Binder 1 QCD Kummer & Zach 1978

Binder 1 Kummer (1974) & Ball &Zachariasen (1978) QCD problem Qn anemarnany aRecmadar denomOCganiiiome Mae 10,196 OBina Warum SO 2DieWhataniston: OkeAnisepaguannates,oatitansEiBumanaeOat BanEOCEk,WOUTT OrnCAELEONE SIPONONTF A, ennai tan. i -Wi _eeevi-Asya eene -Sadana22laenamines Tateeatns weeNan\Oe —JaneaeBe\ocakeeB=2seGashJulgang=Near——pspoathsbyByoaSewen8)Qelatothbewt, a oe 3Gil)=OW)+PbeK FO Gan geddimn waedeeyheOASJosealFTaadtantanislate _eee astnghe EMAtreeos —Hemmatattana OnaanSadcdoaspect Ke KB HV=SE phe i “qoutefeeDusensentence ciwhan, ee 2).danwiochim ol,Soot.qhunenQeopeaatn Wandasaactabiy=lonataauge G00of >. vaste We Ce — Bind 2”QuSD.SAM!, ee Oo.Frigg cuntBsatinAesDayOritionanbaene a KO) hasan asheoatstoned layoulenackees wr!YeCaeterna, 2.——hevade_omouauhie” Deadint doMrideadQOK=KoyKe).MMosguar Titesationandasbgadatee sehucumlupata. buaoneaeunlagroda, ~fn S=\\omdWeWannormomdsarrodele, “larsumar a ae +CemunSTEL. importa: Tasso, Queallioned saetsowsih doingfuslegale,-Vex uleqrade usetommyyonGoulddrtarsal Wrse 6NenaakbiaSonaabt.nanoGaMuseeAja aadenvedancbe asG0,filhacasasAeahenhcbew gottaaasGfs0.falciliasousete~ —-CORewlodeim wee aai"SoEEpe®o02 aeeeewe Soooee SICA rankiscnaOia.UnoatkAajar.Ooo,Oredomammabe,ak ee AidoatheddInisowsomsheded QuckkeWeDoaoa. OES aecer ahe pension,westin) VwwoeMestonmuioae marianARocdsasrs wee O_Sesice “Sithaney cndaine. osmnalksAOodedcobedakies aseAne-tangerdoraolsRor.KLossvisaanayaCyfe,cangiikerdeed? (#6)srpacnciona) >gieniesKER,KU),LTR.Sonnsaoutholsbaewnd. waaay eetpeeteasing TEOarececaus Cfnalagaua decacheneLKpctlie) or AT Quid YarGemuadMWordey nome‘SiefSOM.fated(8AaactDumke),wewaders._bo. Qa_omodeh. Ma nN ¥.eeggaaanl:fonctrinaoeeseae"iatae nasa Wand Naacdlentyorgphwalemaanse Sumcbwie Sileral(ofht) “Mae SoDaath doABsS=cacbombe. Crchuaiessondcbalest BRRKSodoniik.diQuintDuceGahIewrdodlete inilagaals, asdasepsd: sdaroniiage InratiaQaskKs.0aasomMosSAM.Yruudslsteyannie. ak.ovnonenaewaeanAcoonackYhilceSanches a a Sore DL-raanss, OnisonandedstarsvaseDif=VOYA,sy=Xeo(e),30.gna.\roussak.wgdanaauh x:dagroochrad Bo.eee a OFmoat. oeQ=hNaSoddeKo =at anda oo=Mmaasdetbe de stl: c=85Crctacldh “LandKgottaie=t=9005, caesou. olstarmed imMooSAMO madd.cQd=aCaacnanesativebana a _—meWW)s2/POrdWs24,6,6Ssadarenegotaomece AaSakeee, aee ee, -Bids 4,"onsom, soa en ee-- =«CheesaManneJevtonTagan). ee ~_—LO thiegenvandesasadvatDisDhsunlenadLalanalanges on9694-20505.— ——NeSenge Teakcmasughee Grasrvata_courainag) ourRekeraadeudacriar oo.coeaggeMet Sowatsacaleesenmtaelig, aoman.Modis sankpa Rot JePahtbcLCADsIneepiesQe. ——Wg assabedlLowfpQuanfergpasnallsatedvastave): °— a Sl On nase.Cason parasipsaanas) [Sete yYon ureoosbind teak\asitn(eeoftcukGeSosaWadAKEETB, so,OYSetkOeaNarsenes squctnon odAKO. ununeonokeindQuahOrie.wosa c(2)sucksGratLe,Peesa Oar SSELINES death CREO)deePe -easth mak.Yorocomomaluin d suos Mak1aR=O. nor-ssasaneala, plieswassaat.AKEOJoscanta asodovrsai. .ahomatebEOe2SEESTSTEEa=rGFEOVAaFOPS —-—Sespind. Un.TeaCondiosmacananrOnneaGokMaodeeas oath.sare.resa“Kyte,DatTe)0."MaiaQedseadsmonica waeaay Raeale ae —-BingonS: *BeogctEadsQe Osing,dasaeanid Wieanges unainDetDyQRS,ualageenbe© oan.G805sauiladanswage WigAanDoesDat; Oarawa,ese.nnora3yawnSe feburyandaedoaDasa TRUSSMalomsetk onderic.7thoBermSong” boot.FaMace, 20080anycarpe BoAnn ey ssreTeske®Blong"Wsatkotkfocodons. Ma.__RawkQeSereweashyodedn_oaden. DAMMaiaomadoa ——Brboas.p94 DSOromowskinnd OnaOnd=Zecnedichoin woouolebad Voydesean~Tk,d=ENbck20.dlswomsdGeainomySAMvisaDiigs —aneeaanslangyi—Darvostfrcaesk ANamdDrerao DarposnNJanisdoMlatecocatlnndar olasgpalisleie lokcastdcuahgphts Dna.gunkblacksabeinKenan aotsuthWh abit.~—Sanco! hatannass ytiaCaos,1gneteDreier—(seo." swlelned ____, ne nud,OotBoomSampanaanbbacame: a bh ET A-----— akakDeathte\oweitadengstMate)eee Lt reedEEEKEJertonqurasaacled - Ro Le, TTItet, ieWlctstakweBE aWSR OT Oaae Thcoosoerelaaegoatsdso . ——NeOvagoku asariwadDeacandTinkenber(3128).@secadDeakSeSitapood inkoeHODoaialSeguaeAcOE =Jiewe.Adovewos dnQuisweskex lnscanoe moOtiga deusoAsaeO\neughth GesBahlverdedadInoutArrclrSinger(Sasbessant—power)Sees“IALAruth):oo,BailRentedcadiersleat MWQ@asK=0eroblen, Pharccirumtonsatts, Qidelinn +Soaausb tochtodoV0,wonsethppdounoug)UntE2geodES.Se eeOe oo. are anQ. eo “ )SOT EET aSeeae — Tso nung watfpowodckvinSeaVEEnastivnotaannAbent ~~BalLowlerc chihoust.o iatWienot Gonsarsd taOerable—sevsllush osShesdiffestense eqmairen, weaAkS0bitLeo\Soee__-Frodls makerwooLenleing J.BaOnswey,OrLacepsExodVorter. Br SekineBehawinsaathQuaDE BeAealadolUSE ~ThewaDaearnFOPaesSPTOTathADCue eaQatagQek =0),QeAadorescowedBeso"Ydeum, Romane We———BadSdersCRIEDorynena votes,. we en ee Cn) Fa Shane stepoateeeBlak+qmaedKad.Gam eeeWalkVeer aoo RAYS AnlnLalowafShiga a) a0 CC ©Dinas Cromeeauaced loLetsanat vohagGahaxQuaCNS.SoArcot oat_oomaradeack”. Sokdken ac coLS.2.OKSaulg3Fay =~(Hops WAY- Binds be ee ee OP iiaaiatukvioaeapbdn,RASaleOoeeae Rood UTYniten0condeop2SARLpendaskANESo.BickquevanesMessen Aenisefoneniiys we~Tanotedcn,‘batasRsoou dudeSassGaalapayvaslenohoLoaa yesreerYRmn,satzaoa AdeBulle)=2soarsaShasahen___ «TRJoni.Gar,awdsomgulessoeinJaalavaverahcama? Qi oxoolewscmes a ee Se .sacieke—seaegpledlsaailsfos248-4,Boeceayes«Enc—7 ao)-WastofOraarFAOOOOO AFaeWRENS TNTTCeadaca)’ Tete eyo Miespatittes a OR aS FA war=G/eetay Vedacaer 00?SS SY aoe —WilorSaadtedeQaReimbsgpota, ba,JainOiaremauckSYARABadadonyokGeaWAL.denne5TE _cheaos” -Anyone, Soon Seam” 2Domana ol. ne qe, eeBoOuronosqvain, Oho.WALSemasassioo oe _Sieasp2]@seeyj ee TREN peteLA Leinedealey, Oiaiaflyaondric som, Sede owRAS. — a SS S.Jposad: on .coe ee TEARS =GaeG my Saeco odingfpleuadaaegon, sulehachasumctinceg pages Seaede Epa aenglouncecloner LeBe Wan) 2 ee OFshikin.sangoeisndondngn_pk.OunySaasGenaones eae’ YokanSe AOsSonebe Or ware. Roonreading. Nr,—oindfins_otaarse aMeo. 2 Te ptpeQekiay egg ADale vam qd ee —asennad ctotivlale Wasdardakine donBryDore erie evs,SemESTNnrerCSSACoe sefebeNnfaBata —tne ns asoe swseghe_Anaitt (rienfi_(samnensssenty)ARIS eg Gon Seery — Kummer (1974) rr ‘ . (ByesWysieaAustriaen41,313-334(1973) (eo) anae . Jc;bySpringer-Verlag 1075 d . Ghost-Free Nonabelian Gauge Theory* : . By . W. Kummer Institut fOrTheoretische Physik 11,Technische Hochschule Wien, Austria . . (Received October 16,1975) summary . .‘Theadvantage ofthespecial axialgauge n¥A,* =0foranonabelian gauge important simplifications incertainfield-theoretic applications, whichareespecially 4important, isuchtheories areusedas(e.g.asymptotically free)smodels ofthe haironic interactions. WehaveinmindWilson-expansions interms ofgavite. invariant operators ete.Itisshown herethatthefield-theoretie machinery eanbe ‘tevelaped consistently inthisgauge. Appropriate generalized Ward-idemtities can :lwderived andformthebasis foradiseussion ofpossible counter-terms inihe : Lagrangian, Although, atfirst,thisseemstobeanexample ofa(presnmably) .renormalizable feld-theory withnotmultiplicatively renormatizable sellencrgy. in° a theenditspropertiet tuenouttobeextremely simple,Oneobtains,e.g.the“naive” 2 :Ward-identity 24=21)=2(#vetweenthedivergentpartsofthewave-funetion .. ‘renormalization and therenormalizationsofthethree-vertex andofthefOur-vertex. 4 1.Introduction * . NonabeliangaugetheoriesoftheYang-Mills‘spe[1]arenowinthe . :{cusofinterest, because theycouldprovidenotonlt2unification of zweakandelectromagnetic interactions 3)buttheyalsoseemtobethe vwnlyrenormalizable fieldtheories showing asymptotical freedom [3], : .: whichseemstobeagoodapproximation totheobserved scaling prop. .erties ofthestructure ofhadrons, . Ingeneral inthetreatment ofhadronic gauge theories evenmores. :. Knowledge aboutgeneral properties offieldtheories isnecessary. E.., .thediscussion oftheasymptotical behaviour ofthestructure functions =1mdeepelectroproduction FA]requirestheconstruction ofasetofgauge-oO -‘variant operators, tobeUsedinaWilson expansion [3]ofoperatots in - .(inection withaCallan-Symanzik equation [6),inOrdertoarriveat =‘auge-invariant anomalous dimensions. *Dedicated toProf.Dr.P.UauAx ontheoccasion ofhis70thbirthday. = typed Aug 16,1978 “KumertNon-AbelianGaugeTheoryWithoutGhosts,(rec'dOct7h,publ.1975) 91.Introduction. Incertain calculations (eg, deep inelastic structure functions) where youuserenormalization group andWilson expansions, youwant tohave a setofoperators handy .Youwould likethese operetérs tobe,gauge invariant. A problem withtheusual ghost-field approach toNAGTisthattheghost field is notgauge invariant, andthis renders difficult thetask ofconstructing simple invariant Wilson operators. This seems tobeoneofKummer's principle motivations «for working uptheghost-free version, ie,forgoing into theaxial gauge. Itis noted thattheghosts donotdamage therenorpalizesbility ofthetheory (aswas shownbyt'Hooft, Lee,Zinn-Justin etcusing Slavnov identities). Mummers remarks about (2 still confuse me,because Idon't ,seewhay, youwouldtrytodo(1.§7)with(@1.8). ButIdounderstand thatthereisagreat practical motivation fordoing NAGT intheaxial guage because ghosts are amess. Bytheway,inmynotesonaxialgaugegaugetheory, (stored inBZnotessomewhere) Ihave derived thefact that (1.9).is the"@- axiel gauge "propagator fortheguluons. (Terminology inthesense ofthed~Landau gauge). fe) 2.Slavnov Identity. HereKumersetsuphistheory. ThenA=0gaugesurfacedleta function isreplaced byaneffective lagzangian term asin(1.10), andnow thereisapathintegration overthedummyvariable figids [ac].Remember thethe ghosts were also dummy variable fields inthis sense. Sonowyou cantalke about things like "mixed C-A propagatori and soon. Each field gets asource, Jfor A asusual, andKforC. 7 : - Themajor equation ofthis section is(2.7). Basically (2.7) ismerely a statment that.$W-0 under egauge transformation, where Wegenerating functional. Ie,thetheory isgauge invariant. Ilwill forthetime being refer to(2.7) as the Slavnov equation. Essentially, itcontains all the Ward identities ofthe theory. Itsill soon bereplaced with the simpler (3.11) but lets wait onthat. Ifyou take (2.7) byitself with nofurther differentiation and set sources=0, you get 0-0, as shown in(2.9), sort ofanexersice inconsistency. Ifyou first take ad/ad derivative of(2.7), then set sources =0youget (2.10), which says that theC-A propagator does not renormalize. Bythe way, remember that these functional derivatives are giving you fully renormalized Greens functions. Next,itisnotedthatallGreensfunctionsare"transverse" tothegaugevectorn", fe)asshown in(2.12). Inparticular, this istrue ofthepropagator, see(2.14). This seems very obvious tomewhen Ithink oftheprop orgreens functions asTaufunetions, youimagine n!reaching into theVEVTOP anddying against anyA. Ithink theanalog of(2.7) intheregular (Lorentz)gauge cumghosts isthetool used byZinn-Justin. Apparantly that resilt israther messy asyoutake higher derivatives toseeyourWardidentities. (*) 3.Pooper Vertex Identities. AsIknow bynow, the generating functional WorZ canbereplaced withafunction J’=peoper-vertex generating function. Several exciting results dropoutatoncejustbyreformulating thetheory inthisway.: (1)Youcandefine an"inverse propagator" asasecond deriviatve of[,ie,the. proper 2-point function with twoexternal propagators removed isaninverse propo. Thisinverseness isexpressed inequation (3.7)- However, thisisnowamatrix equation inthespace offield types, AandC,inaddition tobeing amatrix equation with Lorentz uvindices. Although theinverse ofthefield matrix 4does exist andis T,youcamnot invert thegilon propagator byitself (App)w, sothetWhy isnot really the inverse ofJaq intheLorenté index space. This fact later appears asequation (4.8) where you have anextra term.. (2)Thesecond exciting result isthattheSlavnov Equation (2:7)nowhasthesimple form(3.11) whenexpressed interms ofJ.Obviously derivatives of(3.11) with respect totheAfield d/dA. (Iamnotsure whyKummer nowtakes hisfields Aand¢ tolower case aandc).Youcanseelookigg at(3.11) that sometimes a-derivative willpickoff,inD(a)andnotraisethenumberofexternal particles onproper Qe vertex. This Slavnov isgoing togive alltheWard identities, ie,itwill relate different vertices likeFO? toJ=“WT;thisbeing thehistorical Wardidentitipy. Theprinciple ward identities (3.15) and(3.16) result from 2and3d/dA applied to(3.11). TheC-Ainverse prop isgiven by(3.12), again youseethat itdos not renormalize. Another simple result of(3-11) isthat thevarious’ inverse propagators aretransverse tok,,justas‘inQED.However, wearewarned thatWy) willnotquite betheselfenergy intheQEDsimple way; wait onthat. ; e -~2- 0.*Kinematic Structureofn-pointfunctions(n=2,3yk). Fromthevectorsnandqyoucanconstruct 4symmetric tensors (including Buy)Itisconvenient torearrangethese into R,S,7,1 asneeded because these tensors haveboth projectién andtransversality propertyes, seenotes fordetails. Théprop-full mustlooklike(4.1) because that isthemostgeneral symmetric, transverse matrix. Similar comment applies to(J.6) whichistransverse withrespect toqrather thann.Eq.(448)isthe"almost- inverse" equation referred toabove, Itforces arelation between (A,B) and(a,b). * There are only two unknown functions. . Since the self-energy need notbetransverse, itcaningenerdl have extra terms. Thecoefficients arecalled a,b,d,e where IHaveadded easinBZ.Notedadbutthe others differ alittle. Youputinthiemost-general-form fortheselferiergy and turnthecrankonthemini-dyson '(4.12). ‘Thenyouknowthataandbmustbeintermsofthese morébasic input variables intheself-energy. Thehatted variables are discussed inmynotes. For the 3-vertex, ohmy. 4.16 looks like the correct Ward, but something looks wrong with later equations inthis section.’ (a)Thavechecked Jim's formforthevertex, anditdoenotseemtohavea” eo) (1+L)f piecewhereaL=(b-c).Also(b)Jimhasa3-qtypetermwhichKummerdoesnotshow. Also, (ec)thesecond-line of(4-20), thennnterm, seemédimensionally wrong inpowers ofn,notthatweeverusethistermforanything. The logic ofthis section escapes me. Isheclaiming hecan solve thé Ward— identity backwards forthevertex, orwhat. Iskipped thenextpartontheJ-poitn vertex. "5.Coujnterterms. First, Kummer makes anIRcomment: unless youdoSSB, youwill have infrared divergences, maybe youshould choose p*=-u? asyour renorm point, we will not dwell onthis. Kummer says zilch about "containment" inthis entire paper hot even aspeculative remark. . Now, howdoyou renormalized the theory? Qfotes aTheorem which says: ifit isrenormalizeable (by power counting), then you can implement renormalization pyadding afinite number ofcouterterms ofdimension less than 4.Somehow Kummer boildthingsdowntojustthetxZ,andZ,""kinetic" counterterms shownin (5.1). Recall that these arethere to"undo" theZwhich accumulates tothe propagator-full when you sum over all graphs, ie, solve the mini-dyson. Ie, you A thereby remove adivergent constant fromthetheory. Thevertex charge renormasseem correct tome, eg(5.2), although (5.3) seems wrong. Now comes aquestion: since this isnot simple QED, you might wonder whether the counterterms (5.1) really dowhat youthink. Todinf out, you first show that asadirect result ofthese counterterms, the bare feynman-rules propagator hasbeenchangedtohavethenewform(5.5).Youseethecharacteristic 2?factors. r*) Dothese inverse Zfactors then propagate into the full propagator? The answer is yes, butKummer really hazes uptheissue. Youhave toputinsome mass counterterms also, itseems tome, sothat the full propagator still has apole at0.Then the xZ)goesintotheAnumberator, and23"goesintotheBnumerator. Bytheway, later Kummer will argue that consistency requires that Z," =1,ie,there really cannot beacounterterm ofthat type. Then Zs=Zn=Z, andthere isonly one Zinthe theory. This isalso the 3-point and 4-point vertex Zasin(5.21), these following from the Ward identities. So,tosummarize, theeffect oftheZ,kinetic counterterm istocauseaZ,"to appear inthe numerator ofthe full propagator. This then cancells against the Z,factorthatnaturally accumulates theredotoaddinggraphs.Te,Z,arisesin the numerator when you solve the mini-dyson because ofthe effect ofthe self energy. This fact isstated here inequation (5.13) where you see the Z's being gvein in terms oftheself-energy parameters. Sothecounterterm removes this 2;hence removes divergent factor from the propagator, ie, you have renormalized the propagator. 6.Conclusions: Whataretheadvantages oftheaxialgaugeapproach? [*) 1)naive Werdidentities aretrue, ie,Z4«=z+.Inghost gauges theWards areI gather much less transparamt. 2).4n argument isgiven astowhay Zisepxected tobegauge invariant (ie, not a function of©Ipresume!) 3)Hecomments that ithasnotyetbeen really proved that S-matrix elements are independent ofn,vector, when yougotoSSBtoavoid zero mass gauge particles. Appendix A:First, Kunmer gives the principle part preseription forthe single and double poles. This isjust what weneed?! tosee ifcalculation iscorrect. I will dothat right now after finishing this. Claims other people got same presecription see ref 27. The rest ofthis appendix deals with the fact that when q.n =0things : goalittle haywire, the power counting seems tochange and you might lose Renorm. Ithink heshows herethatthings areOKandfitWeinbergs Theorem. However, he notes thatn220light-like gauge still hassoneproblems. Thisproblem withnparallel nhas long given the axial gauge a’"bad repuation". (°) ——Sroranay edKoonyarsTersns wee aar=et/@es DOaipitnMe gatlgen = tg = oc 7) Re yen oeast a Spe =LgeBur=aponelat_=gpa=Ro—Mw eee ETepee . ee gSeee ee | a gy .RTE RY OeNS ggg a). wee ee Doe CDRS Maer ey a BE KunTowslitin iste, nS Sn TR _ a ee eee ee Jobo Ls. bff we oePo As ee SS 9c Be RE wee AletA fe) Q-%) = R — a we Tae a nC SE C26©SO a Ce Os) CO TO ee eG ee ee OFC ee a 6eee —-—SowsYonaNes OO 2.Ap=.Ae=Ap=“) jaysOussomini~| a Tp“Te=Thr=Srna athag.oevero (WR) Ayr=Wrensgegen ceee ee :oeTy.Sathana =Uaeane.quasarGoameBee. : De =locumpempagghnuty(8.cnwandewst voce) Bes. as 2patid(5:1)aedWeekSe ceTyea eeleb oe “aTpith82_craekaon! SeQe ee a — a Kumenan, Section .. - - -- —Sex\2@)+GudCeKerced= 9Anos. -- wit=\uaqe" Loe OO- 9pgsBackKumarnasona.bevingi,Mak.BlBentmetro Baseek- ae BahDeeyeed Letancegossusen.YdsSeehon Yas© 2=SrA) wear pred .Youdreaxenyrsa. tee -eeee ae 12m | ~ids ee-Tae)c* =T\\ne “feTae-oo =WBS) SG)jeevoiDeBefyees a ©Gans: ikaariasqeumadnSxonaelaqongNuns.Ursaotavageadh Aounts.eumJensarertdSabaSats eee ~~ —.WO 9d.-Be SScopa. _ - —— Sm SE) ae : Nao,SsdoagamexdownowA Soo --SHOTS OO— FRO beywayowwade oo=\pse +ighus@ahe| —.SQ =lg3.S sy halsolsl re OVRicodannKermanDries,agTRO=Se, ~ tee =k 7Stays C@lS). . whee ey < oe Lo ae ese BO QAR ee 8) ee Ke RMR COhe =Cee =Cae 7 : ere oat ® co=WA)._BESO 5[Sango HOGneseTSR SS aa ot. . Le AG8[8(H)+(wd)CIC SAY=\uaad fw we - onsttc [ESa SURLY DAB Tw WS¢ oeCeatoe-7Asst3 Coe o=SUAUd HR) Stiri ga-whee _. -- ab/3 re .Le.~°F|uC)3xpeeEawhl, .CN akekywcWYtists TT Peace Sea (Ber wi Be).Sistas “pp =° _ a Baebyporrnanraceannen osGN. ee aaa- SLPS :QO -av= siSpusce f _ Be Yura SPSPET. OO - -- — — xstacl. - ee —- INT mes tec o_o ok Rk s{staah =5PS SAT -LekSLABS - Mn LN) =FAYA@ACRKSHETA 8ee =D. BRAG\= BLA)FMC TAL~ESAL =[atte 8]SAR _ RR B=BETALSOY aeee wee ee -oe ak a ASA]Sw SUMMA kyesLetasal]HOSS ]e _ -WoBoaporkemgupebio &OeQeaDeoktorbeduels —| Oo cee otRef ae an nlBW=TEAL [iedheBELTatoso)|_e7 DOBAIAVS ST RASSe-OA Boner!Gouecanmeet aebbgko Bw=Daad, | BoteBERG) (haFe) e Veo moteA=Bsond.Ce“SKscneCeecanek Mo su--Sao|FUNGLelie fa © Sine Tanmuah lastaSonalata BatOD, weg GB)shse\ Mois ite) wisd<c .]olen ____Gonmentst whyisdiic0underthisvariation ofA?Justlook....Iamconfused, I __ thought adding source terms wouldbreak thegaugeinvariance ofyouraction,soa whyshouldyougetdila0?? oe oo an) ©Qrusaangernehairyol)Severemine 1 EeSan8Qy =Sagdag) uh rmssut wyaraner MWe SEoo re = -»Say -- —- -- ee eeeeeeeee . “> weSeay Sls QuayeQ) gs) LBW sO= WAR)POcee 34 Vay - ce ee eeeeee oo “Yes. ThefactthatdW=0hasnothing todowiththeaction s(a)=8(A8).Infact._. -—.——thisisgenerally nottrue,onlytrueforthe$,(A)term.ThefactthatdWeO |___.. follows fromtheinvariance ofthemeasure undertransformations ofAtoA8,ie,9 _.. under gauge trassformtioss» © 8 ©MayAesGA).OkTeo,MAbdonAmbienet a wee perry B\/s yes Oe ~waot 8geeACH) eW=0 — BW as) sw : :(ee eT —Sofe20|)2weave! Syya .--——|-d3BW |=iaG)ey- Sw [ ae 7[sBot=8Eat) RNSEOW.GaKByeWeeColCISO,sevire userymaratTE.> gee a Grey.Norgowackdo(2.1) re |sta—atel enniSelw =o . ©AegiyF/Suia) Ryko Be a alo me, aeay1SW e. yJw & s 5) 2 5,3 o eets So) Bw) <0 7nePoyeSTea)oe——-- a KA woeete es=hat)7eSWY)ataCs”aS+aign 3Ww=o ce weLeesionny*SASgS syeyZBE.Taek o we majisw Sate soo6 ROT °— a -'@2) Sw =SwWieu-s) - . oe WSPOR) aa a_ WeTKD 4) xJowell: eee fe 2 BG G8) =mead papqatn”, _ . WesdadeokDaNgee® Co ‘ mepoe ee eecs .-ASMBUSBa) aSFAUTMIGAG)Soy©SOA - - v eee aS, eb=ASotSUNALKG) Bled] © Swan | 9 eee “jeVu.oebestcil < Openeters pieSAS. =4\Ed)Ries)CrA)Sap eATSAM] © —Va=DaceqerpagehscamadtJanebeCK. unrbulakwwns Ey _ _ a a Nth abt NatemntDavo©prepagebe to i nSw =ottZ\TASI KOWE)ls “SPOR) |,7\}a >© oO Netemat REO) nagecitnr |Eat Boom iaGaadannngage.SRS RuelAdy=bso COMgpyaA2DaySACoesdee. QRH feSenoaaneenDelevargemnackiaayoTddas Se ee Gncel’.WhegPa8bw SN oe ae 8 _——Kugupesn, SectionS— .eee Son Ayatda, One0g, ee a wR BcTa+One fee, Opa +OacTen =),bakAya LS Valuofc,Datssiannaintaat.GeepagainGack adatntiykaaaeTaktomes tates yeshake2220,Dasanak” annenna _pempagactss stuisnnges). Batae a k==3hSte ntfeadnRDangen, bk —. Ss 2SH =-S(SP\ =_ 2 SQoo Scare és oo eeUNS S3T/ |SL Tw aSeTI Swe SS x 5 a eae “2 “2 -Wee ——WetCaanseatfpcronnsi.. CANJpwevsSADwe? . [Asse wsfase. Ve ee ©TELaiNenaunserdarwnluere yorwoh)VeWa01.DooksealsVMamgueg CR) eae Wee a ee NS TAYENakesas)0OD an ae Se a ee he Bee ee RSS BOS Yislen ed A= ssee . eg a Siege gonad dene G3Q)W ge aa _-a- PDhoweonrmnad akSuakeloid 2 ioeneee “Sra *ool) WwTOU ssw t14s)SS aOO 6oo oe awblyphorbt esol resfee to=TAS St“ge =an iGA)" nn A “Dhwsyeaknese, KaaageSDfia=Syobmedh Movde,BS depwatnas 2 — SSP _mnS83)|= : -FS yesel ne SIGS TPL OY™Ge) Wau)6->>|ial =Foe eet oo Ma ang8faies Oo ec =O oo.ae Boyar f——_. a _ss 9.3tiso git wedokumdeoaiwe: 0 ave |VIGeee aei ee BoM Bamaad@ | —--4s3}seBE SPs). ate|. .a..BHO)LLBatoBateRP . — » BaPOdRP@ T.. eee Tah) Vale we CRS aEJaELSTMiGs oeee SS) WG =e. _ ae rn—ovalporeCoy,8)==taeS$Gen)WyeGua). 2 i a <>) - ugCage)==LeaaeTeas)Wogcoe eee a a SeSeeTRG oe age Gi)oxegthepmducre oa1!ates ~~~Tooting, =Late. 22 77k, PsaSaadiediy’ oteSf] 7] TT apa SHES ae ELL SansGFicohajpoadane | 2 aS Se]GSESSLT TGhataguhe Jarad Le 9 RSID ET aS UIREA SoSype Sree REE seg OySanarinse’ Raginef one ware ge? a¢— =zee apetinn saes--—.4feeneeGage AySug Cfold1 SoteShee Gan) —~—-.wok~YokPap.- ee ee Nea asece janada. Mice,oweGt) a Tejot hy}8.oo Baru) enosathasree SoWgMWe:aye=apetbeine cashed, a: ; yA == SSE] gssatus =Gyles)a tae ~WaskaDarmadqgares oresSnYsJuhlprmpagaba 2. hkNptye=tr soMekomWeo Bee? Wah,saatNpbeginepyOe (Oy =AgyctBaycasSCNeWxacNW£DCyWe eae) ©| RadaooboaWR Yaw «OgeeopSoAGL ne Soagin ct V a teh Kage wn KaegeES RO AGBsBR ONaKowWeesuena 2 gayemeat nk EG ak) atnpanel ve EGGkee PG kyee) ER, coataleyhonvemallic troll EGR gee Wake MaeCApee TER Ges)2BGO. es) es) a DyGens Gea, Mea) 0 e=™ AS ©=GBT leSyee Se = GIST ha MneestaddoDakagain 9) Bw A+Bn =Sanvrs a ee Nid ttEAT Baw SkRomeCa)STes ee a Cw ee ~ Get. Oboe -ay Paton |oo. OVD. Dn ee Rom MellyNYaNaw=BoTver SekE weeeeeXE on*(9) -. ne _Baten NatsGrn Fes WFR)ns ee moo agai=Taya oy ta eG) SOK saarme Geek oe eee ee thnSpBeTaetwaantlMeets | a Ce mE OO Aa ~Sahes LaasGanym GojcIneSkee RS wt SSv =LgRace Grau at] GR afReye BO ail oSype Sy —Ma =O ans) . eTSpdSe ~ ogeee 7 .@"| RR SE RS =SR=0 : oo QeNa=QW=wae”) po wae tea as4pad=@ugGev My=nda Ny Tua Rk ApeSo OO= GQ Qaendevrowthy GpulSO -a a2 = Gy @e=Ya Yo Yo) “PaRw=GOr.MaNy =GN(QW) SS EN ree =QQ) Se =O EN |SA Ea oe oe —Q- = Q\a@,.- made No - \> a0Gplaeager sealerwf a. no eS! eres —— »=Gadeapare=O_SpSwe2._| SBBedWesunmmaiige Dee | “Reo aTeo || wS=o gS =90 _ __ ReR Up Ses | RSS RS | AeonlugefheytylSiprnmsas,WunASYORSameae.fmoe xa 4.“VakaGaawadodo RTS. Ya9kBw=ABw+ 852 aA= ARaBS XUcoe SL eeSS see Ou areRSEGTSS)=AaRTsRS__pa Te AQa)s@G0 NS WR. er =Aa: > = yy WAN = amahaee 6Lapae“axeee “Bats RELgegyotGanhyAan pane comHa =again! oo — hayoa)aerca a ag SM ee Za =< i a CongerBalZackYoBums omABebe OiasegpsOa On TILIA Agu+Qo0T=lapGo-qwn'|s AgaDo TTT Esp eye) ©OBL NR] (Caer=w/e) KAONAN | =RTCe) 8 82)Coes eRaera SS ee eee eres eeinare A a \RO ee 2 ars - ee ne TEES Gea a ns a Wtnequneite Spo a 2oe ne _ TS geeeyy Sede ee ee oeee Qe OO Veeps Spee a Se gat] =AeGyeREphGQ =GAGTEVel+=yea ig a —-Q- os bab/et) Leone Or ery)aAL ca(ead(et)of a_St A Sept Yo —_ oe Se a ee oo _._-Kupurnta, Sre_eats [slartiig, page223]. _. ee a omAyn=AR8S DooWhagegemagein oo=Ayeataoonpergagabn ilGeen Te ©BromeDiagaaio: (Bw Cage akingesbee TO “Souk yuccainscinjettaaAyoSkdscyfacnag? ye Aye TT OVCapawa: Pree Se AE__ae _—Que_ANplly=CopaansYorsaan)=apteacrgnga a=BSLeyactteed WooseteafeAadBe “Dy = B80 +8gy=Royale |ne -RB Wally A(gyyDR eS. igoo ORSVER) AR aety =©SSPRE OR) ok ; oe Aste owssage, —_-——— ane ee oe PoeIGSeeBBY TOROS eR Tk tad sage ee a A ~Beeston +Tales=toate|TheBasle—pelTp Camalooske beGeeOmbre “a.3oan= GeWng,CTSMSWaHy ae.Tebgp7aiwef met“Tee(e1)[*)AL ee etes niche feta eeegaps eG. _ —— omBeGao=PGeeyer,o) =SG-eyre) 2LK TR apayOVSTEMea face SO TSB apcoene) agpppagLip)SETS UME Ga) 2 Hoe)besapangye STIG ©— we ng oo ipGere) Lig(4_are esas ery oO RS SPS CE :OaaayyhRAySTE) ce aDat9papagheifalaaietgiyed So.©gigSapSPST TM (G41) Te ea, SSOeye |VowmekALmdgate = _yessreGh TERaaJockpealooEEROTEL ekSour beet qe are SUPha_See_ 1G) oo ane A a __RkTWdagenus.codygo ee ag RSGas) CG, @SOEN = AAAs -— we ee Roum:QasrQuaiandlSplceeUae,OuWeed—Syqn — RU as esseat fet eT©) DaysSuotien an ~~to OO Oy od=pean he NyoywinesAee ke SY aorVaeacocheTO ee A re aAria,OninAsediastioKuntarcnsthadsony. =2 a 7 AL Sale “Saby - H MTV Ae ~~ ne Am Y=Sala) +DyeCa)KaleByGk)Bee)Age Ce —— Re eTabe! albu 8. Lyle) =Byala)+Boye(a)NoyBigg(qkok)AgeBane) SeCR) ‘ot oe ACTIN 90cm ee Poaeae ottYathn:Oe atbah.oo GG) =Ch eC (¢0 _enn“ater L9ers Parr 4ar Q) Za WAT=WAA ALS\&YSTA Gaol RaasASasban espw+dquate @ STBRSOBST GWcpaSage ya en (an a =aSig VAR] alts GeESEAa Cease) conse ee OY Be ee ee, ceol a —_a Sue Stemoa oenfl) Naty Magy)os ee, enna: Se heel a he, ~AptisLosisleapeanee||ee -ae anid ks,wa ween ot —Q- OF aateSoe SB) vata a =faee) —TRS o-=TA Mbt fyAy Aw =So=3 Dat soe Ny oxMaald» ee oo So We=OsAnMYRalGeBsaWy ee BO UESs J aes La SL pe “WOR dpepln Tegne heBe a ast sosaoaSESGe pageaa TSA UsFat as oy ase 2 aS Qatar) so[reAaecryeatst)Se pled} 7 Mygmaguakin: Tyo VaR ©Ga penta TEathease0 TPraocenitiedGal)| eclaDeankanrus ee Han:vantoestans ContTQ eVsland) fos es we.eee .——Tsergs Odo Coat)SpoDa QL roo ee ~ oo vy a)RwTn a>AGE =Seeoh seraex oe eeNS, alySaoaekn ae a et rnoreef oe So,DEweosama Wa Wy a wo =aytbSoediyofan) . —Ss _arsaslfis GTtssae nCRRES SONaQhearAecke,wadoP RES mRtOee ooTES Lm SEOAR+Ase AbRS+step. gs TBO —ERTR SpEATVRSVHARYSATSATA o\A “TARA RYTCH*EVTRATELES] Jer) =PACERS BCE)SASARAT RESATS. aot -2- oeWoospaedi.. weeeeee . O__[81Ra~ goRr=ApiGG.Nee eee oe NL QE NE - nn AE ey —_— _=Rube=reee eee eeeeeeoe RLR= 1.Na$a9ge NeNy =NaNyNog) ceeeNI gh ene EEO OLRS a aed ©3)0,a aea yyFkCO2b So anabbe iee ate wee CeeeTAG) LaeSTERRIA GR SE — otBACH) AG<3AMAA) -¥TSS]GY=oO De eed yeEVERETT S=a a a AkaKADEGro=oe royQa BaeSeeieTpesefia - we iy Satna earalienle 2yineewey woe ced.SeOsaedTyae —~|weO12)Kommarnss, yesJas)Laakee es *) pT p=Te+OEE=Se88)oe©apy|— Laaxodamde ovo cust dvlerwmusd lon (sa ee __Kumman, Sion,SosQransnabigabind ae en OFoaTuataa(s,e)SandAp 4)jp) (BS_anesbed)©(Roma)-€(GoosalPO) 3)poonichgameQOsueGide(BPeebe) aaer(B.9)couladhgonadddoOlenonigaiahobdegikprinbecnal”— Daa: wp —ManysinoniahgangbinedQikhecontacfsaia _ See =OLS Gpaye aan—Dw (qe —Saos eym) —,geaka. -TeAORASOO A ee Was, (BD)atgetDigan, gisTone ——TyeBao] Gao)Tea) hg —=2aLye3.0) +C2)ge 6-=(LsGallgon22 aeJaa Sse De ancefhpuabteandpsabenSoVon a Ce a 5 on LE quBagg=gelak a * Rego bw=guGos — 27a .Be 3 De¢Lag=da(gt-s)- dG@syP “ae TRShaye hrs ERseedST Wanade D,anduntleranddaadsoatconshen OokD=ARSBS.So _.q=.do=(AROSgHyeTSERIES _ Ofenmar9=AamengoMig Se =ARTADARTHARB+OS4.845 age att-\. RAGS BSG)we ae eewe DarlSat) 4=RACs Tae)+BGG 5 £4-AGLQRT ARRsens. WathfovateGin,SLNand©,Bah,sagtonabhylagRouseSIRERAS RR =ABAR +AWRi9Arte delsgd a Ae A\ . . O-Ra isgkXP a ed Rectgsiemens QuAS,CAA CV @lotedt (2),Siew pala.Qaamengongmanana afmcanes 3 bereTg) oata aRE BERnn. 2geegeRB Binder B5)By=Deangeaefens(5:5)aCts)Mawes AL al 6 it FR cat meget a -B:Gn\saatGotssakReagngentnasm 8).—Oe,Desata —Serot_szwalestema allanvealnoarevinme.nnerenumloonmd, [*) ES eS dsacruben)ey aBrdarnptefh A SogqearyorJyadtale.Marecbrongsa intheQsonionsDyers(8ait)5GoaWeEUs aWaktege gh “ Ts, youoaddosonDeeSAB RSISGris RA= RR) AGA GATED) S)) ; ~ TESategate DTE |zs} _RI-BASAST RR+ATEGD)) &SUFEEOEYQ) es _—!. en ee | Ae ye RT ——-| - (-PQataysQ)dn. . oF HD en a oe Tg ee Es oF,eS —e)L: ©Waa 2e2deet Geydafardadagcombaband ierl)hessgall Me aly atbiccaSled SEgeomece upumoneRetaanhWarkSaustitinn, (omgeetinns) (goonsPs)Jo@20, Cie,arialaauge):“s) ©Prppacgter ot, =LORG) =28Dyacg) es Pa Naea=apfea) ronAy(9)=~ ~OHS swels)=lappy +Hut an =~ gpCree) +2Se ©"Daverae’ Propagate: ° my DETypG@=~(gy>qed) Sato =Op=gee wd=o0 os) qreo ©Nyso-Vorker (oerCane =geGah)=CSTayeChote)ee wanes aysCPsfasts)=[gpcreeds +optic.|97 ©VeeftWorkLobb: Ga”Tyg Cestuts) =[ThyeCed ~ThaCrd]gy [isd ©UreFouWander =a=3VagaginstPaste)=cy)+(1324)+(14,23)Pas * Oper wecmeca| cree ae)=-igdC Spy.SaasSHHSy] Wale: Tooth, 3ondY-verlex amsim under Cler2) ie, Cprspnyse)<>Cpa,pa,de) Aims jdowierl lows. ‘Yra4/3Word : . Oe8MedBan (wyad 6pr2950 Nabobasa POLMypeppnCuesta)=ig|©SVsCRB8B)>wo+ce] Comwide deBock Dike “Pox Salle: K=3ceautSpeeCospueyed)©Ges)+con| Faconveninee eitannwits: 2yase aaeaanYap eee)=~agC [SynSyms~Saypes] +Gert) 4@es) ©Rutan.fenoleWenorts yem _Qed-Gex) =0 Jemsogee, Tass: »etemt=we=eerKear 5oteeow=&Res=.Jou|GEE| 68 OowsagEONOIRBd To Oe PR COEL SigSy| rr ySee |agesis©fston)~me —Sp etOaeee Se ATGawa aaapotatoe 7 ©AWS=ie.tEL SisCePpaSgnRetDtPech— TT REE ey we ee TG SSBghy) aaE SpalastSoyalta? us ig(Gist,FORE =CNETEEE —_-oooooSaar ceayegtygb——ag) Seer, [CCS CT ESE YPC00KSaaE cl ee RN es a aoe mend.tonMsso0D. ae Ball &Zachariasen (1978) (Tadd T0260 MeICe,TRBal,oein.yw’ aee isdaMasog.ee 7Vabatachie— YPpng_oSDI=-ahBonapadag«tonnegaiaOaonto ohsatsDeinaahbeOEEaeOSefilerOYEPedpdad i tt Zea clewounarate” ase,Eradlein ‘bg eee uslpPL enee sbaptinsnnbe Foiepeeesee Se secenscanme cineoecer©wenn es a a TG =Ge=p=ewe— De — Cro~\eyo be a Os AS —————_____——__- ---<= (pa —NS accoTRDSDaeFea Seehaeae oe =e hte ashsopolequatian, pee —s ySSe_ooslad MakAPogplaie: do.o(62).ahWie,Jacadengibh Mes byyyNoySashaRogrongtyMenWedasabethERages an Bad reci_vachinteaot4gfe Crete sedi) 0~Dash WandMetena Sx SER feBaral, agraeme aseeeVoWeeficia eadaatge he2 8)sontGuat“Thu.DigaeeSondcodagngaalidinia AacnnaleeHtaalaaeSr WYonabeytedey beakzoOKfoxFR Soe i; v aSe a - a — SaSuit a ta a ——Ghiriemdtiy bedelOctDe bfog _...Fred.claimsthatPT(0)x0, butwecomputed ittobeLobe,eventhoughitieWo and IRfinite. Now Ithink heways that Lobe will cancel against dand e,now—~~“pnat-the jeglue tshrtheregiving us{Iformow)-more freéplays Kiso,hedoesnotreallyhaveamost-general formofthevertex, andalso weno TLORBSRCBIUSEEELAT GO| 27 7 CALT68-647DoE RESEARCH AND :DEVELOPMENT REPORT ce) InfraredProperties oftheGluePropagator inNon-Abelian Gauge Theories * J. S, BALL University ofUtah, Salt Lake City, Utah 84112 and ae F, ZACHARIASEN California Institute ofTechnology, Pasadena, California 91125 ABSTRACT Inapure Yang-Mills theory, the Dyson equation for the gluon propagator isstudied inthe infrared regime, under the assumption that, asinQED, only those parts oftheproper gluon vertex functions determined bytheWard le) identities arerelevant, ‘hecalculations areallcarriedoutinaxialgauge. With anumber afsimplifying assumptions the resulting integral equation for thegluon propagator canbesolved inthe IRregime, The solution displays a 2 power singularity inthe IRfortherenormalized coupling constant g(q ). .. ~~. ' Sinaaleemegpdicing DeepagerantAIA)atKememnr payer+ Supported inpart bythe National Science Foundation, under grant No. PHY76-14907-A01. we .Work supported inpart bytheU.S. Department ofEnergy under Contract / EY76~CO3-0068. -2- I.Introduction o el There isawidespread belief that confinement ofcolor innon-Abelian gauge theories isaconsequence oftheir highly singular infrared behavior, which isitself aconsequence ofthe non-linear gluon couplings inthe theory. From this point ofview one expects confinement tobeproduced bythe purely gluon sector ofthe theory. Quarks, while they feel the consequences of color confinement, play norole ingenerating it, 2 There isafurther belief that confinement isnot aperturbative phenomenon; one must gobeyond perturbation calculations tosee it. This isnot tosay that the numerous perturbation studies ofthe infrared behavior ofnon-Abelian gauge theories have not been ofvalue; indeed they have provided much guidance. Itisonly tosay that one must not beoverly constrained byperturbative results inlooking for new phenomena inthe theory, 3 Tobeginwith,letusbrieflyreviewoneconvenient wayofobtaining the (>) infrared (IR) behavior -that is, behavior near the mass shell -ofthe electron propagator inordinary QuD’), Onefirst writes theDyson equation, which expresses the propagator interms of itself, the photon propagator, and the proper vertex function, One notes that the near mass shell behavior is dominated bysmall virtual photon momenta, and that inQED since the photon is coupled only to (massive) electrons the photon propagator has noIRsingular- ities: forsmallk,D(k)>24/k’, ‘Theonlyproblem isthevertex function. PY The vertex function satisfies the Ward identity, and this allows one toexpress part ofit(the charge form factor) interms ofthe propagator. The other part ofthe vertex (the moment form factor) isnot determined. But in the IRregime, the charge form factor part ofthe vertex dominates. There~ fore tostudy the IRbehavior, one may eliminate the vertex interms ofthe propagator, andobtainanintegral equation forthepropagator alone(withthe 6 . y := 6 jhotonpropagator Zypkasagiveninput) whichmaybesolved toyieldtheusual near mass shell expression. om What would beinvolved incarrying out the analogous procedure inthe non- Abelian case? Suppose, inview ofour comment atthe start, welimit ourselves topure glue -noquarks. Let usalso work inaxial gauge tomake the analogy withQEDasclose aspossible”). Thenthere arenoghosts. ‘TheDyson equation for the glue propagator expresses itinterms ofitself and the triple and quadruple glue vertex functions. Inaxial gauge the Ward identity looks like that inQED; part ofthe triple glue vertex isexpressed interms ofthe propa~ gator, andsoispartofthequadruple gluevertex. If;asinQED,onlythese parts are relevant inthe IRregime, the Dyson equation becomes anon- linear integral equation for the glue propagator. Note that here, incontrast toQED, noexternal input analogous tothe photon propagator isrequired; the 6 equation isentirely selfcontained. aoa Theintegral equation thusobtained isnon-linear; itingeneral may have non-perturbative aswell asperturbative solutions, and these may shed Light onconfinement, Indeed, since inaxial gauge the propagator includes therenormalization constant Z,andsinceg°=28°)oneshould beableto Oo extract thebehavior oftherenormalized coupling constant asafunction’ of renormalization point inthe IRregime, Asingularity inthis behavior is often taken asasuggestion ofconfinement (9)(4), io] Thecrucial ingredient intheprogram asoutlined aboveisthattheonly Oo partofthevertexfunction whichweneedtoknowintheIRregimeisthepart determined bythe Ward identities. Inconventional QED this assumption is justified bythe fact that small virtual photon momenta govern the IRbehavior, re) andthereforeinvariantfunctionsinthevertexmultiplying spinologicalcoefficients having larger numbers ofphoton momenta inthem, such asthe moment form factor, play no IR role. “be : P8 Invariant functionsmultiplying theminimumnumberofspinological photon r°) momenta are all that matter. But these are also the invariants fixed by the Ward identity, because they are precisely the UVdivergent invariants. a Inmassless theories, however, small internal momenta donotconstitute the IRregime; indeed, perturbation examples suggest that internal momenta comparable toexternal onesarewhatisrelevant. Therefore inamassless theory, such asapure Yang-Mills theory, thejustification (ifthere isone) ofour basic assumption must beput onasomewhat different footing. e0 Such ajustification mayeventually beseen tolieinthefollowing remarks. Inunrenormalized perturbation theory, the only mass scale against which tomeasure IRsingularities isprovided bythe UVcutoff. (Upon renormalization, this scale changes tothe renormalization point.) Thus again UVand IRsingularities must be correlated. Ifitwere not for the existence ofUVdivergent insertions ininvariant‘ functions whichwouldotherwise befinite,wewouldconclude thatinvariants which [?) are UW infinite are also IR infinite. But insertions do exist, sowe cannot yet draw such aconclusion, and can only hope that itisnevertheless true. Ifitis, then, since inYang-Mills theory, asinordinary QED, the UVdivergences inthe vertex and propagator are connected bythe Ward identities, the same will betrue ofthe IRdivergences. en Needless tosay,these issues arepresently under active study; inthe meantime, andpending their resolution, weshall simply assume that theWard identity maybeused toeliminate thevertex function infavor ofthepropagator intheIRregime inYang-Mills theory, asinconventional QED, andinvestigate the consequences ofthis assumption. Ifthis assumption is’correct, then theDyson equation coupled with the Ward identity yields anon-linear integral equation determining theIRbehavior ofthegluepropagator. Nofurtherassumptions areinprinciple necessary to (?) a . 5. a obtainthisbehavior. Inpractice, however, theequation iscomplicated, and weshall find itconvenient tomake some simplifying ansatzes regarding the solutions weare looking for. These are not further assumptions; they are guesses astothe nature ofthe solution which can bechecked against the integral equation. The first ofthese isthe following. Weuse axial gauge throughout, since, aswementioned above, inthis gauge the parallel with QED ismost obvious. The Ward identity issimple; the divergent part ofZisgauge invariant; there are noghosts. The price wepay for these virtues isspiriological complexity. The unrenormalized glue propa~ gator contains two independent invariant functions, Aand B,and these are functions oftwoindependent scalar variables q2/A? andy=q’n?/(q-n)*/here AdstheUVcutoff andnisthedirection ofthegauge choice”). These functions renormalize according to =? 2N= API ys85)=BOPPDAgAMEssBOLD» 2 22 BOA08)=2OPINIBgg(4MMOsor), any with acommon Zwhose divergent part isindependent ofy.Wehave sar) =202/18," - 2) Atq2=, therenormalized quantities AL,andBL,become specified 2. functions ofthegauge variable yandg(if). Eq.(1.1) 4svalid foranyM.Inparticular wecanchoose ”=q”.Then (1.1) reads 2.2 |acer. 2 ACPI Ys8)=27MIA, Coys 8) a3) .re) andsimilarly forB.Ifg(q”)hadafinitelimitasq?>0,thentheIR. singularities inboth Aand Bwould becommon, gauge invariant, and confined 6 : totherenormalization constantZ(q7/A°). InmasslessQEDthisiswhatactually [?) occurs. InYang-Mills theory, inperturbation calculations itapparently does not occur; itseems that the leading log IRsingularities are gauge invariant, butnon-leading onesarenot”), Nevertheless, wearesearching fornon-perturbative solutions toYang-Mills theory, and wemust not betoo heavily inhibited by perturbation theory. Therefore our first ansatz isthat there are solutions in which allIRsingularities areconfined tothefactor 2(q”/A) ineq.(1.3). This ansatz can bechecked for consistency byseeing if, when the vertex is eliminated from the Dyson equation, the resulting equation for the propagator really has such solutions. Ourfirstansatz, then,isthatintheIR, evanDrange .5. ACPIYsBy)=2OP/MIAC)Basa@? BIY,By)=207/080) aa e where aand 8are functions only ofthe gauge direction. With this ansatz, the Dyson equation becomes anon-linear integral equation forthesmall q”behavior of8°(q’) =2(4/A7)g,”. Weshallarguethatthisequation permits solutions withthebehavior +(a?)+?fe}0, as) with a>0,Thepower aisinprinciple determined bytheequation, though we have not yet inpractice ascertained what value itactually has, Weshall, however, compute itinamodel. Our second, andstronger, ansatz isbased onthe following observation. We know that acertain combination ofthe functions Aand Binthe propagator has no UWsingularities, Thiscombination isvaya. Perhaps, therefore, italsohasno IRsingularities. Infact,ifittszero,thenthespinstructure oftheexact o ‘te : “Ie Le} propagator becomesthatofthefreepropagator. Thusoursecondansatzis 2(q2tn*), multiplied bythefreepropagator. Wenotethatthisproperty istrue oftheIRbehavior ofscalar, spinor, andvector QED. heansatz amounts to_ guessing that a(y) =yandB(y) =1,ineq.(1.4). Again, ofcourse, this guess istobechecked against theintegral equation toseeifsuch solutions really doexist. Finally, inthis paper weshall make thesimplification ofneglecting the, quadruple glue vertex term, This modification isagain notinprinciple necessary; itismade solely for convenience and the consequences ofretaining itwill be reported elsewhere. (Wenote that inleading log perturbation calculations, thequadruple glue term does not contribute totheIRbehavior.) Ourhope, andfeeling,isthatthegrossIRpropertiesofthepropagatorarenotaltered Le)bythis simplification. onefinal comment. Inamassless theory, where intheUVlogq7/¥” approaches +»and inthe IRitapproaches -@,the leading log UVand IRbe- haviors. are formally the same, though ofcourse the leading log result is correct only inthe regime inwhich the theory isfree. The Yang-Mills theory 4sUVfree, andweknow theexpression forgintheUVregime: Pa er oe sq)=—foret pe1-be"ay1ogap a8) — with b<0. Formally, then, the leading log sum inthe IRregime must bethe same, though since b<0theexpression hasasingularity forlogq7/M" <0. le) -8- . IE,Derivation oftheIntegralEquation (*) A. Notation- Weshall generally usethenotation employed byKunmet®), Wewrite the unrenormalized glue propagator as 6 nn NN 2 2,42 18(que 2B465-EB? 10sys8q)t Beeca"/077,89) nygeee . 0 J @.ay where Nis avector orthogonal ton,:2 xon,-SL= a.nd ia ec The functions Band Care related toKummer's Xand Bby i EVA Bu 78 J cS = (2.3y oF ne q Ineq.(2.1)aandbarecolorindices.Weshallforsimplicity generally eS suppress them; they play nosignificant role. Theinverse propagator, connected toA.,,bytherelation Axs ao-ee 2,ofweer Si?oweR Q. we can be written . 4,4,7Aude 22 . 2142 ./ Tyyl@=- a6,3ATIBABq)“2,(IN1589)+(2-5) where yg beip, anya © : 2.6)/ and cisrelated toKummer's athrough . bo JGe ECE 2.7 Thefunction¢isactuallyWfinite”), ThevectorQ,isorthogonal to @ 4 ad . /Qa -oe : > (2.8) : -9- & TheDysonequation isanequation fortheinverse propagator exceptfor . some additional trivial terms. We define . 4,4,Qa an, VA |z@e 7 ty_%,2%y uy Fy@=Wy-Wy@+a -AYtee, 2.9)q q n where Ml),isthefreeinverse propagator Wo@=- @saa.)+ Wat)= (2.10)Ww wwduty 4Tots)=O Asweshall verify below, dand eare constants, which inperturbation theory are quadratically divergent. The Dyson equation reads H@ =Ffae42°, sake")ww =3 gaat Conky AgrgrEDsgry(Ekta ean/ e. andisrepresented graphically inFigurel(a).Wewritektk'=q.Asmentioned inthe introduction, wehave neglected the quadruple glue term shown inFigure 1(b). There isalsoacutoff” implicit ontheright-hand side; wedonot display itexplicitly. We use the shorthand dk tostand for / tAte?c,(ae) sober)” (2.12) ‘0 AV Bo47ens} . where cyistheCasimir eigenvalue oftheadjoint representation ofthecolor group. Allcolor effects areincluded inC,,soexplicit color indices canbe suppressed in(2.11). The 1/2 in(2.11) isfrom Bose statistics for the virtual gluons. f°and7are thebare andunrenormalized vertex functions: we have vl 0 = _ ARagag(828s) ~Gara), Saya)+TELEpermeattons @13ral ©yt ‘Sca (yr9adr,Ode oy ne Pad ; +GeWarBad, oo -10- + B.TheWardTdeneity oO‘TheWardidentityrelatesfand1:(colepvweot)) IPSgig,49995) (99),=yg(4)-TygGy)- @.10,/ 94904419293) 1999,*Me,“ta?~Toya,“42 Using this and the Dyson equation (2.11), itiseasy toshow that dand e are independent ofq. Evidently aay=mea(ate)=fakne(k-k’24,47) : asd and / 2 i254(aa) aly =aod+ae ¥farece?n'*)a, Ge)=PakQkQA Vk) (2.16) Thespinologyontheright-hand sideoftheseisreadilydecomposed, andone &@ obtains \y /J,ko427,2 2,42 -Ay =a:200$f35a12?sy989)+2802/8sy,589)] > an/ 22 2 which isaconstant. (Here weusethenotation y,,=k’n’/(k-n)”.) Two other scalar equations, for the physically interesting functions in jl,areobtained bymultiplying (2.11) byaayand6,,,respectively. These are =@laI-l/y) +Gd te _sne(RS) ' to rJ3fake2Aigldaygr(KTgry(Kk"aa,»(2-18) and a(2-%)/=2(at)-242(o-) +4,+¢Pao (aek,-k! Pan (ark, as okt ' nn a> \2JaeEye esetga(GOA,geDSPGyrCesk"sa)+e\ ean/ ‘NY*( . = ’ : .“Orsthase,opel, ye)TW.So CN. ome : -u- ce) Inthefirstofthese,wehavemadeuseofthefactthatnisorthogonal to 4,andtherefore terms withunoreyinthebarevertex donotcontribute. ‘The starting point forallofoursubsequent discussion isthe "solution" of the Ward identity. Itisofcourse possible towrite atriple glue vertex which satisfies the Ward identity inmany different forms. Any two such forms differ bysomething which vanishes when inserted inthe Ward identity. But our basic hypothesis isthat, asinQED, the only part ofthe vertex relevant inthe IRisthat determined bythe Ward Identity; hence any part of the vertex which yields zero inthe Ward identity isirrelevant inthe IR regime. That istosay, any two forms for the vertex compatible with the Ward identity will give rise to the same IRbehavior. We have found aconvenient form tochoose for the vertex tobe the following: 7 ar, (414943) = 8 919293 vie"s @, Da3 +85oSara, SF 8192)41793 7%42°43 ey (5, Oo, +a Dp, Gy) 7 - at 2 3843 G42 Ef. o+Laoh224A 192°3|43° (ay) po. tna3(25+2]-ee oP,)aynLaymagen] "age[aya*ayn fy,fSa42_41°43 a)Fangmgngyg| i 192°3|Capen)" \gym agen +cyclicpermutations. aa’ -12-. . Hereweusethenotation £,=£(q,),)=e(q,)etc.,andthefunctions £ e and gare given by 2 J f= +a > (2.21) B=-a(bc) , amd interms ofthose previously defined associated with Il. Let usfirst focus our attention oneq. (2.18). From eq. (2.20), wesee that we can write Agr (kok aay = Pa ark 8551 4 ~ark ootVanaFE).~SE80) kk Agkgt+qnoetnck4Soto!mck|gcyy Koogsk0qekkek! ror W695" net. +Hofermt) gq) qq 4k -(samewithk++k',oo") ean where wehave dropped terms proportional ton,orn,,asthey will vanish in eq.ean Because ofthesynmetry ineq.(2.18), theantisymmetric termin eq.(2.23)simplygivesrisetoafactoroftwoThenifwedefine KokgrSokgt WA x8,=Be(kek')mek J5te +22) , (2.24) otnegek ook kek? gi,=Be(lerk"){uh ’Je.25)‘oo" oO neq . 4 |I6Igt -diggs=Betecenck’,Soe ; JS2.26 rwoa nqek q . -13- 6 theneq.(2.18)takestheform -P(al) (1-1) +@-Yate L : =Afak Bg(KDB,gr(KD Curae {sie+KB8K)+sto}: (2.2n/ One comment onthis expression isinorder. Inperturbation theory there are quadratic divergences inthe Kterm; these give rise toconstant quadratically infinite pieces and contribute only tothe constant dand e terms onthe left-hand side. Further, this isthe only effect ofdand e; once the quadratically infinite constants are removed from the right-hand re) side,onecandropdandeonthe"ft-hand side.Contributions toaare* therefore only logarithmically infinite. Werecall eqs. (2.21), (2.22), and (2.6). Thus a=y/A, Aisalinear function ofAand B,and fand gare linear functions of1/A and 1/B. Hence eq. (2.27) has the form (disregarding the trivial constant terms dand e). =% meOTfaieONO . a) . ‘ik Kets Ag). ke ~ A,(k)A, (k') ke where welabel A;=AandA)=B.Alittle thought makes itevident that eq.(2-19) will.also lead toanequation ofthis form. Thus wehave apair 8 -14- A .ein . a 6 ofcoupled equations forthetwounknown functions. {Thekernels ineq.(2.28) are all dimensionless, inboth momentum and n,and functions of: thescalars which canbeconstructed from k,q,andn.)Onemaytakethese tobek2/q?, y=a2n’/(qen)”, y,=K’n7/(ken), (keq)*/k2q". Ofthese, onlya Am the first isnot dimensionless individually inkand q;hence only the first could berelevant tothe IRproperties ofthe equation. But itiseasy tosee byexplicitly looking atthe kernels that they are well behaved both as K2/q? +©andask’/q? +0. Therefore theinfrared behavior isentirely controlled bythebehavior ofthefunctions ALthemselves. Cc. TheIntegral Equation The coupled integral equations (2.28) are complicated. Therefore let us make the ansatz, asdiscussed inthe introduction, that the IRbehavior isall intherenormalization constants Z.Wehavetherenormalization prescription o Ap@=200/00 (272P 800)» 2.29)PL with acommon factor Z.Wechoose M’=q”,andthenassume that thedimen sionless renormalized functions a,(1,7,8(4)) areIRfinite, andsimply functions ofyinthe IRlimit. Thus our ansatz is A@=untae),oral)=hg a.syl. and2qq/A°) acy) : na«22807,ofbq)=SpeEVzq@7/a) BY) Tocheck whether this ansatz works, itisnecessary toinsert (2.30) into the pair ofcoupled integral equations (2.28) and show that wecan find Z,a, and §satisfying the equations. Wehave not been able todothis; what wecan doisthemorelimited problem ofshowing thatifwehavefoundanaanda8 e which work, then wecan find aZthat works.. : ~15- } : | | &Todothis, wegoback toeq. (2.27). Under the integral, wereplace ACK)by207/47) aCy)andsimilarly forB(k). . Wealso replace fand gasfollows: Sore C2 U8=le KH ek) = 6K) —- + 202/17) 1-% na(e)L-2m -0) .t «LP 2027/8") Miny!acy)86ndfaitho) Then eq. (2.27) becomes 3 Fe -a-im2@7/) aly) 5 ay (LiaYsa20syeces)L Seryeyes 2 20@2/8") 1-yary 8) Pas +.faxzee?) {seFon-1 * ee iy, Lay) BOYD x t Cats) ae E*-—155[ene+°]:(2.32)/: ry,bey?8G,HT a=v pe! -16~ . )t ..vaneOg oP withthedefinitions Qo(asepee), : re) we[se-a)(pr)[Pe-—*staatRea 1m n a 1-y' a . J (2.33) andsimilarly forKfandL,This isstill apparently asomewhat formidable equation; however much ofthe complexity isillusory because, asweremarked earlier, theonlyimportant variable onwhich thevarious kernels depend isk”/a”, and their dependence on this is trivial. The unknown dependence onywill not mfluence the IRbehavior ofZ,except indetail. Itisalso important tonote that theapparent 1/q” singularity isalso illusory: aswehave commented earlier, theconstants dandeineq.(2.32) arejust such astocancel the1/q”terms inthesecondintegral. ee III. Outline ofthe Solution ofthe Integral Equation Weshall approach thesolution of(2.32) insteps. First wemakean evenstronger ansatz than(2,30): welookforasolution inwhich the propagator intheIRregime issimply anIRsingular function tines the Exepropagator. (Thisbehavior, wenote, iswhatactually occurs inscalar, spinor, andvector ep), Thusweassume aeeDEELGaAey-4 V.uvSD" 2 w722 B.D Aandconsequently on - oe * .ty@{eteatnf, “44/3 Jo.»| _Thismeanswelookforwaoiution Like(2.30)butwitha=y,B=1,Then y 4 the integral equation (2,32) takes the formv : -u7- an “ .AT)aya& ie) 1L-Uy=@- ‘ —Ta «CO-y= G-Uy)zq7/n’) ot Cane) = 2062/p2)z.cet7792)+ fax > L (3.3),27/0") we! 74.2 2c? /n)+Uae fax K Ke arPus tapate Note that theKeterm in(2.32) hasdisappeared. Theremaining kernels - A are ° e: aan nn, wonpntXONeN'NN'y=arteae(“ee(6,4ye+ool, so) wea) we) oot“2 we msaf 3.44 and(writing KforK% : Kkor dgkgt as(e-k')odie 0! oo! Kott (Sart“2“=r) a’gek: kK’ ange NN NING) NNN,‘oo: ot+NN ot \. (0:~oes . +of) 3./n a a The structure of(3,3) isquite transparent, though seme spinological complexities are still present inview of (3.4) and (3.5). Therefore, tosee how things work, let usfirst arbitrarily simplify this equation by K simply pretending that-L=k7K/q” =(1-1/y)°* Wealsoignore the constants dand e. Then (3/3) reads 2,,2, 2,,2,1 1 20071?) 2.("7/5 2H"t-TapSak—ea BH2q7/N*) 2 In’) kk! . i247/n?) J +fa——— (3.6)x!? “18 po Wesolvefor2(q2/A7): (~) 12,21+fax202. ee? . /= G7 2(q°/A) 2)2 ard 2 1+faeBEMIZ INDPou Letustryapower solution, inwhich 8%) ot 27a’) =71a) (3.8% forsmall q”.’ Ttiseasytoevaluate theintegrals ontheright-hand side. Wehave @ wa 2 1 Jt i T(ot8) a8 dz 3.92)gi?)HB T(ta)r(t8) 4”5z3(1-2)® aes Thuseg.(3.7)becomes Wht gecis Y nag; 1+? 2@2he. (93°= Eta)tag—__—_ @10)/teae?2@e) ane) tye” .(tay)? raza) gq? Thetwosides match asq”>0provided thatthere exists @positive a such that T(a) Pita) P(2-20) =1. Gap &T(2a) 1a (1-a) .-19- ‘Thisindeedhappensfor=2.dence,asq7+0(3.7)issatisfied by 2,2. J z@?in’)=a2/q?hl? : (3.12) This simple model provides aguide to_howwemayexpect Ztobehave ainthereal situation.” Letusreturn to(3.3). Wefirst expect the constants 4and etoremove from the integrals the quadratically divergent terms ing which occur whenZ+1intheUV,Wethenexpect theydependence todisappear from the equation, upon inserting the expressions (3.4) and (3.5) and carrying out the spinology, The resulting equation isthen much like (3,7) except thatthere willbedimensionless kernels involving Psandk+q appearing inthenumerator and denominator integrals. Ifwetry apower solutionintheIRlike(3,8),thenwewillagainhaveaconsistentsolution e)provided the equation replacing (3.11) has asolution for apositive a, Thus we will again find 2. 2e@In) >Ul)" 3.13) 2 witha>0asq?>0. To illustrate how this procedure works, let usdescribe amore realistic version of(3.3) than (3,6), Namely, let uskeep only the following simple spinological expressions forLandK): kayBGKark Guu)n’qek Wewishtosolve, eq.(3.3) withthese choices, -20- .. Atfirstglanceitmightbethoughtthatsuchatruncation ofthetrue [*) kernels in(3.3) would prevent satisfaction ofthe ydependence ofthe equation, But infact straightforward manipulation ofthe integrals yields the relations peGek ok2y(t? 2 2 o n’qek k’kt eta-ty pote 2Gw1y%)2020) aye3.15)3 y?Jqkqk" a“ ure +G. and10) 2 fByckuark ee aeneqek «i? eo --2@-bse 2(k'?) (3.16)3 O-Di geen “V2 : Thus the ydependence isconsistent, and (3.3) reduces tothe integral equation 22 kek'_ 2.02)115Joeeek 1a ee : 3.17)2)ZC)Z(t") 208) wekfae—4>—0g?(que?) DEM q-ka-k! wet This equation isvery much like (3.7), with some minor spinological complications resulting from ourmore realistic (though still notexact) treatment of the kernels Land K. - -21- re) Asbeforewetryasolution 2047) =(01q)* with apositive, The power aisdetermined byanequation analogous to eq. (3-11), but more complicated due tothe spinology ineq. (3-17). A numerical integration yields a=.52, not very different from the value 1/2 obtained inthe simpler model. Ananalyses ofeq. (3.3) including the full spinological glory ofJand K4s.inprogress and will bereported elsewhere. If, when we turn to the full equation (3.3), the ansatz (3.1) fails to yield aconsistent value ofa,orifthe ydependence fails tocancel out of the equations, then wecan goback toeq. (2.32) and try the same solution for Zthere. Again itwill work, provided that the functions a(y) and B(y) re) havetheappropriate properties. Ofcoursetheequation,analogoustoeq.(3.11), which determines the power aisnow much more complex, and weare unable to explicitly write itdown. The value ofthe power resulting from eq. (2.32) therefore remains unknown?) , IV. Conclusions Can wedraw any firm conclusions from all ofthis? Not yet. Torecapitulate, wehave made two assumptions, one essential, the other anansatz which isin principle and perhaps inpractice verifiable. The essential assumption isthat, asinQED, only that part ofthe vertex function determined through the Ward identity counts inthe IRregime. The ansatz isthat the Dyson equation then admits afactorized solution oftheformA(q)=2(q7/)a(y) andB(q)= 2(q/A*)8(y). Withmore strength, andastudy ofthetrace equation, le) eq.(2.19),thisansatzcanbechecked. -22- SO Giventheaboveassumptions, weareabletoshowthattheequation admits [*) asolution nearq”=0which behaves like \oh? 20710?) =07/2)" a an The (unknown) functions a(y) and 6{y) determine the power; the solution exists provided they are such that the power ispositive. The power, we note, isindependent ofthe coupling constant; itisalso independent ofwhich color group isemployed, aslong asitisnon-Abelian. Ifeq. (4.1) isindeed valid, then wehave v@y = * : 4.2) oT intheIRregime. Sucharesult issometimes takentoimplyconfinement ‘°)Thiswouldsuggestthatconfinement isindeedaconsequence ofthesingularIR e properties ofYang-Mills theories associated with the fact that glue can couple toitself. There isnoobvious relation tosome ofthe more esoteric features ofYang-Mills theories, such asinstantons, merons and the like. But since the solution weobtain isnonperturbative, there may nevertheless beone. Inaddition toclarifying the relationship, ifany, ofthe Dyson equation/Ward identity approach tothese more popular topics ingauge theories, other more mechanical things also remain tobedone. Itisnecessary toincor- porate the four-gluon vertex term into the Dyson equation, and toverify that its presence does not alter our basic conclusions. Itisnecessary tostudy the true equations, (2.18) and (2.19), and check that (3.1), oratleast (2.30), isconsistent withthem, andthatthepower solution for2(q2) still obtains. Finally, ifourpresentconclusions continue toholdup,itisnecessary to (7) : 23+ re)understand moreclearlyjustwhat,4£any,implications apowersingularitying(q?) atsmall q”really hasforconfinement. Weare indebted toM.Baker, J.M.Cornwall and D.P.Crewther for advice andcomments. / ReferencesandFootnotes Fa) 1.J.M.CornwallandG.Tiktopoulos, Phys.Rev.3)2937(1977)5 R.Delbourgo, J.Phys. A10, 1369 (1977). 2, (See for example: W.Kummer, Acta Physica Austriaca II, 313 (1975)s noth J.Schwinger, Phys.Rev.130,402(1963);cas .FradkinandI,Tyutin,Phys.Rev.D2,2841(1972). 3. For example, see Cornwall and Tiktopoulos, Ref. 1. 4. More precisely, Zisgauge invariant.within the class ofaxial gauges; that is, itdoes not depend onthe direction ofthe axial gauge. Thecoupling constant defined byg”=gq”isrenormalized atequalmasses forallthreegluonlegs:g7(M,M,M) =20089" Thisgistherefore also independent ofthe direction ofthe axial gauge, but itisnot truly gauge invariant. Only the "on shell" g(which does not exist due toIR singularities) isgaugeinvariant. Therefore thesuggestion thata e singular behavior ofg(M,M,M) asM+ 0implies confinement isone that must betreated with sone suspicion. Itmay, ofcourse, bethat the infrared singular part ofg(M) as M+0isactually gauge invariant. 5. For aparticularly relevant example, see J.Frenkel and R.Meuldermans, Physics Letters 65B, 64(1976). 6.Wemodify Kummer's notation primarily byextracting factors ofq™*from Aand B, so that our Aand Bare dimensionless. 7.Thatis,weinsert sufficient powers ofA7/k’+A” under theintegral onthe right-hand side ofeq. (2.11) tdmake itconverge. This introduces amass scale into the equation, and permits the solutions tohave anon-trivial dependence onq”/A”. Notethatwecould alsohaveusedadimensional cut~ off, though inaslightly unusual form since wehave anintegral equation, not anintegral, todeal with. Itseems tousslightly simpler touse a masscutoff, eventhoughinperturbation theorythatleadstonon-gauge e invariant quadratically divergent terms. These contribute todand ein eq. (2.9). ue -25- + 08.Themass scale determining "large" and"small" q”,istherenormalization group invariant mass.u, which for large Mcan beexpressed through weweer/bs Oe)Thisistheonlyphfeteatty meaningful massinthetheory.Inevaluating integrals suchaalthoseineq.(3.7)therearethree regimes: 0<k2<q25q2<k2<v2;andy? <k2<A?where A”isthe wvcutofé. Fork?<u?,wecanuse2(k") =(k°)""5 forverylarge k” weexpect 2(k2) tobeproportional to(logk*)"1, because ofasymptotic freedom. Thedominant IRbehavior oftheintegrals comesfromthesmall iregion. pVrwyeereegtRee 9. ‘These forms arise naturally from aslightly different, but IRequivalent, choice for the form ofthe vertex function than (2.20), together with simply using 6,,)asthespinology factor associated withdabiot . 10.Inderiving eq.(3.16) wehavediscarded someconstant (@?independent) 6 termswhich, inperturbation theory, wouldbequadratically divergent. These are lumped with theconstants dand e.The integral ineq. (3.16) still contains anadditional such constant piece. Inevaluating itwewill discard this piece aswell bycancelling itagainst dand e. Having performed this service, dand emay now bedisregarded. 11. Again, ofcourse, itisunknown that there exists apositive asatisfying this condition. Unless there does, the power law isnot actually asolution. Figure Caption Fig. 1: (a) Shows theDyson equation with only the triple glue term included. (>) Shows the (neglected) quadruple glue term. 8 ° (mgr) =(awe)+beim (a) (°] 4wife6 (b) Figure | oe Equation Derivations Indexof,EquationDerivations: | o (2.20) themost general form ofthevertex (almost() (2.23) thevertex dotted inton,forthennequation (2.27) equation 2.26 (the nn)with n.vertex inserted show that vertex (2.20) solves theWard (2.14) (2.16) the qqequation involving dand e (2.17) solution for dand e (2.15) the ngequation involving dand e (2.18) the nnequation, original form (2.19) the trace equation (2.32) thennequation with general alpha/beta ansatz (2.33) thetensor IcallW,.W',,, arising from pair ofpropagators (3.3) thennequation inthe SAmodel (3-4) the expression for L (3.5) the expression for K (3.6) the nnequation inthe fake trivial model ;(3.7)solutionofsameforZ(q),feketrivial fol(3.10) the power solution, fake trivial (3.11) equation which says alpha =1/2, fake trivial model (3.1) the SAform ofpropagator involves only Z 7 (3.2) the SAform of"inverse propagator" (2.4) equation relating propagator toits "inverse" (2.10) form ofbare "inverse" propagator 98 — (0) G2 +@27) .7 . — fe]tSGh=aeatte)DelDre ELLOS a a ae a Rho ~ ithe =Sea il heeGee as Bay}He.ged~Sadegeal+bywofif2_an Ke task EN AAL —tbejotege=(eae+Seeone!pee ~ + gt)Geeta|veMetOle6lenmadeondenables, .eew.kG)Kr Senateparroeind. . --- -- —Swat —ah*).NeFDPABAADa+a&)WyWe\oeae eeeFC ECD SSAvenmanananigava Pe or Gay ‘CYRcay . WossDye BFGeBRE)BOLHhCODEdeato ae Arete)=L059,(Bie~sone’)BCH)DANEeR. ae GruneDnsneok 2*ZKA =.36dommeceRASA,CE).LhebeThemyon.Oe Akedoae Yisugh - wee eee ee ae aeee ng. |shbtpW.gtaastwdudes] a+G8We:GeFeGg DannaseytoUWNeaamee fe] WBik)e Gea)Sere! -ne! ee ged Cle(nn)e?Skee Kin fowardtookcksymmadig Pe. a Soe Be) =CeeSeFie de LG. . =ovabisypavina bie,andr Ceeey wes perv). eee Blige Osae?Negus paseaia So CO oe a rr . — (3) Worlanorimagy aleorXonitinrpevmilin, »witSanBayfa— oe — ken. ax) -- CGMeee Oy=Vee:|emaroc|— a og 30Code:fead scaiGaleCae!dben){. . ae Tleey Fe)TOG ek SO =[Sefkt=FFop .SO i _+gofedeWe+edGwdale{ oe _ ee et xc)aSe — = eel =CarreganMadWSOHSpauseacofeeEf ee wf. |. ae . —AeECR) LL cee -8=Wee t+@Wa@e KY (a) RYCRF) . a Tye Wee Fier (@Me, =Cila(tle —(Kee ~~Qhan{f=Qede-Gele+deGle OemnmeKC) Ske (ea) mca) _ we nee ckWe(eetae@e(Welar-aslae ; ROK) ke (ee) (ka) _ oo YT ~~ cae glee! [Gee] =@eted BAT we)Ss8%=tk@e(Detatee(We?Grek)Oe a te RR LL KEa) ee ee eee Wapiti 8). _ Thate)SaWaeed. ON ga CG ee Ope!(kik)=Stpinteedece - a a ©Lak.Samie)FPQaeHeesege OTEaCa ° =roeove)fFoefemey] ET =SakDyeCe)dae)[KeeFO)4Kegh)+Leeactwee rs J 0padwen Spo LS RS=Kyte) Wa(R)—KyeeC Kaehamapeltoo -fFvaulted So ——~ per : UWS =.5oe - oo -ee acai anne “Tok mW ay wore eS aevee acOe +ty\Bes2556) +@%Lea ae) osSy3)AaLoBRe3+feteatt)sw . a on=SyY=BeeSe+aaol4Cker). a Quskaeemem FRGHED Arie) -S- gyMe))5(GeSh an -SLSeme=ieee)MadeweBMseal—(ovsess), rey AO Wal\e fa. So - thfos8DOeGW)PEsBelegies) LR -SYn y (*)RXSlows=|e(wyelesdal+@&WH)(Fe-e4 PAS openf105,40)_GnFAO|HO J - emu §Oe[HY+X) OYATELFTLk Se - - A v eee (k\t kySON (RTS) f) ae a were ee ee PER Be 2-0 8eeEe]PRES wey (+eoysfign40—Heh).+Gaoya|wet-uF) fo Nee kenkon pwke] aBay pman kwen O- Fsur (ee x Teme: wie kee) __ -- a x“ a -~fe)_(keG Aw@(ek gk +SE AR)awBee(ee—ge =]Comymaye |£00(ee—ke)4S06)ek euteh. | | \sCeeBee] om =5hh oe _ _ ~Toor oe aeeot AindLeckogeinaBieSdbwand? — 2RUS=.CpRe!)800—(erYe) stk)Aswth:WaBonsnmads ovmmodtin bode!Onnkswihelee Biereenemn.Gal _—aliayySee eee ~ . .- “Oo“gp[30-F0) ~Ely-one[KM|).en hades at KE(bs)=eew“ ny|OLFEEP.oft ea)«see a (es)LEwe KATNe™ We)Gmtiin :: .AanSEQ)mad:gpary Seeee b= SFADRenf+Re)-ce+f[eke Cerri CenF) COR KR (eT le En . ow Kn a eee . a coe SO) Pik kaaKikk=penee[> senF ES |zeeenoKe re besklayso a Ne(nekbntkake!apwkee a Leee weSg RR panke theemeCkeg)taEE (om oemk Lpartek| =akke[pee enor (eb)=-ktGet). Gi AO Gey) =-SQ DU . aa.eK) saleea eeeeee So that band AEM) =Kiki ek we Fee a ee NN “Met,ioeMykeLundcomatcbsewok _. _Sf ao a / .|erieADakeg@) Md AOL+pr Le ook KE pe oye na : . ¥%ayiaogs 38btBe) ©OX RM) Lh & - “wad as fe : L oe -=eee|+28=~300)fade[peace =fale)eyweeeeek —pkpt.Spe em eeAge indepabebe-(orit4)=.\-(sm)=0VA 0es 88)eee =AG) [rk] peen).=pkkl)— a akin oad 4d Ges ae)oeoeAeSy “yal(HH) =oken Soadhmsewa an: oe a. _ Debete\cae k,{s)se eb)=?ke o--)= dukegoKEaekekgcse :36)ep.JtRRKYveeKrk!pareee|ve cat). JakeKaz beeeebeQepee oo Le tkeLL) bakkie 2dete[page+bee . ee): LL See oR Lsek!keJ) =-lkeCee) SoKookie arYe) geKEGyH) MOL .ooBE BGbth omFLUE b~eyyy — _— neateadadwak,JYarrdwomMakC220)mls(22)| 6ond€2.22)deca DaWardyCan). ©AMAename 4(2.25)warsonodad’ CoddohdeseSR 8Pryrodara pamettr came. PF ey&Geuin) __ NvsitGroAseSenna) a ee OFoFidsdesads Cm)den . a gara=~E\de(apieasrtron(Oxo) |Taran) qa] feat tne andgn [fll aedBet ES eee Tei Ra=Sa(eySpoke] Othe dkag) aes Te O_=25a[SOS hkRR AE Sedna (218)4.2.16)aNAamde. .. Votist oxamniae <2 pe pe ELED fayene osSakTike])Gccom)8wiiec'sea) a(aehe[Qa awe ew)eTaoe -Neosconsa0KestelweJewel —TresanatoASewnsaramaumdSaNet sme ensonsanSe.qeeeeee.NSon=ifckl Ce LaYSAGB) Gy saneasa _ A) =BBEWe oO eG eT a cee =a GIBBS wlcySgesTaewe; .®tTeSoa] Bait) IN-3e : , Nea-k“k!“s WhAcpreoedts _—aNene - ..mw=_(68 @uy) —_(geenbh eo 7OST Gate % Mead26d)seg —Oo ee ee Lg — _CBeeSetsaretyisfastWiet=Wak5G0 oe eng pre Ger ye _— ay eee ZN BO KERA seSARA SS ON ea 3— wen wed av - - - OES NaequntagtaTp ee aern REORI Oe gu aOo =9Tgayn)TG) SST ABGuys NeRey ee RE Relea |NaeCG To\aTt] Mo Wye= ALS ads a\aefaeeCP EGayT [| = Nae,20+ACKfy eg =arb +ACA-B)| ~2\yeTaeN 5=Nat[asa] Gata ares “Yoagerackdyroti a ae ti Cek= BE sae éege eraCas T=thtalict)TF-4°kL, Se Sas aS —aacteT=, d= BEClot)os at Aastay 9)ABAD pagegee ese =gahy,Cassy ea ; AidjerFess(GE EELSERS KL C218) +18) . . Bwo shuaDpow’Cake) DaeGeddxe(HD4Tee(le,t24)——__—. gene ateDer ye US=oanTROT +A(uy~oe)xeSate1a _.saps[>CatB=9d)[=—atattn =ryoneTo Oe yyaye)“=hsGy[estategat)=Gney). _ oyety &(ee~Gv)=a\ed=Ge] Ce BeJeegtia(n [genase .calen.bens} _Se pe eye-. .. CS~\wa-aig]=-Tonseaanh"Te)coosof1aVT_etomkt Satrat~ofseke aeenserco Sokwnt oedS. 4 we8 voy - a LS(dhbedye) =bayEgeGot}a)WAY#Lateta gay) 2 Wo a@am[qnnatd=wymates=omwq(leh) : SWE ay we AWS=4G)ayLea]=GS TRG) #aLy Oe gat(rm) 6) 0AE=DOSE) —bck7cax)obea se Yfere G48). Sotasnek=Gent(leks)(aah oy, Oo ao—AL.ont a +) Po GEDICED)mt(.ee)=\-gt@-o (=) .. (ws=ve|aeCand(eV)+AK)<2\ oo “pn ay.[geo Sn4GHWDySeecesiglig|— —wee“. ~~ 1.2 » - -=-(axe)!“yfconeCe=4)Sy!-@- -; \ MnSav(Kk) a _ SS GReL. — — _ | - —ek poe ee| SeRAS =EENRe1(BB) Dye(WDret(le)TreyCoH) Sa NN _Zo[BanCyrksB) om\EBB S68AN .ia SCS sharin ~NX. =MCRAE ° Ne ew) Sy So oo Aasvack Q-heme ae - CEcet sXe HpLrg ybDelDee(We)LiAveeno CxS) ~ Semis tedew (eb)Abe)doe(h)[Pros(aay aene ta\dew(Le)Abe)Be|)[Weety |= Redaake( 2:15) Boe US=oyeaayTp?=.say(Ty“TwadYAYvewejo_ .=aySie+A(ws)FeOra),Yeast _RES aid ;=_aye owCRK)Ade(ie)Ade) TrPoetsGuiza) qu. a FARKSera(WR)Ae(ieDdet(keMael(h) . oe et _ _=Podem(ek)Danww)| Jniandodrtagen agerSLAbeyegatesMeeWan)meleesedantie. So8Qasr ikaay CE) —— -—Vohom sanakion, CUI). ee -_— Ooiditkine oe Sasa) ay=CHa 7hp=e Paes) ay Gee ~3Typ=CSi@re-dsaadee DaBASiesomntaeinsC2),saDeencadeaabigdane, wilhGed ge Oo a 2 (2.32) +(2.32)+(2.3)+(34)+(25)b7ae jee.[Pgh O-eS yee TP ooSy=Kei_2no= i.ey 7oow=Ketk)==)aSlikwal . ee ok _BO SITE OTE) =EUG =eg LoSo..Lanude-“27: =.A=5.al ~\UR(EC) peMoe))peUi j ee spe fee) ble eaeay i ae = NN, —.- -. megk. i _. O. ae . — cM WSLS. Le. aYostcagde _— ee .eeeeeeee cee eewag Sa wom J cee ee cee =egGesle)aul()|- amare ce oe ak * a“ re ~ ne “Oe 0... OF UNDE NE Leet L oe cee WeLae\su- =+ me SOS Se)ee BhMise Leet=weCecte)ont!qe!po thSQ Beeeee . rata: a wea ee BaeeQE =USSete, ~ @ ns farbsbe fe . Se ag ;® hiaieivesundegpate.Walrsnt: - ra)“atae) Heatel -ee a AND LYY f(y & meooValef(a) +ee a Nao.gd TVLoRAS+deebyCe Soe ee =FEDER) VER ae — _- te aya \ aakoxBe}RW wsKE ROS | O-- |—@Hlease - oo -Rasins+—oe —_ -_a-.oneee::oO "ows=$GeVh)S232) OkmecegtonMeeewa Gut,akaeVauk prtco Ought —qpleeack Aopy8. oe ee AN, Se LedeeCee)F85-(,8) . ~- op.NSaah wigALE we LeetCRY=gbaGee) ote!(dete ane(eeLick (ange!me awastEE} ae \\a a fleRLeek!Caete aUefcke (tei a . es » Cee )ate! ne(kK!) ante LeetGab3Looe)=s(x)|ak °ee TUEady seen (BEB) Sak ove)LEH)weSenet ed e sw QaleWeal) LeerChik),TaosCee) ne co oeStq€Loe oe)BSLCee) re -©2SteBaja)LoteQedFE.) | Mee Hosvaith Real!acaavesfettur gekskt . Sq HOEYLate) Fuh) Zo a ASH VG) Lee) UideereBdeyneamswetke.asilkSefueLyaly Reores. G1);G8)idanceee 6) CB.AY+B.)+(3.0)==mann somredGur) CFCisbakLeEKATHOU),Ore(S-Dntcomes _. + de. ‘io 7 - ~en 3eo~gy +yada-2he&G6) Deen BQ aey kate! __. oe we beNewsy) 79) re) A&+leauf t De «Med.Unda =Chit) bk ee ne oeea)©fedSaecitneyAte® 7 amweftht. SakORR Acer] a Ste Ska eibgeths aa‘.Nee FONReyOt 7 =(RSMo PO) at : ._ GrasPAs)PH)Ge* a So a Oi oo” ar eeark Oa x = ~\dkh i, = a REDOC (d - eee ee - —“ --| ueBALbine ae - oe Be 79see 2EO. lg-.SO)at FAD La PRY : me i ; ae. [?) a7 GAY (2-2) +@A)&C210) GavegraninOraArialGogg . bane P)oAeaniaa atMagulsadrpal quoatijatediJprogaugeOhana the Bway algonwend aoens _ oO . -ee STAM - whl=Sua aay TSG eee, - ak data. a -7 -ee ax(R\a[Macon)=as|TRO So a cead-|Sae UO. WOT{sye,Bel— 0. MgPAS NyFRee RnUaiene LTATHA,Donyor‘Leo.ORLMidVALeHAY,Sesqpos. 9.Yuet . . a KS 3xda,SOY Wet51)tFoSdanThen,yrwideATA]=\isdfac|©wet ™ =@Bropoal eeese psga ~ > V.@ —---Gaepose, apsedn sa wee ee - _—_ ats t= OwWR a ©.GisVEGTS77 WRWe) =Sat0GAD)365): VAyCh a»WO _Blit Sey)oo : DA 7OMEBGs ©BaDamgoesgitApa ss . “ranidhgaugs oN20,OeTk indagdAwogluke gg “Obstounoolosik MraNAithe7MyeinelunaSk —a Saal =FA RAR See SU dravthgenebeete 8 SIT QeNGAI TT a OSouduedwagd?FamichgageQeJauhlen ~ aS ef(QuBh3885=aeGetta 7 (ey gpyoT GBB~ae4DSL - a ee wpe eeSo £-eMLM qena—.y-e ywor A ete te Tes EAMCQptga tym) BaOaigweWenelaleJeOust popu, So Bar =STCgpeKeeeye) TS PRM EWS Boge TL. SMyemet yedmpe fe 2DAG)AS[AgotBasCaySDGoarm) \@, st —2- Aspsbipn scm+B(misolg][qaoll=leks=dans] : jae Aljes~Axayy+ike=Byoe=BXky (ek) | OF oanAyaggky(xk)~coy oeDEG+me (Dewy =DEKyb —kDyk Dea ~=awLALaeyA «eer DEI Oye ae _.+apt=NACE =Contfa~Oke] ee —+&wleseH aden 7 +byLeeartyee _ eo _S, oe Se eeoO “AL wo.ARV. .ZRA-DEOtVSo 2 Lm Dawe EL O~ Alas Cah=tk~OM)keSO Ae ._— ~Be)wk aDe-De=o. nn:(ee 3.@sor|,; areeaSree Am(weak) 2aGet) . R= ay V4YE=@kyGeek) =Oey “5 Qube KTLayofotakilete-Conytems1a DT oN (me) _Vepengaadby.Ra oo a . =F[gy-syns+WN| . 8&8 yeaieGreetheke)tenm =oor ge ges=a[tran] +bbe =oyooWye=eLyleOS nn _QininravedonVeseraare. weDek AAP20. - —|Sinwnary.s iaDeaoagauge 90K0,ohvaninlepihOarGeigytn1°: weam ca*(ursooduialtspaceUiaqakn”| iL”. oruatteRacnniom agespode 7TYR =—8Japtiniee -ky| |]Waadeubshyoutie.BydinacaBaudkeon weencoseNah. alo Soe cee eee aJAWYCaa—e]gy+tellpe-Gowty15 [Le a . ky” Cay ajo. ||aehemamasptpinging. ndsMidDayoBT a a= — - y oe en aeyet” =lgeT ©fone Neeaye=Wem/ler), OraagpenreonRubio CB) fF Gn ee SBuh ngemits SPgeot Me A — ribaJour. eee SO . J Co\Dyoltth. =Gp=.a_ win:BaUesaldey.Wor=~bgpleek).Shdswe Sates ag =apr ~Soamung ©sine(DViauadifinnh otlao,vandubsacpgndasinat |~=f. peapagatnbby? wens Loses ye cee -| . DoLT=Spam emJerk). . rr an SK Fark] So no Notes a° lo aes ay oo —Mysuchspeandic?..—Ga as=~-PSN +7°EonRies napoursLasenesyne, | -ee Wwe=GyWFApneGoats a Maa) Pptun!dependoater oY wee sy.ory 2Jour|Deenegizevinden O° Bae=Sop, =silage ow Loan Gs=cet . OR Gees tenea Wek — ~frte Pa aa ——— : Mp MpLMr - eee -eee Muchas byt, |LO Low? cee OR a Lo. Cee). Cham lnm tm) hw oo Ge GRE Caidhe:39Same. Gohnenpag e- ©WsGavenayeQed . .s aa nd 0SRE SS - IORSE ee 1 —Ge de OeeB? 7 ee A gyOye ayes aye_ee=\eGy[Wn8GDHAWN Ae Gye Leda oy a = OT egytek de a _ Cgrid+4=gy[moRE»)|FuegooNRxBeayGee Seo jotwe A a RR aplAheh) Alte. egy [SON OF ROA Oe Qe “ae nn auyeo.Se geTha) (Wha Sus) (b)) Bade 7 ay! 2 _bpieky +RE ihOats .x a OvLasts «00? -Tbp,Kyin KeyyWa imYeaoe . “GagneBuccs pigsa=Main) a Capeeky=weke) = aoo Geka Be Ue a RY comore wah. “La. . eget heBeRR) oa OE aEa ay RPWay CRy eyum~ KyMa) ss. en OR at=Rakeaye). 6 nn a QD. ; Finally ayers Fe re Say upnoing@Gat BaWowddensee, SeogowictePanmatt BakvosaodilowUn.Weonbisps. co |CostaningoyDenaGnaDohemdain7|O TO,DeeemsumafevDakTso hD.. Seana |aagpening cystsyn+Keilabhinyan|ieoT fo=GpFARCRY endeof yee” | joo SsGye3OrOwns tye vue Hiaustbsambspomibe LO es a8 fe} ©powey =IronLala+g+ghGals : +Bem] gaypatna gyysl-ol. . PR Cs)aieba=AE oF oAFIGEN WybeByah | Se OR Kem, Yk Eel (Lede deebh,ype, SF OO _ eX eloe)UaCkegay WN Ba) -oye a ce ACCS Gy) Abantaes) qe a -. a |seven) sarea@adlignsgespATO oNGatto) geen ee ee . _. ©OnlyQianggercenseyDeeStescmlkedllako ST Fae) =8e(eseH rsCansheoktBRR So}BnSynCues), Dondenendaiamar Us~ ae - Snr Yaad)Danfom _— 7 Fy Geis) =atpeneoon Que) ¥eyebis ween Ye.Neinon7AeploetBr=Enya;oe | _. _ Do _.1Tyan(yypya\=cacy£aAy(kK)CeACHES) _ .acnren eeees nena Ba — _— PU eho ate a ye ar nn eSCOLONNCSBeasSons](edmiye@ TOT RenCae +QTE) CS rs=[SOFelSAGOPRwSYTOT TL SAGE Ge cae ROE ? TLIO NGee OL ° RGR) onyveeWaTyas Sel aw+SSComon p,2: Oko)wyWageaBe,Alone sR ByGa)ayeWon STEEatYEAHanbetyNy FaQae)mapdyWN aqueFela Whpe+Se(Hk)ay,peat ; ~Gee)om‘)cylSidaSRSabyaSyl ayWa#egGyn,EF SalihayMpa Lakchop es esTF Paha tanyeykeROPeyLT ctor EH[Seco eeLOkshe|YAnemoflere.> pahaen- sey 9 Way ehaeey <0 _ . ECaey= See) BeCWW) +oe(KE) co _ ©[GeyaREed0]OiaasheSyoreunbt.7 © GERCE) eo,same en 7“YoccktheBAOgreg. ee HAe(kk).=Seo(kk’)so4,astndepeandiate | a ed ©Ose, He(ky=FaGok)=.=SeCeyp) mega. | Rede) =Battin)=-SeCett) oo —He,“KOSS IG] ee ©WRGmhMew KYJegee2LTE _—sWeliedr-p el=GPxene OQWid,jeseveringSodBreesdgt | BS “|cn\%Ryley+het|> . +fthely+Gaba] (Teete ve = »(Aer cons aeco SrCehy[Felons ‘|;ea yee.wtkyLSky,~Jolep]aEmel wpb rena t— DFYasadatlemmegpear veCrandeolf. ; ru -\—- CFSA_ficont 22pager . - - TaEaboductinChgchunes ocean qfuanomeplingSDSiugplacERain. Scabot. QuiteYakenn UskAonihgeneaeAbe== stvcheon“Uith latoramenganbanlashiae --—.BnQEO,inceovepectinghacbas,quateoaacsshapeaway) so je BG) = Balk®maornk=o byGet Qyerrnedas .Qe ~ee -/pogeBRO8D. ee foSpee oe . "Baked awaamad Bimanwe,exhublah beyawaualepebomasgee. ©AX] Zen=2)anTpieee Adee: CBraet-eet Saree Se hareSe eet[later5Real.NoWashaloodkA207?VorsarnePoshegyyCanterDussWarkdogGagasqvechin TB.glowpuny.“WesQuamdeckJRQ=gla,-- nde asShed)Yataamewn. gous.YgyAsengyBon bnBamaseat, Brcrandiace. Mats,cernardaoadwarydo(ushcy"Wanddaluraise Oresna equate WNduvergewese Amudvelac toseletR osebooRate. fe...+=ReeDru,WiiSedcamrient DreddWardalOninQuitesseqach.. Wharwrewrit Dio=gparpragegehe, Deenwieendty Qo——dmvanrond Sedna,da.coayeaseabing Lad.Cn).ler)Owe - O_ACL) =OMAgGs) TO GarG0) asauma ala)= Bad. ThunDsnashcofain oo Weanabgaiin befedOak wo . a, EG) @Lagoo 7abe og 2»DockThamanmingonepn,Werwsinp, Radway erfinerst © Oasaiebis Oeauagetenbedarin |A,Biacontained Byes. eo. RGIS gy)=GINA a. warp x= at/cqery oeBEING a)=WGQUOPE) oe husnthe.entalgargsJoodarach,.Gaitsc.Heat Wek F)= BEIM YE AnseAeERYieDVJouManabealeeDEJete Claasiy.OWAaafitomenVanapen.Qaabenyee Reoe esBoE aa Oe Quast8d=mnghack Naafene re Aineeramanbe asomdact WTC a ~2- A.Nsabun.Taraacors.cedh.yibradat,hapapsgagainnbe= se]Sake roeoaYG shfirmvavareteSehar..Uns“univonawhs” onat/ftamdVogeinThunsinceOnceumntannen').wawar:Seberomonhowksbd ved)berg . a Bey =YG RAINTa)=HwCG,oh Urea O wre Xummats AodNes maaan): aradlerSak SainQagpwariondaLieoncart,Lpquae. .a,hoXcanes" Foy 4A ~= +.---—| AaTo=Bo-VI) degen deNalgarg” himaetenGQWas gt he — —-1otgpavwsoeas~a») -@ WE= -1Ge- WEb=Bye cw be®,aoaaVier)paw K“e=VR-AVI-DOR iaDeealnyQaueVVpooks.Vode - CoYaoVastJurena porea — Toa) =—CEGo=49”) Cee .weToa)=To)=Tye)ae“eyeaa: weerdyeant80Jananlaran costa. a : Wardprraeme DDred ated) DeeOiseQeraeerrandUUs Pesgngneee. Ropes de0168 oy eelen, . . pr owea MOE cno.[~~ =je) +Pe @Q->tOoo .a eomea Toya 2)FateThat ASM Due)iTCiena)ne Rance wyott we ok. NVidia |panshgoneDreesFaeyrrmomn netic: ©LTRs) =galqude sepe OT ge B-VaWad: eee oeeneeee (54)CANES Mn) =Marelae) Wah:aaobi):0.WadogyYasy.boRASACaSecele OF,ak BOewin soleTEtraweensinhod Ct). hmeehi | weNi eee DRO de G8)cdeseme DH|,omit ooGANOeAabouhein\Aone. nn 2 NeoBabsaemaghC246)AnDeacmensch dfnheldclackfa. Seofecttace.ch ergy8oeieee legate: aaie.bogenre seer squatone) ogDeegnha aed, Gia)bembales dereyesua oe OK,gudaasenttnd. paik(8)teGahCUE),A)ontse “RAERRGMD)Meet temo, O ae —3— Qtensedvine :(spt) o©-© Oo.- -— oe. a \ > yn- Oye g=.oy[pepeCE | :Q)= vg)cayOO ; @-&)=-GnalatnewegqhQ. eV LL. - @=pg]abdrapa7.GoggKye FO ee -WEL. ayTye=-atquh at=o.Ofowen,Raeaemnanned ouch Py lw.=Odeo:todaVemesegded on nnOa 2.oyeTG)=Gea)Late | gyTa)=FYaeiTfarrG-) Dpekydyorle?) ESeattheaay, Waibated agilenk)=ay.|GedySaFCH,SatCla)ie} . 6=GE am+am(-B SetGA) em Qh yeauWw(a)=“aSakLG) otatance daa!—Gaon |oO ee eee --—--(#). --- ee —-ET . —-se xDroCdxee). [ALCe)Tae) _ ~ aw: Doe =(So-ene? — i Ovinw:TORWS[Seet ae ; _Dye(e)Trelle)= [Sve=Kune ee - - we ee - cee SoWS=ELde.[Gry[Gem oeaCedap _ Ore)|Oye~KeMe~Bayt|By~kangie zipoles eae] ah.—i =aye(Gasmagengia weydoo!\.—_oeOy |kameAxe)Be enPO.oo (Gen) Cen) Co QndbaDaewey dw= ackwerAnWav20>Adrsabme6 Sacsas005A le ne _fe\8|)GannomyKateDode) a Be wdwso. 2 LL Lk Groacleao)akwonWtWWDirMeleawdenshe _ss — ——- HH. — .7ee JaemeCW)Dye(W)Dae(H)|e “Wee)|:e VoasWiden oy .(ae ee TG geEke dy kat weeeetSadem(FE)LO) On)| Bo nee 2TENenheSana(ey“OY wo! a Nd (q-2e) OBG-bE)e - = -Ue s_ eee Wn. BAY aRwBSeey = as,TEEN eae)a(gt2e)Ow) cae Bae omGate) DOCe rr Vas a 2&8 East)©)So=We(UHDES. a.ok)Sendigyabe.. .a wet YH Summary ofSectionII sofar: . - A— ___The Dysonequation ignoring the4-glue is(2.11), Thisequation involves both| ---7.__the glue propagator (anditsinverse)andthevertex.Notethatthisisatensor____| equation. Bydotting this into the four possible combinations ofvector-pairs, you __generate 4scalar equations. Theses equations are:(2.15),(2.16),(2.18),(2.19). ___t _.— Iregard these 4scalar equations asequikalent totheonetensor equation. (?) _.. From the first two equations you can solve for the numbers dand ewhich wecan regard asconstants, abbeit infinite ones. Well there isanimplicit cutoff sodont_worry. net Sointhesecond more dynamic pair ofequations weknowwhat dandeare. _.....However, seems_to methatdandearefunctionally dependent onAandB.Butwe | are going toidentify dand ewith quadratically divergent terms onthe RHS ofthe _second pairofequations. Wewillthendropdandeontheleft,andignorequadratic ___| __. _ divergent termsontheright. Thesecond pairofequations hoth have the form -. (2.28). . - _ * Akeystepinthis’is toassumeacertain functional formforthevertex, asin __ —. (2.20). Certainly thisthingrespects theWardidentity. Ipresume thatinsomesense__| this_is themostgeneral formthatdoes. Thisthingisofcourse afunction ofA__ -—-and B, KandLare_just groupings of"spinology"™. 9 ©akeVaktupetSeater, Ouumilck Ande, B=Dp, . seth AR Diae Me @arya ax). Coe Ae > ~~ Soe ee - : -oe Povsisgomane eerigta,GlasNaniex, - CO©Fading Medea, conbinngene(ybdakeasteleAduttaTL,pi. om rsp prope ————-________] - laigevdpe =Le.@[geamesSe¥O Ran FOS creed aeitbaked _Vembasuisy =JEFE22cjtem)lates. wate -* a ~ — Titan WBS) =JamDECe)REL\gssd—SagoqymtadeinDatesany | re)~L= <pmimall Slebe. -ee ©heDalegtinenBtachuchyrenetMEESpinsS32039faOuse,enkyDuos 2xgreonstata wibyRola \=)tancortibote. — © ©Whdkdoedkshalishale” dgwoning gpmictusequemiby yonweathn . . gluesVurhoh —>Lt=evn. — Al. 8 XK ae _ a..\1®VA =.GS= GO)od(2,2)aanDas.only _. \ 2Kr eMac possite Srboe _— _ Quequarter nk TS=eoVSL’, eyprmpnrlakoe. “prBAWS2Ayko, aghOKQueseostdbtaoovata ae OQyusses.VisadeoneghrpndMilteaos9vn. avay — ne :\eeBe —\oenkaaDraarnrenns. BaJad, ce 34.0.5 — a --~TENS Gace) =Casson Tyas Caeets) oe -Bak saryDaeglakthaoerfab,+ Goneludieon: InBallZachstuffweereineffectdealingwiththefunction £with _.the color structure functions removed. Thus wereally doexpect this thingf _tabeantisymmetric ifyou swtich two ofthe gluons, which here means youswitch = _..=twomomenta andtheircorresponding lorentz labels. Thereisnonewdtolook - wee into thedetails ofthestates. .ce Otecccus ecb EEOee cee a 2 OE Te geweeTatarnoaui, JaaOuidkdsestNineoa S008)suk x———TronSheL8Merguug beneieSO),PubdesemeJromeeils eS: By ee 0 VED GESkane,ca,Kaun o,kaso) =WCQ) ay apyYonah, Thala saat,“Tu tohoaoedrgmden crho, RO CES,hawereesg jhsmeramds,o, koe.)=Req) Mnea Gua To CgRD QeSy TO ESE Gog Bee, R= Wie =dedkSdeWRag, ©RakaRBVA tageen AMASCGRRE) uateea,toweabee Sewamlydeemagy LayBeGrebotsueGAMade O00tanytnang)hated nonestoredade Brgp.war\eyVanpronedebi Bo OOGaricaatteasadubewei. oetaloodRe a dgt(Ente) gt =ag — p=(wiv) /2Tae a ee Masts 1SEe weg aheuCoe SE ig — is. ee ee . 7 _ oe ae oe Sa. ={[Q&e 2%) ok DRE ketch SS Eee ee ge ake ate)=ac.V8)ale\"=telekae - an a oe W. deee -—_—. 44 a ag —_ ee ne Se=)Eeeaodataoce . — J Sen mee ae _ _ “(Ex 8)_@=(Ig,8)=(9,8)-. © a We(8) gwd |R=TRsinsos,SasStak,ose) a BSC dceoosdk oeGow)=.9C0S%fate seis=Gan)samalao (eg)=9Ex.=Oat)=»__@x)=qdee ~Wa) =gh=ECat) = sa —a- (CRW)=Excosee,+loseSima=Heeet)a+t)Licagose=3 O_Goo~ Booed=(Blasesma=Hidbviy)e =ielSinetcose=at © ReapgoinonGustdebpredate . - . ec—= ¥a___(en)=_ {ua =Gen)=iy[RSETondsocoe Nowwrendrapowh evoalale, hat 2 ah eo Cel!)=ExGeSRL=kefgmiel=egoee EGE AIS Gt) aa Gok)=titSia?Rlese a tals _ te ekSs deeseS gt| |e == ge = Gaagk a) Sg _ Ze A(ee) =Fld ep TE =RCE) eee een a 9g ad ° \ =an.&)\ace|Jadgetey’ |BCRZARA)Nay LG) —fu ee = Lh AAC) AG)+OSSGEZECS _ hn -oe wiesaamtsadfobole aaterfor ee Aes [ataetee Avs et[e-atadard)- aGea Ay =offia)xeapteieadeQaeles] neESCCS —Asb[oe-daa a] Oe ee er aan a ee selaa *‘ _—3_ ae);i=|stedsfestavet) |___.cee Le Agbeh A te WaWinonplus,aigmmuley “Aleaquaint|lunspembie Jack. Cheejotsamsaeglify byenalinsDabealfadiye ~Lay=2)SeastaTease] Totcue)awee . —Growettoutmoepleat wasAGeeopmantig: awaywatQeet)qaoneak—-—tia.Qing,«veeee (eee age Lg=CaelnaWa)(ahQEDSSakeSEae) eauthaanaSeanCtsadandoatSlgahcamasuubias trababatodin scandal —asAthydoeQay Aa away =0 SIS eisREeee no ye Si Ri) a_ : SSS de alty adeSay ASGEA) fee=eG a Cn Oe ee _.—_» age ee — aNigGSESayRate=SayteErage ° SRA 2880 2S =aA EE ——L\yy@ =San(eae =ar—Sty =Bo-oess ~ ~=. -—_____ ns we Naty =aE ASGWEL ce 7 ne _—es — 3 —————————_ oe ON) = LegSiaRY .@ nee Be ne Va\= ae Ppeee el a ee ma. yet SEV \\daw=\»\=Sd S=2 -aaC Oeee)eee He oo oeGedtee ~° HeonaahoDBaarhakdwscualesomtving Sumptn Aaoadeado ane. 7 Meola ‘ -. sella. Katt= Eatliewus'| wee E(B ee Qa ekeeae ee etLoeest]we LS EVO) ee gtEee Ast] ©Svatsacl Sadhu gis — yeSeth aetw) ad a = ete) dekgGow4s] _ ee ee -Siglain. page,Svassion B.-ABDspagan)— ee Ocage aaa comma — & a -Be ee a Dyer AGE) =ag EB)=OI)aeae ta, A=ECM GEg)= AGEs) Rejoin degkASA RE9)asnoe Oa AGA) =a. TASACE3CR,9) AG SE) : 50=20g @®RegsckMecloonsioncommealoshostJorua: ae Toams Saosetegxeeksukdek Bat Ga oo Zo w=BOA) a ee - FEY =W@V/2ZOQ. a —AGUS, g)=2.GS3)AeSBF,305g.) Ka) =2G)heGs @Aswad duckdose Maein,B2-6A igae dened ee — _Q- ©Redd nomialgauges.WarenoDadiate v=Fe a:oes -os ooey ee LL. -—WorBCaeaetheMeansREgouge, te,"Uanemnamobirsd”. Neo, ia - 2wn.Beg Bake BS BEIT gy RE RR Rg —Re TTTT "Qu Haan IngkDLLOGg),Bitegece_.Sek. ievachepSKyMatanieor soc.wD.Oaarsaas suchaooansadeAarsine dormtwash ce oe RCMENGga)=aCDSBE3OD5VS.g) —amdagain SCxxx)=AsoGEC=Dfaonyomtue dg4 ©Nenmallly, dkingNeYeaotAven oweauntgua= VeaRyeesaKade —tGa ekwie, UFO) =t=b@s BOEG@ SA) Pa ——sae_angbasse. sunasgaanedLabGe)surfs”DGS.7See @Possike. gum\aQewes eeeee See LtGy eae ee te Tt GES 9,90.) =KGa) cua <Se[Geya“myconvea _@Golanpda GobackdovisaageiedDe——* TYRGN) Saxgeeeedeh Mohwge *AnSra,oa dak. OtaDacedyoosandee ee ACh0Se0) a? —3- SowaaMornakwaeee ee OREED =OY=VOWEGOE,pO 7Bile Naandeg GEN =26K Woven Raka\e4 Oy Noeheen tpanch SRS EYaeaedBR 4 a Sa Bao BeSpe * coww\=se2nauDuarcandsegueQeede -ee SnPSaeCtasaC 60 SS pre‘daseeaeR.»