 6-H)940)} 3 >A om) =~Gg)tLawardee[SRO3G.) le) 3 SesENED Iq9(6,4) SA%s) Bil305) 3 3 ny 3 by=~GSY AeLanateds |SA)3,406) GAGA2.5Oe) SKor 335.x)| N32 ON| cy OH Me (x) » (3)amanads N69[3.36620] ALO[aeg09)|ALGS|3stvs} — :.43 ay Oe' 0Cenoh#ZW Os = wae ad, a =“Ss.Co:CageioeCagesCastghy==eee — -3-- > * TESloopveSoe sARR a (x) Ae OW Ms 3)+Vandndys Rioay8a4oon))ReoYaso|Moola (0%) Woke+adersraill save\)colaMadeshbra+(Qalledduaty qh,glum)oarLR 6 8 — (AB),(A,(2-6),B69, BRD,(B.S) (3)depgnain A. 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Bow sr(Ly= So BBV:wgk[es)\ @sfe)G&S) GBN)Sdeve Oust jrSeas5Buds (SUR\witl)-» e% peGay ones DB(YD) sere RUS=O Wag: [SGgBS\=0 cktagenmestFOVK). ADan,Deoombertom DakDhbnyokSus,O06Basecond Yamwar(3:).So >» **),>jpQhAl»a.-Sv(Se~yeitanensfoll) 07 U N ° » wopS \ . b vrbeg=GE) \x33:Say7ROHVhEAesie[were ry 8 Dandi (AS). oyBYpindSKwealdospond: TMG =YayS69FeHey) ThawOkeShyKekeOiaahAmokeea huedeme.Silke6: DLancy ~adSaysoe” Wes)| =Yay(8Sees}[O8Wess) +SaySees)[TTAOy)] =YaySorgy I4PHO) ~[xm Hass LSHes" ="a NedalleTrnnaguiseaeSearwehuskin, 2 a x de =kss[Seat-yeARo] HBHeol? Asp ELNG] =deeeekbuoleoua oxBSJN). JockYom mgtEasKARO THE 7 =*done.@a)reetonyad e) Qovine(AS)(39) : ©ooGauk(08)et)4uaeKSSObeghhe Jajh-Wand ade (AS)aestalehompope:Ka onear bsgorwig SE](VRHOD A(RV53K nigLLLCPMOP(Swag ER Wao2ayxeedotbyaably, so =dgtht(ayBorg)|HPos)+JWon)seKol, 6 ottasain,<We\soSeoonence,ajA045 @Wow,portsBPeamboOeLeledogs -ebay. =igte¥\ay[9WOrs)||-Fg £-Ko)§G-3)Loe ed ~YHA ~igTESaybaryon] BNBef YosakALY adOev[RWSAGA? LISGa)aenLS(AK)/ a fe) a danwatin. oh(3A2) . aa, ol aN ©eosoat Y=[bt-gh¥e) -[Q-3| . 2a,5-1 —_ Guevuegk =§«G=[L2-4REN] -pT Neo . SSoat\yt¢-(42-24-29) = - ~\.(feT- BoEe") Me By 2Sey =@hy BeBeySo-\= 4" Sys FSe~Ay = ye 4 gue TAfo) g--k TA Quer 3. + . e) ws . ‘ 5.79 0Derivation of(3.10) : Me Boy ga ay Comment: Ihadalotoftrouble withthisbecause thereisalotofstuffgoingon here. Finally,.however, Ehave made itworks- * Coe ge Wewanttoshowthat(3.10)duplicates (3.8).Thefirsttexmof3.10obviously isthe same asthefirst term of(3.8) socancel those. Itremkins toshow that terms 243of(3.8) equal term 2of(3.10). But,notice that either side of(3.9) isexactly term 3of(3.8). Sohere iswhat wewant toshow: ([(.8) term 2] +[RHS of(3.9) ]=[(3.10) term 2] (*) Normal order: whyhave theydone this? Themeaning isthat allthose d/dJ's sitting inside the::aresupposed tonotactonthatJ?guy,butaresupposed toonlygoto therightandhitontoW[J].Youwonder: whydidnttheyJustwrite theJ?onthe left anddispense with the::.Reason: dont forget that D,contains aregular derivative dywhichactsons(y)soyouhaveaproblem: whichever wayyouordertheseguys,one ofthederivatives wont beright. Youwould have toback-arrow thedyinside D,. But, Icould still fixup(3.10) byfirst partsing thed,inD,over ontotheH 0(getting aminus), thenswitching theindices fromcdtode(getting another minusonother term for ceverall minus). Then 3.10 could bewritten like thist d, Fig © KeeDaLe-TRt SagHayOPER” WGBa)bw=e Theprice paid forgetting ridofthe ::isthat nowthewolor indices dont match correctly. Butnowyoucanreplace HwithG”andgettheindices tomatch, soyou get: et atteat. *¥Fix DS/ Cea ag abTse yas.EraSayKD, GEGosKaiubl=o . Howyounowgofromthisto(3.13)isastroyIdontknowyet;)so letsgetbacktothepresentproblem.IwillkeepthenormalorderingformofGpaIknowwhatitmeans. dswwehawk~ —> Proof ofabove equation (*): Look at(3.10). Forget forthemoment theaction ofthe outer operator Dd onthe t...d: .Then the Ddgoes rightg into the integral and goesinsidethenormalorderingandhitsontoH.By(3.6),thismakesadeltaand fe)the net result isthis: -+ D(id/dd) J+ However, Iclaim this isexactly the same as(3.8) term 2. Actually there you see~D Jwithout the normal order. But, the term inwhich thed/dJ hits theJvanishes bycolor symmetry (just didit). Therefore, allthat remain istoshow that ,when you let the outside d/dJ in (3.10) hitthe explicit Jinside the :..: factorf, this must yield RHS of(3.9). Itdoes,asisshownonpageattached tothis,notveryhardtodo. e Aside: notice that [d,, D,(id/aJ(x})] #0. Itmatters which order these appear in! ThatigwhyGt; forexample. -— -- rs 2 ? ~ . x _ = aa —A ec,Boy Ll BS, ws — a ~e- or “< .ata “o LI ise\am eC Ore 65 2 UeAr 3 OSgt&SK | Zeem} Sf. av SS ° “ Lol += Fe?) Ste ge, ilJl oe <a 4 e. ‘Z 5 ) ov a)2 KEEN 2[le 4 asf o oa \©. = \ :aj | | ry 2 fe) NN E NS Z a 2NG ISIS Zz Aan fe) SSA AI T2%34567890’111213we” SSseugiayim og SN] A ;NNAITEIN @) Figure15.Schematic ofRestricted Aperture. ve SignErrorinLeeZinnJustin-I_ ‘e)1,First,noticethatyoualwayshaveaconvention onhowyouwilldefineyoursources and how you put inyour gauge fixing term. There isalso aquestion ofsign ofmetric. Ialways usethemetric which implies p*-n* inpropagators. SodoesColeman. Also, Coleman shows that ifyou Wick rotate correctly with this metric, you get: dix=idheg Thus, Iamable tojustify the sign ofColeman's gauge fixing term: a. i\K[S@- HORT] e =e Qk SeLt 5\ &Ja “y +i\dkbaaGS) =-\dhe [ezaN andthen theidea isthat, ford)0 atleast, this exponent isnegative andvaguely speaking youhave aconvergent integral ifyougoover toEuclidean space. Anyway, this isthesign ofthegauge fixing term used byColeman, andbe1ZJ-1, andaloinAbersandLee,WhenthisiscombinedwiththeusualL,=~(1/4),FR, fe)you get the following bare propagator: a—Wyle)pdbye * kee This iswhat Iget, Colemtn gets, Abers andLeeget, andLZJ-1 get(see 2.11 ).Ie,we allhave that g,, hasthesame sign asalpha. Wedonot all agree onthe sign ofthe source, but nomatter for now. 8.Facts when Iwork through 12J computation indetail, keeping exact track ofall signs, here iswhat Iget for myfinal result : ate eo}i8 ne ebbe, Rep lizalS|ee, +VayKYW ER6lyx,f = < & yj=0 we[3*SeyToseVERTGx,&)]WE noms: (2.13) Thisismyequation(3.13).Therelativeplussignbetweenthetermsisconsistent with fe)LZJ's equation (3.10) because you get aminus when you switch the action ofDfrom J over onto H,etc etc. So, Iamwilling, without justification, toclaim that the square bracket above equals 0. Iwonder: does myresult agree with Slavnovs result? -2- On,a-pageattachedIhaveexaminedSlavnov'spaperandhaveconvertedhisresultinto fe)afunctional result soIcan compare itwith mine. First, Ihave toreplace his source with J=-n because his "source-sign convention" isopposite L2J's. when I.do this, Ifind that Slavnov's result has arelative minus sign between theterms, whereas Ihad arelative plus. BUT that means weagree because Slavnov's {isminus the4used byLZJ and meand everyone. Tey . Asemov =Sigs : Msrevnov 7~Stzd-1 Thus, myresult agrees with Slevnovs result. Conclusion: LeeandZinn Justin have thesign wrong intheir (3:13) .Everything else they have done iscorrect, except typos Ihave already marked. 0CompanBovaRealboleeVY.Parnesvase SalVOrrd+5 (98) ©w=\aviae)©Boaters Yerragutt eesd)oeODwateae *s 2) ale -!‘ iSO=\way-]2 Ox\be.OLSsg¢“poRengy| g =MaLOOT Baty C6. @ox«aS o=\ue20]+0820)yeygDQCae Noa,Aaglace (dz): SeWewrtaon 5 4 a, sveo=fotaalLeip tafe1GaNWA WavfooraleDniaDockkeeLID,RFY=—-T.frou: _Se STE +HERD ie. sl48yyy 7|o-FBPay)SeogDTAWICH Oniniasacle) (18),BATQuia'senwogorrde ager3! RE.MoWITS. oe ‘ ~~1@)A.mer.\Janyshhomgeabel, @Ws)s QnckGane abn. mswereatk.9asknokwadsfaeae,ATA NMagiaeyeaLaeconergetstaterCeavsschyMageeo| TI) =role seodar QainAwe. BW)=Meafkgegen. Ayo=gangpeop &n=3 alec, ca S_phy : 0fund X~Space =PoYp\~)motwd =)le-spac " ; D Gay)psode«*[email protected] =hyo, = a® .: t Qa“eee.XSpact,geekctfsmngyDyer!: Zy=~Z5() [proaga beydiamine Wad} someleind WedWBasa teat] . ay phe ty wm0 FIsTO)HO%,‘\=0audgh 3-glue bos [ght :Tete Ee Sitnee N _ Connie ofrpedoing, FP. ~Foscingy ak(aydam ob bey we »3THUG] Ssh) =YaadyReLAelSoe) Cex A) : =Vea“f-g,MeequentGaGy)=QeSGRy) a Qe bsMULSl Qader5ergend anydaub, \ = = GaGoy A)=-MaGul) VecomAurtak -\wv Tv HaGony= -&IW Gs)=Gacy) 7 (oe)YoBAp=o, WanGHG Byt theway, weknow from ourpropagator-identification page that this function W309) isthedressed ghostpropagator! Recall fromBDvolIthatanypropagator isthe Greens function ofadifferential operator. Here that operator is dD. Aswritten above, Gisthe amplitude for propagation toxfrom asource,at point y. Lee's remark about "outgoing boundary conditions" merely refers tothe fact that you want the positive energy component ofyour "wave "G(x,y) toradiate outward from thesource, torespect causality. Innon-rel theory this just means G=0 for time inthe wrong sequence, and inpositron theory its more complicated but same istrue for each energy piece. LeeeZa Ke) 8 SPONTANEOUSLY BROKENGAUGESYMMETRIES. I... ist tty,W.B.Kibble, Phys. Rev.155,1554(1967). 18g.p.Feynman, Acta. Phys. Polon. 26,697(1963); ‘“purther references canbefound inRef. 8. (unpublished).‘3p,W,Lee,Nucl.Phys,B9,649(1969);J.-L.Gervais 2p,DeWitt,Phys.Rev.,162,1195(1967);162,1239andB.W.Lee, ibid.BI2, 627(969); B.W.Lee, Chiral 1967). Dynamics (GordonandBreach,NewYork,tobepub- ‘My,D,FaddeevandV.N.Popov,Phys.Letters258, *.shed). 29(1967);KievReportNo.ITP-67-36 (unpublished).- 163.N,Bogoliubov and0.S.Parastuk, Acta. Math.97, 2g,"eHoott, Nucl.Phys. B33,179(973).5) 227(1951). 24)slavnov, Nucl:Phys.BSL,301(1971)‘5yN.Bogoliubov andD.B,Shirkov, Introduction tothe ™R,Jackiw andL.,D, Faddeev (private communica‘TheoryofQuantizedFields(Interscience, NewYork, tion). 8)11959) ‘UA!slavnov, KievReportNo.ITP-71-191E (unpub-‘ge.Hepp, Commun. Math. Phys. 1,96(1965); 7HZories shes. deta Renormalisation Springer, Werlin, 1969). 25Johnson (private communication). My.Zimmermann, inBrandeis University Summer 2K.Symanzik, Lett. Nuovo Cimento 2,10(1969); » Institute inTheoretical Physics, editedbyM.Chretien Commun,Math’Phys.16,48(1970). etal. (MITPress, Cambridge, Mass., 1970). g.gona-Lasinio, Nuovo Cimento 34,1790(1964). e 184.Slavnov, KievReportNo.ITP-71-83E (unpublished). PHYSICAL REVIEW D VOLUME 5,NUMBER 12 1sJUNE 1972 Spontaneously BrokenGaugeSymmetries. y [i\Perturbation Theory andRenormalization ; tle) BenjaminW.Lee )e National Accelerator Laboratory, P.0.Box$00, Batavia, [linois 60510 } andInstitute forTheoretical Physics, State University ofNew York, Siony Brook, New York 11790 i » and !Jean Zinn-Justin®2) InstituteforTheoretical Physics, StateUniversity ofNewYork,StonyBrook,NewYork11790 4(Received 10Macch 1972) ‘Thesecond paper inthisseries isdevoted totheformulation ofarenormalizable pertur~ 3) bation theoryofHiggsphenomena(epontaneously brokengaugetheories).InSec.1,were formulatetherenormalization prescription formasslessYang-Mills theoriesintormsofé ‘gauge-invarlant renormalization counterterms intheaction, Section Iilgivesagroup-theo-otic discussion ofHiggs phenomena, Wediscuss thepossibility that anasymmetric vacuumisstable,andshowhowthesymmetcy ofthephysicalvacuumdetermines themassspectrum —4 ofthegaugebosons. Weshowfurther thatinaspeclal gaugo(0gauge), allunphysical fieldstanbeeliminated, Section IVdiscusses thequantization ofaspontanoously broken gauge the~ oryintheRgauge, where, asweshow inSec. V,Green's functions aremade finite bythe renormalization counterterms ofthesymmetric theory (ihwhich thegauge invariance isnot 5) TeBttenoously Decker). ThoR-gauge formulation matos useofredundant fields forthosake ofreaormalizability. Section VIisadiscussion ofthe low-energy limits ofpropagators in theR-gange formulation. InSec.VIIweshowthattheparticles associated withredundant 2 elds peoullar totheR-gauge formulation areunphystenl, i.0.,theydonotcontribute tothe ‘sum over Intermediate states. he 1INTRODUCTION tionwhen chiral fermions areincluded inthemod- | _ fhaspointed outbyVeltman," andmorerecently |Inthispaperwegivearenormalization methodbyGrossandJackiw.?Thisdifficultycanbecir- Aandaproof offiniteness ofrenormalized Green's cumvented inarealistic model ofelectromagnetic Fordefinitenessweconsideraverysimplemodel_problemfurtherinthispaper,butpostponethe |inwhich SUG) gauge bosons arecoupled toatrip. discussion untilwedealwiththerenormalizabil-Tetofscalarmesons, Thereisanextracomplica- ityofarealistictheoryinasequeltothispaper. i . -1- LeeandZinn-Justin II.Perturbation Theory andRenormalization written March 1972 ie}1.Introduction. TheseguyshavefiguredallthisoutintheSIGMAmodel,endarenow ~generalizing toarbitrary NAGT. Actually, inthispaper theywilluse2simple SU(2) color group with obvious Higgs fields coupled inasyouwould imagine. These aretheonly matter fields they will use, Ithink. ‘Thegauge wherein theGoldstones areabsorbed into gauge masses viatheHiggs- Kibble mechanism iscalled theU-gauge. The claim isthat inthis gauge, unitarity of thetheory isobvious, butrenormalizeability isnot(after all, youhave massive vector propagators screwing youup.) Butthen there will besomething called theR-gauge where therenorm iseasily proved, but inwhich unitarity isnoteasy tosee. Iimagine the authors will prove renormalizeability inthe R-gauge only. Then thegame istoshow that thetwogauges ereequivalent, andIthink that isthe topic ofthe next paper, No. III. Asanaddedcomplication, authors allowforthepossiblity ofSSB.Afterall,itis SSB which creates the would-be Goldstones. Discuss this below. II,Gauge-Invariant Counterterms.Inthissectiontheyshowhowyouaddcountertermsto re)thelagrangian togetafinite theory. Finally thisisclear tome.Itistotally un-obvious whyyougetafinite theory unless youhavestudied say#.Itisjustthe usual multiplicatiye renorm deel, except ofcourse you have toconsider both theglues andtheghosts. Good discussion ofchoosing rencrm ‘points. The both Wards areused toshow that only onecharge gisneeded inthe end. Ithink the Wards will beused again later. Ttisnoted that ifyou define renormed sources, you can get tenormed Greens bydifferentiating the renormed genfun W,which now ineludes those counterterms. It isnoted thatthecounterterms, including theregulator counterterms discussed inlast paper, are all gauge invariant. Anewversion ofI3.13 iswritten (general Ward) butthis time forrenormed objects. Ihave notyet checked this result, because Iamhurrying toseewhere this series ofpapers isheaded. III. Group Theory ofHiges Phenomena. Aquick review oftheusual stuff. Start with agroup symwith Ngenerators. Assume theHiggs field hasnonzero VEV, sothegroupof the vacuum hasonly Mgenerators. Thedifference ism=N-Misthenumber of“would-be fo}Goldstones" (Goldstone Theorem: theremstbemmasslessparticles. Te,moftheHiggs" gealars must bemassless. )Then you choose afancy gauge sothat the Goldstones are eaten bysome ofthe glues, soyou get mmassive vectors. This isHiggs-Kibble Mechanism. Here all discussed in terms ofclassical fields. Co ~1- kusQuantization ofHiges Phenomena. Takeitslow, itallmakes sense. First, write dom * Lagrangian including Higgs fields. Higgs self coupling nowcalled )andthere is.a Higgs masscoungterterm. Wholethingisagaugescalar.WhenthisLagrangian is'combined with|, lo) gauge-fixer andghosts, yougetaction called S,since parapeterd, 18still inthere. Major point: write down thegenfune forthis action, putting insource Jforthe glues andsource Kforthe Higgs. Then the Ward Identity (4.5) 4strue whether there is SSB ornot. The Functional Ward doesnt care where the potential has its minimum, Now wewant toavoid imaginary masses inour Feynman rules. Obviously, then you want toexpand your potential around its minkmum, asinclassical theory. First, write each field asasum of"long" and "trans" parts, where these refer tothe direction chosen bythevacuum inisospin space (these have nothing todowith long/trans with respect tosome k”asinagauge propagator; thegauge prop always haslong and.trans pieces inthis sense). Then, imainge that "long refers tothe sor3direction inisospac e. Soyou expect yourHiggs fieldtohaveanon-zero VEVforB=Thedifference of$,-v which isthe VBV iscalled ‘Y2 The long part ofcolor vector Ais called A. Sowhat are wedoing? Toget sensible Feynman rules, weare replacing the field ,withthefield ~~;thenallfields inLwillhavezeroVEV. Thenrewrite action interms ofallthese newfields. (only onenewfield here). Newaction is(4.15) plus (4.16).IhavenetcheckedallthetermsbutIKcould. fe)Next, ifyou want todoperturbation theory you pull out your interaction action and claw infunctional derivatives ,asin(4.17). The non-interaction action can then berecast intheusual way into propagators, asin(4.19). You seethat there are5 kinds ofpropagators inthe new theory. So, wenow know the "sensible" Feynman rules for the SSB broken theory. Great. This isahairy combination ofmany ingredients: 1)functional formalism togetFeynman rules foraperturbation expansion; 2)Yang-Mills theory coupled toHiggs scadars, all gauge scalAR; 3)gauge-fixing andFPdet ghosts stuff; 4)SSBbreaking possiblity so fields must be shifted. Now ontop ofall this stuff wehave toask: how doyou renormalizes the dam. thing? Its really noproblem. First, you are supposed tofiddle with the Higgs mass (counterterm) tomake sure that, inanyorder, your field Y*really haszero VEV. Ie, the tadpoles with one PSI field mst becancelled. See *term in(4.16) tosee that this mass counterterm isinaposition toaccomplish this task . Soghoose that mass counterterm asjust mentioned, then doastandard multiplicative renormalization onthe theory. Here I'll just list things off: 8 . ©Weges GTB=EGR) aeLaotabeep ° & Geet “Apa2SAp x) Got:“QuBi [BiotdohdoRL not —4~ | ) %|daohTyg=2. 0. : : a a WoAsaSngleevar: eds)|VRPTeere oO.YoateDaggnn|AequodvecdineC=Ts)hex devos, ;DyAta.NSWaggawanker .alkeswer\y29)va — ge cae Vigqe ° . i Sowedefine siz Z's inthis standard way. Then wecan take the Lagrangian and replace allfields andcharges with their expressions interms ofrenormed fields and renormed charges, then weregroup into terms plus counter-terms. The result will be (4.27) ‘which shows only the counter-terms. Notice that the FPghosts don't get mixed around with other things. You can imagine adding this counterterm lagrangian tothe thing shown in(4.15) plus (4.16). so . Comment: Now things are all set up. IfIwere.at this stage with a.simpler theory, theremaining workwould betostudy integral equations etcandfindthat-if youiterate order byorder the theory really isfinite. Ie, Iwould want toshow that the multiplicative renorm scheme really doesyield_a ‘finite theory. Icarried,this through for(9;\BDcarry ‘4%throughforQED.NowitlookslikeL.and2Jaregoingtocarryitthroughforthis le)complicated :gauge theory which has everything: non-abelian soghosts, Higgs and SSB, etc. This Ihave got.to-see!!! ot fe] “A 2mogplchsp}omeliad| 5.Proof ofFiniteness. Iamvery confused but will jtry tomake sense out ofthis anyway.Usuallywhenyouhavesourcesyousay:computeaspireandthensetsources=0. @);Yownesays: setsource Ja0,butset,source KV iyIdontyetknow. So,atthe point (0,8) forthesources, dZ/dK, =someHiggs field value v,;which isnotzero. Gallthisv,M,theHiggsVEVdealpointinthe%direction, fine. Thendefine field“F asin5.5 and rewrite Lagrangian interms ofthese new fields asin(5.7"). Notice that this gives just the kinetic and mass terms ofthe lagrangien; the interaction terms are thesame asin4.16 with vreplaced byournewv,- First, notice the significance ofsymbol m: itisthe mass ofthe trans Higgs fields. Thus, these are the guys that would bezero ifGoldstones (start with 3,break 2mdownto1so2goldstones; goldsarethetransversals here). Sowearecurious to see whether m-0 or no. Meanwhile, wearetoldtoascuneapwillbeprovenleterthatJwiltbeproportional Vtothevector$sowecansaythat=,sosymbolcisthisthing.Ascgoestozero, whyou return tothe usual sources=0 place todefine greenses. AK) Small detail: thelinear{*terminournewlagrangiancontainsextraterms weandturn souttobeasshown in(5.9). Again theidea will betoadjust the mass counterter. tokill off all tadpole graphs, asstated in5.10. as fo) Now,offwegointopropervertes. Theclassical fieldsarecalled 0,B,,Be where these last two refer tothe Higgs guys. You can form the proper vertex genfunct and then its func derivs will give you proper vertéces, assays (5.18). Then comes the step Idon't getyet: ifwrite the proer vertex forthe theory with v=v,and its apower series inv,and the coefficients are certain proper vertex functions ofthe theorf with v0. Lets accept this fact for the time being and see where itistrying tolead ujs. If5.20 istrue, sowhat??? Well, 5.21 states how you renormalize thesymmetric theory. This paper isassuming that the finiteness hasbeen shown forthe symmetric theory (symmetrci means noSSB sov=0). Ie,you can choose the countertermic Z's toget afinite theory. Nowdefine the renormed objects shown onLHS of(5.22). Then you get (5.23) which isnicet: ifthev,arefinite, then theobject ontheRHSisfinite because the proper vertices PIofthe syhmmmetric theory aare assumed tobefinite. Now there isadisconfintuity inthe discussion, anote added inproof isadded. Trytopick itupagain, They want toselect avalue forZ)sothat (5.2h) istrue. Thus theyhaveslightlychangedthedefinitionofZ)(byafiniteamount).So,at7-0they ©) want theKiggs prop tobe=n"? (and mbetter besome finite munber!). Wenow start ontop ofsecond colum, page 3147: soo -—s-— . This colum isjust about theGoldstone. theorem .Ihave derived (5.26) from (5.25)andthusIagree,thethingsareparallel.Withthenewdefinitionof2),I| fo)also agree with (5.26) andthus (5.27). "Spontaneous" breaking means all sources: are zero, sdwhenyounowtaketheKsource<=cftozero,youseethateitherm0or, v,=0. Ie,either youhaveGoldstones (massless Higgsgs) oryouhavenobreaking of the vacuum. . Lo But that was just Goldstone stuff. The mgin work ofthe section isalredady done and iscontained in(5.23) which says: ifthe symmetric theory isrenormalizeable, then soisthe theory ijvith SSB present! Itisassumed thet one already knows completely how torenormalize the "symmetric theory" iethe one with noSSB. : Infact, Ithink that evenwith c,held away from zero (sothere is.a K-source present) you can renorm the theory. a) a 7 | oi] : . : -6- Recapofthefirst5sections ofthispaper. Letslookbackandseejustwhathasbeendone.InSection2theylistoffthe a) counterterms youwould addtotheLagrangian torenormalize anormal theory with glues and atriplet ofHiggs with normal mass. This section does not properly allow for the possibility ofSSB because iftaken directly, itwould imply imaginary mass inFeynman rules etc. Asweknow, you really should define new, shifted fields ifthere isSSB. So,skipping over theHigge-Kibble review section ,wécome tosection 4.Here, they write down thesame action asabove fortheglue +Higgs system. Then they define potentially shifted fields (the PSIfield) andsetuptheperturbation theory ingeneral, allowing forthepossibility that there isSSB. Equ. (4.20) shows allthepropagator in the perturbation theory, for example. Then onthe next page they show all the counterterms (now written interms oftheshifted, physical fields) youshould addto achieve renormalization. Ifthere were noSSB, then v-0andshifted field reduces toregular field. The 2's which render the theory finite can becomputed interms ofthe cutoff. These are “the Z'swhich renormalize thesymmetric theory". These aretheZ'syouwould useif v?=positive. NowsupposethereisSSB,howdoyouknowyoucanstillrenormthetheory?You ro)have abunch ofcounterterms written down, howdoyouknow that thereally make the theory finite? Answers youderive equation (5.23) which states: theproper renormed vertices oftheSSBtheory canbewritten interms oftheproper renormed vertices oftheregular (symmetric)theory. Sochoose theZ'sthatrenorm theregular theory, and then your SSBproper vertices will also befinite!! Thus, thesame Z'swill renormalize thetheory regardless ofthesignofué!1! Oneinteresting point which Ithink will beimportant later isthis: the Ward Identities havethesameformnomatter whatu”is.Whether ornotyouhaveSSB,the Wards look thesame, Moreover, since thecounterterms areallgauge~scalars, they have noeffect ontheWard identity, thus, inthetheory with these counterterms included, your greens will satisfy Wards ofthesame form asthewards oftheunrenormed theory. Inother words, thefinite, renormed Greens functions satisfy thesame oldWards we haved been using all the time. So,Wards survive renormalization and SSB! So,these sections ofthepaper constitue a“proof” that mgm SSBdoes notaffect therenormalieability ofatheory. Ifthesymmetric theory isRenormalizeable, then soisthebrokentheory!Notethatthisproofiscarriedoutinanormalq~Landau ro)gauge. However, as(4.20) shows, youhave "redundant fields" because youhave both goldstone Higgs andmassive glues present. Onewonders ifthese massless Higgs fields dont mess upunitarity somehow. IntheU-gauge these goldstones arecompletely gone so uniarity iseasier toshow. ses: ~qA Section 6:Lowenergy behavior ofpropagators. Inthenextsectionauthorswanttoshowhowthenegmetric+goldstones +Faghosts fe)(all massless "particles" present inthe theory inthe R-gauge). all cancel inunitarity ° sums, soyou never get anamplitude having apole atzero. Thus, inthis section they want tofind out how toexpress the behavior of the negmet and goldstone propagators astheygonearmassshell (ie,220), because massshell isrelevant forunitarity quaestions. Theresults areshown in(6.10). How are these results arrived at? They write general scalar-function forms for three "inverse propagators" asshown in(6.7). These are related tothe propagators through amatrix equation (6.6) which reminds meofthe similar thing inKummer. Thus, you canusethe same symbols A,B,C,D inyour propagators (these are scalar functions). Itseems that C-O identically due tocolor symmetry somehow; from this you learn that forsmall k”,thefunction A(k’) =k°/x where yisthegauge parameter. Theupshot isthat the negativemetric propagator contains function A,whereas the goldstone prop contains function D.Sofarthere seems tobenocorrelation between functions Aand D, but there is as shown later. Section7:UnitarityFirstexampleislookat1-particleintermediatestate.Theyshow fe)thatthegoldstone cancels exactly agains thenegmetric (ie,thelongpartoftheglue propagator). Inthis example ofcourse youcant have FPghosts because they canonly oceur inpairs.7 Toshow this little cancellation, use ismade ofavery simple case ofthe Kurd Identity. Thus, very important that this identity betrue for the renormalized theory (and possibly SSBbroken !).Here iswhere theWard really helps: showing these ghost cancellations. They explicitly consider the case oftwo massless and then three massless. Ineach case the Ward allows them toshow cancellation. They dont have notation that allows ageneral case tobedone, but seems clear that itworks. Thus, the only poles atzero will come from the "phtons" ofthe theory, gauge particles that did not acquire mass. The other massless particles always cancel out. Also inthis section they claim toshow that S-matrix elements cannot depend onparamter alpha. Comment: Inthis paper, authors show both renorm and unitarity entirely within the Regauge. Theymention thatthere isaU-gauge where goldstones arecompletely eaten, but they never use that gauge inthis paper. \e) TheFamous 2/Zratio equation forNAGT inCovariant Gauge. 1.This equation appears as(2.2) inL-2J-II paper. Ifinally understand where itcomes from andIwill derive ithere. The only ingredient Ihave notyet derived istheWard Identity which appears asEq.(J.15) inL2J-I paper. Youmust understand that this identity isderived using thefunctional formalism from acertain Lagrangian, namely theYMTLagrangian with F.F andgauge fixing andFPdet. Thislagrangian doesnothave therenomm counterterms inityet, sotheWard Identities yougetalways relate the unrenormalized Greens functions! Basically, thisWardIdentity (scalewiase) reads: onc eneAY =6% as) A 4 7 2._Glue. Propagator. Eventhough there areghosts toworry about, thelongitudinal part oftheglue propagator isnot"renormed", which means itisthesame asinthebare prop. Ihave notyetshowed this, but%Z2Jshowed itfunctionally (and sodoes Kummer later in theaxial gauge). Thus, your unrenormed propagator looks like this: 8, yWebk~bute)’eR yew BpoO’ eCap-BET@)+e [ium Apv(eS=ALCa—AX Paty Raley [psa.veGy7EVTE) bly jam Que akAaa Voy|~ *° eoytJA=Bd9{Tw=ATS K=2A %=Bmx gd Renan omsthead : a ahKH) = Ko) =-te acyVEH- 1>ApGr)= -klgot + 3.Ghost Propagators Mae)=“LE “n=¥. QBO ap aehe Ae zu AG)= Le) Uc®=4 5meee o ee Ron.Cond. (-a*y 4.Triple GlueVertex. ° st Qa,; wa BO B=gyBR eCrtenies), Sesympk@penaandOe Sooascamtengimgcheh togeneby?adeace eMglvacins (hula Tt)AQED), ° 3h, Faun —\ Ae Hae ao a>haWea gy)We YsGras) OSong . Rosaceaecob thePE5awpopdad comualty . = anneeSn ve 2gy=2525a Sua sgis~~. Sothis istherenormed "charge", thetriple gague vertex coupling constant/. Ifew details erenotyetclear: like, what happens tothose extra terms? Sure, youcen exclude them from your renorm condition byprojecting them out, butthey arestill present inthevertexitself.Moreover,duetothepowersofmomenta(powercounting), Ithing fo) youcanshow that theobject above called (other tensor forms) isafunction only ofg,anddoes notdepend onthecufoff. Thus, itseems likely tomethat the term gets multiplied byapower ofZ,andtherefore will probably becutoff dependent intherenormed tertex. Somehow, itmist turn outthat these other terms never contribute, toanS-matrix element, just tike the"gauge terms" inaglue propagactr. Later Iwill look atthis problem inmore mdetail. Fornow, wehave chosen acharge gandhave insisted that thetriple-glue vertex bethis number g(times bare tensor, muisumttx modulo projection) atthesymmetric renorm point. Earlier weinsidted that therenormed propagator have “unit resideu", ie,that Jal atthe renorm point. Butisthis possible? Actually, ifyouhold g,fixed (andm,=0) andyouhave a given cutoff, thequantities Z,andZ;arealready determined toanyorder ing,and aredivergent. Nevertheless, since propisrescaled exactly byZ3,youcanstill insist that Jel. However, gcomes outasitwill. Butnowreverse itandinsist onag}then B,gets forced tosome value. Fine, fe)52Ghost-glue vertex.ZpRaw : Ys=geBeYe>(pativiecnatime) \somgeAimsOgapgenrn vnVatXconghiig. oes —3- oyee hy ran ‘ . X=GENQSNX aries,one,gobo SeOwevadlec oglut 8~yt Shp \Zhes safeBs 2&=YX =(Bz Ys4)% +bn, Ni oyBeoye RescnCond?Sn2z»?go)=os =3k o Again, suppose g,=fixed. Then allfour Z'saredetermined, andgand€come outwhere they may. Or,youmight choose avalue ofg,adjust g,»then youarestuck with whatever @comes outinthewash. Youcannot set¥also. Luckily, asweshall see, =g. S.ABbly Ward Identityt:. y j -= -. ~ wy Ok SFeet| osWwalzrl+(sel[22 x] Gotepsn gk: sy Bow » Rok te: =DsAL: xho\=\b th&-4% Ok ees, -=wey =P ae |y-8). Neo Lock Wack so a ~oie . As % VyBege=B20 2sg . as" as Qe, serler DeeeeeeLE Als 6 . TheWard Identity interms of--renormed objects must besatisfied simply duetothe.fields scaling. Ie,Unrenrom Ward implies Renorm Ward. Then yougototherenorm point andyou find that g'sarethesame, andyougetthefamous Zratio fact. Perhaps alltheother tensorformsprojecttozerointheexplicitWardidentity. Obviouslyyouwanttorenorm fe) your various Greens functions atthe same point soyou can use the Ward torelated constants. Vek he 7.TheFour-Gluevertex.Wecanobseryethata7T=Pforthisguy.Wecouldsay fe)that atits sym point the renormed Tis equal tosome new g’ times thé bare tensor plus other tensors, Then we.would apply theward (4.17) andwegould then conclude thatg'=g.SoIthinktherelation between the3-point andtheJ-plint-is the gameasinbareperturbation theory. Ofcoursetherearetonsofothertensors -aroundthatyouhavetodosomething about. : 8.TheLagrangian counterterms. Land,2J showallthecounter-terms thatyoumst addtoimplement the above renormalizations. This istheir.(2.1) ofpaper IT.Lets see ifwecan't verify these things: a By vey4(a) SAE YA) pga cntDueseDok ja=2%O ; - “loo :. Qasr A=WOLeama easeatlesQinorghe, : Note’ that*they show renormed fields andrenormed charge intheir lagrangian. Itreated theory'in this wayonce, AllyouHave todoisthis: start with thébare lagrangian fe)intermsofbarefieldsandbarecharges, thendotheobviousrescaling andallthe terms will drop out. <y ve Sat amARY_wveAeALaya ont Seon=~§RSA RORY BERY+qhsi, ° + yee tkA=BA gg=Wg QaTetonymt)ole=Dg ayya Bek Be ayo at7 — 5Lees ~42s[SALVA -AB.2sRxh'\ 2(3#) a ah +qh=~ads(LYM), s 1.AA RYa4Beog(ASR) ho 2s Reh*)aRY(ALx) semyCA (Ape All weare doing isrewriting the bare lagraingian interms ofrenormed charges and fields. A‘Thentheclaimisthatasyoudoperturbation theory, ineachorderyouwillhaveno divergences because they will becancelled bythe counterterms. Asusual, itwas good formetohave done this first inB|ar)dgwell sofar. Seorsine.yAviva”(6:26)proms(6:25)- “.. 0, -et\y[oteZe.yeySL _ (S18)soys a\ae(HOFe+Key2a,Wora Aguy3,QncwakTso.YeaassmSolan4femme. 7Sk ty beSSReo leo ee oe Quan DB/Sisso0 wanDIOSeDao .. Seve giety (On€Yel Seog ttoaii\ adeDY OL tobe Uy . a :.=€ASeQace Keape. ooo= SK)X.,SKK) Lo, . Quek Rok BW. WEBS we =eaie)Lo —_ SK.Se)~ .. WSS8K\y-¢ oa Lo gb < so,onne ae Paeg : ”oneT eS SakBMGs) Qacraw ta5. mo can :“8o=ettySeye SaxBons) oe SLY Vwedicnsy |‘actum mde: AeseyECR. BRSquantaYssag.4? On Ssoe a /N=X dxdes): 7~- T= BeSSects ; Sh=Ale) sh (quaged’) Oo_ _ becl@y = Oy eel rn)gh--CO ; | an aye if fe] QO 81 input AND gate, whose output generates another forty-nanosecond gate, which isnow used asthe “wall” output. Obviously, awall output occurs only when the required odd, even, and total coincidence requirements are met. Pig. 22indicates themaster coincidence System where the individual vall pulses are routed and meshed. Each wall has aninput which goes to various four-input gates asshown and also tothe wall unit. Again, switches select the coincidence conditions tobesatisfied inagiven run, The outputs pfthe four-input AND gates are coubined inanOR circuit, whose output drives two umivibrators: a120-millisecond UV used tohold off the coincidence system long enough towrite the event ontape, and a100-nanosecond UVused asthe master coincidence output. Emitter (o) followers couplethisoutputtothevariousunitsrequiring it.Onthe input aholdoff from the recorder isalso provided toprevent enevent from coming induring the recording time orwhile the recorder iswriting aninter-record gap, which occurs inour data every 24events, This is an easily-varied parameter. ‘The wall coincidence information system, shown inFig. 23, amplifies the wall coincidence outputs, feeds them into 400-nanosecond gates, and then into two-input AND gates, where they are required tobeincoincidence with the master cojncidence output. This eliminates spurious wall noise pulses from being recorded. The outputs ofthese AND gates are amplified and routed tothe readout logic system where they are used toset RS flip flops which are later read onto tape ahead ofthe pulse-height analyserinformation. \e) Lee+Z7“ e ~ ne a Bad 4 : 8 SPONTANEOUSLY BROKEN GAUGB SYMMETRIES... II... 3155 ak - andsinceanyvector tobecontracted withyorvoftheabovepropagator maybeexpressed asalinearGombluation ofp,(P,) andg(g,)weseethatthecontributions oftwomagsless particles totheself-energy arenecessarily oforder p,p,, disregarding ldgarithmic factors. ~ ‘10nleave ofabsence from SPT, CENSaclay, B.P.2, 8.B.Cutkosky, J.Math, Phys. 1,429(1960). ‘91Gif-sur-Yvette, France. ‘ON,N.Bogoliubov andD.B.Shirkov, Introduction totheIM.Veltman (private communication). TheoryofQuantized Fields(Interscience, NewYork, 2),Gross andR.Jackiw, Phys. D(tobepublished). 1959),3p,W,Lee,Nucl.Phys.B9,649(1969);J.-L.Gervais Mp.Higgs,Phys.Letters12,132(1966). ‘and B,W.Lee, ibid, B12, 627 (1969);B.W.Lee,Chiral RK.Pp,Feynman andA.R.Hibbs,Quantum MechanicsDynatnics(GordonaadBreach,NewYork,tobepub- andPathIntegrals(ioGraw-Hill, NowYork,1985); {Tehed) S'sehwinger Particles,SourcesandFields(Addison47.W.B.Kibble, Phys. Rev,155,1554(1967). Wesley, Reading, Mass., 1970). 5,Bladman andA.Klein,Phys.Rev.191,2369(1963). SB,W.Lee,Phys.Rev.D5,823(1972).2 S5,Weinberg, Phys. Rev.Letters 27,1688(1971). 4G.‘tHooft, Nucl. Phys. B35, 167(1971). 1.Appelquist andH.Quinn(unpublished) 48D,BessisandJ.Zinn-Jastin, Phys.Rev.D§,1913 8,DiLandau, Nucl. Phys. 12,181(1959); J.D.Bjor- 1972).ken,doctoral dissertation, Stanford University, 1959 “85,TxGlashow andJ,[Mopoulos, Phys. Rev.D3,1043fanpublished). asi. PHYSICAL REVIEW D VOLUME 5,NUMBER 12 1sJUNE1972 Spontaneously Broken GaugeSymmetries| IITEquivalence . Benjamin W.Lee ‘ National Accelerator Laboratory, P.O. Box 600, Batovie, Minos 60510 andInsitute forTheoretical Physics, Slate University ofNewYork, Stony Brook, NewYork 11790 and Jean Zinn-Justig*InstituteforTheoretical Physics,StateUniversity ofNewYork,StonyBroob,NewYork11790{Received 10March 1972) Wediscuss theequivalence oftheSmatrix intheR-andU-gauge formulations ofspoa~ tanovusly broken gauge theories. Wegivedefinitions oftheU-gauge Green's functions in terms oftheR-gauge ones, forbothAbelian andnon-Abelian cases. Based ontheequiva- Ience theorem, wegive «senormalization preseription oftheU-gauge formulation. 1ByTRODUCTION Strathdee* about theequivalence ofthetwoformu a lations. But more importantly, thepresent work Inthispaper, wewishtodemonstrate theequiv: gives definitions oftheU-gauge Green's functionsalonceofthe$matrixintheR=andU/-gause intermsofthewell-defined R-guugeones.formulations ofspontaneously broken gauge theo- ‘This paper isorganized asfollows. InSec.TI Ties. Wehave discussed theadvantages anddis- __weconsider theequivalence ofthetwoformula advantages ofthetwoformulations inaprevious Hons fortheAbelian model considered previously. paper (paper It). InSec.TIL,wegivesome illustrations oftheequi- Weshall carry outthisdemonstration byexpress- valence andformulate therenormalization pre~ ingGreen’s functions intheUgauge interms of scription intheUgauge. InSec.IV,wedealwith those intheRgauge. What weshall show inthis _thegeneralization tonon-Abelian cases. paper isaconcrate realization oftheremarks ‘tisempirically known thatthe7matrix forthemade previously byWeinberg’ andbySalam and Abelian case.computed intheUgauge isfinite." SAY Iee and Zinn-Justin ITT, Equivalence written March 1972 ABaDoySectoaseade7 01.Introduction. Theywet asimpletheoryintwogaugesandshowthatthe “S-matrix elements are*the shme. IntheR-gauge there areredudant fields wandering around but renorm isstrigignhtforward. Inthe U-gauge the massless Goldstone fields have been absorbed onto the vectors and unitarity iseasier toshow. Somthing tells me this paper was not done with quite the effort ofthe previous papers. 2.Abelain Case. The sample theory here isscalar QED. There istheusual electric charge, andthere istheJscaler self-coupling )jwehaveafield theory withtwofields and twocouplings. Ifthe$field hasimaginary mass, wegetSSBsolets assume this is the case. inthe Inthe R-gauge you just shift the field aswas done inthe last paper.. Choose usual Lorentz gauge, doanything youlike.Write downthegenerating functional Wp and doperturbation theory. Inthe U-gauge you dothe little expon thing and choose gauge asshown, then the goldstone vanishes completely from thetheory. Thefinal result isthat somehow you canrelate theamputated Greens functions computed inthese twogauges asshown in(2.15). Themainideaistoshowhowyourelatedtheperturbation expansionsinthe fe)two differing gauges. Itisnot done very well inmyopinion. 3..Renormalization. Does this only fortheU-gauge, since Iguess thelast paper did AtfortheR-gauge. Rewrite Lagrangian interms ofthe U-gauge fields called UandRHO. Define awhole pile ofZ-like quantitys inanambiguous way. State some renorm conditions forfixing theZ's. Write down aperturbation expansion fortheZ's. Not‘very interesting tomeandtooconcise, justtheopposite oftheir usual fault (ie,tdomichdetail). : wh. 4,Non-abelian Case. ARehash oftheabove ingeneral, case where youstart with alarger group andbreak down toasmaller one. Yougetaresult which says that T(u)=T(r), modulo some Z's. Igather that the ratio ofthese Z's isproven tobefinite inthis paper sohere istheconclugion: finiteness ofT-matrix elements inonegauge implies they arefinite intheother gauge. This isthe"equitalence” that they have been waiting toshow. The Wards are not used that Ican see. The main use ofthe Wards was toshow that theR-gauge wasunitary, togetallthose ghosties tocancel outinunitarity sums. Gomment:letsjustcoolthispaperuntilitscontentsareneeded.Inowknowwhatitis re)trying todo, sothat's enough for the time being. Compara ASwewogangs. 0,‘Sugperewea,LosadbecdoyQeUsguage?Wewthdoha:am 2.eke Swdy=8p =3|=Srastaqasy Ms)58,C Bydv oayeeaten: JemsCin). Omtuabanbond q So. JSRDy=CApAd) =Yiaaytdyfay TEA)ApAve Bndreasea Mast?diGrinkQuikB=An+Ughtinkpat5,weYm onokay SngyWamanaran, yorvarCUO)&gk weSy=CAA = )Wem Rasen. ©OwWactnherd wv Beegy~SH=Sod<hr bene95 fe)=RP=oe=KK) shy 2 &AR B=ad+(sw idomometetomyorkXK ee)louie . SohereisIthink thepoint: theWYandW*gefunes canberelated tooneancther as done in(2.11). Asfarasglue propagators areconcervned (transverse), they arethe same ineither gauge. But the scalar propagators are different. Thus, ifyou define ZpandZ,ineachgauge separately, youwillfindthattheZ,'sarethesame, butthe Zp!'saredifferent. Thisiswhat (2,12) and(2.14) aresaying. Therefore, consider what happens when you renorpalize each theory toget finite Greens functions. Ignoring the implementation ofrenorm counterterms etc, you imow that the end result issimply that you rescale the fields with the correct Z'sy and you take allcharges torenormalized charges. Iguess inthese twotheories wearecomparing, you conspire toendupwith thesame renormalized charge)buttheZ'saredifferentg for the scalars. (Actually, easier tokeep)» fixed andthen saythat there aretwo different charges \gand\xsandalso2°#2". Soifyouwant tocompare renormei finite gens functions, allyouhave todo 6isrescale yourfieldsproperly, whichisjust"wavefunction" renorm. Theglue'rescalingswill cancel between the to gauges, but the scalars will, not cancel, hence you get the nice result (2.15) which now seems quite: reasonable, buteach theory still hasits own charge asfar asIcan figure. Lee4Zinn-ustin {wv O: POINT TRANSFORMATIONS INQUANTUM MECHANICS. II... 1049 formulated thetextintorms ofthelesscustomary point ‘Again, thistsnotthemostgeneral system forwhich oo transformation, inorder tofacilitate thecomparison ‘Lemma 1,could beproved. More genorally, onocould .withpaperT(Ref.3). writefeyoooa81+atoteooo)001HLek - 4g.Carmi, preceding paper, Phys. Rev.D7,1038 handsideof(3.12), a(979), which tshenceforth referred toas“paper I.” "The main restriction isthatthecorresponding Lie ul Itseqlations will bequoted, ase.g., (2.3). + clement pf+fp+ghas'a nonvanishing domain inZy, 28 ‘Professor Donald Newman, private communteation, operator with range inZz,andthatthisremains eofor - ‘Thistshindsight, ofcourse. AtthetimepaperIwas thoL4oproducts ofsuchelements, = wrltten, thesolution hadtobefound bytrial anderror. "Robert Hermann, LieGroups andPhysics (Benjamin, - . ‘hus theequilibrium-thermodynamtcs (lo Doe NewYork, 1968), p.139. (nJe"Al|n)) cannot beevaluated exactly. However, if Nz,VanHove, Acad, Roy, Belg. Cl.Sel.Mém. (Series =| thosystem isfirst transformed bythetwoBohm-Pines 8)26,No.6(1951); Bull, Cl,Sei,Acad, Roy. Belg. = | transformations [D.Bohm andD,Pines, Phys. Rev. 2, (Series A)31,610(1951). 609(1953)] andtherandom-phase approximation (RPA) ROS,+fd+8.Plathid+8)“PSs+Fab+By,withfy (end,possibly,also[G.CarmiandA.J.Lock,Phys. =2US-Sits ete a Rev. A5,1447 (1972)] thoBogollubov approximation) ta ‘Those results donotseem toberecorded intholiter io= used, theresidual interaction issmall andthetransfor- __afure but,judging from their simple nature, theymust = mation (3.1)will(€{t&schosen insuch awayastofal- aveoccurred toalmost everybody whohasencountered =| fllltheother eriteria ofRef. 1)describe also thethermo- thisalgebra, = dynamics ofthesystem quite well (Ref. 1). ‘within theunderlying associative algebra (with re- 7‘hisdefinition isobviously notthewidestpossible specttoordinary operator multiplication) thealgebra |- generallzation ofthecorresponding definition inSec. Il, canbe spanned byabasis oftwoelements only, e.g, ,but{tservesourpurposeshere, Aampeth cetpandB=, } PHYSICAL REVIEWDVOLUME7,NUMBER415FEBRUARY1973 — fe) Spontaneously Broken Gauge Symmetries{1V| General Gauge Formulation = Benjamin W.Leet , BI {nstititeforTheoretical Physics,StateUniversityofNewYorkatStonyBrook,StonyBrook,NewYork11790 [al and ze=| Jean Zinn-Justin. =ServicedePhysiqueThéorique, Centred'BludesNucléaires deSaclay,B.P.2,91GY-sur-Yoelte, Prance =Received 30October 1972) * , ‘Theadvent ofthedimensional-regulartzation procedure allows thestudy ofrenormalza~bilityofspontaneously brokengaugethoorfesformulated inawideclassofgauges.Wederive andstudy theWard-Takahasht identities appropriate fosuchgauges. Aconsequence oftheWard-Takshashi identities tsthatthephysical §matrix isinvariant under avariation ‘ ofthegauge condition. Asremarked before, since thevariation ofaparameter intheRy H gauge formulation shifts themasses ofunphysical excitations, theabove result, the¢inde= “4pendenceofthephysical$matrix,impliesthattheunplysicalexcitations eanmotcontribute 4tothesumoverintermediate states,establishing theunitarityofthe§matrix.Wealso oe givethe renormallzation procedure ofamodelformulated intheR,gauge. an 1mxrRODUCTION ofthetheory inquestion. Ithasbeenobserved? wa| eo thatintheso-called R,-gauge formulation invari- ry‘Theadvent ofavery powerful regularization ance ofthephysical $matrix under thevariation procedure forFeynman integrals ~theso-called ofagauge-specitying parameter (i.e.,£)implies iy .dimensional regularization’ ~permits ustodiscuss theunitarity oftheSmatrix, that{atosay,that y =.intolligently therenormalizability question ofspon- unphysical excitations donotcontribute tosums , Opreo Peakecagencorinformated in overintermediate states.‘Thus,theabilitytoor- » fairly general class ofgauges:* Thepresent paper __milate quantum theory ofspontaneously broken - fsdedicated tothederivation oftheWard-Takaha- gauge symmetry inageneral class ofgauge condi-shQW)deuiiosinsuchAVE,WHICHcanbetions,inawaythatreflectsthegaugeinvariance used (0prove therenormalizability andunitarity oftheaction asexpressed through theWTidenti- LeeandZinn—Justin iV.General GaugeFormulation. written October 1972. [e)1,Introduction. Notethatthispaper,although calledIV,waswritten7monthsafter theother three papers. Intheintervening time, tH-V dimregcame out. Younolonger had tomake long-winded comments about gauge invariance ofcounterterms and soonte get the Wards tobethe same for renormed greens functions. Themsin idea ofthis paper istosimply state theWerd Identities forageneral gauge rather than any particular gauge. This derivation isdone inthe first section. Other sections summarized below. 2.TheW?Identities. For the first time authors gotothe compact notation (1)where allfields including thegauge guys areputinto samevertor J;.Thegeneral gauge condition is(2)where F,is, Ipresume, anyfunction ofthefields that youwant, and ajisavector offunctions atsomepoint. Notethatthereareasmanygaugeconditions asthere aregroup generators, they index. However, asnoted later, when you getthe gauge delta upinto the action where itappears squared, you observe that the dimension ofFbetter not bemore than 2,oryou have non-renormalizaeable LeGRAngian! Next there ensues the standard discussion ofthe exponentiation ofthegauge delta. Thereisnolimitthatanyparamtergoestozero.Sofar,then,wehave(11),This oO column ismuch like myownnotes, andthefunction Hisjust taken tobeagaugsian though other things might do,Ltt &MAdSo deny wayviewmt! | Next, they observe (for thefirst time!) that thedetM canbeinterpreted using ghost fields with [dcdc] functional integral. Itismyguess that the authors did nothave access totheFPpaper_ (unpublished) attime ofearlier LZJpapers: theFPletter does not mention the ghost fields c.Maybe Feynman's thing does. Nowcomes alightgning fast derivation oftheWIidentity: recall howtedious and simply awful their other derivation was (pack inpaper I). The trick here isto utilize some kind ofconstraint (Ihave notyetstudied howthis works). When youvary subject tothis constraint, outpops (17) which istheWard infull generality inany gauge!! Quite apowerful andgeneral result! Yotice thatM”!appears inthere andthat this isthe ghost propagator, sosomehow theghost propagator isgoing toappear all over in your specfic ward identites! This derivation issoshort andneat andgeneral that Ishould nowprobably invest some time toseewhat Icandowith it,maybe make some general notes onthething. Their previousderivation wassomessythat,justnotworthit. zgNext,theyspecialize togaugeconditions liney&inthefields(asareallthe re) gauge conditions whose useIknow of). This leads toresult (20) which Ithink isjust aepecial case ofthe general ward orsomething, not yet studied it. -2- IIT.Consequences ofWard Identities. With thedimregdeal, youcanrenorm your theory intheusualmultiplicative way,andyourrenormedGreensfunctionswillsatisfythe (eo) Ward's too, andthese arebbtained from theunrenormed Wards simply byrescaling fields and sources and charges. what could beeasier! The rest ofthis section shows the following result: ifyou vary your gauge conditions alittle, the renormalized S-matrix (called S)does not change atall. This point was alittle confused inpaper III, Ithink. This isavery important result anditisrather easily proven: change thegauge,ildorit, change yourS-matrix! Thus,once this isproven, itnolonger seems necessary todothings like showing that theR-gauge and the U-gauge are equivalent, and soon. All gauges give the same S-matrix. Butperhaps thecatch isthat this isaformal proof. ‘Thedimreg issupposed tojustify all manipulations, but maybe something does not work. Idont know that the Gribov disease isbut itmay beconnected somehow. ‘usRenyfomalization. Hereg isanapplication ofthe above results. Theidea istoshow thet, ifyouchoose agauge which depends onaparameters, then ofcourse youknow right away that the resultant S-matrix isindependent of§. Thats what was just proved inthe last section, S-matrix isindependent ofgauge. So,gointothisgeuge,whateveritis,Itturnsoutthattheghosts,the oe)negmetrics, andthegoldstones (called %)allhavepolesatPen*/g,+Theargument nowgoes asfollows: since S-matrix independent of4,these ghostie particles mst always cancel byhoot orcrook, otherwise there will besingularities which depend on §intheS-matrix, andthis isimpossible. This iscalled theRggauge because itistreated intheusual waywhere you havethese redundant fields likethegoldstones. Multiplicative Renormalization worksandyields afinite, {-independent result byargunents similar tothe Regauge analysis ofearlier paper II. Then asyou take<-»0 the poles ofthe ghosties recede toinfinity andineffect you haved arrived attheU-gauge. Similarly, $+©isprobably theR-gauge. Thus, the two gauges have been interpolated inacontinuous fashion and you can watch howyougosmoothly from onetotheother without theS-matrix every knowing about its gauge choice. The particular model happens tohave anSU(2) group which iscompletely broken 80thatthere are3goldstones called X.Butthere areoriginally 4Higgs fields, the fourth iscalled ‘Yand isagauge-group scalar soitistheonethat gtsNZVEV. le) About the constraint onthe gauge functions, and Appendix Stuff. fe)1.Equation(1)showsthefirstorderchangeinfieldsinducedbyagaugetransformation byanarbitrry gauge function gy(x). Normally, onethinks ofthese gauge functions asbeing arbitrary functions. However, youcanmake useofthis arbitrariness toyour advantage. Suppose, assuggested byequation (7), youmake your gauge functions g,(x), what Iusually call 9,(x), depend anthefields! Inother words, pick apoint x". Atthat point, theform ofthegauge function orrather the value ofthe gauge function depends onthe value ofthe fields atthat point. Suppose youknowthefields atallpoints, f,(x). Pickagauge. Gooffand trytocompute the object M.Inageneral gauge, Ithink this canbeafunction of ga(x), 80ingeneral Wtisafunction ofthefields andg,(x). Tonowinsist that (7)vetrue isaconstraint ong;itdoes notdetermine g.Itforms some sort of equation that gmust satisfy. Itlooks like ahighly non-linear condition. HowdoI know that ag(x) exists that satisfies this contraint? Duetothis constraint, thegauge functions g,(x) areactually functions of thefields gj.Ifyouweretochange fhefields galittle bit,youchange the constraintg, andthusyouchange anysolution totheconstraint. Thus, write g,(x; $,)- Oo2.Nowconsidergoingfromfieldsftofields#®viaatransformation (gauge)which satisfies the constraint. How does the path integral change? Wewould like tocompute thejacobian togofrom dg,toagg variables. “ “ %\IQA 3er+) weberht) edie=ak(3)aak|8g+Tyga+CadgrkS$i:3d eaSSAC) Xe) oo=ak(tx)= © ~& whKeAkOnderew3° -x oa “24a= \ «iaWou=2{tkgatCexbeMYG3A sa) bx 4 vsone re)WoesQe@).=ea ~\SH>Bypage+MisOe~o9baa)-\W)«pe3 ers AS SF PA8p —_- 3.Yeeuninemilydanny)aDaddar,rom(uitmoensbous,wawrth, ©uoqkVat,WeWake, Top(Mg) =.So© Ts.og[Cb @)eGbye] (8) BE=ack(WS)<aah(MSSM) =adk(MY +ROW)- oOareoYLgohRN (aa)y. Vwwos)=aetaal'—Beta <fLoeonLa) =S{Quor\ =ay. ain T=\+e.reeUBSAL(AN) Qudkcommeof gtAe -bexde+OeTA(PSR)ASesame |Qy, fa) (Les-4) yre)ProofthatEquation(Al1)iswrong. 1,Remember that for the gauge fields the symbol .represents aderivative onthe group parameters, asinequation (1). So,the claim of(Al1) isthis: Tt (TRO ©9K : ‘a t SawNGy(wae)=ePALpal i(You) *Vey) 1ASw4Yeu) () oR ot Sah " .GaVes—GasDee =bCua’Doe [aiteaigwo(4)So-3\\ Natdénously,War,sosrepamreh 8 *x £ aGh|SidayGude|-Go[SYigGorAs ‘wieiCei|SaediyGaAs) Aker vw 8ie) Ga¥—Gad =iCyad” 0) ee 88 wads *. nig{GaoeS~FaoGufsalehgfiCeheFaw4As(2\ WacoordWasainAdasqvetion dott, IMtokHL4weeayia geagerhy AOS x of gx .(orGaGatGostAS=eLiiheGasthy oes sx 3 . & .ow EE-CF -2+iGu& a SSDute.Mee Ord (i)pe.BE *by 6 ¥ Wa Ga ~Gy\p*= ~Gea =+Gi Oe”w a =Ge SO wt ve) Therefore equation(All)issimplynottrue.Itmightbetrueifbothsideswereto.actonafunction that waseither 1)independent ofx,sothed's donothing, or 2)4santisymmetric inany pair ofhanging indices. But neither ofthese isthe case the way they use it! ' More ontheConstraint ofLeeandZinn—Justin.-4 o)1,RefertoFPJacobian notes.ThereIwrotea“gaugecondition" assf(f,g)=-@ a,where a=some number (one color, etc). Suppose youhave afield Jandagsouch that £(f,g)=a. Ifyou were tonow make achange onthe field andkeep gthe same, your gauge condition would nolonger besatisfied. Forexample, £(#%1, g)fa. or, £(6+d, g)4a. Bygoing tonewfields, youhave “left thegauge" that youstarted in.Eg,suppose youhadanaxial gauge nA,=0andyourotate yourfields byan aribtrary rotation, then nolonger true that n.A=0, youhave left this particular exial gauge, although youarenowinanewaxial geuge nA,=0. 2.However, suppose white youchange your field from $to%+df,yousimultaneously change your gtog+dg. Ifyou choose dgcorrectly, you can maintain the original gauge condition!!! Thus, youarecorrelating your change ofdgtoyour change off. Ithink this isprecisely the "constraint" they are talking about, lets sect:: 3.Choose your gauge f(f,g)=a sothat, foragiven J,this thing issolved byg=e theidentity (zero forparameters!). Always consider itasparameters from now on. 0Then we can say: z2 » 3 ' £9)=FRAY+Gd)829 Sewon dg eet >FAQ) =a+ge)5 =0434 fa. -\ =qs Fag -(C8S\ 8ane “8 SS wou Weenangus SPowSa[doyOot) cmdAa)nd But, when yougeneralize tomany colors, this “solution forg"isexactly equation (7) ofL2J-4. This equation (7)tells you what gmst be(gnear zero inparam space) in order that the gauge condition f(f,g)=a continue tobesatisfied. Sowewrite £(d,e(0))=a true even as$isvaried bysmall amount. 0 weygeebh“6, weslnbet)mervy, Sets Analysis ofSecondGolum, page1050. ‘@) Startatthetop.TheydefinetheFP-object, asIcallit,andIthinkthisthing"4s really also afunction ofa,soIwould label itAyala]. Next, they correctly observe that, when evaluated forfield onthegauge surface, theFp-object isequal toaJacobian object, hence Eq.(4).Itisthen easytoderive equation (6). Inequation 6,since youhave agauge surface delta, itdoesnt matter whether youputintheFP-object ortheJacobial object. They putinthelatter, called detM. Nowcomes thebigquestion: howdoyou"exponentiate thedelta"???? Iwill accept forthemoment that dAdetM isinfact invariant under their constrained gauge transformation: note that theJacdetobject isimindx being talked about here, notthe FPobject. IknowthattheFPobject isinvariant, butIdontknowmuchabout det[Al] asyou move Aoff the gauge surface. Similarly, Ithought Ihadshown that dAwasinvariant, butthen that proof sort offell apart. NowIambeginning forthefirst time tobelieve that thepieces are separately not invariant. So, assumeing that dAdetM areinvariant, look what you candot aSta) fe)WS=Y\ag)atwla, © S(@)~0) ‘ 9asl=\Ladlstate]<SCE)-«)5/Afwomatte ite~)BaadmleTSEMD-2) Rewer Soe, shiaums)Grnabu , Bur FOY)=FQ)+way ~FQ)+) es MW SoWels SaalakMla)osTCP) =Caan) =Wr, Youcanseehowthis gauge transformation waschosen: thegauge function Fchanges bya constant(independentof9!!!).Thus,youhaveproventhatW(a)isinvariantagainst fo)shifts inparameter atofirst order. But this istrue atany point a. Afunction which haszero derivative atallvalues ofthearguement isaconstnat! Thus, W(a) iscompletely Antependent oralll!UnliketheFP-object! NotebythewaythatdetMdoesnoteven -2- Nowtherestiseasy,StartoffwithIntdaH(a)=1.TotherightofH(a), le)some integrable function, insert Wk=W. Thus, L=\eh® 2wave wie) wadegda. AW)=Wah) aero adn) F(FA)-») 7 ot SXS) 5.22.99 c =H&8C akMAYH(FO) : Finally wehave acorrectly done "exponentiation ofthe delta"!!! And notice that it really isthe Jacobian object and not the FPobject that goes inhere!!! Iamheppy with this ifIcen really show that dAdetM isaninvariant! TeeudolesrottaeoxDuceOLgagebeat.|] an 863M Derivation of the Ward Identities. fe)1,Ihavespentagreatdealoftimeonthisbusinessofshowingtheinvariance oftheproduct dfdetM. Thewhole point ofthis wastobeable toderive the Ward Identities, sohere goes that derivation. . 2.Once thegauge surface delta hasbeen exponentiated andthesource turned on, wegett RSAC)w=\radne & Sau) =si)-ERs o0% (2SQ)ahh [Ens eetel, Remember that this holds foranarbitrary gauage andarbitrary number offields. Thus there is&field source foreach component ofeach type offields including the gauge fields. Asbefore, J,isa“supervector". Now: ASA) 0W=\Tasyaakia) © va§AitabedYe&byCET.Growgged worse eeenuddroneW- Vowwarwwommner syiSer’) : WeSagat Veo antorsdedeok Sa) ASix)o=Yuysna) |© © Qa A Sag) iSRSet) 3S) WReCNWA)ABRBE=SOTASE i = SyaQ) ReSETG) _SLE)0=Ylagadn)joShGk38.-\ 0 aort 9s : ~o= \la oxey VS_) pq dx.Yea ) \(Sea) 9Nn od =8S). 2 Gat a ‘ BSa@)=ES-42hey-g+TAVK 6~° +SG) NXfoMengenet gh Bf=Few =a(ReAQ)~= Wo =Dee see pelo . S. Baaay =YEBAMO+TeEnaataCRi}epDp. 3.Iseehowthisgoes, butIthink Iwilldoitinmyownnotation tomakesure Idont get confused with where there are and are not integrals. So: wu Sala =3.8)ax[regi -4R&S) | SSHB=Lax[EOBRO—HASRG] 6 Xs, SsS&@)=-reDaSeo STO)=SorVEOBQVaie) =»BRAS) =\arde VECO) VIG Sr). Y“sVR) Yorts) = YEA] 24) pos(slg ~Me@lion =(mma)Leeds)Lag, fo) = Quan obec: rs) sae~-4[Ddebion ~~DYATMAR =She DaGy) Wee), Ae) GseDoses BSails) =Vax|Ho){4ayaaDees)Tee WF ~Fad|MER =‘bage TOC4)DoD MGA) —JanHG) ~LanaaeSeqC4)DaeQeWeGs)AO~LaxFateDao) = Bg)C8 Wee(ex)=AeALS 0 Sax{eaSSDA Weed-FecalNecs) OninceQuverti danbeanbrvbiany Ae). Japorkerdan LotBel)=he. You . SSP=dk:\Sm{TRG EKODaggMLE»ssaet| WS,geadebs iS)O=Sioaknw) [Saeree{LagsGenrgnnent a-fl 3 NS 4 =SuedeySwMNefgeBIgegdatsaytieley)—Ref re)Ithink here allweneedisthat} beindependent offields J;itcanbespacetime function. Theabovemustbetrueforanyfunction NtTherefore weconclude: ~y- Sit a o\ OBSasne™ FEO~HYagaeTHDaadmeanl =0‘ ‘ ‘ Bemerecomglet: N ASMA) ~ _ SideSakHQ)& THe)—E\aeFitsalga; 8)MeeGed) =© Here weare taking care toshow allfunctionel dependence onthefields J.Since these canbereplaced with 1/i d/dJ wecan say: ai) ay7‘y br(Hy)5\ayeels)Delsrs2os)Mae(a5te)Wit=0 2eaeWLI=Vlas]ath)expt[sw\bh-£h@)1 This result issimliler tothat derived inLZJ-1. There recall that they derived the thing without using the trick gauge transformation. There also, only gauge fields appeared, and covariant gauge was assumed. This result isvalid for all gauges and inmlti-field case. Ithink probably this isthemost general Ward Identity you can have! This isanoperator acting onW[J] togive zero. Eventually this will lead toWard identites onthe Greens funttions themselves. i 1 Fuadkin .cae OvREVIEWDVOLUME2,NUMBER121sDECEMBER1970SMatrix forYang-Mills and Gravitational Fields E.S, Eaaawy avo1.V.Dosw Physical Leben Insitute, Academy ofScenes, Mose, USS.R (Received 19January 1970) Amethod issuggested (andapplied totheYang-Mils andgravitational Seis) fotheconstruction of thegenerating functional (Smattis) focfields posesing aninvariance group. Theunitarity andgauge independence ofthe Smatric onthemass abelareseenexplicitly 1.INTRODUCTION which fact reaffirms itsunitarity. Itisshown, further- ce.more,that,takingtheadditionalconditionsconsistently [HEREhaslately beenconsiderable intensifcation intoaccount makes itpossible toobtain self-consistent inthestudy oftheories partially orcompletely eauations forthemassless Yang-Mills fieldinthepres-invariant under non-Abelian groups oftransformations. [°°ofanexternal source. ‘Thisisinconnection withthediscovery ofvector “TES.t"IVtheFeynman rulesforthe.gravitationalmesonsandtheirclassification intomultiplets, withthegai!areSomstructed Incovasiantgauges,‘Theserules”useofvector mesons toaccount fortheformfactors ofCSrncide withthosesuggested inRefs.2,3,and 5,Tntheparticles, andwiththecurrent-algebra approach. Inter-framework ofourapproach wealsoobtainthe5matrixmediate vector bosons areintroduced inmany schemes fo,noncovariant (Dirac) gauge, forwhich theFeyn- ofweak interaction. Animportant example of@theory manruleshavebeenobtained byPopov andFaddeev* _ withanon-Abelian groupofinvariance isthatofthe{titymethodcloselycomected withthecanonicalgravitational field. formulation ofthegravitational field.Inaddition, by =Inthepresentpaperaprocedure forconstructing theGurmethod,theequivalence oftheJmatrixin’co- =|Feynman rulesisproposed fortheories possessing *variant andnoncovariant gauges isproved.zuige group, such asthetheoriesofthemasslessYang:We'.ethefollowingnotation,Greeksy»,dy-.-and fo andgravitation fields.Ttisknown thatsometheLatin,7;#indicestakethevalues0,1,2,3andtional (gauge) condition mustbeimposed onthe4,3,respectively. InSecs.ILandITE,gj»means the dynamical variables inorder thataconsistent quantum 34:r'kowskt tensor (4,——-_-) andBiawean theunitfieldtheorymaybeformulated onthebasisof@tensor,Bythesummation overrepeatedindicesisevery- Lagrangian density invariant underalocaltransforma: here’meanto,b,-debo—-osbs; 0,020/02";=9,945 tiongroup.Incovariant gaugesthiscanconveniently 9.9,4,,InSec.TV,gusmeansthemetrictensor,andthe bedonebytheuseofLagrange multipliers. Thebasic finkowski tensor isdesignated as6,,.Theusualsum-ideaofthemethodproposedistochoosetheLagrange mationoverrepeatedindicesmeansa=Yip!Obrultiplierinsuchawaythatoneisledtofreeequations Weusethesystemofunitsh=ow1inSecs,randTILefmotion fortheadditional field.Thisfactguarantees 14Sr1gcg-1 inSet,TV(where fisthegravitationaltheunitarity oftheSmatrix inphysical space. The Constant)Feynman rules obtained coincide with those proposed inRefs. 1-5.Thediference ofthemethod undercon- y.GENERAL THEORY OFCONSTRUCTION OFSiderationfromthatofRefs.2-5isthatwehave|"SevNMAN RULESFORMASSLESS YANG-succeeded inobtaining asetofconsistent dynamical MILLS FIELD. GAUGE INVARIANCE OF equations completely: describing thetheory. Ontheone §MATRIZ . hand, these equations make itpossible toelucidate the teason fortheadditional diagrams toappear, and,on _Inthissection thegeneral theory forconstruction of theother hand, guarantee theunitarity ofthephysical aunitary Smatrix formassless Yang-Mills fields is Smatrix. Section [fisdevoted totheconstruction of considered. the§matrix forthemassless Yang-Mills field inarbi- ‘The classical Lagrangian foraYang-Mills? field has trary gutuge andtotheproof ofthegauge invariance of thefornt theSmatrix. InSec.LLL constructing theFeynman Lox)=~2Gye)Gue*(2) a) TulesintheCoulomb andaxialgaugesisconsidered on. - —_ ‘hebasisofthecanonical quantization procedure. TheHereGy."isthefild-strength tensor,Smatrix obtained coincides withthatfoundinSec.UL,Gy,*(2)= dyAy*(x) —3,4,°(2) +24,24(x)A,%(x), (2.2) AFFeyuman,ActaPhyolon24.687(M60) As*(a)=forAye(2). (23)oOB,PonceaniVNpope;yg.eter285,30(og);The/*¥arethestructureconstantsofthearbitrary Gi,hemandl.Fats,EAPrepr,Rie,1904nite-dimensional compactsimpleLiegroupG.Thefo *S.Mandelstam, Phys, Rev, 175,1604(1968), 7C.N. Yang andR.L.Mills, Phys. Rev.96,19(1954), 2 2841 Bs =a - 63.04 <Fradkin and Tyutin rec'd Jan 1970 ©coments tispaperoccursbeforetheLeesequence. Iamonlyinterested insection IT which discusses YMTh. Later sections discuss comparison with Canonical quantization procedure which Idont care about now, and also gravitational applications. Section IT: 1.First offthey write outthe legrangian forYang Mills theory: noother particles areputinhere. TheyshowtheEuler equation thatthefields tensor F.,,° should solve, andthey also show ageneral identity that anyF,, solves, regardless of whether itsolves theEuler equations. This isthefirst time Ihave even thought about Euler equations for YMT, sowas worth reading. See gauge notes for derivation. 2,Next, the usual problem with quantization isnoted: you cannot treat all field components onequal footing andimpose the canontéel commutators: itjust doesn't work, You ere forced toassume agauge condition ofsome: sort. 3.TheirgeneraltechniquewillbetoinjectaLagrangemultiplierfieldwhichdecouples fo) from allphysical fields. This method works simply inQED; theEuler equation forthe fictitious field effects thegauge condition. Thesame general idea, withm certain variations, also works forYMTh. Ihave discusséd this technique onaseparate sheet, seealso notation translator. Theneteffect isthis: thecovariant gauge generating functional is(2.37). Theimportant facts are: (a)There isnever agauge-surface delta; ‘thegauge fixing term appears inanatural wayfrom thegaugssian integral; (b)the FPdeterminant detM appears, not the FPobject which isinvariant. Sothis method ofderivation isasort ofalternativeto theFadeev Popov group method. Itgives thesame result! Theficitionous fields aretheghosts, andthe usual Feynman rules drop out. 4.Next, they show that theon-shell S-matrix elements areindependent ofyour chosen gauge surface function PSIwhich Icall f.Ihave nottried tofoldow their proof indetail, but seems reasonable. yest 5.Next, they specialize results fdstst totheaxial gauge (npoints in3-direction only), Coulombgauge,andthentheyconsiderthealpha-Landau gaugesystem.Inpassingthey fo) show that theS-matrix isindependent ofalpha. They dothis labt deal byusing afancy constrained gauge transformation: itwasthis section which helped meunderstand how detMdfisinvariant inLee's papers. I I Bn Adiscussion ofpage 2843 ofFradkin andTyutin 1,InQEDyoukmowyouhavetoimposeagaugecondition likedA=0toremovesomeofthe fe)fields from consideration inthe canonical quantization procedure. This can beimple- mented assuggested in(2.17) byadding a“Lagrange Multiplier" field term tothe Lagrangian. Adding such afield just adds newEuler equations, andyou‘sée atonce how the gauge condition appears asthe Euler equation inB.The field Bso’used turns out tobeafree field. Thus, ifyou don't include B-fields inyour external states, they will néver enter the S-matrix because they dont couple toany physical particles. Inshort, they are just atool toget you correct quantization. 2,The idea istosomehow dothe same trick inQCD. Ifyou try itinthe same way you run into trouble becuase you find that your.B-field does couple toA.Soanew method isneeded. The. method isdiscussed atarting at(2.21). You beiig. bypostulating alagrange multiplier field Band you add certain terms totheTebreneyen asin(2.23),Notethataparticular gaugefunctionhasbeenchosenin(2.21). Soyou have added anew field tothe lagrangian and you can now get the Euler Lagrange equations (2.25) and(2.26). These equations areeasy toderive. The trick isgoing tobethis: arrange for field Btobeconnected toancther field Bvia anoperator Dasin(2.24), andchoose Dcleverly sothat Bcomes out being afree field. Then you can ignore field Band you still maintain “unitarity of the S-matrix" fisical. Examine now the sequence ofevents asyou godown the fight hand colums.Ifweapply.afatD,ontotheirettermof(2.26)wegetzerobecuaseofthegeneralidentity(245)Thus‘thelsecondtermalsodiesandwehaveDRB=0asshown in(2.28). Wercan define Q=DR. Essentially, this isthe transpose ofthe famous objectIcallM.ThenQ°isjustQwithA-0everywhere. fe) Next,script-D igdefined astheGreensfunction ofQin(2.31), thisitis the transpose ofmyM. Ifyou then make bold-D the combination shown in(2.32) you gets @B=0=afp] =afa"7a%a] -0°s-0. This last equality isafield equation forB,andsince Q°does notinvolve A,Bis seen tobeafree fields, asdesired. Derivation of(2.37) So,stick-this lagrangian with theBterms added into your functional integration toget agenerating functional asinfirst line of(2.37) where weintegrate over both the gauge field Aand the free field B. The first thing youdoisreplace: a 1GB=det(dB/dB) dB =/D/ dB. a1 (aay where Dhere isthe bold D,soD™ = (QS). Qy. ;yougetwhatisshownin2.37.~(@e)QsWritingthisdeterminant asexpo Note that the order ofthetwo operators inthe trace doesnt matter! You now are left with atrivial gaussian integral invariable Bwhich you can dousing Sidney Matrix Theorem which yields the term Ihave circled in2.37. Irecognize that term asthe exponentiated gauge surface delta function: sonow Isee why they shose those added Bterms the way they did. Also, Isee something that atleast resembles the FPdeterminant, which Iwill think about later. : The point isthis: the authors have obtained Fadeev and Popov's result using theLagrange multiplier field. They never factored outagroup integration oranything p Like that. Ie, its just adifferent approach.fo) GardwakeDuckhsGuaFPAKmatOeFPobyck|MokogpemeHH(259), Opec as 2 ye Woe:Quinsauaion.(220Ca),(28),C25") | OE EE TFT ea ee rn, a Tee SOs eee oe _—3Mba wo a wee ee eh oe ne De ep ce s Conarstviad Glooit Taro . O95Geonrding foFET,GasaaQWngoberdatyoroakamthooney WLbdrDumDybee DaDraDringMakdebeBoB®. “PindVaiDavis,Sowaowldohe ~\—iv eo D= (ar)(3B “ly St \ < 5 rD=(38M) Cn") a=Lggelmay -eeeran -iyG&iNRIA)|eae ee oe Q Youseethat this first factor issomehow removing the“inessentiel part" ofM.Ie when youtake adeterminant, youboly care about theA-dependent piece inquantization, soyouareallowed toextragt anyA~indendent operator youwant from M/. 0 Notethatincovariant gauge, R"=d"sotheremoved factor isjust(box)"> which Iamused toremoving. 2.Now, when youwrite this outintrace lognotation youget: = 3 \D\=exp(aL (Q@)}) « a9 ~oR.|Aye oe\)ae > ’ om =1@Veoh} BUA -Py Fa= \> Tyree a\ 9 ewanonga[4(CIG-DMSF)C-YNOVC-o] Hereyouseethefamous minussignfortheloop,andyouseethatwhatever at happens tobe, itisthe ghost propagator. Infact that isHust what FTequation (2.33) tells us.Like(p*-m?) f(x)=0ismotional ,thenpropagator isinverse fe) ofthatoperator. Sonowweknowwhatisthe"ghostpropagator" inany_gauge!!!!Obviously for covariant gauge this reduces tothe result Iknow. . godockETmolatur doLeck alvin 6: ©Onenseabney HI:wyadhgeWhiemedLATDronBanwtselBubQnmerewit @OyamyBkGpGuoePomme JP Greae. btOHoy ode nsAmn * w iN0 Go=+|We, OWeoLekagerAvielen St) To(2) aVARY =+y¥CaveAwAy weobec& ra=4)4 NAY - ypeee). Congenebo :Na=3Cave () ANowLook. (us).Qow achcewr .YrsRratAe<awk7De=dDn—hyCoreAca¥ ah. 6)Tort,Lodob(03!) , “abWy’ flew :o .sA~imxVea a hs EHP ehwm => je£-€ . ©Yue,dey(29) . SM)wedati dt““We TeOne=0s-i@Ge =Sa-i Ge - . =Sab&_~Cate8. °=Sab+CowSO, =>|;.pe 7yace ( eitUe=iC©. 0)- Qu(2,3) oneMotCirewrt( . 6 weOC)saya {~(&)aw @\34 uw=,G)6. \n GYXAY Se ann ; ;U =6C oo Apes seekndOyo9 Ahd Fas §s4 GR=vitiwh+ L eadv" WwGAsoN=-CCabA=#6CacAW ads A |=BVPcayAg Ooenkbamce bySormsay .AL .a ~\ i>A~AVA tyAW whby 2 ; ” : al)oar YuncmuctSage| °al Ne (& af A=®UAT ©ON )ov)NS 4=~&(wy Ogen,Weavncacebed wilh(2-5),by©. Conclusion: either mysystem isinconsistent, ortheir system isinconsistent. For thetime being, Iwill assume theirs iswrong. Thus, Iwill notead their paper "close" on notation.‘ i) Slavnov oa NOTICE: THISMATERIAL MAYabSawyARD IDENTITIES INGAUGE THEORIES PRoTeCED BYCOPYRIGHT Lav)ers OME17U.s-copey oOEk. A.Slavnov eit Generalized Ward—Takchasht identities areobtained forgaugetheories ofthetypeofthe {Esse Ressless Yang—Mills field.Itisshownthatallthedivergences oftheYang-Mills theory .25, canberemoved bymeans ofarenormalization ofthecharge andthewavefunction.ee {RES Arelativistically invariant formalism forgaugetheoriesofthetypeofthemassless Yang~Millsfield|BBexasfirstconstructed in{1,2}.Inthisformalism the$matrixisconstructed fromaneffective Lagran~\$eian, whichisanonlocalfunctionofthefields.Despitethis,thegaugeinvariance ensuresunitarity ofthe1g848matrix. Theperturbation seriescontainsafinitenumberoftypesofdivergent diagramandthetheory}fisjemtherefore berenormalized bymeansoftheBogoliubov—Parasuik Roperation [3].However, toensurePSSenitarity oftherenormalized $matrix itisnecessary totakeintoaccount theWardidentities, whichguar- igmeegaugeinvariance ofthetheory,whencarryingouttherenormalization.{202 \.Jnthepresent paper weobtain Wardidentities thatrelate diagrams withadifférent number ofexternal $gBelizes andaccordingly renormalization constants ofdifferent vertices. Weshallshowthatalldivergences 43#iiean-be eliminated byarenormalization ofthecharge andthewavefunction ofavector particle (forsimplic~Ageulty weconsider aselfinteracting Yang~Mills field. Allowance foraninteraction withotherfields does iaotleadtoadditional difficulties). Ward identities havebeenconsidered in[4].However, in[4]onlyre~{editions ofthetypeoftransversality conditions forthematrixelements areobtained andtherearenoex-Y'sslicitrelations betweentherenormalization constants. - {E2585 Themethod ofobtaining theWardidentities developed inthepresent paperisapplicable,to.anygauge UgSecory,includingthegravitational field.Toillustratethemethodwederivethewell-known WardtewiaesEielectrodynamics inthefirst,section. Inthesecond sectionweconsidertheYang-Millsfield. S32aiioldthisinvestigation wehavebeenconcernedsolelywiththeproblemofultravioletdivergences and oO haveLgnored thedifficulties associated withinfrared divergences. Togivetheexpressions wehaveobtainedpaisPéll-defined mathematical meaningitmaybeassumed, forexample, thatthesubtractions arenotmadeon\giliemassshell. Ofcourse, thisdoesnotsolvetheproblem ofinfrared divergences andthis.question re-. ils opénatthepresent time. 7 fentesElectrodynamics ie amiesa‘ThegeneratingfunctionalfortheGreen'sfunctionsinquantumelectrodynamics isdetermined bythe‘Mpkectonal Feyninan integralSeb.: A rae £ 2m—fenif{ [[Set. era) ve ‘ +d,++be]dr}adap, ERIE 2)Jshegaoge-invaviant Lagrangian oftheclectromagnetic andelegtron~positron fies. TheYeeC/20)(0f Ay)?fixesthegauge.Inparticular, a=0corresponds tothetransverse gauge’a«1,toNileFeynmangauge7ThoGreen'sfunctions canbeexpressed interms ofthegenerating functional (1):. BRS - = ozPos:OTA FedHED=BeaHREETDnets %= __ ___ ——— Poorer A:Sw@hlov Mathematics Institute, Academy ofSciences oftheUSSR, Moscow. Translated romifsereticheskava iMatematicheskaya Fizika, Vol.10,No.2,pp.153-161, February, 1972.Original article +Wlenitted June 23,1971. se!|©1973Consultants Bureou,adivisionofPlenumPublishingCorporation, 227Vest7Sweet,NowYorkSo|MY.10011.allrightsreserved: Thisaniclecaanotbereproduced foranypurposewhatsoever without - Permission ofthepublisher. Acopyofthisarticle iseveilable fromthepublisher for$15.00 ape .- *” Sees iBT a Slavnov rec'ddune1971 A Inthefirstsection, theQEDWardidentity isderived usingthe functional formalism. Itisallvery simplg, though Ihave notcomputed things. Inthesecond section this idea isgeneralized topure Yang Mills Theory. Using theconstraint method, thecomplete Ward statement is(20). He then differentiates this various ways toget desired relations. (a)asusual, thelongitudinal part isunrenormed (b)thesimplest non-trivial wardis(25) which relateds thegauge field propagator and vertex tothe ghost quantities. Itisthisidentity which produces thefamous statements thet2,/Z, =2,/Z, .These 2'sareclearly defined astensor coefficients otcertain quantitys. (c)theL-pointWardisequation(34).ResultsarefoundforitsZ's also.wot : The major conclusion isthis: these Ward identities relate the Z's and-reduce thenumber ofdivergent constants youhave toworry about. Theprnof* ofrenormalization isthen simplified, Iguess.- : Better:whenyourenormialize afieldtheory,youhavetopindowncertain PY things: eg,youhave todecide where your 3-vertex isequal tog-These areciled the renormalization conditions. TheWard identities just tell youthat youcan't end should'nt independently set renorm conditions that are relateds. fe) Comparison ofSlavnov Notation tomyNotations Hunting forErrors? fe)1,ThefirstthingtokeepinmindisthatSlavnov's symbolWisthenegative ofmine.This just follows from the sign chosen inthe gauge-fixing lagrangian. Looking at Slavnov's Eq. (13) you can seehow other things are related: Save ome. 8 5 8 .a Ap [gouge Sod. aoe, ob é el Saree reek .Dy +idw /5.1 as —Xie . ob osWS Gly) We o\bo <)gtbs . 7 ro)Wy Cg) MnCO) ; .@) ye@ [sssnens 2.Now lets looks atSlavnov's equations one byone and tie them tomine. ~ Eq.(20): This isthe basic Ward Identity. Ihave already checked elsewhere that this agrees with mine. Eq. (21): Ithink Slavnov has omitted anionthe left which Ihave filled in. Now Iwill convert tomynotation asfollows: first change sign ofalpha, then resale wtts toget: Ry at +h* =j2— 3, “ ~ £38 &>es|(«WASgCGroH) ~ we x rn ae : 2a-ghraerby =-$4 -£CiCrOM). 6 aorashy 5EGR)-8 ») And this ismyfirst-level Ward Identity, all factors OKexcept Iamnot checking bolor label ordering. Eq.22:With alpha sign change andiincluded asshown, this agrees with myexample 1 onpake 5e, exactly. ©) 5.23: havenotchecked exactly. Theobject D,isFPof1/k”solooksright. Eq2k:HereSlavnov hasthei'scorrect. MovetheitotheRHSanditthen incorporates into">togiveghost propagators. Thisismylevel-two Wardidentity, see page 5a. Bq. 25: This iswhat Icall the Slavnov Identity. Ifinally proved it onpage 17. Eq.26: Goback to25first. Slavnov's Dis myiD,sohehas iD... =Gsxitiher According to(26), hisG-=D®. Sohisidentity should read (converted tomynotation), pb’... =-iD8...Myresultshowsaplussignontherightbecause Thaveky=-ky-k,. Weagree! Qq27: Iaccepted this toconvention mySlavnov Object gamma uv. Eq. 34: This ismylevel-3 Ward identity, but heismissing ani. Page 5b. Eqs 35: This looks about right for my4~derivatives onthe J-glue. The terms onthe right are from the disconnected parts onthe left, ifyou like. See page 16-19. ca) Oo