Renormalization

Renormalization Bogoliubov &Shirkov Intro totheTheory ofQuantized Fields Chapt 4$-Matrix Alin2026: Introduction totheTheory ofQuantized Fields (1959) byN.N.Bogoliubov and D.V. Shirkovis consideredaclassicandfoundational text,butitisoutdated forlearning modern Quantum Field Theory (QFT) asabeginner. While itsmathematical foundations and historical significance inthedevelopment ofrenormalization group methods remain highly regarded, moderncurricula favornewertextbooks forcovering contemporary techniques. Comment onRenormalization 9.5.78 8 IthinkforthefirsttimeIseehowthefinitecorrections comeintoplay. You start with atheory andyou first make sure itisrenormalizeable, ie,that at worst dj=0forallinteractions, taking proper account ‘ofvector particles. Next, youuseBogoloibov procedure toregulate allfeynman graphs youmight want todrew. This isaprocedure forremoving divergences. Basically, you only have todothis tothe finite number of"renormalization parts" inyour theory. Thus, eg, inQED youhave certain objects like G=full electron propagator, D=full photon prop, and soon.Te,when you write the symbol Dyou have inmind that the divergent piece hasbeen subtracted out. Depending onhowbadthesuperficial degree of divergence isforaparticuler renormalization part, that ishowmany subtractions youhave tomake;. These subtractions aredone atyour chosen renormalization point. Thus, inQEDyouusually subtract thephoton progagator atk=0sothat thephoton pole stasy atzero, andsothat Z,=1forexternal onsheell photon (ie,thentherewillbenoexternal photonrenromalization effects.) Once you have cleansed the renormalization parts ofyour theory, every~ thing isjust busy work from there. Tocompute aprocess, youmust compute only (eo) theskeleton graphs.However, inthesegraphsyoumustincludefullvertices andfull propagators. Andnote that these guys doinclude various corrections asyou gooff shell. Thus, thevacuum polarization isthere, ie,there are corrections. Similarly there isacorrection due tophoton jumpting and end vertex inCompton scattering, say. Ofcourse formally that -jump isincluded inthe full vertex, but you still have tocompute itassuch. Ie, you have tocompute itsfinite contribution. This isofcourse the subject ofradiative corrections. InQED, theelectron self enegy hasD=l, soyouhave todotwosubtractions, sothe little derivative appears. Photon has D=2 soformally you have tosubtrac t including second derivative. Ithink the idea ofsubtaactions isfinelly sinking in. a ~98VarFiQn ee oS aceSO re Ce SSEAntaguih: dace he, Reuse20Apsmomelagte, ne, Se Se Ses ont oR ewSM eee GD GEE we DgRe eeag. .eter L . woeee ee aYeseannteaddalenWiboogerLapamgien taecaslshange”) Rags srk iealtechangeatlSoreag, ohm Woagska)\\>2aGnmgi) mm Sa Sen >NO |ee Po eoay aD weeKe tt “aa Deg Victron seasoete GagsMy GeghWy en a Dai a/Bogoliubov andShirkov-~-partialcontents Chapter 4:Removal ofDivergences from the S-Matrix. /2h.QED, Second Order (electron selfmass, vacpol tensor) J25.QED, Third Order (thevertex, Ward, gauge invariance) on™ 26.General Divergence Removal Rules (theR-operation) 27. Analytic Properties ofCoefficients (skipped) v28,Classification ofRenormalizeable Theories. (thesimpled,rule) Chapter 5:Applications ofthe General Theory ofDivergence Removal. /29.In-theory (Hurst-Thirring) /30.QED,Part1:General FormofCounterterms (Furry, gaugeinvar...)31. GD,Part2:MassendCharge Renormalization, (FullGreens, miltiplicative Renorm..) /32. ED,Part3:Radiative Corrections, Second Order (Compton example) V33.ThePI-N Theory 3k. Schwinger's Equations for Greens functions. > {5ALGYe. OroeG0ecadsondin, Dhgankotin AVEAwigetLama a Onaga =FB salescmsarne Banc deadoeglDh Asmapdan Jumeionr.quidemAenatacibe eeina Sete sagCOS)aud_atg(SE)boysutahekedas ~—TYheds wandaeabies aotgg uo Se Wy Nileee]iGtdebel =armet)empEBA] aaah. cia 183, £64)=(CodeathSunaCoJuteSete)oe “BaenikeonapaieGay —thak o unset imgoseace >.Be)ostiespa Gavin:faalanaisgasgction BR) =Os CUGm36) "Qaim heSalewingSipaadepdedk Acubeead tee =20)~-C224s\Whe —Te,2G)statete, auld:dapat SoeaesMetad\Tonfom. On Dagege ToS dinaquedantinadsdquadenbicaSly OveSinaOLMm a TO) =Tyo. 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Letusconsider the renromaligation ofthe electron line, The bare theory lagrangian is this: xy: :L=PCswmaay ie,BK “teee Now, suppose wecalculate the renormalized electron propagator only tolowest order, ie,toorder, «Letusstay close tofam soweneglect other terms in&(p) other then the constant, Then you find that: . . (omanpsp=at ae = ol 41g 1F-Me, pre orm Wun Sa NxS(t.) =matrix inDirac space. ° ve ocale & ~R=BeetleOn,|Ae f=(c(AB)-2m4 Wr fo x Notice thatA°andB°arefunctions ofe,andm,.HereA.isaUVcutoff and A isaphotonmass(regulator forIRproblems). 2,Our problem now istocombine the two terms shown above, Byignorimg any terms +» ofhigher power ind, ,you camconclude that =. foxpet)=—£-- 2 Youcould thenineffect lookatyourfunction a(n) andchoose m,*such that aogm,=m,thephysical mass, Thenineffect; youseemtohaveanewpropagator propagating atanew imass. You could goontoshow that! therésidue ofthe propagator isshifted because of3B°(m,). Inthis approach, you seeclearly that the mass isshifted byamount A. However, thevarious functions like3Barefunctions ofaandnotm.ThisisOK,but not very convenient. Notice that we have not used any “counterterms" inthis approach. . Notice also that the (new prop) shown above isexactly equal to: te ee a a ekfen, . a Ie,ifyouinclude selected higherordercorrections, youcanarrange foget° exactly #inthedenominator. However, Idontthinkthisisauseful vay to think ofthings. 3.Let usnow present abetter way ofdoing what wehave just done. Simply and trivially rewrite the lagrangian asfollows: R= DUB —LeRAYSmBae Susm *) Wehave now arranged that the bare propagator has physical mass m,even though it isabare propagetor. Also, wehave added an "interaction" which, ifyou : like, shifts propagator back again tobare mass. : Now, using this new lagrangian, lets goback and compute the lowest order shift intheprop. Weknow, bytheway, that §misorder’de just like #,so donot forget to include the "counterterm" interaction: Craspap)=a EN =—t- oa ™1 \ % \ = + . a Se weg-m fomzea ¢-™ em Wehave now slightly reargagneget the theory, because now when wecompute Z we get: JA.= ArtBG) A-¥ - gee=PP(om,wi) B=wm) (>) Although &has the same form asbefore, you see that Aand Bare now functions oftherenormalized mass m.Also, thepropagators arenowfm instead ofpm. Of course both these facts must be compensated by the counterterm because we are computing exactly the same (new prop) object asbefore. Wecannot get adifferen answer. What isnéce isthat now, inthe counterterm method, Aafd especially B are éualuated at the mass m. . Also, the three terms are now very easy to add up: \. Qe Sam-Cropp) =—|4[Beem +d Dom G)pm|B(e-™) Vin qo \ A \. => abe) ts + :pow(ORO ER Nowwechoose m,suchthat Sm+ A(m)=0,Thisisinfactnotthesame condition asbefore. But toorderd, itisthe same, You have tokeep your cutoff inthere sothat a"term oforderd," isreally ofthet order, and notlogarithmically divergent, orworse. a?) Having sochosen themass m,,wehave eliminated myfrom thetheory and wefind: \+B8 =. Cnuaperg) =ran soYy=\+B® -2- 4.Thus, using thecounterterm apporach, weseevery quickly that Co a)theelectron isarranged tohavethephysical mass/ b)thepropagator residue hasbeen shifted byamound Z,=1+B. Donot make the mistake ofthinking that now that you have minthere you camneglect graphs like ~2&%, 11these graphs arestill present, unless youusethenewpropagator which includes these effects. Ie,Iamjust warning +younottothink graphs likeUS with physical massmare illegal. They are OK, but are now included. 5.Sok inevery feynman graph, you will want toallow oneach electron line all three possidlilities, ——~ ,A, ana“#-. Buteachtimethese theee appear youcanaddthemuptogetS,therenormed propwhichis2,/(f-m)« This thing also has the physical mass.Nowwewouldneverconsider agraphwiththis:rhe, or=X .These dont make sense, orifyoulike, arehigher order still. ‘ 6.Since ultimately every electron line appears between two vertices, you can seethatyouwi}1wanttogroup \%, toeachvertex andthiswillcontribute to the charge renormalization. é8J.Inthisspiritofaddinghighertermssincetheydontcount,letsgo back and reinterpretnt the counterterm method imthe BD manner. Wehave: (wopg) =——+ +(2+en.+-)wo!+k +Gee+eR Ichose the wrong terms toadd. Here ismore what you want todo: =wm! (auspag)= pe Ot (BR +iP Ox :tak Ihave selected avery small subset ofthehigher order terms. Thereason for choosing these isthat now you edd upand get: =Dtplertalp +DESEO ~oo Iblew itagain. This algebraic series iswhat Iwant. Inpictures itist OFGuopg)=—(tise )y(Be+etee*4) tebe, So, once again, Ican add +ese higher terms without doing any harm tothe order ofinterest. Itmakes iteasy tosug upthe series toget answer which must becurrecttoorderofinterest: *)(mopt)=Fondo oveud =p-an2S Nowthetrickisto,choose myinsuchawaythatSn.cancels against the constant term in.@,. This will cause your mass tocome outatmasdesired. Consider: . A=Ax BG-w). \. a: ~\*t& (were)= - = a i... BomK-BG) Ye ~~\\-8) to Sofine, wegetthesameresult asbefore. Itdoesnotmatter howyouwrite something ifitisthe same either way. Refipfmalization ofQEDtoorder X. . . : Gnéethecounterterm ideaisunderstood, therestisrelatively easy. Co Eg,wegooffandompute thégraphym»(withmeveywhere) amiwefindthat theregular barevertexisenhanced from1to(zy-1).ButtheWardidnetity *shdwe that 2)=Z,‘and wehave already computed Z,. Thus, noneed tocompute Ze Now eventually you‘want to see the complete renorgalization ofthe vertex. Todothis, you simply add together all graphs oforder Weasshown onpage 168 ofBD1. You trivially add all thése things up and you find that the ‘net effect isthatthebarevertex isenhenced byVem What was23?Hadtodowith theVacpol. © .. 1.Vacuum polgraph. Firstofall,when:youaddthecorrection ~“O™ toww, (again with physical mass monthese electron lines), you find that the photon mass is not shifted away from the bare value of zero. This is so because the loopTyla’) hasnopoleinayq®.Thisinturnfollows fromuniterity, because there are nomassless intermediate states you can put inthere (like Goldstones!), Infancy gauge theory, the goldstones generated bySSB are just those massless states needed toshift the "photon" mass away from sero (see Bernstein review notes). A Soasidefromthesmallq”correction (whichyields theUhlingpotneitial and 27m worth ofhydrogen Lamb shift), the effect ofthe vacpol thing isto renromalize thephoton propagator to(newprop)=23e/a? .TtisthisBe taken toeach adjacent vertex which causes the{% charge renormalization noted " above. 2.Averyeasywaytolookatthings isthis: Zyand23arethenewelectron and photon propagator residues. Consider this way ofthinking about the renormalization ofavertex internal toafeynman graph: —--me [+no-|\~ —--- aete This symbolic figure represents the sum of 3x3x2x2 =36Feynman graphs. One as order 4,(thebarevertex), 6areoforder 4%,somoare%,andsomeareeven order %,!.Thereason forwriting this asthat what weessentially have is: os Te,theboxwhich is(++ bm)yields amultiplicative factor of2,7" (this igthedefinition ofZ,); eachofthelinepropagators yields itsJZ; sinceyoutakehut@cthectfutt sinceyouwrite2,=$&(Hana takeonefactorinr?) each direction, = Bythis method, weimmediately gettherepult of(8.56) onpage 169ofBD. You can then say that the fact that weadded graphs ofhigher order does not _ Change things because we are neglecting them anyway. This approach has the.advantage of applying to an arbitrary internal vertex. Notice thatwehavenot computed therenormalization toorderqd because, even though weincluded 36graphs, wedid not include such things as++Thus, ‘ourresult isonlygoodforthisorderQo: Ineffect, we"divided "byYEz foreach electron.propagator, because this isthe factor that goes off and attaches to the “external" blobs which Idid not show. Thus could just as well be external particles. 3.Important point: you mst maintain and hold your outoffs atsmall finite values while'you dorenormalization ineny order. Sofar the only countertern : Ihave encountered is the simple mase counterterm. *) *) — Renormalization ofthe electron propagator toarbitrary order. ‘ 6 1.Needagoodnotation, Iwillassumefromthestartthatweshiftedto 7” thephysical mass inlagrangian andaddcouhterterm Gm. Itwill turn outthat inanyorderofrenormalization, thequantity Smwillbeoforder%,butwe will compute itto higher and higher ofders of accuracy. Now, here are some definitions: 4- -— +s =A+diw “Ee =4a. +A= &) Be. a RL A OL iB =Se =BOELD) Bm =SG2)48m CO es =A RE. =St Obviously, Box(3) areallthird order proper graphs contributing totheelectron self-energy part. Box(1,2) are all graphs oflst and second order. Notice that the simple counterterm graph isincluded inthe Box(1) object —itisaproper graph. 2.Next, wecan define the fm renormalized electron prppagator. ‘Notice that you are always allowed toinadvertantly include graphs ofhigher order than you want without harming anything. Thus, wehave: —@- +—++ SH Notice that wenever deal with Bgg(3), but only Bgg(1,2/3)+ Thus wecan shorten the notation byusing instead: QQ -O-=— +bSi)= Qe=+Se+ . D+ —O-— =— +A+ EE +D-DD . ~. Eg, inBgg(3). (innew notation), wemst include Box(123) iteratéa 2and 3tinres sothat weare sure toinclude all terms ofthird order. Obviously weaccidentally include térms ofhigher order but that isOK. - : ~~Infact,sinceweareallowedtoaddsuchhighertenmstoourheartsconten wecan dothe following: * -Q =—4+ +h s+Ce fray Byfaking these things upinto infinite series, wehave made iteasier to sum them up. Thus we have: Er" Gi Or, immore familiar notation: \ D@\= or pom— BUR) Se 3.Now aslight digression, Consider anarbitrary self energy graph that is partofsomeBox(12...n). Exactlyonthemassshellthismustbesomenumber, [*) afunction of the mass ofthe electron. Expanding around this point, the lowest correction would beoftheform B(f-m), where wesetp2=m@inside coefficient B.Wecould include ahigher order correction (fm)? term, but easy toshow that this doesnt matter because we are interested inthe final propagator very close topam. : Thus, ingeneral wewill find that: . et =AD4BO) =SK) Adding things up wewill find then that: =Ae £0) Gorm =Ga) Bee Ba We AQ) =AG)FAQ)* A, Obviously A(n) isoforder n; and A(12...n) isoforder 1,but iscomputed to accuracy of order. ne | 4. Insert this lest result into the n-th order propagator: \o Der)tk (+ee») hers Bom =Alin)—Bare GE») ~a e-M abBee Aten), 26 Nowtheprescription isclear. Ifyouareworking toOrdernyyou!éhodse 6$m=-A{1,2,,,n) gnéthem2,+b+B(12...n)s 5.Look what happens. Sule) =—AQtor) =-AGt8s) SAG) =Baa) -AQ), Eachtimewegofromoneorder tothenexthigher order, wehavetoadjust Sim bythe amound A(n) which @s oforder n. Je, the adjustments get smaller and smaller. Buteachstageyoudohavetoresetthecounterterm massSw,. Similarly, each term youimprove theaccuracy of2,(ie,of14B) toanother order of n. 6.So, how doyou dothings tosecond order? First you compute Box(1); this tells you A(1) and B(1). But dm(1) =-A(1), soyou now know the first order mass shift. Your residue is 14B(1). Next, you examine the graphs which make upBox(2). You now know A(2) and 3B(2). Butthese areprecisely the corrections needed tocompute dm(2) from am(1), y and2,(2) from2,(1). Thusyouareallsettosecond order.le) Inprinciple youkeepgoingforeyer. Presumably thepartialsumsA(12,...n) andB(12s..1) converge! °Obviously youhave’to ‘hold’ 11your regulrtés andcutosfs fixed fortheentire process. Things like dm(n) wil]depend on.thse cutoffs. Only after the entire renorm process are you allowed tolet these cutoffs go to- inf. . . 7.So,wenowknow precisely howtocompute dm(n) and2,(n) toarbitrary order ne Let us move ons Zor) =Ao\-+ BE)(g-m) AQte r=AGV+ACY +eAlw). fo Say==AQaed | Dey=BOmm) = oe) p-™ Renormalization andtheGeneral Notion ofaCounterterm. CFLsMite tonowwohaveenoountored &masscounterterm only,Wefoundthatifyouconsidered: 4~ inQEDyoufound bytrivial power count that this thing diverges, ie, iscutoff-dependent. Thus weadded acounterterm —*— chosen just right tocancel theeuroff-dependent constant =—h.. Then we gotohigher order andgetgraphs like “@-. Weaddthemallupinsome .order, andthenwejustreadjust theoriginal counterterm Sm/PP toexactly *cancel. ‘Thus, toanygiven order wecanarrange .tohave afinite propagator; more accurately, wecanarrange forZ,tobefinite when#=m,Infactitappears +thatwaalways makeadjustments sothatfo,=0.I'mnotsure“whathappens to.Zoe . . .Now,ifyouwantyoucanlookatthevertex Devinthesameway. ‘You findthatthegraph }diverges, ie,isafunction ofcutoff; soyouintroduce acounterterm aftheformSEMFAY tocancel this divergence. Thus, onceyou haveincluded thiscounterterm, youcanignore thegraph >sinceyouhave cancelled it,andyouonlyneedworry about Jw. Ie,youhaveadjusted the “charge countertern” sothat Nye0,inentire analogy toadjusting ‘themass counterterm sothat 2=0. : Fe)-Agyougotohigherorder,youhavetoreadjust 8togotthisfecounterterm*tocancel against AMcomputed tooraér of.interest. Are there any oter counterterms meeded inQHD2 Potentially, each n-point function maynocd acountebterm, soyoushould check them. Lookat--UL .Had this loop integration been infinite, you would need acounterterm ofthe form \¥4- inyour lagrangian. Then QEDwould notbe"renormalizeable". But assimple powercount shows 4/6 yitlds convergent integral. Less obvious isthe fact that Delbruck F&Kdoesnotrequire acounterterm. Soitisreally the fact that, inQED, the only counterterms ever required, are ones which are already inthe lagrengian, that makes QED renormalizeable. Ie, toany order you can kill off divergences. ‘There are ofcourse subtleties like overlapping vertex insertions etc, but the basic point seems clear. 2.Nowconsider g"theories. Trysome#3, Doyouneed amass counterterm? —O- Yes. Doyouneedaecounterterm? G-= 4/6notHigher terms areevenbetter. Thus, inthis theory Idont think anything isinfinite except the mass stuff. What about #4?Mass counterterm? “Q- zach loopis4/4=logdivergent, so s you youwill need amass counterterm. There arenon=3graphs. Look atthis rom ne4graph:YX=4/4=logdiv. Soyoudoneedat8hf4counterterm. Maybeyou need acounterterm for n=6 ?Then theory isnonrenorm. The only graphs you might drawissomethinglikethis:it+Butthiscontainssubgraphswhichhavealready been"counter~termed out".Ie,thegrabh justdram iscancelled by onelikethisbewherethedotsindicatethesearethecounterters f4'-QJvertices, Thisleavestheregular graphw*=4/6whichisfiniteUV.Icantfindany$4graphs withna6thatFeqire couterterms. Graph mstbedivergent andnotcéntain reducible subgraphs. Bytheway,“contributes tonaseot-term. Howwemoveontof°,‘Rather thanlookatthelowercases,letsgohigher.Letne6andconsider XO =4/4=logdiv. Atonceweneedat#counterterm, butthisisonly#4theory! Alsothere isS-=4/260weneedaf? counterterm, This theory isterrible! Inthis theory, forany even number of external particles ,oroddforthatmatter, thereisagraphlikeBs:’y de, aninscribed regular polygon with nsides. You cannot isolate countered outfactors, soyouwill need acounterterm ofevery order! { Ingeneralthereisarulewhichsays,ahove¢+youwillneeeinfinite number ofcoutérterms, andfor$4andless theory isrenormalizeable. The notion ofagraph being "counter-termed out": inQED, eg, you dont ever compute thegraph WO ineescattering, evenifyouwant order ¢4This wouldhavebeengancelledbyamasscounterterm graph,exceptitisagaugen°) soskipthisexample. Youcanmbit { becmee leftvertex hasbeenadjuste toinclude this already. Ifyou like, this iscancelled byacharge counter-term graph:poneo 7 Ofcourse itisthe infinite piece which iscancelled out. Iguess some finite piecemightsometimes remain (Uehling potentiel etc.()ye TheMeaning ofaRenormalization Part. ~ 1.Recall (ALSection 18)that anyL,youlook atischaracterizga’by anumberoO §=dim(L) -4whereyouignore thedimensionality ofconstents ofcoupling. Recall that &=N,+(3/2)Np +Ny-4,where Nz=.number ofbosons atthis vertex, Nyisthenumber ofderivatives inthecoupling, ete. Here isapowerful fact thet iseasily proved (AL18.5): regardbess ofhow complicated agraph you have, even ifitinvolves several kinds ofinteraction lagrangians L;withtheir various §;, thenintwospecial cases wecan relate the superficial divergence (ie, the power count) Dofthe graph tosimply ‘the number ofexternal particles: . case1)“noderivatives case2)IfsomeL,doeshavederivs; &=0. No,Isaid that wrong. Theforma isthis: . d=n,8;-B=(3/2)B, +4 _Often ithappens thattheory has&=0forallcoyplings;, thiggorresponds toGimensionabes coupling constants; forexample, thisisthecaseinf4,QED,and gauge-ghost interactions. (n,=#vertices ofFaetype). 2.Sowhat?“Assume forthemomentthatSj=0.Thenyoucanse¢thatasyoumake 6 largerandlargerGreensfunctions, youimprove theconvergence ofanydiagram.Thus, if§=0,only asmall number ofGreens functions will have DO. Example: inQED only "renormalization parts" are ~@~ with D=2;(gauge invar reduces thistoD=2?seeBD);MD" withDetl (but=00byFurry); @ D=0, but gauge hakes D4; all higher photon parts have negative Datthe start; ~S vith De41; and4+with Deo. SoinQED, after gauge invariance isaccounted for, the only surviving renormalization parts are m®~)—®—-, —8-, def: "renormalization part" isaproper graph with D¥0. Ingeneral, these are the only graphs for which you will need counterterms inyour theory; these are the only n-point functions which diverge. Atoncewithout looking atanydiagrams, youknowthat¥inQEDdoesnot ~ require acounterterm; itisnot arenormalization part; D<O. 2ythe way, D<O does not promise you that atixgrank the diagrom isfinite; itcould have avertex insertion; but inthe end.such things are countertermed out, soDAO really does imply finite inthe end. &Example: Ing4theory(agaim§e0),~@- hasDe#2;-O+=Det]except=0.Then also M{hasD=0; inthis theory, youwill have only twocounterterms. 2.Suddanly nowafact becomes very obvious: anytheory which contains anuy with aBg &50 msthave aninfinite number ofrenormalization parts! . -Thereasonisthat,nomatterhowmanyexternal particles youhave(themore.?) the better since they tend toreduce D)byadding enough vertices oftype i with 650 youcanalways make afeynman diagram with DY,0. Infact there is nolimit astohow high you can make D, In_such atheory (onehaving aSre )every n-point function isa renormalization part and youneed counterterms ofevery order; (Iwonder °ifnon-polynomila interactions like e” could help here?). Example: °ormore; Isuddenly realize thet equation (3.5) ofColemans "notes isjustwhatIhavebeensaying here. Another example: °weakinteractions with yy kind ofstuff; here &=+2,youaredead. * 3.Digression: why doesnt the Wboson resuce weak interactions renorm,because / b=H(t, Tw" hasG=0,dimensionless coubing, eto.Ieit1loks good. The reason isthis: theW-boson iscoupled toanon-conserved current (in wh) particular ttheaxialpart), sotheW-boson doesnot’giveyouBE”foreachpropagator he-butrather E°,Thuswehavetoreadjust allthoseformulas. I'llmakenewonesWY"—nereforthiscaset .” " .6Ty=Aheme "bad!vacaboasQue. . e Quan B= Y- QE -ATp-Olw =\Q-ATy-Te, Woo = etTw)+te-Ttl se . - H=UTerx NEwshdp Wet =QTer\ry+dEe NU+y,~ale -Te . Neoweedaohort nid;=WadmgudmbermDesesett) onBurm byven =Eg+ate fun Mw=EyeXowa ach sep site. So,D=(ti~Es)+R(miw; Ew)Blnfr-Er)=Wy+Sri; em =Ap;[exam+34.4[sia—te(ayey a) . adi - e _— — —: EGatan Qa, Suk 3,(ordwe+Bh+d-4/ Oran DsSHS;—Sy28ZENA Ouch, now you see the problem. Amassive vector boson eneters just like a regular bosons buteach appearance inthese equations ismultiplied by2!In partioutar//¥e nowseethatthe“Y4rWlegraingian reallyhas€=+1;againyou are dead!/ Agood result toknow. 4.Conclusion: whenyouseeaninteraction lagrangian, figure out€. S90 you can throw it out the window because it will have an infinite number of renormalization parts and counterterms. : ahulyy. fe)-QpersSvenfolWateBBW) ee CeoOChadian uponqegh B=seanacementa 8a 8|WE cmt itchyp ee Faget femme ee ne eee ee MeeDeut -Alas creepT|eAnde tines Sraanons grntaaDannie qumpeyetn A=Baader ele UsTovesesoba gea B03, Sims Beth tamotody oe @MagRarer deYds [bs Awe neers | atat ©Frompdea 0)PEOMAE MOG Vee aa eA a8). RAS gen RSPg Arkod BOa 8) fA) Bok)snSotsSS ©DERaeda eC—DEO ey), Mee ——26O ome -—Soparggatn: d= K(Gaka'g—vegt)tg.&GO)we ne ~ Boapeama Qe a —-gask®shy Pornaaapasenan casedWe_sosidand. Fataxcansaght _ co oDPE aahgkoT cee OS Sean ae Naa Reontoadenoe oO i a GAs 10H)+a TigTap) —__ibG@) = —WansyoscontA wee eee ~oo Y S Rowe) d) OCn) | ee AD Qs oe a et) geAGud) Geet) geBQH) coe Oe ee gt A SRNFoe7-70)a co OD agOw)=aOO vo ee ns=|gEAPCs) ceoa sae Sto oe “Sinaigon edAVEyesgamaonsaachsOakeSLVSa, Dar yorrons ee SeeSS BGM SkDege@toeSP —-e-ONFORFouF_aeFE a coene i ~CoundoNewn WithokOOS eoeerg a a eeSSSa . \os.y oe Sit=ndgeoSe a _ —2- TG) 2b) Ate VigtGus Bee Be ——wees2g©,TEtana—Kom,andao_atleafne.Nakeivx Aodke SnSon l_So. p a Gt) = - ay Se Se mn (Ew) =TG) Cg) TE ARGS a9Tele we a A aS 9aeOT —DusenGratin WabuoodratOruaaYulalts paik > B= LEYSBaphaeeygpak Jourbaaused? Oo2 ©) a NG. GEnSTGGe)AT aps) SO Sinaia OonabaneTE=.“ ontQaeat gee ee Skea BG) Sod eh an eee ee © a“as ~~a- RectoOseCyaan:Kanade — —-——______. ——-|.AG) =artes] AeeiowrlrGkve\l shaw aR =gyosSeemoeMendpageCnn youtcoauscngnawedDune0 TA adegutagecnh Macy.FLusitlper. “Deck datequod Styaledyn? ae --et a DoodownadaleTWdomamepergngalsss @ vs ps taawe =Satcee AMD BD “Whee FA)SGT SG ane chosen aBSCcaPaaS ConcaOS a)wn ee a As bigOOS ED1G. RCSS — . steady sdonainengtnacdabePltoperyegadnn soona OFBoommains LeWem TY St anee ee — To eS ares iD(a)2SAg=i9 gee Te Se eee a aphestyseuss CT AETV WeetSe =[eteTOe SUS CRER eea se fists 40 a Sg Vek gx —Ueopd Te):vaisWoodyoncomdeenOakgpleossoyp cents MackSo -Stanviapodedound[oooeQDI,ey} oe soe og1 a or oSQa =BebeTG0]faaMeme.scl_—— —Rowshoto compat satanManey sla GN——Crvoaatont) de ghTH) pack DG) yo=e a Tg ©Quo?oenatonsoamtfoiling!Bascodsmoaaloce ena,prmgat| “ergaheHRSoh.aneye__iaWe)=o. LANsGerCENTaCaeolgave.~eeoehleee hg Se © in 0a") Cosee.an,#0), oe eee ee Rate Det me _—EGY ==Dag.casa 4=9a os B= FOTO a R= = Te) Orasendade, 7 yp geSAS TSteSegGe) % -2. a 6Vord,“Auwnabied doyeda’2 oe a YOw=2sgp.—ania. gh=0Co a.SG Sansa geeree2 SS da . TRE), =2,Yams Eoandgem. reneeeerTETinRENE STP? at7 _MinewghQus,sopuaradockName‘feposhTT(a’ea!Ww\=0, -_80DRyma)dhspreMaderAas|(some, SN)ae Lo. oo &snor Ae oo oo.QosomeLoon 2,omycame tAsluasicak nna iarandeptsndolRHR ade ee Toudlit, Uavanout ,woeVaz inn ~Ronacmsdizad TenagaGuerlain:shingecoseallEE eelee =Bho) fe The ge BD Ome,A)Suonlody \esallhf WSS awk =TO) CAoEETinfecbornidy, Oyatovalspaheg ietenmadGO”.poreanwwPETTI|peeverrsceien abosomelad___Ce\nouststoS\andMakooQarorBio,“MasOreng"ve Okdeayootamckmaadined youOita Kepmasssll._Serre eee ee ~~Ronaas aug:onto akDachabisedanoycanada,untas Takamlanadany“SanaprreationTC2)soma”GEO SaORUPOOTASaPTeaPOUND LRTTOSNOTEIRED Me ae eh ap ——2233 en wa en ee a a _3- —DemwolkmTe:222 ne 0.TRG)sT=Te) Tso Wyteat (Resto CnaJanchon to Ryo eSgeeMSl"RetwoLL. ne (2 weeee Noo sesnact: UpsO7R=HaeteemntOh=DeBase+hleissekel Stay gayeypaE Seis Oh EEE Soin Dest Ramssegtaced &val,@,oadWale)SLES, gystinguaSrsonra ——andSsoweyasJaansenate TSP,Ponde,ypoenigaadiockebeFRascadeearmYokedeldelSookcktheuilegael sepantin omg3 gu age = eeNyse VEER Gee egy Ge a 2 PET tie Gy] 2Ra&2(sb,0e,}ec) SBE REST) ar€,~woteoQua,eq,R=Kod,FeFe)aadSom.Now, CY2c.om inane steed lonYDOnctxOW,UAAginn =2. .62th inmoeeee-Menor 2=O(G\ ondtoeer(el 22 Qsdua4,cunmn.gowkKe an e Batecamnincel unajstsayMackVF ye(3 aengl.dskomy haonOymeeeHila ee Ce Qa,pada. Ae)=&dM).Te,Alt)arank4AsSeaded: © ensecLA cccng,SLowanatrc frsuplesrh svat 7 a= Band AQ)St Sy a0aayXeAOE)SL,yeeoe oralanrcreny | Uskpov,SEase aaa”ELsqwsttbetheask AGE)=psSO, S30ypeSancosceleds AuMQuir_posvhue asoByy - Ts =Qecome. taoDPE)ceaaakUyae ae Oe atta eS Yaa So_isgand.X= (4) 255$3Q)jee Wage <a =< OO Gran Beek Lawyee)=t . TAO =1SSO EB oudAets(enon)samnmcal She |_gipegpes SeBascosgichdy agedBhsitefis OC agted Nu Gade osaisWBSO=I TNebak 2XO) maceemaba —Tistogdh neBln) RamnbndugingSaaneaOg252208) who donee yondSan Rate Ute NRakedottyam “O’otetnaa dein.tenckWed"yoda aAunnslgaknpuke, SERED qecnde/beu SssijeaOuRe?eyfom” ——teQEDysdeMer. Se ee Gye eS ge SG éaS. ~Ae An ee “Reidy OC7tte OT TIS eyee ee Le .ae eh es a ‘aaafot. nent ; AeendsqMata OeSON SG) oiabn inthgebogiegeeponassge Yustonke SouegtUp RGATewahLR —SnSoashay Inst S000 2 - S a er——SuRROSE: .-- eT ge ge esO ene tsao . EET) [a ve+8 . Ee Oo NsMoin ekseven58 weSeeee Rd ==ECO a8 ~w- _@Nooensider NAGTFa. masses. Dak afo}oatabi,aerentersreaeeQosneLe oe—Ory ger ==CQeg- Pee, aesareet a—Apso a TataaDoannaunagonichgong,Psbees. Joudedak AWin havonollphepboevmnamnee adMa-snail, ohveaarn tha_anenasymm stan pt. ee [Wad sepOuck aha(autna) Qe,=Teale Sue stgyoweabe—pt,weuhAadQhok ats utwewthSandQok Ng aT CS PQDT TEE (RGayaGER Aya aE Nate 9k304geTF EG)SsSG)AGaah&Qh ChealingoalSTESYelSsREL arTaecnccockedaheapetubescege A Be =De WeepraneSg Diyo Bimal to ©ete ayyeTbareAene 7we LQ PY.a ey ee a a | ° AtalN ‘A.comment ontheelectron self-energy, vempartsh tof° fo} Thelowestorderexpression forthisobjectisstatedonpage162ovBDvl.Powercount,showsthatDal,de,thisthingislinearly divergent. Thus,evenafteryouperformonesubtraction, theonge-subtracted objectisstillLogarithavéallydivergent. Thatiswhy2,iscutott-dependent, ‘inQED.Details telow: geCanc!L£yorwt . aa 2 ee@— =1ACW NY) |tae . aa rweg o—D—o =bSeCe Fat,88) aree Dip > SECPyeNY)= \ -Bre 2(20Fw ON) eo) 2CR=Zor)Cm)LCE) ateQn(A) Won: BG)=Oputawoesakrm,iechinese Bm(N)soZwlooonanKR+0onyQaoA Qe: . “a ;SeCfew) =A = I-Zt) Ch) [=2/8] . Here youseethat Z,involves thefirst derivative oftheself-energy wrtmomentum, andthusZ,willcontaindos(eutert) “Togetheup,recallhowinBStheyhadtosubtract twicetogetafiniteobject. ‘asechewgageAwaeplck ~ example 2;inPtheory, powercountshowsthattheself-energy isonlylog-divergent.Thus,Zathiestheorytheonce-subtracted self-energy isfinite.Thus,in9your2,willbefinitetoanyorder,unlikeQED.Inthis”theory,the‘constantinyour expansionfortheself-energy islogdivergentandisthethingtobecancélied bythemasscountertera, soyoustill havedaninfinite’massrenorailization, In oe} GEDabove,théconstant, islinearly divergent: inGEDyouhaveaninfiniteelectronmassrenorm, andaninfinite charge renora. InPthaye'ss onlyafinite charge” renormalization inallyorder.. . - a\s\nr. Redo theelectron self-energy business, stating ellimportant facts. le) Tvillbeusingthecounterterm philosophy, s0define: wee eatry _.; @—=3LAE AY) +ACS,8NYBe)|=12, ie ee La + +ye oh Se (SykesRS) =i ee RCeUSS EcXCR SLES) Here Ihave separated theself energy into twocbyious parts. Alleffects ofthe acscounterterm areutintethe,secondpart,30,thefistpartLeindependent _ ofthemasscounterterm « “ : : :~~NovisgoComputethefiretpartoftheselfeneray;pavercount,showsit.ts Lingariydivergent,Del,socontains“(cutoft)+aalestere)+finiteterms.Compute @_ bethsbjecte through sonedefinite orderinral‘to-40‘maserenormalization, yousimplyadjust,theParenteransosages, Boe ae_ _ Zire AEyoNP)=Inaadydosh=—EE, NNSe) Baowaikontin, By=Treod)|ardQuamyoussnGckSm(A) joSwan me NK iw ordre. Thus, toanyorder youwould expect thetthemassshift,dmrequiredislinearly divergent in'the cutoff, sincetitstaeventrueinTove}Sider:Thus,you,might PP Ses Oh regent IISRUOMEII RAD6feGea ROSEMehPaAUR, Meresaywearedoligalinearmassrenormalization fortheelectron.Alinearlydiveygent, Mapsretoinilisation. ieShedoingthiebyailing«couterters toshelagrangix SEtho,saneforma0«termsstineady thera,Sooke,weore,Tonotalincabhe Tt‘thisregard. Rather,ait SOUR, OMpMeaehe gswlYay BAeance “"“few,"toourorderofinterest“Teounewehaveselected dm80propagator, hasphysigal mass.Haying selected du,wecannowéxanine Z,(seelastsheet)’ tre ‘a MeeVer Se Byala a(tpteRYH520(ebpatosNS38),uiache)voles. )CateBeg(N)da Sey(KY he Iamclaiming that thefirst part oftheelectron self-energy looks like this: 6 woe phel) Baktye tESB hy)=BSN) +ews)Zi(sors)Chee >A Kh >AN ote Ie,Iamclaiming thatthelinear termhasacoefficient which deesnetdepende ommomentum q.Toseewhythis always happens (Tthink) consider thelowest order loop integral. Togetthelinear divergent piece, youtake allk'slarge and _yougetanintegral which doesnotdepend onq,sothere. Thus, yourselected dm willgetridofthemassshift forpnearmaswellasright atm.Thisisall tosaythat youtanremove theleading divergence Just bysubtracting. Now, ifyoulook atthelowest graphs involving dm,youcanconclude that thesecond part oftheself-energy also canhave itslinearly divergent part removed byasimple subtraction. Therefore, thederivative willhaveonlylog(cutoff). [ASeered markings]. “Soyoucanconclude that inQED, theobject 2,isreally logdivergent, so youhavelogarithmically divergent charge renormalization toanyorder. Te,you fo)wouldputinacounterters lookinglikethechargeinteraction, andthisgérerated amultiplicative renorm which canthenremove thislogdivergent charge. Sumaarize: inQED,theelectron massrenormalization needs alinearly divergent counterterm, theelectron charge renormalization needs alogdivergent counterterm. -4- Now show how multiplicative renormalization removes cutoff dependence. SQ@\= 20)=<6)+Zon)+Fonite-B®)~~ 26) ‘$-m- 2 =o Joy) a Qawrdde ox ;E@= Gw|zon +| 5 NffctepondOG)=0 BarRepaka pul3. So S@y= 1 Gr) [\-Zod -CGD] fom 4 Waledusk—K(ww) =Los]=Za. Newa0cmlaamonins yeeo)tn “\Sey =2S =Legis__(2). 6 L=2)= Ol\Fo Nowcomes thetrick: everything iscomputed tosomeorder ine,”. Eventhough the cutoff issone large parameter; westill have anexpantion inthesmall charge. Thus,making eerors onlyinthenéxthighest order, we'can shiftthenumerator into“denominator teget:_ -_lest”), o Se)=Vo iceGa) L-Lzeezoyl-cqy] Go) \-e@ ~~Tee ON re. ey 8silenced ok| Ofcourse aslong asthecutoff issome finite value, C(p) will depende onthe cutoff. Butthepointiswecannowtakethecutoff toinfinite andc(p)dsstill finiteandwehaveleftarenormalized propagator whichissomefinitefunction ofP.Ofcourseyouare’nowsupposed toconsider ifafunction of¢andnoteeinaccordance withthemultip renora program. Ineffect, hereiswhatwehavedone: Oo . See)=tt —_______—g-m- EG)~G9)&| .Tw sulhache) Dew,Bm)abaady =o. de Repeat thelast fewpages nowforthephoton propagator. ©) thecosentiat diterence nowisthespinology. Tfyoucutoff inagougeinvariant manner, theself-energy will have theusual gauge invariant form (covariant gauge) andyou just dont get anymass renormalization! Well Ibetter saythis more correctly: there could bemassrenorm ifPIhasapoleat'q’=0. ButinQEDfromunitarity youknowthisisnotthecasebecause, thefirstsingularity inPIisthetwo-electron normal threshold. NotethatinQCDthetwo-gluon threshold falls downtotheorigin (Se,thecontribution toPIfréxi'tio bare-gluons withzerobare-nase); this‘inQOD youcantsayaptlori thatPIdoesnt have poleattheorigin orsomeothersingul- arity, 4oinQODthere maybemassrencraalization. (Maybe toinfinite mass) Notice theasyumetry between theréles of2ssid-WT". Botharescaler functions; bothareineffect log(cutoff) diver'getit, oncedmisadjustedin2toKilloffthe linear divergent term.But,werequire A(m)-0ascdndition thatelectron mass doesnt shiftawayfromitsphysical value, andZ,isrelated to2'(n). Dactoits ditterent spinology, forthephoton thecondition fornomass shift awayfrom 0“isthat@T(q")=0asq~0,46nopoleinPI.And2,iscotmectea tnWP(0),andnothing ofinterest isrelated tonmo). 6 awGn=ily=4Chgy-ev)TT) AEdankcone. D2 D=O. odD~o =Deo De(Po=—Bei wsB=HT)©Beg.dint.eTVET) ~\ VesBOG =-se|EQ]#3las— eLema ELA Asbefore, toorder n,you seethat the multiplicative renorm is subye, equivalent tosimply subtracting PI(o)downstairs. Onecanthentakethecutoff to| 6 infinite andhaveafiniterenormalized propagator forthephoton. = Oy. CeTa)=Bow[GRAS] =ooteLekfae CA)frake Aro]nq loarOr tee OO, _ 3, ekTg te eepean cee OFQc=3paos=(t= 4—_§— es eT .osenaaoanwashwage OmTy a a ag Uae~- aye _ Siajoseen peSeat. ©Dhowwate DsATBAait gummyQe —_ewatr atr|=Te Sa es ee ee Be aes =(QCSYTSTAT SB.=[ATE STelt 3 82S =IMTAHT= ©) SSAQer)=@)- ar ae ns eee——&Abewds G-ehem) A= leQem we eee! _ ¢ OO MnaGaanapd gahunacnslAanannnalgg ye of w= OE aee 0A| loo = Qs ReBo a Cee a ee ee a ;eee ore oe(e) .vweneeacey“aaa SY). —Ciagech:geaesualYaee P= Gav—gut)ekgngy re Se ed alaslr, _—__2.__D= \ weet oe _Sl f-@S55S. ve : apelin coa — ee a ©AaDyeigem Dankoattenwoes testele sank: TS” Te ERIE we bn Qo[SsWeSal+ Be Bephgeetala BaDatatypes QosFromeADy Mav ©WeksankTOsingsalepeckosinwaliMeahaka a ee_AdDynwe=~So=raGD) -one —Naoonsdseat2)oe) WlomndoMehQed .a)_GoeRepea— eGp=he EdaResaomentaeHebd ioyannce ce ANAcnn eeROW OPUe ee OLMegadade.TT,Tatogee BDChapt 19 Relativistic Quantum Fields Bjorken &Drell 1965 Chapt 19:Renormalization o13.1Babeductin SingssatinOuchpistusection pansionsouseeWiSuite Siz ea =Fn a Ne) =ene Se yaeS KS Man sugeedcieshowindhecbae 1.3erga Sgualeonas, |Moestintinn=tomksoadiadQsKo 18.4Shascton ChaleSrgansiane ClaviinsDeaookaySecasyouseelsesseslapaing BOS aagonsdete dowast9skalatanqangle setpaaatioe in0184 -opanara SchinTWQhepaBSKU] Jasasoranbece. Ure SEPgwe7A —nMsintthe aanea a TawOdgetel Vaan Watannneaude}Ts“nlyarmse waa 2hener, pondcon. SMOMsouse peesantaalsa. loge 2Onsioner aah.Aragrooniths psavaShated ee Yost,WandShtasrasdhopasaDlacka=o-. 1Wokacaa3=ifheaeAo ©IG Anosondpafanstag ee,Muayangh -——-——______--—. gtSAG) a a: a fan 2G) — Qanoua, siaieeeSe ZaGo)pufmaZnsehenOkayaa—NeckACen,Bom=OsetesodcpnchtaiSaomygusteandsa...So,Resersaa. 0DScmsabertenan, Wackord (=BaY Fa)dodosswaorr DeeBee2 & Sag Meme. -2A(gem hazelee BRL Se De Bae SB.STOsac\. gM) =o. es ee a ianne AR |AER Wed eo wastenuevas teSB Week© ORG asites =EEKGS,BSRan)SReDEP———earngpinad ong ne appnts DeraVedonQeVimcartin opogincrh armas” amyootee,So~~Seog Dasach KiaSShwe aomiesSlategpngion+sweaton”OF,Se,Pe}__Wecen —amgcomremaatben qoesoldes SS FGM GpasRO ——~88.Sammon, wea OFBipmnacaver Comend foi mee enn cae beeSe GB) SatiagpeKRe 2 a TS) oe —Miensentonetotinalnamecyoor aehakion43Deeg2rod88.OakDae Desh sedoreDecentonanMestre Deanie Gt osAAG “Costncetasin teBalesmundila Qannnanaligaleie Ope eeanenate SoeMoeWicksalleSo:Sadiaaas BSswordol 2ageo ee cn Sdn Set hes - 2 % “etSe -MkWkSitecolnadcovobtepoeiida OaERgaslidngn a —1aa0 VecaaA Qaeengenc Ea ASwwergeece EaannAahwnbhia, aguglenga, ReasuggeSicSankayee - Die compued JpnandQEXnsmowe paihondovrmediagaosisnn with =PRAadatagenaMaas.Gneaigiing, “D1©Ciena ils) — _ Msi ee ee coe - .aa m e —Weoaronan.dkAon.page.B22,_.-The_rest_ ofthissection isabitharder thanthefirstpart,butatlastI —— —-think.Iunderstand whatis_going_on here, LookattheFeynman graph infigure .194324 Ifyouletallmomenta scaleout,together._(_yom go_ontin.somearbitrary _____directiow inthemomentum hyperspace), youwillfindthattheconvergence ofthe. -..graphreally isthesuperficial Dyoutrivially compute, hereNe-2,BUR,suppose.you _hold_4-momentun 1,fixedandsmallendlookinlargelyregion. Thenyou __. —lose.convergence. powerspresent inthe1,loopandyoufindinthasregion om_____“tube!_of thebyperspace. thatD0.Je,thissuperficially convergent graphhas — .adivergent_subgraph. ne For this graph all the interesting tubes are shown infigure 19.35. Bach tube. _ ..is_justa_region wherecertain integratiom momenta areheldfixed.a donly the ———-test_(snlid tines)ofthegraphis_cansidered. Youcan_see thatthevertex subgraph --.anditstubearetheonlydivergence problems thisgraphhas. _____-___lietinbergs_theorem_says theobvious: a_graph istrulyconvergent ifit..._ —--converges superficially andalsoinallthespecial tubes, ie,allsubgraphs _______st_converge._Thus,- figure19.32flunkstheteste.—— eo. ——-liow_look_back_at_page_311. Notice theZ,sitting-inthevertexequation, —-..This divergent thingwillbecancelled bya_subtractionmadeinthe.secondterm- om _the_right ofthisequation (d),Ie,eg,whenyouseea simple vertex of.lowest -—.-order.{hawing Da0),.yousubtract to.get.aresultant. Deel.Similarly.the_Z, which. .appears. intheVacPoldyson equation (f)reminds youthatsince D=0for. .._ vacpoll PI,you.also havetomakeasubtraction there. ee Now, I-gather thatyouhandle vertices andPIvacpol subtractions inthesame _ Way,so-they_omly mention thevertex, LookagainatFigure 19.32. Hehave earned thatalthough D2overall, effectively D=0because. of.thatvertex. ___-..-insertion, To.fixthisupweshonld takethelittle insersion andsubtract_vithin —___ the. graph, This "fixes up".the_D=0_tube_in figure35-and_changesthat_tube_to___ ~—essentially-a_constant-with-respect. to-all-other--loop-momenta(like-1,),~you—.— ~ - may worsen the-divergence in-other tubes. To-see-the effect ofasubtraction you --—-. -drawa-dotted- Linearoundthe-vertex_insersion-and-shrink-it- toapointandsee— ~~-—-what- youget.-Examplet- Fig33A-hes-D=-3—in-the-tube-shown,-butinFig348-the-- subtraction cmses-anew situation vhere-D—2y-tey eseconrergent-(out-atinr—- a): 338) BoyouansIePtwithDe-I.Fig33MisstilltherewithDe-3,batthesutraction ‘addsnewsituation 34AwithDo-2. Thissubtraction doesrepair thegraph of19.32. "O)__ taeyou500otubemarked*whichbaoDen3sButywhenyouputSpavortox_subtractiontorepairthevertexinsertiontubeandloweritfromD=0toDel, ____you accidentally bringup.the*tubefromDe-3toD=0asshownin37B.This / —___of.8_subtraction within anoverall vertex insertions 0-1+Asimilar problem 4sencountered_yith figure293%38.Seepage329.:The—~ --—_ttibe_(b).hasDxo,Ifyoutrytofixthisbyavertexsubractionyyou generate._————... ..ph(@)which‘causes another /De0-problem =ce . ———--— -Tits,zou.see:thatwhen_you: haveasubtractiom majéwithin either a —--—._ .Vertex_insertion orwithin aselferlergy insertion, youhaveaspéoial problem, __24think.this: sameproblemcarries.the.name: overlapping subgraphs. ——-. -Tis_sectiom merely drawsattention tothese-two problems, butdoestot .{tellhow-to resolve them2g a -+.___By_the way,Ithinktheideaofshrinking downtoapointofapieceofagraph ——_+—™stbethesameasB+Sso-called generalizéd vertex @and.their operation Re__ —___—Thein Reoperation: mustgo_inandcheckeachsubgraph andsubtract whereneeded.=___.But theytoomusthavetheoverlap problem ,leter-solved Ithink‘by,Zimmermann. __.._B9s]1 Proof thatRenormalization Theory isfinite, Inearlier sections wehave_ _. ___Hoted that after the miltiplicative renormalizatiom welave awell-defined _ __iterative shoeme forcomputing things inpowers pfnot_e, butphysical e.In. ___. Volumte 1authors showedhowyougetfiniteresults toorder-e's. Todothisthey had to.doone vertex subtractior, andprobably onePI.subtraction, _ - —__....... The gist ofthis section isthefollowing inductign proof, Suppose youassume . __that thetheory is"finite" toorderesWhatthismeansisyouassume that forauygraph ofordern-2,thatgraphisconvergent, ie,hasnobadtubeswith_ -. D20+Suppose thisisassumed, Thenyousomehow constret graphs oforder:n,and ~ wou_ask: intheprocess ofadding tvoextra pdWers of©,dowesomehow create __ .-——_divergent subgraphs somewhere? Ifnot,thenn-2implies. and you-have‘an-| -—induxtign_proof_that 40eyorder.n’ allfoyriman grabhsarefinite, ‘ie,haveno_ ~—- -—Givergent—tubes.—.- ——..._. --Hinat,_and simplest, consider thekernelK.Pickoneoftheskeletom graphs-—-.. iu_the_skeletion expansion forKa(mst_he order lessthannsince wearenow__ =~ -bnilding a_graph of-ordern).Nowtake-thisskeleton andputthemeatom,ie,aewer-on_full_vertices andpropagators. Claim: since: theskelton isatleast order e, -——~—ne piece ofmeatcanhe of.order larger thamn-2.Eg,.no vertex youmight addcan—- be-more_than-order n=2inthatvertexes a _.—.!.Nowsitbackandexamine-this Ksohtibutiom youhaveconstructed. Bg,if|ieéembwaideee: ie arene =~ Kh - ene == 7 - "_____ EeveinfacttheoilymeatIputonisatonevertex. Question doesthegwaph *,80comgtgnotéd ‘haveanydivergent subgraphs, ie,badtubes?Firstofall,there ___. arenoproblems withim anyofthe-meat-pieces because byhypothesis theseare "finite", sincé they mst ‘beorder n-2 or.lesss In‘the abave example, the + __vertex has been subtracted woheeBeles Soe __ ._. Them.you mstconsider graphs whichLieoutside themeatinsertions, But it _Asolaimed 911suchsubgrahs Kavenegaitve Djtheonlyproblem Dgraphsinvolve ___ ___. .Vertex orse1fenergy insertions, and there areno such thinks intheskeYeton __ ___.expansion forKorforanyhigherGreensfunction. (SAFER TY --2 Thatconcludes theinductiol proofasfarasn-point functions withn.GB.4__+areconcerned, Next,wehavetoanayze thevertex. Atermoforderninthe ~-__Wertex isgenerated from1-2meatypartsViathevertex integral equation, Again ______the meatyinsertions themselves causenotrouble byhypothesis, soyouneedeo .omly_consider subgraphs -which involve thelinking fermion. linesasshowntop_ _____9fpage-332eCouldtheressomehowbeaubintegration involvingtheselines _____ tiawillbe2badtube?By.exhausing all,potentially dangerous possiblities,____authors whowthattherearenot.TherearesomeD=0graphs, butthesewill_____..__beremoved bytheoverall vertex‘subtraction inthisorder,eg,graphsoffigure41BwillbeDaprigir tooverall subtraction. oe __.. .+Finally youhavetodothevadpolselfenergycase.‘Thisistherealmess‘because nowyoucanhaveoverlapping stuff. Ahugeamount ofworkisneeded _. ite: dothe iteration proof here. Ihavenot evenreadit.Intheendtheyfind____ ______ sthatPI isOK(after overall subtraction ofcourse), andergoDpisfinite. __ _....Similarly sincevertex["wesfinitebytheabove,SpistomviaWands __ . Avid thatistheendofthisinduction proof ofthefiniteness mixkke QED _ computed toashigh-an-order-as-you want. ~---©== ‘WatGal teCong,BHaldaatiafin)Brae—ms OED +OS Ge OO Bicone (epee)=Rien RigoaE eR SESRSRg po Se Bess eseep=ate] Sp er_sessdodfoisload Nok. Lnaduekin -aee iapgSoe GaeaLOMNO Notes onsection 19.10 April 13,Friday 1979 fo) IammakirigthesenotesbecauseIthinkthatfinally,afterallthesefears,Iunderstand what isgoing down'in this section ofBB. var) 7 Thefirst 5pages ofthe séction isoldhatbynow: identifying the renormalization ” parts ofthetheory ands6’on. Inprevious sections therenormalized integral equations have been setupandtheauthors have finally comé toface theproblem ofdefining aconvergent integral. This séction isthén good because itgives examples. Thefirst example isthegraph shown inp322. Onpage 323you‘see the possible subintegrations. Each oneisassociated witha"tube" andIfinally knowwhatthat : means, Iknow howtocount thepower ofanysubgraph. 4 Frommyownnotes onf,Ifinally understand whythesymbol {'fortherenormed vertex isactually thegraph minus thesubtraction atthesymetric point. Thus, I finally understand theorigin oftheequation onpage323.Since wearedealing with renormalized integral equations involving objects likeT1,eachvertex isreslly a vertex graph minus asubtraction, the subtraction being adivergent constant. Thus, you have toconsider inyour analysis ofdivergent subintegrations those which arise from the subtraction itself. This isthe idea ofthe dotted box, see page 32) figuresx. Infigure athedotted boxisimagined asapoint (since just aconstant) andyouthen fe) powercounttherestofthegraphasshown.Figurebjustsetsasidetheintegrationwhich isthat constant. This figure bwill then cancel subintegration (b) onthe preceding page sothat the resultant object isfinite.OS Although Ihave not gotten that far inmyanalysis, itseems that the rule for self energy isalso asimple subtraction, but egofthe scalar function PIfor photon self energy. Thus you should also worry about dotted boxes around self energy insertions. Again, the rule asdiscussed onpage 324 istoshrink these away and pretend they arenet there. Next example isthemore complicated graph onpage 326. Iamnot now worried about the disjoint cases. Basically there are three subgraphs toworry about, page 326 which Icall there 1,2,3. Graphs 1and 2are cancelled bythe subtraction graphs, one ofwhich isshown onpage 327 graph (c). One then worries about the new graphs present when the innermost vertex isconsidered asits subtraction constent, ie, dotted box onthat innner vertex. Only (b) and (c) onpage 327 interest me. Graph (6) cancels graph (a)onthepreceding page. Graph (b)ismore interesting: when youshrink the inner vertex toapoint, you then expose the other ,outer vertex divergence! Depending onwhat you are doing, this might becancelled byadotted box around the outer vertex, [@) butthatreallydepends onwhatequation youaredealingwith,ie,does*appear,or what. Thefinal exemple isthepage329analysis of4thorderphoton selfenergy. Somehow this appears in:auph a.way that each side 4ssupposed tobesubtracted.Thengraphsa,b,cshowatoncethedivergenct subintegrations. Thenqandgshowe results from putting in:separately asubtraction constant oneach side, liker my -+.lntimes born.loopin&notes. +Jhen ¢shows theactual subtraction integral which cancels b,£,shows’a similar thing which cancels c. 2 . Somygh: for the section. Now wegoontosee,how weare supposed tocomé up with such objects. . : . : Showing thatthevertex iterates upOK(partofsection 19.11) ie)Considertheusualintegralequationforthevertexwhichyouusetoiterateup topower nfrom power n-2, Youwant toshow that your order nvertex isfinite, after its single subtraction overell isperformed. Howcould itnotbefinite? Theoverall BaxD-0will obviously be“tamed” by theoverall subtraction, sotheonlyproblem canbedivergent subintegrations of somesort.Luckily andconveniently, duetotheiterative method, weknowthat ; thevertex andthekernel themselves cannot contain divergent subintegrations (Logs) | because byinductive assumption theyarefinite completely. / Therefore theonlypossible problemematic subintegration must"runthoughé" the connecting electron propagators assuggested byfigure (a)page332.First, consdier howtheline runs through thevertex ontheleft. Itseems clear tomethat theworst caseontheleftisthecaseofjustapoint vertex orasubtraction boxlikethat shown in(b).InsuchcaseyougetD-0contribution fromtheleft. Anyother “path” through somevertex graph ontheleftmakes negativ.e D,liketheD=-2Ishowin picture onthe bottom ogthe page. Solets take worst case onthe left. Similarly, Ithink Icanargue thattheworst cased paththroughXontheright fe)makes D--2 (nypictee intheleftmargin shows acaseD=-3). Somehow there isalittle theorem here that asubintegration path through some object cannot beworse inpower countthantheobject's cannonical D.Ibelieve it. Sofigure (b) shows some problem graphs, ie, these are eubintegrations which amount toD=0. However, there isanother theorem which Ihaveyettoconsdier which \claims this: ifallyour external momenta "flow through! asubgraph (see footnote on page329),ie4fyourexternal monenta areallattached tothesubgraph, thenthe divergence (ifany)ofthatsubgraph willbelowerwed bytheoverall subtraction. Thus, thegraphs offigure (b)page332fallintothiscatagory andarethus _reduced fromDOtoD=-1bytheoverall subtraction, 7 Thesameapplies toanysubgraph which involves allthelines inside someK.(Last sentence page332). So,theonlyremaining problems arepossible subgraphs contained within&whichdonotconnect outtotherightasshownegonpage333.Thesubgraph is supposed tobecontained entirely within boxAandisthus "insulated" fromtheexternal Lines ontheright, andisthuspotentially dangerous since aDOsoarising might not betamable byoverall subtraction, according tothejustmentioned theorem. ~ How@ver, thepossiblities inthisclass areruled outonebyone.Thelastcase CO) isshowninfigure page333bottom. ButitisDelbruck andgetsineffect anextra-y "rescued byfour very welcome powers". _‘Thus, subintegrations which gothrough the electron lines are all OKand the overall subtraction removes all divergences, and iteration works for the vertex!!! ! Showing thatthephotonself-energy iterates upOK.(partofsection 12.11) O ‘Thissection begins onpage334.Thecrucial starting stepistogetridof ‘The Dyson equation which isasymmetrical and has that bare vertex at,one end, and toreplace itwiththeequation (19.67) which issketched onpage 335byme. Whyisthis step soimportant? Intheasymetrical equation, youknew that subdivergences attheright sidewereremoved, butthatthere still remained buried inside subintegrations arising fromlines ontheright vertex moving overtothe left vertex. Ie,these subintegrations were unexposed. Thehope wasthat somehow S theyexactly cancelled withthe.logsintheoverall 2,(or25)factor sitting out front. The advantage of.writing the self-energy inthe symmetric form isevident: nowsubtractions have been performed onboth sides, sonothing hasbeen left out. The only complicating isthat you have overcounted your feynman graph, which iswhy you want tocorrect with thesecond term. Inthis second term also, allright andleft subintegrations have been automitically accounted for, itturns out. "Of course, withthisintuition, youhavetogoandreally showthatthere are nodivergent subintegrations.The trick for doing this istoexpand each full vertex inakernel-expansion, asdiscussedonpage337.Thenyouregroupthetermsinyour (oe)self-energy bythe number ofkernels involved, asshown in(19.77).. Soconsider thegeneral term which involves nkernels. Some ofthekerneles areburied, sotospeak, intheA's, andoneoftheK'sisshown explicitly. Youknow ofcourse that this entire object hasoverall D-Owhich isremoved bytheoverall subtrac tion, butthequestion isthis: arethere anydivergent subintegrations ofthetype that cannot beremoved byoverall subtraction? i Any subintegrations that cause trouble must interlink more than one K,because you know byinduction hypothesis that any particular Khas nodivergent subintegrations within it,toorder n-2. Thegeneral n-kernel term isshown inthepicture onpage339. Theprocedure istoexamine allpossible gubintegrations andshow they arenotdangerous. First you look for ones involving only the pair ofelectron lines called 11, ie, you look foradivergent subintegration which iscontained inthefirst twokernels. Tou quickly deduce there isnone. Then through tortous consideration you finally conclude that nosubset ofthekernel groupings, say1,.++..1, contains adivergent subinteg- ration. Asusual, you are free toignore disjoint, subgraphs. Then finally onpage 3k you are done with this proof: the photon self-energy does iterate without introducing logs.Thus,theuncalcelled subdivergences attheleftvertexinthe ©) asyunetric Dyson equation mist have exactly cancelled the divergences inthat overall 2,factor. Toprove this directly would bevery messy! This xsmyfirst approach which led to amess. M,Baker &chooskyu Lee 1977 a pee 2 feed 9ARS 3 y.PHEYSICAL REVIEW D VOLUME 15,NUMBER& 1SAPRIL1977 a , J Overlapping-divergence-free skeletonexpansion i| Ainnon-Abelian gauge theories*. i4 Ar M.BakerandChoonikyuLee ..alla ‘ amanPsttencyWaningsa,Waser85” We (Reesned2November1978) er . ieie Sc tet aprerlaion. oftheSenwingerDyson ensions. ofgunig j . Tisshenre ponethai.ed9ampleprecedeforelingnomads Wesseen eu wigrin anolseesaSarinpsteerie 14 PetusttionethsnawshowtateYangMiswerenerfucenane Fi7§ Sere tnetionwevielStnsBytheWard-Taahah etyaterof a ; , 1vtRoDUCTION ghostseti-energy.‘Thenbyfurtherexploitingthe,HlFBoostoekanasht identity,weeandetermineboth ieee‘Theproblemofoverlappingdivergenceshasthavectorself-energyandthevectorthree-point iyroatiycomplicatedthetheoryofrenormalization, fonctionsntermoftheghostself-energy,the thae:‘eveninthesimplestcaseofquantumelectrody~ ghost-ghost-vector vertex,andthevectorfour= Hii:3namics(QED).InQED,therearethreebasicver-pointfunction,Additionalsimplifications occur vili |eaten electronself-energyfonctionsandvertexfunctionssupewicifiniteandtheghost i aneeaaiGhotonvertex.‘Thefirsttwofunctions self-energydoesnothave289‘overlappingdiver~ i,Head tneelectronplatoTrgances. Forthecaseofthegences.Thioccurspertuse,inhsCaves Wea electronself-energy, onemayavoidtheproblem gauge,twomomentumfactorsfromthefirstand (ieeebyinvokingtheWard-Takahashi identiy,butoneIstverticesontheghostlinearereallythoseofat?cullneedsadetailedanalysisforthephotonselftheexternalmomentaofeachsubgraph. Aaaenergy? “Theskeletonexpansionsobtdinedinthispapex— NaeRe)Denpite ever-increasing interest innon-Abelianase gaugetheoriesforunilyingelectromagnetic, weak, ibipedwtstrong interactions? theskeleton expansion aite 2 rn veMillstheoryhasnotbeeninvesST {; Wlse:tigatedindetailbecauseoftheenormous complex- ifase dl i H.iHl|~ityofthistheorycomparedtoQED.InpureYang-: boopky ey i4 Mills(YM)theory,therearefivebasicvertex i PoEy mt ise4functions(inthecovariantgauge)whicharesuper- i iiH H HiiHolallydivergent:thevectorself-energy(quad- LCook | i Hy3ratically divergent),theghostself-energy(linearly oad TA Strerent),theveetorthree-pointfunction(linea? Ooccanenneeh USlydivergent),thevectorfour-pointfunction wo th ilakeGogarithmically divergent),andtheghost-chost- . aycoevertexfunction(logarithmically divergent). pesos MieNitottheaboveexcepttheghost-ghost-vector aeen pony tong Ca eeeeeeemnlicaied overlapping= dy : irca eScxgenceproblems,Sometypicalexamplesare i :Ta Ung§ShowninFig.1.Bytakingappropriate derivatives H iofbothgenet ahsrithrespecttobxternalmomenta,wehaveSvC~ H bp bebe eyFe‘ceeded insystematically decoupling theoverlapping iieee Lesseeeeeed iy‘ divergences imthevectorfour-pointfunction,the te) « eyBa ootos three-point function, andtheghost self-en~HAND = trey.Forthevectorself-energy, weusethe ra:1.Sometylenol ineFerre WaneWard-Takahadhi identity*toobtainthevector FeeTAseanthedottedlinessepeenet iitdWard-Takalath Kyiromthevectortheee-plst theFaderFoorghostnes,Thetebotsee 2)vertex,inenost-gostFector vertex,andthetitevariousdivergentsubintegeations Faa 13mT ess eesyaryta eee aw) — Rowrshee _ oe a .|“Quen Eig.omaieofNAGTithsomoresbappinng dtuseguuase’ —__. fo)PROaro_(ny..4oe — as wee. a. be eee eee ee — -BeQahe asada koe fom QED sear,TodayisFeb3;1979.Iamrereading “thisection forthesecondtine,,Hore ig-vhatisgoingon.First,recallfronBjorkénDrelihowyourenronalized QED rakinguseofthevariousintegralequations.Thatisthe“conventional”program __wenaafancieralternative iediscussed thet,avoidsoverlapping divergences, _______ sahaxdoesitwork?YouwanttobeabletoconbutethebasicobjectsPL, _SIGMA, andGAMMA,thethreeinteresting lown-point functions. Thefirststepis _____ta derivefigure9whichshowsthefirstderivative ofPIintermsoftheother — basicfunctions plusthekernel K(which isnotarenormalization part,andwhich ___haga_skeletion expansion), Theadvantege offigure Qisthat therearenooverlaps _ _.. «.-.on_the RHS,although theRHSignowmorecomplicated thattheusualRHS offigure 2. .=...So,do thesameforthederivative ofSIGMA, seefigure 11.Again, there are__ .—no_overlaps._Notiwe thatthisfigure involves another n=4kernel, onewithtwo| two.photona..Now,insertskeletonexpansionsforthetwokings Oe inteatnosvanmoanathatR=K(D,8,6H04), 30,functionals ofthe wo- ‘Thus, you-now_have equations fofPI'andSIGMA' which donothaveoverlaps. __ —_--—— Their-overall_integrations havesimple logdivergences soyoudoasingle subtrac tion. —_______Now weare_reazdyh torengrmalize thetheory intheusualmultiplicative manner. ——_—_—_-Define -Z1,22,23-in-the usualway_and putabaroverthenewly defined functions. _ —-- Notethatthe scalar function PIcanbewritten interms ofasecond derivative ~-___ef PIas-in-(20),. andyou.can_get. thesecond derivative asin(15)bydifferentiating _ _-—e-single-oubtraction-nneeded inPI.Thua,definePIp_hy subtraction offPu), | —_ "an-arbitrary-renorm-point. Notice from(21)and(19)that29isdefined temake. __ ~~ the-photon-propagator-have- unitnumerator_and thepoint..q@su%, Notice theclaim ———--— 4n-{23)-thet-PH(0)-can-only_be-constant.Y timesgyy«Then(2h)givesPlpin terms ——-— —of-our-PE++_. The-peint-of-all- this—wasto-show howyou_get PI,fromPI)''«_ ——-~—-Now-the-pair ofequations-{22}-and_(2i,) telyou-howto_get B(orderntl) _ ~--if-you-know-eli-the other-funetdons—to—order a =-~ --+“Next;~the-argument-s-repeated_forthefunction5_=electronpropagator. And O— arsotorkOe—S0-you-ened-up-withy away_to-got-fron-orde>-n-to.osder-ne1using— -—+~the-renormed-integrai- equations;—the-only-divergenoes-are-the-usual overall ones... ——~" WhichCanbedumped bySimple single suvtractions. Nowitsoteype Dh Raa Weonesora sonitechoatsfudheCe,“puneYneTa® =-@--:__ oeFQ a ge eS ge OD coee . Teleco nenaan ooaeQamagOgithgoontoJS = gBSE, &Gg Gh)a Besiwons DEE ea —SaesRG =GSTGSATESSIS. SE,TOY _ee Bae Ee elem, nse2 RinaavatsMeSeemiheaeosnesedges ASeis qoute Aslate adn we + ++Sp Sg Seey etoeeeeeHe -| Sern ee _—oo bgRossa ~ Shans yomnaat-suggened tetadogoniaslabinarOnOhFe2ubade neoceeoasedep parlilsanes canoagare Standen deO60. FE GedaDrain,isogOaaain,Macebaosabehbeala “Deecishencs = ——rimsatcr_ qual pe (ObetnAWWgongreedsbadgaleocuorombapginndid Solsep. _Entreguetion, Recalltheproofofrenormslizebility ofBDasgiveninBlorken andDeelle ...Youwrite downabunchofintegral equations, andyoumakeskeleton expansions forsome. _ basic objects. Thenyoumultiplicatively renorm thebasicfunctions andyouendupwith —____renormed integral equations withnearly noZ'sshowing atalliPutting piece’ together, . .—-¥ou_get_a-way tocompute things toorder nifyouknoworder n-1,andallintegrals are_ Afinite, so_theory_can berenormalized..Themaincomplicationg involvedintheproof__ =fvsaresthe.occurrence ofdivergentsubinegrals whichcannothedisentangled from_____*—one_another,je,overlapping divergénces. Whenyoumakeasubtraction ononedivergence, bX~Fonhurt?theofherintegration, notclearwhattodosThenightinarish 16pages section -—19.11_of BDdeals,withshowing that_you canhandle theoverlaps. ---— —Baker_and_Lee_havefoundabetterway+bdothis,Theycleverlyfindawayto— show that.QEDis.renppmelizeable_whichsimplyavoidstheoverlapproblemaltogether.—ALl.— —__theyreally.dois_construct_skelton expansionsforbasic_objects which,though more--— ©)-compricated,simply.dont_have-the- overlaps.Not_really_all_thatfancyh.—So-perhaps—— — their method-is-the-best_proof- ofrenormof-GED-and-would-replace-the_chapter-in-BD-—- ———_—_—They-then-_go-on--to-generelize“theirpreof—ef—renorm-te-Q6D,and-general-NAGP.— ~ tothegenerstparegaugetheorytheré-ere-ait-idndo ofoveriep-probiensy 96-showm-—— Gy ——nr- figured; —In-Sectiton-3-of this-paper;they-show-how-to-do ity~~— —~-——--——How? En-QeD-there“are-6-basie-funetions-to-deal with; {inciuding-ghosts;-this~ ~ ——--ts-not-axtei-gauge}+-But-you-also-have~two-WardTakidentitiesto-work-withs ‘Thesein--~- ——effect-abtiow you-to-determine-two-functions-in‘terms-of-the~ bther-i-and-simphifies ©~~--~ ——the-probtem:—Then the-usual-renorm-probiéin-is“stateé- and-pushed through; assuming the—— rant $“these-needed-skeletor expansionsare~computed ;‘very-messy-indeets—————-+==—go-E- guessthe-main~part-of-their-work-ts-to derive these” overlap-free™"~~-— — the-appendix goes-or-to-get: onemore“skeleton expansion forthe 3=gius; one ——ahiel- wastotnested tirthemaintext—because the3-glue waselimitated bytheward's. ~~ ~——But“they” think“resultmight-be-useful-myway>Thts-eppemitx ty15ofthepapers-33~————— ——PhysRev-pagess To ——>so'tesptte tietecimtval coriplextty; I-think Tyetthemainpointsthispaper ~gives-a- proofofrenorm ofNAGT.Makes“useoftheWIidentities; andmakesuseofthéir ~ ~ouroverlap=free- skeleton expansions. ee ee a ©) Multi-fields . 5. Gonterits ofthis Section. 3 : aa - 5lvComment:intheprevious:sectionIderivedtheInvariance TheoremforfaA{dett] first incovariant gauge, gauge fields only; -and then ingeneral gauge, but gauge Fields only. Here Ihave generalized tomilti-fields. Ie,youcanhave asmany fields asyouwant along with thegauge fields, andyoucanchoose .anygauge youwant, ie, thegauge condition canbeanyfunction ofallthefields. From theinvariance proven here, you conclude that the generating functional isindependent ofaand this allows the‘texponentiation of.thedelta" soyoucingetridofyourgauge surface delta andincorporate it,ineffect, into your Feynman rules. NowIamequipped tostudy arbitrary gauges made bycombing fields toegether. Contents: \ See1:compare gauge field transform toregular field, then invent anotation 6 whichallowsbothtobeexpeessedagone. | Sec2:Withmanyfields,redefinetheobjectMopar)anddochainruleonit.This lo)nowinvolves type sums. . Oe Soe Sec3:Combine type indéx with component index togetsupervector index i.Write things inthis notation. . A Seci+Invent-a generalized D,, object tofurther shorten notation andtoimitate thepurefieldcase.-Forexample, writeM”=(-1/g)D'Rinfullmatrix*“+ : sense. Note that some matrices no‘longer square. Define Rami Smatrices.:' Sec5:Compute dMr(x,y)/df,(z) functional derivative. Usethis with theconstreint togetaresult formatrix S=dg/af .Thenusethistogetexpression for dJ, variation ofthe full, all-fields Jacobian. . . . Sec7:(Getformula fordM".Insert generalized Jacobi Identity togetthisinto . simple matrix form. Then compute d[1n(detM)] andobserve that itcancels dJ!!! Appendix 1:Attack.... herewasmysearch forandderivation ofgeneralized Jacobi Appendix 2:Faster Derivation ofJacobi Identity! Result compared toLZJ's incorrect resul,tAlloftheirappendix. re)Appendix 3: Derivation ofthe Gamma Identity used inAppendix 2. Composite Notation toinclude all types offields atonce. \e,C@)conmentsIamgoingtotrytoimitateLZJ-lasmuchaspossible,butnottothepoint where Ilose precision inthenotation. First, from myownnotes, Icanwrite down infinitesimal gauge transformation ongauge fields ornon-gauge fields: q a ‘ LoseAg=Ayp+|CaeAy“5SLB|ee [aamag as icfHe. hoa feiGy Ce [aapedn - Inthis last line, youaretounderstandx that theobjects G°arematrices forthe representation. which describes thefields f,.Forthegauge fields, Iamfinally convinced that theabove shows howthey transform nomatter what fields they are coupled to.Recall thatatonetineitappeared thatthetransformation might be + dependent, but that isnot the case. Letmefirstreplace the¢vithG88) inthefirstline(soitlockslike thesecond), then I'll artifically enlarge thething soIcantreat theLorentz and color index ofthe gauge field asacingle index: 6 a¢ % oie ) fo ) Ava=Aaa*GiGascha Awa*ze&3. Se 4 ‘Behe LG, )\ \ee eww .“Ge S _oa(-SSwyargal =See(okSak e=8[-Chl -Swi Cal. ” 3 a Somyonly "docbbring" istodumy upsome Lorentz indices onthe adjoing group rppresentation soIcan treat the gauge fields asifthey had one index. Notice thatthere arenospace integrations intheabove. Thespacetime argunent xjust sits ineach function, Lee and2Jgoontoinclude the spacetime label as part oftheir single, composite field index, butthat iswhere Igetconfused. That isthepointwhereat thenotation ceasestoaidmeandbeginstoconfuse me.So oO;aint gonna doit.Illstick totheabove. Sosummarize onnext page: i => % © e 6EH =&&+\Ny&@+A:@]a soy pospSanhard aenyeadd Ry 3, Tard, Tey] $: Ap © 78 wieM C6) iPCS) e ev .. = vyK ° ae) 1)Number ofvalues index oftype i,jtakes depends onkind offield. Example: forgaugefieldandSU(3), indexitakeson4x8=32values 2)However, index eintheabove always takes ontheusual 8values inSU(3). Thus,youwanttokeeptheseindextypesdifferent! 3)There are noimplied space integrations ! @)Next,letsfindouthowtowritetheFPdeterminant andobject, aswellas the object M,and soon. Todothisweshall start withagauge function F,(f%(x)). Theonlynewidea now isthat this might beafunction ofall the fields, not just the gauge fields. ButIdon't: think thisreally makes muchchange (eisacolor index). Asbefore wecan doachange ofvariables: % fem=BCRGA) > OWWe) =TT Lge Qen=Peelen en vanY=|$==akMIS , VSym = AY VSsGe) ~3- Before writing this out with achain rule, itmight beuseful togoback and introduceafieldtypeindex.Forexample,Imightrewritetheabovegeneral oO form ofagauge transformation asfollows: y & ®pe @@) ©&, e be=She+[IY ie+°Xo| ee Exsentially, this equation is"diagonal" inthe type index soyou don't really need it, though later itmay beuseful. Once you know the type t,you then know the nature ofand the number ofvalues ofthe indéces i,j. You also know the Gamma object. Earlier, this type index was implicitly carried bythe i,j indices. Ithink now that itisprobably agood idea toexplicit#y display the type index alway. So, wecan now doour chain rule asfollows t 3 3, 4 yd Wains)=VECO) 2sr(TEL)(2tet)- ee Yee(4) ae VO.(2) Yee) ro) Nowthingsaresensible: inthechainruleyousumovertypesoffields,because Fmay involve all the types, then you sum over field components i,and you integrate over spacetime asshown. Now lets assume that gisvery small inthe above, then wecan uge the infinitesimal formula above, andweput#®=9infunctional arguments. So: S —>Weld; =Ade. [VECQO) [ew OeeGims) =gay i Ty&@+A,@| 8aSes)Ve so Next, this can beconverted tospacematrix notation asfollows: VRQ) (th,& Wb vied = [Fe] [Nt TA he 6)NowImustadmititdoesappearusefultoputthetypeandcomponentindices le)together. Solets define this: e ®*\ ‘ Tee=SeLTS ‘ ey %Ge=O&O+|TreesPROKuoSe) b WMa@l=2}veo] meLah+E = eeTb VeetdehiNx And now you see that everywhere you sum oni,you are also going tobesumming over type index t. Thus you may aswell combine them into asingle index: (t,i) now becomes just (i ) fe) ‘Thenwhenever youseeasumoni,youunderstand thatyouaresupposed tosumovereach component ofeach type offield. Sofinally Iwill rewrite the above: ¥ ebO=&@+ltXO+Ko|&.) By. \Nna|=(9) vei| Vd VS. VR) oe iN = a(S~[SA ey Ve = ‘S 3Eachrepeated i,j,ktypeindexistobesummedover.2)Bach such index istobeunderstood as.i =(t,,1) soyousumover fieldtypes, andforeachfieldtypeyoustimovercomponents. 3)Inthisnotation, allthefields arejammed togebher intoonesuper—vector. -5- 3,Now Ithink there islittle point inwriting this fancy object again and a again, soInowpropose toinvent thefollowing symbol tomimicthepure gauge~fields noation: ee rea (De®)=Cy{VyLat+1Ax |——_,_ ¥ ~ % bO=&)-ryDe®oe) Mug =[SRA -2)|D009] : vee 1. ; ®Nowwecanevengoonestepfurther: youseethattheonlydifference between .ouroldnotationandthisnotationis: :: mo, lo) a)the derivatives A_aresimply absent for i4gauge index . . b) the first index onDetc takes onmore than the color values. Itismyguess that these changes are sominor, that myold proof ofinvariance ete-can becarried through with little change. Lets write: : oo: (may=-5(Setld|[wo ee| Nowthis looks exactly like ourformer expression when wehadonly gauge fields. Ofcourse nowitmust beunderstood that matrices like Darenolonger "square", ie,thematching "color index" intheabove product isactually asuper-vector index. Solets run through our basic stuff inthe invariance proof and generalize each statement. For example: -6- For starters, wecan-define matrices RandSas:before (butnowthey: arenot square!): ye| YOu) v ay Ve =YS@ .Ts =Sa@sy =SidLeeO2eiogVy i\aney a9) “ i ve acall ={slx. Ei S . teoerene~ 4" 7 ie =[RI Ye = . an Rada=VEO fo)VoiGs) J ; Tfyouwanttolinkbacktothecomrient gauge,whereF=d,A"depended onlyon the gauge field, then you get: Ralsey =Regs Go=Segey Ryn Gd oo ~ . : : ~DIS} FG» as as = =Bel=-Sapq SyBS) = HereIhavesketched Rasanon-square matrix. Thenin aA theusual covariant gauge, itiszero except for @square portion which isthegauge sector. Nowwecango.on towrite Mtwoways: notice thatMhappens tobeasaure matrix, although itismadeoutofthieproduct oftwonon-square ones.Notebythewaythatyoucannever‘haveD’=~Dnowbecuase ofnon-squarenesst : OO wa)'=-+[Slo wti=-+{d1[8) : |ty3lgsllyl wo gSOR| cH '| °| ' RY -t- @)NowusingtheexpressionaboveforM"wecancomputedg/dfduetothe fe)constraint as follows: x Wade =-Ee DiGedRien)” >Vike)=-+\e(ExtasyRal4Delos(Pata) Vee) 3 TF;()/ WOAPAGRS Ra, ~ ‘ VDulst) ~Vdvib) 2GayVikByeBea) 3-2) Ve) Va;e) Here Iusethefact that the/.thing, which only exists forgauge fields, isnever afunction ofthe fields soyields nothing tothederivative here. Also, look back atformula fordefimition ofDiptoseehowigotthis,andnotethat5isalways“anti syminlower indices (look back atdefinition!). Therefore thefirst term on the RHS above is:_Sea.2)"Rule ©Sanisom x“Hk DiGn ReVe,e) 04) ~< ~¢+\Dissret‘\Ro») =secondtermVA8 8by Nowcombine these terms tostate theresult inamanner similar tolast time: | YlaGos)={Seok ~5BjDux)awRa) V&@ &s) -8- This result isnow inserted into the constraint condition as follows: o=SaySey)Wns.» >O= BCVea(y,x)te Ves) Wea(s,x) vs “)VAs\VaSey Read va| ’ 2 a O=-HOT Read -FLRG~) +SSes) WeGx) +. Ne =Isliw] =taLigh +&(igi) , m ma vnF=a\aySela)Pu(a)V_ .o) 4 V4.) Ryawary, , a>te \ DaGy =cyLIQ&Gy«AvGy] > » maDeb} =GayL0g&&)—MiG)] J)geMeeeey8gaegeancyhin ~s Dols)=Gs)lwebls)©Ro BAG)=GyLhedeny~I) a 4 ~S =63)Ledels)+MG}=Deals) fo) Thus,ifyouhaveonlygaugefields, youcanreplace D"by-Dwithout touching the colorindices. Thenthisoperator Fmatches theprevious operator Fexactly. Notealsothat: r=eoeeSape -9- Thus, weheve extended our previous result tomilti-fields. Inthe limit ofgauge fieldsonlythisresultreduces totheprevious result. Nowasbefore, youcan 0 solve for dg/d) asfollows: S ayy MM il-{wal2ltgl+Fig 8) roroy ye or we The transposed statement isobvious, noneed towrite it, see before results. ©nowofcoursewearegoingtoputthisobjectintosometracetogetthe variation ofthe field Jacobian. Iwill now show that the usual form ismaintained inthat operation: a » YDa® =(3)¥ySey) . V&bs) 3 ¥ \ .-(48)-ty-74loll-wes \e) ab = ate ly sx \ Yo=>[dlestag)ue.|v=1-3i{(Di|sf Aeove The result looks the same asbefore, but ofcourse now the trace isinthe larger super-vector space since you arecomputing aJacobian from all the fields toallthe gauge-transformed fields! Even though objects inthere arenot square, you can still interchange order oftraced matrix product. Iaminclind totranspose both objects inthere togt” — + av.-s{Wisk 3 = = I . \ + ie SS ~\ ©|sv-,6(th)ttlig~tafww) A -10- @Wenowtixmntothecomputation’ofav”foruse“in“thedetMvariatioideal.~To COthis end Ifind ituseful toutilize the previously’ derived formula onbottom ofpage 7,except Iwant t6switch theorder ofthefunctional derivatives in thesecond term (back towhat itoriginally was). Thus: vv vSWAG) =\eBaoalYWeels5]a Vd;@)SL, . = . =\k[5hstDeGt)&6||Se>TkRG)-FDabs)Y_Rely) ~Vee. Bikeund+ me ‘ ‘ .:se(AX)Oe)PeeRe a(h Te. +EVERGDaOM Re)=CE[RalTeLR). Qoeoed Wenn , rn + .Buy) JawewTiens Roo =FIVE Rely?3 45 3 ~ yt = + Ay bmel =4(OAFBel)++Tel,Reeh Notice that, inthe second term, the color index bisleft hanging inthe sky, so itisnotyet possible towrite this thing asacolor-space matrix (forthepure guage you could becuase you could just rotate indices ).Nomatter. Obviously thenext thing wewill need isageneralization ofthe fancy Jacobi jdentity. EanfodeseWK(she Av=emitxaaniv) an : » [elle] -Tedtds) =es)tk OfcoursetheRHSherevanishesgenepd{ly.Onlyifiis.inthegaugefieldsector, re)andeventhen,onlyifizb,doyOgetsomething ontheright. oO -u- Inthe section which follows this ("Attack on..." )Ishow tortously that [e) afkexthefollowing identity-is true: . .. [:%29,Hoo==tealDal +Walteali | Rewnte: b S< : |els=(Wreath -10fwai)| Mynotation does notlet mewrite the object ontheleft asacolor matrix. Now, wegoahead andcompute d{1n(detM)]. Since wehave notchanged the dimensions ofMsince last timé, the formula isexactly what itwas, namely: Sawekny =Te[tee Leet] ie)Wissen, wehanadrownQe: : . Rat=4(LAGTg)+5LOYcoal) -peal 4 4 Raguck: . ngcoe isadl=git) +4tealteayy +Ween Que: a ml beea) seca)=4¥(L071fotgUM) 0ee — wes K(Yed=or, can.Sos : [SsSuaanco | aie fe) 1.Myquickkilleffortshavefailed.HereiswhatIhavetoshow,butwhatIam not yet able toshow: ee at | ’+ v fre. 1| efUDel,ByEBSAEW=ef(MeaTyUReV Ie, once Ishow this, Iamdone with the completely general invariance proof. What isstalling myefforts right now isthis: the useful Jacobie Identity which made things golast time doesnt work here, and refuses togeneralize! The problem is that the Jacobi identity Iused last time seems torequire acolor label onthe entity[pe], whereas intheaboveIamfacing(00),wherejisacomplete super- vector label. 2.Sohere isaplan: take the first trace above and break itinto two pieces. Inthefirstpiece, index jrunsthrough thegaugefield's sector, andinthe second piece itrunsthrough therest ofitssupervector index values. First,hereisanexpressionforthetermswhichdonotinvolvethecolor le)sector: .iS .At 9a,Maths&fUMeaRell AESA rae a 3.Now examine the jsumwhich isinthegauge field sector. Ifweput j=c, then theindex imayaswell bereplaced with dsince yougetnothing ifitisoutside the gauge field sector. Sothis term yields: » +4 ‘ “A tetUWelYeCRANE, teFUTURLone © i== -t& 7 . < < Ravada: Piel -(WDlPter~tea LStpl aws oan) Pianingoolaahodherpar ten! tat ps sot -“ilteatALD) ie)yO “* T4pe Coeat - NanasgtHf(Wh)hyCeeterNe(OATRBLEw) =2- Sofar, then, wehave taken the left hand side trace inthe original equation O andhaveconverted itintotthesethreeterms: Sye + 2)SsMelk+(tentedLeedfa 1) . ive 5, a » ete(Dedllodatte) ) < + - 9)~The¥eTolWDRas\nd} Notice that wecannot yet simplify the last term because the correct formula forM!inthepresence ofmilti-fields isthis: Aa .ay Gy)Urteed=7LdaRA So, anobvious idea thé comes tomind isthis: add toterm c)above the terms fo) thataremissingtomakethiswork,thensubtract themoutagain.Thus,weshall replacetermc) with“)+): 2)_et)+2) < v “1ay~Thete(teaDante) r& vv y=") a)+teZA.te(toaLOG\RAME). ’~ = ee Qu: « waeet\=—Tee(teaTestwsy) »C3) &=~The4CleSag): 6) ae) ro) SoIhaveforcedthisthrough,mimickingthepuregaugeapproach. Wearenowleftwith (a)+(0)+(@)s 3. Next, lets trytoforce term b)into what wewant toseeinthe result: o + s x - = bEToe}tea ThCadtus gen T S Ara ae =AfWedteaTKTRATMETE - =(ei) ~Z{WM eae =2)for But term (b1) isour desired result. Therefore wehave toshow this: (@)+(bz) +Grey =o . avert) ca s re) © 9Fehh1Caco Ril!) bac awe W=oFall» (tase}Tes)Uw)paver> ce es3SeTaN&(codtdWHE) Things arereally confusing without type labels, butIamgoing tomake aguess which Ithink iscorrect. First, inthefirst twoterms above, change from sum onjrestreited tosum-on-i restricted, effett isexactly thesame. Then jis acomplete sum. Add these two terms togets ‘sSs bat aN @+O)=5Ze{Mls-ateTees)Ug(ME\) ann, e =CareNeJemseaegehen 3 . cont ayo~oyZpCe W(Uaesg QeME). “he 0 Butnowyoucanseethatthesetermsexactlycancel(c2)andwearedone! What about this commutation rule? Consider: Me Me L aaee theIf,e\=40°€ eNoutsewJIAWap. eyxO): We yy oi # . as spe . &= LEGGE) =+OG) ce))roe\cee,v© Pryor. of SSMSadekLiga, Thisisastatement within Hilbert Space HY,butyoucanextend thistoe direct sumstatement inthe supervector space. This then gives the "conjecture". Finally, lets rewrite thelast line onprevious page tomake sure you see how itcancels: P 6 a (a)+(02)= Tn Th an 4 0 y= EMAL.) Comments: this was avery tough theoremy toshow. Clearly there must beamore concise way tostate it. Nevertheless, Iamoonvindd that Iproved itcorrectly. 0 Faster ProofofGeneralized Jacobi-like Identity usedinMulti-field Invariance Proof. 1.Myfirst derivation ofthis wes awful 4page mess. Now that Iknow what Iwant, here isfaster way toget it. Desired result isthis: » Ss + + <eal =-Meel +i0he y ° Toverify that this isindeed correct, take each Dand expand itlike so: < < [Died =cy[Teetee +1A5oe\| Tv » bIa=Gyeth -1 Notice inthis last thing that when you transpose with capital T,you switch color labels back toregular form, and the derivative changes sign. Insert all these things toget: cSyb&he»e fe) Lue)(hxTe+Teele+Toyte) ‘ © =o ®. «e fe ae +Used tl—1teat +Riateg Ifyoucanshow equation (2)istrue, then youhave shown that (1)istrue. Now, infact each line ofequation (2)separately vanishes. Thefirst line does so duetoaJacobi-like property oftheGamma objects which Iwill prove onaseparate sheet. The more interesting problem isshowing that the second line vanishes. The trick here istorealize that this line isreally confined tothegauge field sector. Forexample, thefirst andthird terms vanish unless index j=acolor index. So write thisindex asd.Next, youobserve thateachtermvanishes unless indeix 4 isalso acolor index. Soputitinasf.Then thesecond line reads: ic b © e e ASWiesel Tay—TeeTodTAL+fueCAPYIsad =o* < ? Baths Tote NeSet Cedlae| aPasdavLeelee) =o? S , s > ‘ole TegTotocy +PeeCocytae)+PahLae)fes)=0? » S 2PStWedscayear-tatterf =© -2- 2.AttempttogetthisidentityintotheLeeZinnJustinform.Obviouslythething [o)istrue for any color vector offunctions 9,sotry toabstract them away. IfI functionally differentiate eachterm with respect toparameter 0.(t), Igett » DeGit) Seay Ty =~TeSout)Dethiy) DyGAt)Sea) KE Ifyou multiply back byO(t) andintegrate over t,you duplicate starting point. Now take the above and just integrate over t.You get: SenayYe(aeDyety =-TreDaGasy +rDy(3) Re Carle ee Reastes: «TyeOeGye) —TH,S605)VarDyeGt) Le)=TeDeGd WateckavodheLemakd\ -3]: eae > < mgTE[RedebsMO]8) —BFSQLak[Tedx@+o) Spae e-=MeeLede) +MA) TG) 4§ » gke=Eis Rot%o-o -Wren VQ)4 ij é (4 rownyk:BRR “HNG)Car ory “x By k k Yr. 3 bY ¥of NkLieay+ KeylSq-%) ~Me[Ryae (X6-s)le) x “ <a=Cee[Rb aAol f-) 3. Now remove the deltas and make this anoperator statement: OT boy es ¥ Spay 4 ConeRely +A) -meta 0|=Cael geeNed| This isthe corrected form ofLee ZinnJustine equation All intheir appendix. The whole validity hinges onapiece being missing asinthe above. Their result is simply wrong! Derivation of the Ganma Identity. 1,Inthemilti-fieldsetup,thefieldvectorg,isactuallyadirectsumof oOfield vectors invarious representations. Wehavelabelled these representations ‘ bytypes. Thus, for example: Now,eachfieldtypeis (: meansasetoffields %&>(qowae thatbelongtosome 2representation of thegauge group. Thus, wehave, within agiven representation, the’Liealgebra: (One“hs oe . ee = OE ee 2=Gee) When youfirst write this, youonly intend that theindices onthematrices go overtheir appropriate values. But,youcan"extend" thematrix Gamma‘) tothe full superspace (direct sumofHilbert Spaces) bysimply making itsero everywhere outside thecorrect region. Thes, forexample: . co)r_ys 6 Thisthingwillbeasquarematrixsomewherea a onthediagonal. Obviously’ withthingsset 0upthisway,thematrices goingwithdifferent WSfieldmiltiplets commute. 4‘o Sometimes wejust say: matrices indifferent Hilbert spaces commute, butreally you have tolook atthedirect sumalgebra toseehowthat comes out. Here weareexplicitly implekenting thedirect sumalgebra byextending allmatrices tofull size andputting in zeros. Now wecan define the "full gamma matrix” asthe direct sum ofthe basic gamma matrices. Ifyoulook back ontopofpage 4youseethat that isprecisely howwedefined thefull gemmas: asdirect sumoft-gammas. Thus, thefull gamma matrix isinblock form. Itis’then easy toverity ourdesired identity. Ofcourse iffollows atonce from atheorem Ialready know: ifoperators satisfy aLieAlbegra, then direct sumoperators also solve it,QED. Buthere isaproof anyway: PAS @ 1= 2 St ast estUM=Bet 22(py. Zh2 wtnet tt aFhe Jee S bus eb vejVyNiNyye=VeeTheHl Old Notes Nota© EW.3/23/9F Index: we; 1+Various levels of aQManalysis.CS2.Gell-Mann Levy1960notes. 3.AFFR notes onSigma Model. 4.Gauge transformations, section: a.Behavior ofA under finite gauge xform. >.Convention taken for"g" andSt c.Infinitesimal xform ofanyfield fand&, 4. The conserved gauge currents e.Generalizeed Pauli Identity for regular structure constants. £. Proof that "covariant" derivative isreally covariant g-The conversion from global tolocal gauge group. h,Another form offinite transformation ofA. i.Generalized Jacobi cross product theorem far structure constants. j-Explicit realization ofthe adjoint rep generators with str. consts. k.Proof that Fi, isagauge-group regular vector. 5.Variational derivatives of§andVtype;chainrules. 6.Fadeev-Popov Determinant Section a.definition ofgauge surface £,(A%(x)).b.def. ofAg(A) asjacobian c. Proof that this jacobian isagroup scalar. 4.Extraction ofthe infinite group integration factor. e.Calculation oftheFPDet.asdetofMap(x,y). fo)f,Realization that str. consts. must betotally antsym; inten, ofo,+Def. ofgauge cross product; trace rule; Dot-Cross product theorétl. g-summary oftheFPDet. calculation, nowusing C,4,notation. 7+Feynman Rules inthe W(J) formalism a.general formule b.examples inf@~theory, vacuum bubbles only c. the combinatoric problem of “counting” agraph. d.example in#3with external particles e. ageneral feynman graph 8.Conversion ofthe FPDet. toGhost Fields; explicit form for Landau gauge. 9. The exponentiation of the gauge delta function a.replacing delta with arbitrary function intrivial example. b. slightly fancier case, c.the actual case: includes short review ofthe entire dgextraction! d. proof that (dA) isagauge scalar. e.statement ofS(eff) caused byexpo ofthe delta. 10. How the FPdelta rescues QED; the {-Lorentz gauge. 11.HowtheFPdelta rescues QEDinthe’Coulomb gauge; useofYWvector. Compons aD. 32,Feynman Rules foranygauge theory (gauge-ghost partonly) P a. gauge propagatorie} b.the3-gaugevertex (withpolvectorsadded)c.comparison of new and old method to interpret "Feynman Rule". a.the 4-gauge vertex e.summary sheet ofghost and gauge Feynman rules (Landau gauge). Various levels ofapractical Qicalculation, 1.Hereistheproblem: youhaveanon-relQMprobleminvolving aparticle CF ime narnonic oscillator well. How doyou analyze the problen? Compare the analysis with same particle inaCoulomb 1/r well. Think interms ofSE, propagators, field theory, any method you have ever heard of, 2,First go tonon-rel SE QM. You have appotential which isknown. You want towatch apacket bounce around inthe potential. Ie, you start off with @wavefunction attime ¢, and any later time you want toknow what the wavefunction has become. The most economical way tothink of this istoconstruct the Greens function G(x,t;x',t') bysolving the differential equation. This isthe “propagator for harmonic oscillator potential". The answer isgiven inAbers and Lee ifyou set J=0, Ifyou then goontoset frequency W=0,thenyou Will get the famous free-particoe propagator. Once you know the propagator, you are done. You could even get the thing bydoing apahh integral. 3.Now lets gotorel QM. Now you tend toleave the single particle sector decause you can pair-produce and soon, Shift tothe relativistic Hamiltonian. gee «YOU could goontocompute the relativistic propagator, but Iimagine itisrather © nosey. Hespieces which gobothwaysintimebecause negative massstates now possible. This isnot the free-particle scalar propagator because there is aHOyotential onall the time, and not aweak one either. So, again, knowing the relativistic propagator and keeping energy low, you would get agood answer toyour question ofwhat happens with that little packet. Note that the Rel treatment ofthe hydrogen atom islike this, ie, rel Ham plus the potential. Ofcourse the concept ofpotential isonly meaningful ifenergy islow. 4.Now move tosecond-quantized field theory. The rel Ham becomes the Klein— Gordon equation, except you have apotential V(x), not V(f), sitting inyour lagrangian density. Problems are never done like this! However, the ur potential can be simulated by going toan interacting field theory and adding @source term, eg,having g°+Jf. BUT, Iknow ofnowaytosimulate a dur? potential byallowing somekind ofexchange. Isgluon exchange supposed tocause akrpotential inQOD?Iknow that thephoton propagator 1/k° causes thé1/rpotential, butnotatallclearhowyoucouldmakesomeotherpotential. a 5+Sothepointisthis:youcannot doaharmonic oscillator potential probleminsecond-quantized field theory. (2notconfuse theHOexpansion ofafreefield with the problem athand!) read 3/12/1978 Gell-Nann/ Levy 1960 . ce) 1.Introduction. For+mom decay,answerworksoutright,Bermancomputed x radiative corrections. Forbeta decay, however, you have Cyand c,todealwith, CVC(1958) tells usthat Cy=1. This leaves @(beta) =.97G(muon). whichGNedd Levy find troubling. Later Cabibbo will explain this in1963. What about theaxial hadronic weak currént? They know that C,=1.25 and that this current cannot beconserved because pion mi-2 decay isobserved and involves this ourrents divergence. u . .Theywouldliketobeabletosaythis:, dA=jaw whereaispiondecayconstant. ThisisBq(5). * e ‘This formula issignificant because ityields the Goldberger-Trieman relation which isamazingly accurate.- Soauthors ere interested intheories which give rise tothis Eq. (5). 2.Charged PionDecay. Bytakingthe(0/.../#) and(2(---/2) matrix elements ofEq5,authors derive the GTrelation asshown in(11). Their result differs from original GTR only inform factors; they make abig deal about these extra factors. Claim: any theory which allows pion decay will have such aGT relatdon with some form factor; the only question is how fast the form factor varies. 3.Noether theorem, Weare reminded that ifyou have alagrangian and some specified infinitesimal gauge transformation onall the fields, you get the Eulerlikeequation 2#.ox%[25)rg 2%) Thus,ifyourlagrangian isinvariant undersomegaugetransformation, thenthe ©thing in‘the square bracket will bethe related conserved ourrent. Various examples are given. Bg, inthe "conventional PI-N model" the three conserved currents form the isotopic spin current. The problem isfinding alagrangian and agauge transformation sothat you have asimple axial weak hadronic current. The vector current isjust the isospin current byCVC. Ifyou take the standard PI-N lagrangian and try some obvious axial gauge transformations, you always get unpleasant axial currents; nothing with divergence like Eq. 5that you want. 4.Gradient model. Sonow lets examine some models which doyield equation 5, and which therefore imply the GTrelation via this path. Gradient coupling model isobvious modification ofstandard (Bjorken-Drell standard) model. Inthis little model, the current divergence isgiven bythe pion mass term and you do get Eq. 5. Weaknesses ofthis model are: (1) the axial and vector gauge transformations seem unrelated (though strength same to1.25, seems wrong; (2) such atheary is nonrenormalizeable. Ibhink this arises from gradient bring down kfactors. 5sThe@model. This isthe section that makes this paper famous. This model iscombination oftwo earlier modelg, mainly Schwinger. The unshifted lagrangian isshown in(36) interms offield@ .Basically this isan0(4) orSUpxSU,chiral model, though suchfancy language isnotused here. Lasttermbreaks . symmetry and also generates your desired equation (5). This lagrangian shows nonucleon mass term, but the shifted lagrangian (36) does. They say nothing = about why tuxuxuyiauxis the sigma field isshifted, but Iknow that itisto3 ~—stininize theeffective potential. Someaxial gauge transofmration isstated. The precise axial current isfound. Bq. 5ishappy. Strenghts ofthis model are: itisrenormalizeable (no data given) and both currents vector and axial come ininthe same way (over) Adrawback ofthemodel isthat youhave toexplain this newfield 6”,some scalar particle which could decay into two pions. Tadpole diagrams exist inthis theory (has todowith superconducting vacuum ofcourse. The fact thatncleonmass.isgenerated isnotedag."curious". . [email protected] model. Third andlast model they consider. This isjustarenovation ovthesigma model inwhich the@isnotanewparticle butisrelated tothe pion somehow. You still get Eq. 5,but not renormalizeable. LsOperators, Avague effort towrite generators asLie algebra, Still no mention of chiral and ‘SU2 xSU2, Iskip. 8.Conclusions. Seeking models which give DJ=PIand thence GTrelation. Three models presented, each has its problems. Want toextend todS=l part ofhadronic current, want tolearn how torenormalize their third version, Whole paper isbased onthat Eq. 5;making explicit model inwhich itistrue. AFFR onthe Sigma Model. 1.Whatreferences doIhaveonthissubject?First,papersandtextin“the corCurrent Algebra book, Includes the original GellNann Levy paper. Second isthis > AFFR section intheir chapter 5.Third there isstuff inAbers andLee, Ky other current algebra refs dont seem tohave much. The Bernstein gauge review has alittle. . 2.What igthe significance ofthis model, what are the main points? Firet Iwill Seview what AFFR sayandthen Iwill answer that question. 3.Group Theory. Youknow that you aretalking about thegroup SU(2)xSU(2) because this isthe usual thing. Ie, this isisospin ;xaxial isospin ifyou like and will be connected with vector and axial currents which of course are the hadronic weak interaction currents, and are also presefit inbeta aswell aspion decay. This group istle same as0(4),, which ismuch like S0(3,1), probably afew minus signs inthe Lie algebra, Thus you can think ofthree rotation-like generators (the regular isospins, charges called Q), and there Will also be three "boost" generators, these will bethe axial charges Q. ‘ SincethisgroupissomuchlikeLorentz group, youseethatstatement (5.4) really defines the transformation rule for avector, ie, for a(24) rep. The specific comrels for any vector are then given in(5.5) interms ofthe . vector and axial charges. . Now, theidea istomake thepion triplet apart ofan0(4) or"chiral" 4-vector. Basically, this lagrangian ormodel was designed todoPI-N physics; you have toput the pion into some ‘multiplet, and the 4-vector iseasiest place to pute it. Notice that ifyou restrict tothe isospin subgroup, the ‘pion goes intherightway! ro)Soclearly (5.6) gives you @chiral invariant lagrangian. : Where doz you want toput the nucleons? Inorder togenerate the desired vector and mek axial currents, you had better put inte Land Rdoublets as +shown. SinceaP# (albeit anisoscalar) is4th component ofachiral 4-vector, you cannot have nucleon mass inlagrangian. Thé chiral invariant coupling to +nucleons isshown in(5,14), includés usual PI-N-N thing asapart. - 4.Currents. Tothis poigt wehave constructed acompletely chiral invariant lagrangian, including PI-N interaction. Recalling that the Noether transformation isalways determined bycommtator ofchatges with relevant fields, you can go ahead and use Noether to get your conserved currents, The vector curment is the usual isospin current and &does not appear atall. That isbecause @ has In0 sohas noisocurrent, The nucleon and pion contributions tothis iso~ . current are what Iremember, The conserved axial ourrent has afamiliar looking (by design) nucleon contribution, but the pion-sigma piece isnew tome. Note that since current : axial isconserved, pion decay cannot gointhis theory. Note also that C,=1 in-this theory, Also, pion and sigma have same mess asrequired bychiral invar. _5sAsnoted byGellmann and Levy, you-can convert this GAG theory toaPCAC theory simply byadding aterm ag’ tothe &agrangian. Recali that ®is4th component ofchirdl vector, not ascalar, hence you have broken chiral invar. This ‘breaking term then causes CVC same asbefore, butPCAC inthat DA=—aPI, the famous equation (5) ofGM-L paper. This PCAC ‘equation ofcourse leads to OT relation. 6,Tadpoles. Weknow from general ideas that going from CAC toPCAC causes allowed pion decay. How dowesee this directly inthis model. Answer: by adding alinear @term tolagrangian, you allow possibilityof@decaying Oo directly into’vacuum. Thus, graphs which were not allowed before can be g made allowed byadding tadpoles: Similar toidea ofCoulomb potential allowing pair production etc. Example: there isnoWw@ term inlagrangian (5.55) so@cannot decay into twopions, Butthere isaqfefgg term, so once you add the tadpole term, you allow graph like shown inPig. 5.1 (b), and sigma can now decay. Sothe tadpole graphs ieamechanical way tounderstand what the linear term does. Obvidusly such particles going into vacuum indicate some trouble with theory. Tadpoles are eliminated by shifting the sigma field sothat potential isminimum, Then (0/#"/0) =0and nomore tadpoles. : When you dosuch ashift and rewrite lagrangian, the earlier near-total chircal symmetry isnowcompletely non-evident. Pion isnowmassless,€" has some tas, nucleons have acquired some masss This isthe amazing aspect of SSB. “Secret Symmetry" as Sidney says. 7.Itisnoted that-you can add extra invariant terms tolagrangian toget "Cytobedifferent from 1.This aspect ismodel dependent. . 8,Other bonuses ofthis model: you get all regutar results ofcurrent algebra, since this model isanexplicit realization ofcurrent algebra. Eg, you get the PI-PI scattering threhold conditfon, andthe PI-N threshold condition also. Iguess no~dne doubts the validity ofthese conditions. 9«Nowtake,anoverviewandseewhatthismodeldoes.¢ re)‘he Sigma Vodel has these favorable qspects:~ . a)realization ofcurrent algebra, hence soft pion theorems etc. b)-isrenormaizeable since d=4 inunshifted lagrangian. . ¢)gives the PCAC-Equation when trivial linear @term isadded. a)gives vector and axial currents which look right for weak interaction e)that linear@termrequires field-shift toeliminate tadpoles. Thus, this model isexample ofafield theory with SSB. The SSB has these fringe benéfits: vl. Allows nucleons toacquire mass. . 2. Allowsand@!tohavedifferent masses f£)infact, the linear term breaks 3generators, hence you get 3Goldstone Bosons, the masslegs pions. Example ofGold. Theorem, 10. What does this modql not show ?Itisnot agauge theory because they never -try toinsist. omlgcal chiral 9(4) invariance ana thereby introduce gauge fields. . There are novector fields atall inthis model for strong interections. QCD omthe other hand isagauge theory model. ' The récent: interest isthé combination ofaSSB theory like the Sigma Model with aGauge Theory “like the Yang-Mills thing. Together, such atheory allows +the undesired Goldstones tobeadsorbed onto the massless gauge paritlces to *get’ massive vector mesons. Higgs. a BACT e4©CusidenFCS,3,d). 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Brrmora dalack Onina: BaW=-Olly &w wen =Ay2 =aedug asWwen =ondaroh rt ©Beck&(8)onquorwindash.WeOakQajawebersourmattegal OOceciatugsFutemmeie16.6.)=FayeGe = ~8E seeVET=eGC..5.CL-ieGe)GeCl+ioeG4) =Ghee laGg)=G-wdur Ge =Grit gcerG |prNonBening . Quo, Gv =~BPiSSh1 -~QS)S =LQG. Woo SumC4)proude Gay=TeriSacarGe|A +rr[~Q8.6] 6 =PheriLeKh-5(88)Kk 3sa=ApeiNanKnee~FOP Aside:oi . eve ‘%.PSs| omyoa idee ar “dou! Jew oh wookQusled.me des “8 u AAd)= Ko+ityeke -+(e) @®Wackardracmrosusd eunrute ?Vena , !az a|shale = YespayOs e @®Sasd Wok enoraak daisiecored: Bou~1S-SH Jrmsemsgacendataa, umadiok SQA\E Bde=-£5-6,4\ 1,3) ES- Vai: idee, aLR8l, =+e). Qua: Sam¥OxR 2S ‘Doobendssole) :. Sane=hédea=Gach c) SoidanRier =~Calc Os=~Rxs : Qawirte Deeyak. OS ce a =0Wardnqsaatige Grinboo1tysheqeaug.Coaiden >Vesey a en _=(ho:Ay16a,64)=GM | “ya JARs RO oo 4|AOC =-COR=SRE |I pb ne wae [8st Oatnastinig,iekeenooohsigremtbuy de ee roe niSas_ai((Gte)&_Geke=Cachjsoa ‘|—Oe ettAGoaamokth ohovcle,Muck.Rs(ExG)AXE)CSkebesoeme_.tr errsaaitasachs,Speen |ee ot We nr | Pn ee ee SinoQtomecadtauiwelueUeemenad, ; 6 ; ©WesieskUeaoburn: SoaiSEH BQd) =-1S-Fpd)-1Q,8)-Sh 8h.=BxA,- 4s Wawigs80anonGea SCMed) =-S-S(Ha). S&= -KySAylp BAY= Ba)~igF[WAY +ABA] =BGA)OBS~igS.[BKb-bag4) 63gGR Liss Q\ =~. FHA) —[iS-(SHHHNR -4(GALES) b els, G\erE-®)-GRED| =~18-E@,8) -4S-OE Re . =8% yc)=o =SCO) eS) 2.Comments: this shows that, under local gauge transformations, the group vector Df transforms exactly thesame wayasthegroup vector $.Both arevectors in the same representation. . 3.Notice thatintheabove stuff, youalways keeptheGandgsthings inright order becuase their indieces are suppreseed. 6 3.Comments ayput local gauge stuff. Westartwithalagrangian L(g,4,9)andwepresumeitisgloballysnvesiont ey This means that L(Uf, U(a,h)) =L(f, af). Ie,theobjects fandagboth transform asgroup vectors inthe same way under global gauge transformations. This means that Lmst bemade out ofdot products ofthese vectors sothat Lwill be aglobal scalar. Ifyou consider instead local transformations, you will find that although §isstill agroup vector, (a9) isnolonger avector; forUelement oflocal gauge group, the above invariance of Lisno longer true, But then you repair itbysaying: L(g, U(D,g)) =L(f, Dg) because nowDp isagroup vector, e e 6Qaslvaving 4Qro.FwsttsGougeRone, ©Suggs yorwordsboead aseaterJpnarenTaiRk. Baws, yordeashgroducla ake. s=(ADR +HY Raericordh wworrources nrnannre. (A,8)=s(A‘B) =s(UAus). ©WoOnsite BakDbaeGofd_oR) smashes|gaugeane:Qngde gongeGavarronce au LLRRN =L(V, TAA) le) Seqdeclily ,YondQsorttodawasinsd)DreqlideakgangegpeegcorkOwmedsack4Duardahprrdacle @WanunsoyQaonyngmache"Ueoman: pac *y aSha-iSES Fassfeogangsqpumg,wtrearpureDakOpt)aalas’ :aanashy LeDuneway Us Bie =3.4 13,01) io} s -aSl =3Bj=BLS =-S-@) 4 r . BOpd)=BAA)=[NSC] =1SGY.)8S7 @WssapaieVeinbywallingaemvanaak dann oes ea ea)SRR waeTheSxM-TBS advan ovduskbop,SeAemromork Ue, Secasougeuoanmncr oi L(Va, Tad) =FY, We Gon. @Woice OkBeincudhaMaemechdtheobecupO \prccrrarKeQroveweir purse we,Wap igeHal=RHE RS aVEN=-WE FYiy=BA(Wi) ~15SHWS >dnapoder [EH'-ESAT EOE|° NppameheBamepg Nah ButedanaieJoAy o OSbruetans aST=UEDA +QT. @Suggeyt onViagmpJaVspuialcoAkVELondMages Year =0.Sugpose allerIshDuquanratasWeeDee porns KLGG =NB hLG)=o Loadenadwaderbakcondidese Oehu,Weadopindonvag. ‘ a4Shown. Qrdasihy omastedk ityDinabgdeaanddoce6. QWiaJadecadetforRowhioyoaallan dyel)SUs, AL =RATT +alse), Seiad,WAGbeaaraldonomaninJobwodjovdsug— Aradd)Vierar(uniikany) wet oon Re._Fehlsve.|.ee Souaeak: [asRade+Sehiaew'| eo”. aap ELALah ausgangrrachnof Lora grng, | oanasFaspt6Davin DeckARGO) +aie=O._~Tea(ORGPE+GRA-GHA+ OH-GAI waewee fdsDeeg ne ep | ie ee ee a ia sommes. we en ceetEh oenentadee _|—posagreeeayteki |neeiP RBCS “eatas)<e ets REE Sp SeEEE aaeee ~— Gndurenies ae | (Gee [Cn et en Re) SapisdCuwolns onMyrCupNaSeGop | IG.o\= Ge | NoidaGGA GG MEG aesLA. &@LOS-GLOL- Ya. 7 QitaeaSaasQuey,OEP: Serge Roe nn 0 gmSsYokiisaqrougwack.6 ©DasemeWok Fo=phoVe~4%CK»). Warkbodemsok 5 - Sho=SxBe aves.DiateWaoavackeyBeumalrre, (mache©adiowkag). ©RasadahaanMeoB38 ;S@P= VER) =@BH+SQH)-4 LS aSG.Bho=SQAN-LOA,-) +4COHMxA +4HXOA) =osha’ +OA)4K -Says-SWE) +398+4[8BYBN«satsrhaeA — Yasindeth =Bx]RA,-WA5(AA). Cundt i: eeeEyAG]«Ax, +pAp»(SxA) =8Arh) g 1,4BeGixhy) ©Avx@xR,) +AyeGn) =o - VKOrieLefraDeedes erreepobadr wu| ; 6©Coulan wo agargcenotwee Ver ©GiandaySR=SehYoanobjetoneln e Bropumel, BKaSEAL Ie, ak=8G) B=-1G wk@4y=XSpradagege: shenSh=Elites | =[SRY ee9. ° e aad BAe’ More on Variational Notation. GB ‘lL.ForthetimebeingIamgoingtostartusingtwodifferent notations forthetwokinds ofvariational derivatives. Thenotation SAW) hasalways meant and will continue tomean avariation inthe function A(x). Ie, ateach point there issome change inthe function, If “£” iseither afunction orafunctional ofA,this variation will cause achange orvariation infwhich Icall Sf. Ifyoucompute Sfinterms ofSAyouarequickly ledtoaderivative which Iwill nowcall S/SA.. Ie,youjust divide both sides bySA. Butthis is notthe"functional derivative" which Idenote V{/YA andwhich involves removing an integration. Here are some examples: (0) @ ® t@=K C=K F(RY=3,AO) FIsVajReowe)QY= wA@BA) SR)= 2,(8A)wa aut=Gagrhtinisttesy =SeayGaayher) |B=QuahGyGran)uta) ; wt =SABA)a,Fes) a re)Se=rA@) 3VE=aA@ho)aHe=dBA VAS) xos SAY) «3 .YER=AcoSees BEaresnag)‘S Ke) WA~~3,8@5)VAL) Inthefirst twocases, f(A) isafunction. Ineach ofthese cases the©derivative isafunction, but the Vderivative isadistribution. Inthe first case one derivative isjust the other times adelaa and both are determined byknowledge ofA.Imsecond case the@derivative isnot determined byA;itdepends on which SKvariation youchoose, Eg,ifyouvary A(x) byaconstant shift, the §derivative isidentically zero. Onthe other hand, the YW derivative is determined completely without knowing what OA is. Inthethird case, fisafunctional. Inthis case the$derivative ismeaningless, and the Vderivative isasshown. - 5 Footnote: notice that the )derivative isthe only kind ofderivative whichfo) allowsyoutohavedifferent parameters likexandy. -~2- 2.Wehavecommented onthe%andVderivatives. Whatabout theregular @®) _suzaanapartial derivatives which wedennte by4end9?Howdothese fit into the pidture here? on o Nye A R(AA) =OxA STAY=NagNoyay) cnnaarti |BLYPN|g dA OA . ON lax . “s =3t Bteo Bbow :=n <A ar \atia fSQxR) BSewe,AE=anminglase uA=onan oA The notation a@A(x) means that wechange Aatone Joint, namely x. This gives usnoinformation atallon@(d,4). Thus dA(x) does notgive usenough information tocompute dfinexample 2above. Incontrast, Sf) specifies achangeinAateveryx;thisiswhySYisthencomputable. r)Soexample 2isvery interesting. There are 5kinds ofderivatives you can define offwith respect toAordA, ‘hefull derivative af/aA isundefined. The other four derivatives are all different! Inexample 1, all derivaties are the sane asshown, except the Vderivative hasanextra delta function. Incmtrast, inexample 3only theUlderivative is even defined! Moral: make sure you know what kind ofderivative you are dealing with. Application: whenever you take aderivative with respect tosomething that is xdependent, youshould notuseeither%or&,forreasons above.Consider the Field Equations and the Conserved Gauge Currents equations; ®Bh-3 58 =D232 43k8d) =WS-9,MS_ i)3,8) ayBe'oe%t)Se ve VRS) \booughveegottthDhowl Aatiyat. =ws=Ya3&=o. ae 3p) ve & Naeakvateden VarassqotorYenFLAC] youWomeAR=ARAK. Ts,sramghr ©Lege. dedRax ©Vosramgle Orype,yoraomnstdapYuhArruda ,So odd. ELALS@S] Qaim: |BE2¥tSA. 83 SK By 6BekShBry, HeHS-BBY, LM ®Rsom ease§[Ato], BorinQual WA=SaVQDAE .Ya) VAB® 990) ) Ba a= \ay|BBSe0) QuayYI=JaznN) =sae)3)(83859) =Vig.|SeeBQREFS. le) 49 fYAR) VED SRS(esoe“YO ous . 318.76 we aYhassnprant &OregaugedubbaPunclinn whi.dabevmninat, ©FiskeouadanOhuasgonaycraton ACAYO)=@aston,WrokdoorMnsow”Peat adackonodesdxAXtuckx, pisofaddwedn AG).MeydearJuddeauememuched R= gdA.OndaoasadLUBAjieodyomsudsgy, WELslygaugeomadtcut Tua,Oriecadishinnshed olderyordocewikodyontqamasAdebedSGBdvesad “Quan”Qua: —. A Agsickyntoow arr OareassachinQurougly Drdowndde wadtah dwhdehud. .Plansaed. ®UocohsemeS.GNS) =©fon9,inowenAAC.Oru3=30Y=3(AS).Yewa=prroumsur Jnam.Que, vat=[Ses uh\#\-mxdsleumnwodk. MadeOakT=|B)anefodAG,ooT=(AC).Bik adeoaGren dg.soSLAC,g\.Bukg&TedAsawe ensakdounrctose AYgrrvieich ongamngeconckobin So BTS =sie]Tag.e9. ; @Wwak=k omhdoTonbodedee,(eyrmaube aunapgropiale APIAB)Jaapatewihaguh.Bi)A=Oefo} anadurrals |BoARoanwntauchdoingpathsvihagal, ®Direuespacabinn whowlere amburd dkxi.Cun amshudkandJoss, TY@)= F[8]Trace D ws Diaienav adoulle duleumnrowr awWemn soesenceWm Weere agreebuinn spat. OrinOring,innedaGuachinch J,Ajayikcbneoagena Fakes] =asl)Tagen. QnconniapordrgdoableSawshale a7 TTMMe]=SULTd[o,(t0)-4]Cor ee TrekmabeJ4. Veoogphy,alydga(A®) yoboksee.Ie,Onn «gan Me On. ea? _2- o* TSSagoo[noi=4043 oundaafesadadg, fooaranyvabagat? Wimudum=aden 4ccoondWatdspacebeime pode. ®Vaso:|ai]=FG)deagayeqegout'| Rut: $6)=SagFCUANS) eadok -SyLAS=TAdgSLAY =T{Lag TILLMAN , Nara Yooncwvasure’, -&Wj. ©WadsewWowWekwolume duo. watt ASYuye =Yay. ataTagod{hinty]© : 4SEAL. O=Wie): SLAotalE3& Bye . Ree @Qsoom Suny SK)=San $08) o =SERIAL 2 j a, 3S) =q ¢ RX)=TISaefSOATLTTSLAS] ec @, % He sealer . selon d iSTAY =Wrage) SuaastaTeale aSIA\,-dure What‘has been shown here? Whenever you have alagrangian which isinvariant ander some internal symmetry group (gauge group), thepath integral will always have this infinite factor. (Note: even ifgroup Giscompact, you have tointegrate once for each spacetime point, sothat makes itdiverge). You can physically see where this factor comes, from. Inyour dAintegration, the integrand is constant when you integrate over certain directions inA-space (ie, when you integrate along group orbits). Why did wehave todointegration ateach spacetime point? 1)this makes the aetersinent afunotionsl,, something whichcensppesr ineide apathintegral; 2)@y the x-product will-allow us.toincorporate the delta into amodified action. 4 &6Wegan: ALAL=ak[eseassh se Meyy—= WR .Codonrabe WEJaad.we4)Wants) jouk Maeo= V@ 2 YePhe.Ae pectindaa Y3003)yTA@sent ‘ude. ©Pronygugtum,—We@= -MasoA-@ -43.0°R) =Satidoto- FFea]We) rons O=Yristej -MANO- FI]seg SurSwed, Bese soweAwdolf: VAM 2padiMoy S643oqeVS()l7 8*|<) SYeamyageDay - 4cesgf eyty WANG) Meebo =Sis[pS h@Berd-805$309); TAO* HooWKFwSwgQh: fo) are ree WAC) ery) =YL)a\pe |idaA.-*)He)+P3qayOA) (us‘ oe3 ey4)VA 5WaGa)=(-1SaAy(s)Oea>\VAG) AA)= 34.<0 Shs GR) =VsKOM s'e9) =Xots BAW 9.Ox) =|VE@ Ab, OPsey Lato ;VAD (*) Coots Wie De oben! Maer) =\SaMe(4)By+BaF|S) Somdon (us)ALS VEG 03.3 Sto aMWRG PoeBM ree O ys0 aa) a t 2Wasos)=[isAlenaSayKe|So-s) ; QeBomte « Cin)AL. Oo$&_the©sdwchue emstante weWabomnt wdlalivn, Warn Van Mak CaChan«+Chcs Cxah=0 a) Arh=CogCe. (5.22) Cie=Cage=bbw, ankispn. Gen we opt San=CreeOgre =ColeGes 0=(Gelge [deg =tw|GeGal onSorel ey66.So(Nt) soge: CheCahCyAChgCosCpeChOf|Che=o («Dee er wah - : _an = . BalaSeu=CamCutCoo ComCogCoe ; ) ; e=Wyk:LeakJenn w a) —CysCnenCam=+CaiCoreCrem=<Vspk Core,LandoGoin =—KSnFpen =~kCay2s =WeCyek. C= CmRade =—Cm Gand ==]G\wa(Crjnd =—dace[RY =—kHe Queweat QThre~\kCie=o a) => Tac==Cue. Buhleo, =__avTarCagCAO ° =~(Cha kelhe =-deLEae), poe[GO =CoeOpt=W3e M—dear[G666=CoCrQey=Toe=%Cave \Galve= Rue =Cre _ () a AOSaereBeast,sword16,6)=AngeGotearsen "Sue Sonigor,BiteMranmeclosed Mk, Qa 8 —-one~~Darosipih 199,Shiawosdwn eres ane GY = Se anne =Thue CC ceTowets gylie ai ep KE mioAeyhs 2 aw SNeaiNe SaSAS\eA C20 Wesdag omoddApow SAD a0sgtatESR=00ASxh=o -_Shia rsquinsa sadacuasyse awL Sy ByeSpNieee La TSetnaaa akg=Naapas ArAzseotien ae OP Ee [usShas oxacbscnuctaf Gye gee eea flay ontisy! |.—On Ee sashsg TT Bisphmeguagetla,S06506),gea———Lomation bane counleaSurageCette tN)sonaiet TeASSESS ARS TT CT ae® ~4%_ 3 ol ®Bandacde: (Ge=a2[omepaankt —_ eke: ;- 3 RAGGL= 4)GAG - .=CARiaCie==Gaye =&CisCie : BS toGrr}Waomns, 9g=cast: CegeOnes - Vinoaasunnt Osh,OneasdByDawvddalloos Stooges 4deeWen) aatowtahion 0en, DoFromanita gaugedbdromedormack MaGd= tleHe+4CueAMTea @FaleSardineguingwoeSask =FLERE ay RUWA- 3hi-0 =SK =Yayti 39s'@x) =VAM. -3s.) SeVAL) Pers) Onan \iS) a OYoy ;Ma.Gs)=[SysCoaKnead[Sas3.3wa iad |335 AG) \s =oo+Jato+9,Geyo4)JeyCS) a 4 ~ =-<[Bs8a=9Carhs@ de)B-) Tanta an “_: u 2 ‘ a” WG=“4[|Sathy—5CaA-@alee. | Yor:simenWaod)wae9[gordaangeDasae4WL yydomiong sceMAL Osvensty, moupdheonee teal| 5 B+ Undustomding, Bupremane Radso>onWIS|formation . abo ieIw Dk[Rw <Tel whl= Sue - Sit(tSe) SALLY S| -\ae S ySayAW)AG) Samm QeeskeearoaeahmmManefF(GR)S =F(a)e. Udit dydan “Gioos Same LaRune F(a) ofa e% Yaa,tidepadrudtigul bobeaak4lor sNah aw :SateLVS, wh\= ce »NL] 8 iGak[Lede =ayeaSAAGNIO) vue aL] =Slade =€ w X. .sGehirdse) _~as\dedsa deny) whl=© )e Comments: Ihave shown elsewhere whyitisthat W(3) canbewritten asshom .interms ofthe free propagator. The real question atthis point istoexplore indetail exactly how itisthat this boxed expression gives you Feynman rules for your theory, Iknow that the only right way todothis istostart with examples. There are indeed other "statements "ofthe feynman rules, but at the moment Iwant to understand this statement. 8. Erongis4.Vatawodanvoade NW,yrcredgatwanvwnd,Wwbletee Olan,dakeFQaeny, ,a gtotMbate)dies)Nd wih=L®watt) S oniO oGaioy Leeig qegue.pogrwll=4.ut sueroddoValea duce, :~ELQedadofc)M0). 1g Comment: recall thatthere wasalittle theorem fora£(a/ax) oe, there is nosuchsimpde theorem forf(4/ax) eooningeneral £(a/ax) e&=),that theorem only worked when exponent was linear inx.Here wehave aquadazatic exponent inJ;theonly wayyoucanfigure outthederivatifes istoexpand the exponential! . =glSaeed WWWaCadyBOAEs)TC i) 3\SaveySw\\2:(auto D6)»)\ ‘ ioe)=acidLaxled4%] =<< °Nat Nsynshopwilhot" iag “¥ We\rona,Queodnud OsodASodgadearmeusmgrnpl, 8)Qed: ohanti2> , . , 1 fae 1, 2 iv 2\a[idal -Sitka. 2 2 oe AtQtse)BHT 1?) 2) . .4 -a- = (os yYVDW0° &Yanan Reaemares Fb CHAT [Kpadeieas SOMOMOBY06-426-4)may Ingeneral you have tochoose the term inthe rightside exponential which has the same number ofJ's asyou have J-derivatives acting; other terms give 0either because tofew J's (deriv ofconst =0)ortoo many J's (setJ=0 atthe end). Veo LD = WL Ydedkde IEVIWMe) S(e-¥) bCe- ¥2Sal1 Jdoduade (a-¥)86+%) DV = . a ~ aye)Seto |WeaSpadeTATE)AG)OleHe). }+YH\dadkTOYOAGLOHH) | VV Jan)=HQQWdaTE)ACK)O&K) OFlt42-2‘NSdeTE)AG-%)OG) }4-2JanF@)&G-x)004%) ye,ilJ=BA AG dex)‘ ANU) BCH) SKXe) beeNoy we Sewie}§cayCD[seDOr¥L)GeKe}ULabir}ocieny\ Zed. . . ~ . aA wt uv et=CACOSondeeOrme) GCA)COSaabar)| ab vie ia _ a 0-be +46 O} ‘ Inthis theory, there are only two gxapkaxmkthx vacuum graphs oforder two. Second isproduct oftwo ofloops found above. Note $factor! YawDedeokonde&: - astSande, ELE ea e kt3 . GED[Baeaescanaat WWW) TE)S(8) 3hAfa-&) d¢e-d)d(e-S) Ye4,tomatoe Shan: Sk> 6SYBadlededddeSEYHEIE)TA)TC) SEOeleu) at =, 6SStadededhTEVTEVTS)HA) Bw Be+)De (A=X) eo GN sadeeddFEA\TOIONTO - BCA) de+A)BAK) Wo3) GyZYRITOTEYTC) B(a~%e) O(6~%) a(a~%) . 4G:2\daitede SOT) : ; D(o-¥) BO) 2(6-%) +g. WeS\tadde TOITW TE)BE) OFKYDM~K) o tad —3- * Qevowily wawadaLaalate Ahlekead: OsGAYVarddy 22D’, \Naaee dyeTn)-HO) Dest “To- Gye) OlY4a>4y) oyea8):Weabk—oo: flrosed Noo: dsY=Sag. cdye560)TGs) ; *\6DCH)OCs)Alis-X} Ws)=SagidaTE}THs)|Sly4)04-49)D(Ks-¥) AHwlymsr) 864.-%)oly-V)h DAL] =Vaytge¥GSes)|eAGSEGsha)SQ) a FAA BM) O(4s“%s) AGer%s) $4ACHKe)(4.-¥s) Uys>%)| Sa=aaydoe)[BNLDHX)OyAK)Os>X) GN B(YR=42) AHXe)O(%s~¥8) UL AGm4) ACK ¥s)OK ¥a) AVL DUHr¥e}B(Y.-%) AKe>¥) +YURS (GA~¥2) A(ye-¥s) BKa~¥s) FYLOO) Alyarve) O(42-%s) \ eG+ O*SeOP+ re) } QhguddinninWereteSena”mohogngle .° SABO YayWaAG)AGA)Oey)Ge—4H) 3 OrSE US Ge : ° 3. 3) 6 LoGrae) Oy)5=4)Oe>Ke) x4 CaGinsu) 43>) 492s Ka) ax2 ALSYe)DYs~Xs)oOe~Xs)xt BGR )O4sFe)3(Ye-Xe) cy Oe2%TaQynde) 0s~%s) SeKs) ° ,~* <i YES KE) PPR gute Le2Lela) D()s¥s) 4(%e~Xs) $eee24Lobrte) ons) o(heo¥%s): ~do2%) ol\~¥s )O(%2~Xs) KLSOUA¥) OOKS¥5)b(MeKy). ZSJacke GYD QAs Ms: 44 ODE GEa * : Ss, . ,1? Da : anea .t ~4- . Inow have atentative dethod tocount each graph. Consider éach propagator tobeabox. ‘The number you get from applying all your J-derivatives isreally ©_temutterofwaystomutX'sinthevores,,Inourexanples woworedoing#theory, soeach coordinate appears exactly twice. Ifthird order graph, then three coordinates X,X,andX;, Youkeep aparticular graph inmind andthen you ask how many ways you can make that graph. 2 keExample: oyExample: YZgodsspeDrangcmdonda#prongs As Le - . . \?Ww WwW 8)VetocngwagsbopohmmapiadXp?ay. 1 LYSaneceftaunth acta mM \)Wea gh pode & WwDY )WaanaanweysSogutpor)Xoe! WD BW WD 5bys? 2 ©Que tank =ayxde2 BKck wilh4-1 (4) )ht PLOe Ye.8yy. Gow 6 KS) Suppose youhave theory which isorder n,ie,f”theory, andV=munber of vertices. Youwill atonce ge}factor 1/V$ from each exponential (?). . vo. iy an? \eenenee on &Y)Les e-Z 1eaBsaivs» wT aye load.dm=wlsomeWD Se a. (2) 1 tsup)Ci)© Obviouslythe$factoralwayscorrectsforfactS“#(?aul (o)that you can put anXima given prop intwo ways, sonow drop this fact and the$. Note that m= number 6f propagators. cow=wathbapagle: (ae). Viton ee IthinkInowknowtheweigh} ofanyvacuum bubble, Tryoutinstheory. a Woon %: O WR wo ) C) Mmm lavS13) 2 [OOl MDM arrayVay zUe = : 3> CD WRB wert 24. a6 6 Sowhat have Idone sofar? Considered only vacuum graphs inf”theory. fosee Feynman rules, you have to 1)take only that term inthe derivative expo which corresponds toVvertices 2)take only that term insecond expo that has same number offxetx J's. 3)thendoderivatives inallwaysandyouwillgenerate yourgraphs. e What about graphs with external lines? Here, the difference isthat you take the term in second expo that has anumber of extra J's equal to nunber of external particles. Letsdoexample: #*theory, twoexternals; twovertices: TPArypor C=WYfe *oat .iy . 3a afiSMJAAP): =NavedieSH)=He cay,wt ALEK) DUK>Ky)BKK) S007Xr)|. Ors —YpagagabreyucatanBem4snddDede Okaywage aveteneh y'sOhegh 7 > = 7 . . Ma: =varbrere(woul=onesbinlrnachin,m=&penqagrb wachidioase poreWM=Bagewel podoaba_re i\=ord4Quang %. m= § te 4\Proms VA VF) Ytyndy %, *aa) Sas? .y pny SPB YMBLID) oe Oped) * Qe) Ore)S ne w xm imm .&)NStedew AmeVRS--STR) DRWOKRW --0(-- —- eee yam aw ™m. sony,Som="om+VT)fo\eancn3;wll Soinatheorywithouf derivative couplings andonekindoffieldandno gauge group, I whow tocompute any graph from the above. Idonot yet have areally concise forma for the combinatorics but bybrute force Icould tell yuamplitude for any graph insuch atheory. This has toberight because weAderived this. Ie, itmust agree with Wick method, Idont have tocheck. 80, only aaplies toboson fields; need extra work iffields anticommte! low. ery ~7Qosyporwonbiadn 4ha.das. (version 2) Gd©Fadosvigrorange boMushsle teepou: \axag)bt WtygO3G =Sdredygg@da664)64-8) Poagigr n=45 [Nano =44 =Yardya6)Os)7=Yaxdy 30F@) Itissotrivial that you have tolook carefully. Here, you are allowed to replace the delta function with anarbitrary but normalized function G(y). The reason ofcourse isthat nothing else inthe integral depends ony. 2.Now take above example and construct afancier example simply byallowing y=y(s). Then the above equality becomes: S\\axasXG)(aslGS)=Yardsga\B\Tye. Again, ‘the replacement ofthe delta byarbitrary normalized function @istrivial. You arenot saying thedelta =G,You are just saying that here the replacement isallowed inthese particular integrals. 3.Now lets move onto the actual case ofinterest: a os ‘\araaKoon), 1\%5(Es [Hag--]=04dg ‘{Saces ic=\agarran Soya){Stent |BIASus SE aed ‘+; eoER |AHA)OxSagarcan Koga)Sfftrcalf |“|#8 Hereagainyouseethereplacement ofthedeltawithanarbitrary butnormalized : function G, Again itistrivial because you just dothe gintegration and you see that nothing else depends ong. an 4.Theonlysubtleyhereisthatonbothsideswecanidentify thegintegration with amltiplicative constant and thus cancel itonboth sides, After you do this, the equality isthen less obvious! Toshow that you have amultiplicative constant, watch: Sowa: Aonsoma AG)4WWgemrak gmhoneAG).Recah GrarasanOudkcwascator: ; ax! REAP >deleaid’ »v~=\Fl= shoal Gris ANmaams dhdds dhs.Stine AG)=WQVAQ) andYUL=1ancesemumodalarn rongyrakdA=afte)=AACS). Bram’ enataamumnduon yor yergk0costed GxWp). Fs} .ASA ASTV),NOY Sowmas: VA) ae =e Ae, QuaackinteAAomgrGresepha.SodhedardieM3). WS=Sagatrad Sena)GFF(AMR|aaus| $ VanA=UA. Qua2hP@ve,nga)=FAL. vaKey GHASY =>GEA veAle) AWG IAY=BA ake come baME dt(AG) QueWS{Sag)yearFagan)o]Aah\Ay|oy wtus + : \snfstonemstenkobenpatebigaperat » ~Q- &. ans=Kay)aeeasSenn)SERS\sf VeorvswsGs)Lrwenlofueauieracastak : LaenBea)24(A)STIAT =Saran Sepa)ele)SER,Ven —— Now you see why you can replace the delta d(f) with anarbitrary function G(f). Itisnot obvious here because the determinant is/df/dg/ and that makes you wonder because you dont see any ganywhere. Just put the gback in,nd it becomes obvious again. 4.Bverything wehave done here really does not hinge onthe presence ofthe path integral. Ie, all true for regular integrals here. How doweneed to modify the above toget topath integral formalism? All integrals (including the dg(x)) are path integrals, and all delta functions are path-integral deltas (including the intermediate integration dawhich wasuded toinsert @(a). The path integralization ofall integrals ofcourse renders the jacobian tobearathercomplicateé bbject,butwehavestudieditelsewhere (ie, a.westudied how tocalculate it, and how ityields ghosts). = Wenoted that G(a) had tobenormalized, oratleast normalizeable. Ifdaisapath integral, then wemight take Gtobeaguagsian #hnce this isthe one path integrel wethink wecan go. Iforgot tosay that inpthintegralization, another change isthat all . theabove functions becomes functionals, soinparticular G(a) isafunctional. Wetakethisgaussianform: .EXwGreary ~, ASTHAY ~2aIman e G\= © This isthe exact form taken by.Coleman. ALtake the other sggn and insist that you take limit#-90, Isee noneed for such alimit. 5.Thus, the net effect ofthe exponentiation ofthe delta isthis: you ada to your action acorrection: osa SunelA] =-aeRAY(AY ot=anata, (Asah)qqunnlars Looking back atthe original origin afdelta (f), you see that hhere are asmany f's asthere are gauge group parameters; thus you get the sum asshown over a. HowPadgev-Popov rescues QED. ‘ 7 = 1.First, write down the generating function for some photon-paftiéle theory. io) Forthemomentignorethematterfields, sincetheproblem ofinterest hasonly todowith the photon propagator. The usual start-off is: . pv) i) . S-YkaRF|=(ale AvP” =Jase eane’ y =Sd4 - ‘“ = v =Sak eAOB} =2\skAGS]qul-gala. Ihave not yet specified agauge choice. The next step istoadd the Fadeev Popov correction. Since this isan abelian gauge theory, the FP determinant isindependant ofAand can. be ignored. However, you dohave to include the expo-or-the-delta correction, which ist 2 = -t . .Sm=—dSHO FOR=kgangsendshes 2.Now for stabters, Iets gointo the Lorentz orLandau gagge. Notice that once you have made the replacement ofthe gagge delta with this gaussian factor, thedelta isgone soyoumaynotsetdA" =0wherever youseeit! Notice thatthelasttermintheoriginal action above is+£(d,4")(aa¥). In particular you donot want tothrow this away.. well, maybe you Yould. Itis reallyinthegaussiancorrection thatyouarenotallowedtodumptit.But fo)for now keep inboth places. Then with the delta correction wehave: S+Sa = \akAco]wl-(-2)30]A® Recall thatd isagy finite real number. Now the operator isinvertible; in kespace you have: > z -= ae May=ayoB=%)Kuby ysa \ ve“i aL Xi ~(08=Lqu-Gee)BY (wo=FELay Cra) That this issoistriviel, mltiply them together. Once you have aninvertible M,you know from the path integral formalism that the inverse operator has the interpretation ofthe particle propagator. Thus, inthe "@— gauge" ofthe Lorentz gauge we have: ZolTM@N]) Sy=Dw@s)=ETD vrdtaatenguage. Powe COT, Iyiky wor|Dwlke)=CO|gyGeayAE weLar et 3.Now, looking back, Iseethat Icould have firopped the(4At)? factor inthe original action; intheendthis just corresponds toaredeFining ofthe parameter %. Vole:Ky.Dw(e)=©odywhent=O,hedieOe ” . 5aon+SrtinyBikg 4.26.94 adds aaah Choa 7—Rt,-o:>? ats aaen —@QsunitaMengloggeonDespageseanWarshaikwily“OUF]saBanthasiaiaitDraShpeteanlaqalnjay,"iSOa aTHSreeERAS) |Ss eT ee een eee ee lo eae —Wak30.QueG18)>Toad FE0age —AfG\T Qp@Ad ES= COBANISCHLTAOAG) ~_- oo ran _Semeetaareres in — O QSandetal GES msdeagp2 Se ss a emcee ceen _ e then-you-musttake thelimit-alpha «0,This-iswhat-Kummer-did.And if-you-mean -.-by-Lorents—gauge-er_Landau-gauge-o “Inatovorthet«8deporp-to-propogaten, anor - —Thuoy-elpha-te-a-temporery téol_whichallowsyouto-invertthe-objéot_M-te |— : ee ~ ar Ateuieso oftusSeta ponaden. . Oodwsa»auatogy,Qedsak2wangaYLRomTAAL,ayahdo 3 48) 480) -Saye =ins xSagSG) =A ASC)=Vag:Vege —SG@ars) =JageS)Bqcg AySCg) =sutenl Qhweoonahnde eck! RS) SQ) . Saye sy(Sag\xWow Tees|.No_ GandSakadecomgac sppSoSoy=foe DomesdineVadvtceasenenineorS38Gg)deweAone. Wars wl)Seade. woud oe”Sa TT atanal RSG)1S@) Sige o=Saq\y au©Go|eee @Now,Soarernstwecoma deeGia,Quardejedts arsquab. Rat,. ASodwarbaeraaccor, Arata woodanitlas,squeak. Wua,Supperswedannonaacuank,Chan ;’ A[SQys2-RE KAS}9x Jag SOTsey Sage 6G.) Quad,Jeans danvadwer: ASQ ASQ) StS GeesBard+Yop©|@Ar@o’CG) nata.Seeker. ABdun,Ouqeogegn yorgkodeBUS]dlrdemGoaoeateTAYOreYenOak(Grd)==von,ede See -2- fo)letsnowputthisdiscussion intowords.Ifyouincludetheactionwithoutsource,itistrue that you can replace the delta function byagaugsian Gand the reshlt is not changed, whether ornot you drop the overall group volume factor. However, ifyouthen “turn on”your source Jandcompute upsome greens functions, your greens functions wil1bedifferent! Thereason issimple: with theeource on,you cannottrace back togetanequality. Thepresence ofthesource breaks thegauge invariance oftheaction, Soitmatters then which youdo:delta orgaussian. This then explains why nDyy =0only inthespecial case alpha =0.Itis acceptible tousegeneral alpha, butyour greens functions arethen alpha dependent andareonlyorthogonal ton"inthespecial casealpha =0.Sofinally Ihaveresolved this problem . Why isthe source term gauge breaking??? = 2 JiAaydan vata so Ss=BT R 7 \ooR=orangutan, soSAS SxR.- 398 23 \ 6Que THA mnie ganga senor x QED inthe Coulomb Wauge, via Fadeev Popov. le! 1Westartasonlastsheet,changingousthegauge.Sincestillmmmabelaan theory, nodet correction. Sojust doit: iacereta cowlewleae Now rewrite the delta correction impseudo-covariant notation like so: -o=3K~43(wR) wae Mn=O;0,0) Yar=+1 Crna=igsApp.omypvabice, Sosromwe Oweconnedron deve: eat - ®hn iC)Gay=LQarcury) =QaXY- 2Q A) +Brn) arly) e (0)a) ® ®—Aaa area. —vagy2apEK =1°.(rv4H) This last change makes the correction terms each symmetric. Write now as: Ek|-22+badca) ywCeaA Qhoahar, Lay=tA{po@8)[rE]Qa)=HarrWYN)+yyertiN 6 Saa pull)=guy~(1-8)aoey—2(oA) CR)+t C8) Owe comwire on Wr=(Regalye~2(BaoGnled)(ky-Im(n-4d) Sopda cmerauid, define Be=Be Cr'e) z . — a Pods&vbewah: @d= Ke(oY =e) =F ° Cr) =O Rabo ByPm= Kalo~Crk)[kyalest] Thegame isnowtosolve forthe inverse operator andthis will bethe photon propagator inthe d-Coulomb gauge. . What dowe know about the inverse operator? 3miereatvonient votersaroundarekandp(phasreplaced ETA). Thus:, Ctlpo=Fzye+Alpe+Boppy+©Liaprke ppl. Wenow have the problem offinding A,B and C.Brute force time has arrived: e Wehave nowfound ourinverse propagator: ») a \ \ CWyo=Tege+Tep (eR)pe—(trae 8 Ip ey ee P Yubrkup) e etAL)=(aad)kuky+(7k)kywk, s|qe (aie)Bly+re)(pen ep) aauau howe p=w=(ky Rowse omenwent -Coulewle gauge - (2B)ey+LXPyploenk &) =4,|-3p-+e)ay+EOPykremke) |=D = v [MaKe TK eon cao ce) n” =pladmey: 8 IfIsetd=0 this duplicates exactly the BDpage 79result, except Idont get the Coulomb term. Ie, inthis method itseems that the cancellation has already . taken place. Comments: inthe canonical quantization procedure, you only quantize the transverse A, fields, andyou leave A,unquantized, TheCoulomb interactioncannot becafried bythetrasnverse fi$lds, soyoujave toeddthat inseparately. Inthe path integral approach, you never make this artifical separation, andyoufindthatCoulambhasalreadybeenincluded. oyood- Odeo, 70 Oren que Ket Cad syonyt|Amoltsleet, “Ms(Xo=gp ° =less-ye+PSgro+Akako+Beaty+eUSiPatiat\: Must write out all 12 terms: Y 4 ~~ a va 4 aanGyo=eBub—FePade+ACReoABIEGay+CLRadytkrpn§k a a — a~AX¥Kepler-8Fgy-CfKlee-<key ae 1 a .~Betties-Smp -Seek S&Sphe Regroup toget tensor coefficients: Gps]+pes4gpe—pic] +onyaie+ekOFce] ; e +toBO-br-2e\+beacAPce\ .>®=o “1 a CaRF .=gp[i+klo|-e-sseer|-aeSEI +otaLote) aia HA Vtpt a a(A)apy =guldsbeLCE£y-Atl> A=eH) Cerebus: 2gt eAneR ©)=e+e) © Q=O0 C= ¥F GaugeandGhostPropagators andVertices (Lanta wy) jpw-pehe 2CHBee ghostpropagator Crag & 4ana a=44gadRasghost~gaugevertex fs gouge propagator ihanmnnnntge —CDBai|apn--8 ASi k e triple gauge vertex ‘ Quai (RR). +oye ea“ =+xKinks,QyyeIpsSs days : by =(02,39) 4-gengevertexJ wot=akgCiaCaer|SyupaYryts—JapsSys Ass ~Cvgty)(14,28)‘ Indioes téke 4,denote group-lebel (color); contake nvalues forhparam. group. Indices like [ly denote Lorentz vector index. _Structureconstantsare£48=fy,=iCy,=totallyantisymmetric, 6 &=gape coupling via d,-ig GeA, prescription. Ghost isscalar fermion, so(~1) for closed loop and nodouble-dotted propagator, also nodouble-undotted ghost proph Gauge and Ghost Propagators and Vertices &--y--h& =(-4)Be ghostpropagator i.e é Ceee K host—gauge vertex4pe oe=+4S406Ns8gang 4s gauge propagator ahenn tleCBA]gynD4uk,| K ia triple gauge vertex LeerTO =+5ScieOpp(RR),>wx| Japs ity =(239) 4-geuge vertex J iy,ts =hgomensSyypteYops~JapsSysigs +Cages)+(14,23){ Indices like i,denote group-label (color); cantake nvalues fornparam. group, Indices like Mz denote Lorentz vector index. Structure constants are fp=fapo =iCgp_ =totally antisymmetric. a g=gauge coupling viaa,-igG-A, prescription.Ghost isscalar fermion, so(~1) for closed loop and nodouble-dotted propagator, also nodouble-undotted ghost prop? WAY Gomment_on theeffect ofgauge-choice onthefeynman rules. oO1.Yourchoiceofgaugehastwoeffects. First,ifaffectstheFPdeterminant, ie, theobject M.Since Msets theghost rules, youseethat anyfeynman rules involving theghosts will begauge-dependent. Iguess that both theghost propagator andthe ghost-glue couplings are gauge dependent. Second, the gauge choice affects the gauge delta which inturn affects the gauge-fixing lagrangian correction, Since this isusually quadratic inAYfor linear gauges (which iswhatyouusually find), youseethat thegauge choice will affect theform oftheglue propagator also. 2.Ontheother hand, unless youusesomewierd non-linear gauge, your gauge choice cannot affect thetriple-glue andfour-glue couplings, since thegauge-fixing terms will then onlybequadratic inA.Soforlinear gauges the3-glue and4-glue couplings are always the same. 3.Intheaxial gauge, Misindependent ofA.Youcandoeither oftwothings, they are the same: a)youcankeeptheghosts,buttherewillbenoghost-glue coupling, soineffect &theghosts completely decouple from thegluoons (and matter fields). Thus, you 7 canthen just ignore theghosts since they cant affect anydigrams youare interested in. b)youdontevenbother to"create" theghosts, youjustthrow outthedetMright at thestart since itpasses outofthefunctional integral since itdoesnt depend onA.Ie,ifisjust an(infinite) constant andyoualways drop such constants. Oo a