Renormalization Group papers & notes 1978

Renormalization Group - papers (Particle Physics) Notes Phil Lucht Notes 1978 Bogoliubov &Shirkov Intro totheTheory ofQuantized Fields Chapt 8The Renormalization Group Alin 2026: Introduction totheTheory ofQuantized Fields (1959) byN.N. Bogoliubov and D.V. Shirkovis consideredaclassicandfoundational text,butitisoutdatedforlearning modern Quantum Field Theory (QFT) asabeginner. While itsmathematical foundations and historical significance inthedevelopment ofrenormalization groupmethods remainhighlyregarded, modern curricula favornewertextbooks forcovering contemporary techniques. ~ 510 THEORY OFQUANTIZED FIELDS thechange ofvariable CHAPTER VIII (up)—my=y Ceeeea leads tothe expression ‘itees ‘TheRenormalization Group(sepGg)= TS| (¢) os Tr’ (41.27) ; where §42.TheGroup ofMultiplicative Renormalizations incooheeety Spinor Electrodynamics v=fdy eo=—1+O'(e). 42.1,IntroductionBycombining expressions (41.26)and(41.27)weobtain Weshallnowturntotheinvestigation of@methodofim-«final expression fortheTlectronGressfeacon ietheBlock, proving perturbation theoryformulas whichisbasedontheNordsieck model: renormalization’ groupoccurring infieldtheory; thiswasalready 1 ’mentionedinChapterV,Theideaofthismethodisextremely =e. (41.28) simple,andisbasedonthefactthattheindividual termsof m—(up)m perturbation.theory expansionsfortheGreen’sfunctionsandfor where» thevertex parts arenotinvariant with respect totherenormalizat a3—a) a tiongroup,whilethequantities themselves areinvariant with p=wr TT(3—a) (41.29) respect tothisgroup. Therefore, forexample, intheultraviolet | region perturbation theory expansions turn outtobegiven, not (a=1/187 isthefinestructure constant). interms ofpowers ofthesquare ofthecharge etbutinterms of Oncomparing (41.28) with(41.3),weseethatthetotal powers oftheproduct e?In(4%/mt)Green’s function differs from thefree-field Green's function bythe Starting with these considerations, weshall trytotransform factor theusual expansions into ones invariant with respect tothe ,—)f. renormalization group.Animportant technicalmeansofac- ™ complishing thiswillbetheLiedifferential equations correspond- IfthecompletefunctionG’(f)isexpandedinaseriesinpowersof |ingtotheaforementioned renormalization group.«,thenineachapproximation weshallobtain logarithmic terms Ttshould benotedthat Stueckelberg_andPeterman (118).WO)_pela i. 1alreadyremarkedontheexistenceoftherenormalization grouproe =fllo1-£gay|Le|tee andon.itsroleinquantumfieldtheory.Theyalsopointedout™ Qn ™ ' i thepossibility ofintroducing thecorresponding infinitesimalcharacteristic oftheinfrared catastrophe. operators andinthiswayindicated thepossibility ofconstructing ___Itwillbeshown inthenextchapter (§43.3) thattheinfrared : theLiedifferential equations. singularity oftheelectron Green's function intheusual spinor Essentially thesame renormalization group, butfrom a electrodynamics alsohasthenature (41.28). different point ofview, wasutilized byGell-Mann andLow(48) ou 82508 Bogoliubov andShirkov, Chapter8ontheRenormalization Group 8 First, thebasic and simple idea ofthe renormalization group (group of miltivlicative renormalizetions) issetupasin42.2. Itjust saus that, taking Ward into consideration, ifyou slide allyour Greens functions andyour charge atthesametime, nothing changes. Yourlocation, sotospeak ,onthisslide is recorded bytherenormalization point parameter \.Ihave made some notes already onthis business, seeRenorm inQED stapled section inGeneral Field Theory binder. ksareminder oftheir notation, thefunction disthephoton oronagactr mumberator function, whilce aandbaretheelectron prop numerators. Sometimes these two are treated together ass. With some straightforward dimensional fiddling (all ofwhich Ihave verified) youquickly endup,eg,with equation (42.13) which says: thequantitys eandd each devend ontherenormalization point, butthecombination e*d isindependent of\andistherefore, reasonably, called theinvariant charge. Thenotation dy refers tothe usual d,ie, dwith the renorm point chosen at\=0.Thecorresponding charge inthis case istheobserved charge e, Now, the basic "global" renormalization group equations are (42.10) for d 6 and(42.15) fors=aorb.Theseequations areverysimilar tothosediscussed byGell-Mann end Low, but they seem tobemore clearly derived and there are not somany differnt functions. The pair ofR-group equations isthen rewritten as (42.17 and 18). Itseems tomethat, inthese equations, (x,y,t,e~) should beregarded as independent variebles, anddandsarethefunctions, eshe of3variables, that youwanttodetermine. Ie,e*isnotitself afunction here .Nowitturns out that you need some "boundary conditions" inorder togetameaningful solution ofthese "functional" equations. Anobvious fect isthis: when theparamter e2 (the variable e2)isvery small, youknow dandsasperturbation exoansions. Now, bydifferentiating the global equations they end upwith some differential equations (42.24 end25). They call thése Lieequationas, fine. Here ishowyouare supposed tosolve them: first, yousomehow figure outwhat d(x,y,e%) isforxnear unity. Thismeans k*near. Ityoukeepe*small, youcanuseperturbation estimates theri forthefunctions Jand/{. Then youintegrate thepreceding two equations tolearn d(x,y,e*) forgeneral x,perhaps xcorresponding toasymmomentum. AsanalSde,itisnotedthatthevertexhasmenyinvariantfunctionsandeach Fe)one has anequation similar tothe above equation for sord. Now lets goright totheUVsolution attempt, page 524. Asymptotic means in effect that you can setvariable y=0, thus simplifying things. Thelie equations arethen (43.3 and4).First, equation (43.3) isintegrated toyield (43.7), aresult familier from Gell_Mann Low ormodern BETA functions discussions (later).In43.7,ifyouknewthefunctiong,youcouldsolveford,verhaps.Letssee.(*) Asafirst step, they quote aresult which gives d(x,e) acaperturbation expansion through order é,where x=p/n? islarge. Thisthing iscalled dy because itiscomputed with theusual %:0.Notice that ineffect this thing isnotanexpansion ine2,butine@logx andxislarge! Now they take some steps Idont quite follow. Ipresume that (443.9) isthe large xresult, notreally valid forxnear 1,Buttousethis d,tovompute J, youneed d(x..) forxnear 1andthey aregoing touse(43.9). Ithink they ere claiming that this isOKto“logarithmic accuracy". Sothen ¥isasgiven in (43.10) for small 2.. Butheway ,notice in(42.24) that thevalue ofthelast arument of¢ is07d,nutjuste”,soreally allthisstuff (using perturbation series for#) isvalid onlywhene®dissmall, ie2issmall (myETAinnotes). Thus43.10 “pives B(z) forsmall 2.Nowplug this into (43.7) andwegettheresult (43.12). Nowihave”to ask: whyis43.12 anybetter than 43.9 277 Itissupposed to be “better” somehow. The conclusions that BSdraw from this section are alittle disapzainting.True,theyhavesolvedforthefunction d(x,e*) asin(43.12), buttheydont & goontoask: what does e”look like athigh energy, insome sense. Itseems tomelooking at(42.14) that what youreally want toknow is thebehavior ofthefunction d,(x,e,*) since thiswilltellyouwhatthecharge does asafunction ofrenormalization point. Iguess that iswhyGell-Mann Low have twofunctions sand s,, akin toBS's danddy. Soalthough Iamable toderive almost everything here, the conelusions are 2let down. However, having derived 43.7, Ifeel Ihave derived Gell-Mann Low equation (5.9). Inthatparer, symbol eisrenorm charge ande,isbarecharge. Recall thatiftheir PSI(x) hasazeroatsomefinite x-x,, thenlarge acorresponds toafinite value ofthe bare charge. Bare charge ofcourse means the asymptotic charge. .. Iamstopping onpage 529. Authors goontotreat theuse oftheR-group tofindbehavior ofelectron propnearpean’. Theresult is(43.27) .?) Sxerovas: Adenine C4215)ENRS. Oe 262(Raha) =2Ohat} Se Keke Noead gistRaaaaih Geb G) adaaa arr we et)=a.olde." aOat) =aba) ab aa eo van CANOaTay = —4-feAeped| AGEOEAG9EAGye)) ——— ~Qusstuin sonGy Jo eR Ce ee . nn eSWA oS RL. -—_. —__._- 54. C5Gage) ——.. Mien GeaWiWEY ctaaalyegenngadIngen ee es)mec) a ee a TBs acaeayQt erenea ACT ween a yyet)me £ Sy]>ee eee oaeee)"<7:©ne a Srna 2do(N32) 0 a a. —Oateray =RYdx a——aroAe = macy =Lang Sv)Co 7Mesemneds BA) 1Neneyazy ee A Net kk Kaa boceoe—— dfs’) eR) “GixtGxco a) Alse aOn :Sy Sy ee fhsideeQtr %Eb Pad2kAG eSaieSY)ge Cn a, <n —— —Opuauain dd (aa8 ©Bibatkyooamd& yecade Gha% so Ce Oe) oe(ag) PO 210eae -=F *ghar)Beate72Meejo Callan 1970 ry ee eeSME TOSS SSCKSSTeBese meer rnaseMe QodLe Lg, 2 CONJECTURED SET OFEXACT BOOTSTRAP EQUATIONS 1341 < PA intemal lines. Clearly thecontribution ofthispar- tiouswehaveaprescription forcalculatinganym-legged = ‘ticularsingularitytothes-channelabsorptivepartisamplitude.Wethenhavehereasetofexactbootstrap A ci TraaXTat}thatis,precisely the-body intermediate equations written inclosedform,withwhichonecan =Yq.statecontribution totheunitarity relation. studyquestion’ ofexistence anduniqueness ofsolu- =Has; ‘Thusitseemsthattherequted singularities aretons,andwhichpresent abasisforsystematic 2BP present; whatwecannot yetshowisthatonlytheseare approximations. =present. =aa Finally, ifthesituation regarding unitarity canbe E ae ie jOWLE] 5%satisfactorily clecedwp,weavedeineatheorywith ACKNOWLEDGMENTS 2 : allthedesired properties ofatruebootstrap theory. We Wewant toexpress ourgratitude totheAspen Center =LAhaveawell-defined setofequations, (3)and(4),foraforPhysics, wheremostofthisworkwasdone,andto 5Fg setofvertex functions andpropagators. Theyincor- ourcolleagues there,especially M.Gell-Mann andB. =(283 poratecrossing andanalyticity, andiftheyhavesolu- Sakita, formanyinteresting discussions. E 2 —___ia , &. & euystcat euvisw D VOLUME 2,NUMBER & 1s OCTOBER 1970 . . Broken Scale Invariance inScalar Field Theory* * Curtis G.Cattan, Jx.t = Colifrnia Institue ofTechnology, Posadena, California 91109 .: and z Insttte forAdsonced Study, Princeton, New Jersey 08540 : |(Received 4June1970) = 5 Weusescalar-fild perturbation theoryasalaboratorytostudybrokenscaleinvariance.Wepayparticular = attention tosealing avs (WWard identities forthe sale current) andfndthatthey have unusual anomalies = - ‘whose presence might have beenguessed fromrenormalization-group arguments. Thescaling lawsalso = $ppeae toprovide arelatively simple wayofcomputing therenormalized amplitudes ofthetheory, which = ‘ldesteps theoverlapping-divergence problem, . Pn _INTRODUCTION Jeadstoasimpleprescription forcomputing there- - tke .. normalized Green’sfunctionsofthetheory.Finally, :- [BEgectecbation theoryofaselt-interacting scalstinSec.IV,weshalldemonstrate aninteresting connec ; hesimplestavailable modelfieldsi0,betweenthescalingJawandthepredictions ofthe : ‘theory, andaconvenient laboratory fortesting new renormalization grou‘ideasinstrong-interaction physics.InthispaperweTenormalization group.shall beconcerned with studying theconcept ofbroken scale invariance within such aframework. Weshall I.BROKEN SCALE INVARIANCE sfindthatthemodelcallsforsomeunexpected modi. si ic iesitis 3 ine" ° unexp' Insimple canonical field theories itispossible toiSeationsafourideasonbrokenscaleinvarianceAttheintroduceanacceptableenergy-momentum tensor!tametinehe_apocachsugestedbybrokensaleGwinthefaloving.proper(3)O=ts eeee aINEcutedproportional tothosetermsintheLagrangian having ’So een ‘ehopethattieeewatdimensional couplingconstants (suchasmassterms); luminationoftwointeresting questions justifies yet(4y’thecharge,D—./-d's So,formedfromthecurrent ‘anotherpaperonscalarfieldtheory. S,=Q,.x,actsasthegenerator ofscaletransformations, ThSec.Iweshallreviewthegeneralproperties of—S&—Oee¥aa 8 WA scale invariance asabroken symmetry, leading upto [DG),@)]= —id-+=-a)6(@), a theiden ofascaling law(theanalog forscale invariance ofPCAC low-energy theorems). InSec.IIweshallsee wheredisthedimension ofthefield;(c)thecurrent5, howthegeneral structure ofrenormalized perturbation satisfies d*S,=@ sothatitisconserved when thecearetheoryconstrains theallowable formofthescalinglaw.n@dimensional coupling constants intheLagrangian.andforces ittodiffer fromnaive expectations. InSec. With thehelpofthecurrentS,anditsequal-timecom. TILweshallshow howtheexistence ofthescaling law mutation relations withfields, given above, oneisable *Work supported inpartbytheU.S. Atomic Energy Commis *C, G,Callan, Jr, S/Coleman, and . Jackin, Ann. PhysinwaderSectSPAT GesnditheRARoce) 32,48CIO) x a OfteofScentiteResearchunderContractNovAPOSR70-1860,|#M.Gell-Mann,UniversityofHavcaitSummerSchoollectures, 1dilral Shan Fandom Felon 1969 (unpublished). F t i Summary ofthe Callen paper. fo) Theso-called scale-Ward identities derivedinearlierCCJpapersarestatedinthe first section. Recall that these state that: acertain differential operator S acting onaGreens function G”yields F,where Finvolves theVEVTOP ofthefields along with 0,the divergence ofthe dilatation current. Ifyou have scale invarience, then this current isconserved, 0-0, and F=0, This formelism was invented toprovide aframework for desbribing the breaking ofscale invariance bymass terms inthe lagrangien. Inthe next section itisshown that something iswrong with these scale-ward identities because they predict something which cannot betrue, Namely, you can show that consideration of§G"=F"forthe "basic n"ofthetheoryh (ie, forthe renorm parts like n=2,4) requires there tobead/d\ term added tothe operator S,aterm which does not come out ofthe scale-Ward identities. This analysis requires that you simply pay attention tok renormalization theory and how meny times each renorm pert is subtracted. The result isthis: even ifyou have naive scale invariance with nomass terms inyour lagrangian, thescaling lawreads $G"=0 where Smust contain thet extra term. Ithink this equation S$G"=0 islater called theCallen Symanzik equation. The extra term issometimes called "implicit scale breaking", nice that itcan beincorporated CO cosimply. Aless higborically significant part ofthis paper says that there isaniterative method ofcomputing theG*which allows youtoavoid alloverlap problems. Butthe main result ofthis paper and the reason itisfamous isthe statement ofthat simple scaling law. This law implies that the lerge momenta behavior ofagreéns function (iewhere masses dont matter) issomehow correlated todependente ofG®onthecoupling constant .Itisnoted also that thescaling lawisnotrescued merely byallowing the scale dimension ofthe fields togoanomolous. You really need that extra term in there. From mypresent viewpoint, Ithink the origin ofthis extra term isclearly explained inthe Abarbanel review. 6 — ——Camend. OFBitesSteBasioneOsUsaha socaSLpapiousonteoathUSOEq)y0.Bat)jngampravantire goate...Qnas cottneadeWarde. Curate ——wonasineS) oeee ee ee dB.Nwink_(s)Senne @%Seat arWtoeype uandh sus.gist;. fl)Massevilloggemrncntus BAMassvosanthe Meee Goma Couchots ooSoatiogLa,ghGepucket =NEScanesdysabe —Anwaleagn Mad,bycnapuhing? Daeratclajede Wowtananelenoanti,aye tea = Swe oe ee ee Hosen avcanyret. gash.Y Sesulhadk ok(20.ess eeai epee,Benete—=~ Rewscdon dash. -— _ -oe ee eeSas (Se AprewaneQayeidgeh,pedape ponduonbhee,Oedosanah anAGsate arsWe"Sedat eek ——Nte aedoerailingensiaubhadtsoXaamgonedopa ©Cou Mest iaprsanendDak=ote2GPESeeahem iadadeanass yO, OEaaah ee SEELEY EG) sepp TS eeBONS ed — 2 S a sae aWaSpomade onscare See QaVrenmmn Med B=2/9Ga dalek ce —oRaksetee B=p@Bpaot2FOB. 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BakDvaundMokSeo8eunvevsnee eleoSaye M(d-4)oe) Ce. fan)=O) aCov,Js.fosf) Yowoum, SakenadgullDasdoblec: . Coeyro'(Q,8 “=)ae fe) =OYaahsog)OEHa =XO fh) o‘ (wn MWY 6 :Benge OVO gOBh) | TCwoahqueseat,uavanance;Yarndtliteerm suskea}. oh[D=messisorteBasoe(HM) ) om. gChise,+Pan)=x?gi“Copsnter Ma) Yaoopalabebodksido.OmRaftgik0.Omaigkgit —D=1 ~0 a)Dete)+O°)EO,Me) XHf fan) 2 (o ae ” ‘ PSscompoke $GOP,day.=Mra) SEAR Pe) Uh =966%, f)/IO 4 BEE, atYEGwSBI» BEGG.) Ae ow ReM: ar Ag,=AapSocamas =ZeRSGOpds ted~=fraud) GheeuneDror pon usr O=DNB (OPsPrr---Prod) owl +NTZWHCOO,«fd. \2fiahvj&(BARA, a)=o, YooakDEVawkyouhowedorivah Con 2 ) Duwwobdod, Coldans® Eq.(2).Lonquntal| GobackeboOhCOSpeer+Maybecomes (11d)wrloQW), Arse Zao ow(yaad ADefasdooak! APejtop ped=[Orde z.qe"(25\GO)Per) VWnwJakeDrehaces+rsqloret, LPP pany=ATMO:Qpa) ZG tnd Qua,wanok =1EGre-fs) EPOfopert)+RECPr)=[YOr)amaya?€. .i= Pani,6 .wl AVCIR)=[Sm ZePBT GQgs), elt: Ss Z _@B oe &) os a[anes<=Go|CQssp)=AVES Parr)® FinallytwaBownCatanF'sC)L(2)|Onda vereadBsBoadoveKraremerllSAAoegease Age2dACoraur,bakrouseabracidy) LlamSooamncksqemeonWeWankaasoaccentQu AvwPree, SE=0o My Catan (FD (pusadQua) Seg. al OE | =dpBG)+y.REG+)a] baraaa >=JaBQy#-Z4dBae) nes Taenaeee.fxs) DSTCOsPn)=keCor]Bikefe) =[Yn-26B]en,Bap PE(p>few) Sesmaducasveud(4): omp35.Kw]G~ce a)olrere] =F| Symnzik 1969 : ‘Comman,math.Phys16,4480(1970) ModeswithSymmetryBreaking a ae; a allowsnodirectconclusion concerning e.g.whether, switching offthe-y2 ;sourcetermwhileleavingtherestoftheLagrangian unchanged, thegroundstateforthelatter would betheusual symmetric orthenon-sysmmetric " Goldstone one, although this‘question ismeaningful. Renormalizable Models TheGoldstone situation caninanintuitively appealing waybe .+rn 7 illuminated intermsofthebehaviour oftheground stateenergy densitywithSimple Symmetry Breaking tsafinetion ofthesourcestrength, Thediscussion hereto,fecliar in I,Symmetry Breaking byaSource Term theclassical case,carries overwithfewchanges toquantum fieldtheory, whereby, however, aformal similarity tothetheory ofcondensation K.Symanzix ofYang and Lee (8}isnoted. . Deutsches Elektronen-Synchrotron DESY, Hamburg InSection 1,thewell-known one-particle structure ofGreen's functions ispresented concisely. InSection II,forGreen’s functions Received September 30,1969 involving current operators thathavesimple commutators withthefields, Ward-Takahashi-Kazes- andRivers-type identities arederived. ;oaLagrangiandensitywithinvariance undercontinuous groupftiner|IfSectionIIL,theformulasofthefirsttwosectionsarewrittenforthe :sansormstionsfhefaserastortnerinhedewacded,thesymmenys |specicaseofasymmetricLagrangian densitywithaddedtermleat i ingeneralreducedandthecrrensassciaioa withtheoriginalsymmetryareonlypartialy|inBostfields.Theserelations areusedinSectionIVtoobtaintheBPH 'Conservedthetheorywitouttheaddedtrm&enormaliabl thheywihtht|renormalization conditionsintermsofonlythenumberofparameterstermalsoisandtheneededrenormalization conditionsaretheessentialcontentof thatappearintheunrenormalized Lagrangian. Thesameisdone,withPaeaeincainBonRideCoveytna)hecealoedcouplcrthrierargeutar wig|somenecessaryprecautiontoavoidspuriousinfrareddivergences,in ; respecttothenonsymmetrc limitofvanishingsourceterm,aparticularGoldstonemode,|SectionVfortheassociated Goldstone mode,i.e.thelimittheorywith andwith respect toproperties ofthe ground state energy densityasafunctionofthestrengh |vanishingsourceandspontaneously brokensymmetry, whichmay cofthesourceterm.Inducedandspontaneous breakingofadiseretesymmetryarealio|alsobedescribed directlyintermsofamanifestlynonsymmetric La- | treated, grangian. InSection VI,therelation between thetheories withandwith- . . outsymmetry breaking source term isdiscussed and acomparison isIntroduction madewiththebreakingofadiscretesymmetry.Theappendixcontains ; B.W.Lee[1]hasdiscussedthesigmamodel[2,3]fromthepointof|thediscussionofthepropertiesofthegroundstateenergyasafunction view ofrenormalized perturbation theory, inorder tohave availal ofthesource strength, which areobtained using results ofEuclidean modelthatsatisfiesPCAC!andallowstocalculate inaformalbutcon.}quantumfieldtheory.sistent waytheamplitudes forprocesses involving nonsoft pions. ‘Thecalculation oftheGreen’s functions involving acurrent operator ‘Weshallshowhere?thatforsuchmodelstherenormalized perturba. }will,becauseofthetechnique neededhereby,beincluded inthesequeltionexpansions canbeverysimplyobtained iftherelations, stemming |-papef,whichdealswithsymmetry breaking byatermbilinear infields.fromPCAC,betweenvertexfunctionsofdifferentnumbersofarguments ;‘areexploited. These relations yield allthe*renormalization conditions" " * .requiredinBogoliubov-Parasiuk-Hepp (BPH)?renormalization theory 1.One-Particle StructureofGreen'sFunctions ,intermsofonlythatmanyparameters astheunrenormalized Lagrangian Wewishtoconsider thePoincaré-invariant theoryofamulti- has.ThistechniquealsocoverstheGoldstone mode*obtainedinthelimit}component localhermitean fieldA(x)describedbytheLagrangian density ofvanishing source but,since itdeals with renormalized quantities only,=—_—_ L=L(A,0A). qt)7Ref GHdiscusses thesigma model and related models from thepoint ofviewof ; applicationstopionphysics. Tothisendweconsider, following Schwinger [9],therelated theory 4shortaccountwasgiveninRef.(5) describedbytheLagrangian density 2SeeRef. [6)and references given there“Ref.(7)givesacomprehensive presentation oftherelevantmaterial LsL(4',0A)+5A'" (1.2) 1Gonm th Pn Vb 167 Summary oftheSymenzki Sigma Model paper. a) Probably IshouldnothavereadthisthinginasmichdetailasIdid,butitwesfilled with interesting manipulations. Mymotivation for reading this paper was simply that Iwas reading the source papers for Callan Symanzik equations, and the Symanzkk paper said itwas taking its notation from this Symanzik paper. This paper ismainly interested inthe subject ofrenormalization inthe presence ofSSB,something Idonotwanttogetintonow.Thetool Sym.usesinthisdiscussion isthe scale Ward identities like those general onges (for any conserved current) that Isawderived intheCCJpaper. These Ward identities arestatements about theG” greens functions andtheir relation toVEVTOP with divergence ofthe conserved current inside, ie, the same type ofequation Callen used inhis paper inthe special case of scale symmetry. SoIdont really know howthis tool isused, butfwasinterested toseehowyou can restate the Ward identities intedrms offirst the generating functional G[J], and alsointermsoftheproper vertex generating functional [A].Thesetransformed statements ofthe Ward identities get fancy names like Kazes-Rivers identities. The connection ofthis tool toits work is, Ithink, that the sigma type models break their invariance byaterm linear inthe field (you can think either ofscale @_ wreaking somehow, bytmainly breaking ofCACtogetPOAC). Thislinear "signa" term looks just like asource term, and this source term you recall isthe basis ofthe _functional approach. Sosomehow the functional approach gets used toshow rénormalizability. 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A \ ByatYr PPo=SQ+peerog rr oes| 30 ¢+@@O@Qe+ +O» ~ NVawvefedromQaladeroeollnctaadrclemadsrbri @Sapper1THLaearneng:Mase 6@QO-@® 3 Pe 9BeBe PBsi+oreo Guda&way.Bbunlxan, uot, Te,ADusheroma rglacedby®|bnVv, beBD eepakabamoveSob. 6 Reus . Oliow Gv) . COs =YaySeasAGDast. Geaw ro) Gah =LayBG) AW+KENT ah CS Tan, Ohomuk: BOLE =Stydey AG+3.2G) (s) Veorswstis Gea, GOS) =GOD) -LUA%i[AO+*G@] @) CamoinnG),(2),UX) 6a) nea) Ce)34FSM YLSCA =Voya3)Ala)Co ~ifaoyroo) &TIAL (s\” MeovasBb)doardock OwArkin peed Yh: iA)=NTA)-2YaySOM) 7 WoeAGPEs) Cak1)>COy 3stAade unto Yoo.presen Seud(5).Ccseat ~iAMWRIA—GoatUTA)+{S4enAy) Beayia f-—~— JyRS)-GOAAW/=Soy@@GO)Re9)Avy) / =iv6H)AK GrA Q YY LTAQAM +BMT e =SAT—GEM AEOJoyCOAL).= 4“\p.TNH~oF!+YayGoreTCOMY) zs- ndye “it HH AQ (*) @ \ Ss LL 0\Gia)=BoA)~AGEWal-i66)THTA) =) SING,A)=AAT VIA Rieu )Weenishandt aWakaacnCefuGaCAVMage. Gs): dyeUAL vem der fERT ee&Ge) wedaddy poureee oe e+ Cee), OBdrtsdag wtWotinenefmut Gea) REG, Loy), CAS) Tow) oe Z Z Pr);aC27] \Guu) )&BIE) Geis) I@AL MBGAL|YATAG) |O~-- Sok&aemake 4Moret , . Ged My) worQhiy), CLGgh=Gers)Leity6H)TOY) ° Wed WHS Gods wtPre a -doga;At'yLesap, SHE) Gans) =Calas) 7 4 RE2)~GaGa)+SayGey)@Lim6073) Que _ .Gyr)iOO) IG0s4)=Cabra)+iGy)it6). Qn YdsPEGEGs)Galea)Al)=Yslion GO)Aly\].——-_ ° SSoph =46G)T6055)AG). 2 we N25266)26013)Abs)£3.07 =~i(Asolo Ri ~ >haP-claAcQeRial—NacatGon)Als) cDuaiaSeeEACEI)- oh “Wr, greShia, wee a aa ~EGAIT PLAY~~00aTEADayCas,CO)TEOIAG) = a*=}- LAEDXAYLOT RELAY ° Symnzik 1970 Q 6226 K,Drul,R.Haag,andJ.E.Roberts:OnParastatistcs ommun,math.Phys.18,227—246(1970) nN 411.Dopioher Haas, Roberts1 bySpringer-Verlag 1970|.Doplicher,S., Haag,R.,Roberts,J.E.: Fields,observablesand gaugetransformatic‘,Some‘ianBe is,observablesand gaugetransformations!I, q2.Yang,C.N.:Conceptofoff-diagonal long-rangeorderandthequan i PapeRevinamitSsSomegeneralproperitsofpra-Fermifedtheoy,|SmallDistance Behaviour inFieldTheory &~ : 14.——Wavefunctions ofidentical particles.Ann.Phy d Ce i q icalparticles. Ann,Phys.St,337(1968) andPower Countin; 15,Wey,HiTheclassic!groups,Princeton University PresPrint1946, 6 |K.Symanzik q K.Drant Deutsches Elektronen-Synchrotron DESY, Hamburg RoHaag qJ.E, Robects ‘ceivedMay12,1 b 4atafrTheoretischePhysik ReceivedMay12,1970 |er Universitit 1.2000 Hamburg 5,Luruper Chau Abstract, Forinfinitesimal changes ofvertexfunctions underiniitesimal variation ‘*verChaussee189]aenormalized parameters, linearcombinations arefoundsuchthatthenetinfinitesimal 4Rangesofallvertexfunctions arenegligible relative tothosefonctions themselves at /bngemomentainallordersofrenormalized perturbation theory.Theresultinglinearfirstfeceracta! differential equations fortheasymptotic forms ofthevertex fonctions are,‘quantumelectrodynamics, solvedintermsofoneuniversalfactionofonevariableand {fectunction ofonevariable foreach vertex function whereby, incontrast tothe renormal-nongrouptreatmentofthisproblem,thuniversalfunctioniobtainedfromnonasympto- .feeonsiderationsArelationtothebreakingofscaleinvarianceinrenormalizabetheories ‘ ieseibe, : ! Introduction i ‘Thesmall distance behaviour ofGreen’s andvertex functions in , renormalizable quantum field theories hasbeen extensively studied in : formal way viatherenormalization group (1,2]and, with equivalent . results, some other approaches [8,9].Hereweofferanalternativeap- ‘ proach tothesame problem, which appears toberather more direct ‘ itleads toformulas that areexact and become theusual asymptotic ones upon a,inprinciple controllable, neglect. Westudy theeffectofinsertingoneextramassvertex,orageneralized }mass vertex inthesense ofWilson [3],into allFeynman diagrams for allvertex functions, (Such vertices aredefined asthose forwhich the simofthemass dimensions ofthecomposing fields, ascalar and the electromagnetic field having dimension one, aspinor field dimension : +|three half, aderivative dimension one, islessthan four, e.g,two fora salar mass vertex andthree foraspinor mass vertex.) Bysuch insertion, i thesuperficial divergence Dofthecorresponding Feynman integral is \ reduced? (e.g,bytwoandbyone,respectively, forascalar andaspinor | massvertex).Reduction ofthesuperficial divergence, however, results |indecrease ofthelarge-momentum behaviour bythecorresponding ‘ power ofanoverall scale factor’. This relation between dimension of See, eg, Ref. [2]p.321; theindex «(G) wecall D.Also ibid,p.341, 16Commun.mathPhys,Vol16 y— | ; "QSamypabins” omprrsh reference Symanzilce fa) ThispapergetsresultssimilartothoseofGellMennLowandCallanandCJ,namely,that the Greens functions (or proper vertex functions) satisfy certain differential . equations like i.17 ofthis paver; derivatives are with respect tothe "parameters" ofthe theory, mass, coupling constant. Recall that Callen showed the need for the extra coupling constant derivative not normally included inascaling enalysis (ie, ascale invariance analysis, eg, the scale-Ward identities ofCCJ). Onthe other hand Gellmenn Low got their asymptotic forms for greens functions from the global renorm group equations like d=ddstuff, same asinBS. The difference here isthat Symanzik derives his equations using the functional methods. Insection Iheconsiders gandendsupwith theusual Callen ~Symznzik equation. The key ingredient isrealizing that certain objects have degree Dpositive andthis allows you(requires you) todefine various parameters like gandm.These then appears inthe CSequations. The CSequation inturn can beused toget aform for the asymptotic behavior ofthe greens functions, and you see the deviation from simply naive scaling behavior, de, thelogs. Later papers atthis point combine with Wilson toexpalain scaling results at SLAC. 6 Nextsection repeats everything forQEDratherthan#4.]" Appandix Iguess shows howthenaive scale invariance predictions arewrong. Atthe moment this paper istoo technical for me. Iwould rather get back to the Pagels QCD review Istarted. Once Iunderstand someones derivation ofthe CS equations, Icangoback and seehow everyone else derived them intheir ownway. -~ lo) DarineMeeFawnnsa,(2)€.5)(1-6) OfereetokOkewayoordoperder Drie:ik, Ocm-\uygoSThEOTE] _eieCiS)YasSE GT =og(AkBES)) GI. QheeUprrenenty woskBAsea judhhoDL,ababsapprcwe Yates b=fEhw romeac, QSrqpecr AL=somegtVeaCoe esuterleur .YroLouu./ wow anexGs)QYde(©.3)soe4wp Fact by rant Aw,provabty 3wihchoybyame04,anhFascemmmrlge b>PurBV.te,AAnaQre(5)gowe prow(x3). @Y's aboot (8.6a)- CC)=costont(iSte(4ont@RD)) OO) x(\+Saow22)< =AGO) \a(4 ont) 6) =amd, SeGx(5) awermfds (cba).Te,Jueam-stad © @) fetta BeacrcsiadwiltteocaBedatirediamSaotageBE -ontghivyas1B227C8) —A.GollMounttea PRIS,Roots)- aBaqatinloal +Shinkor book,(1953) Le—-Ain.pagerfousdancedra9mdacesnap, ________ - ee -Peagpriund ste: Beee bwin ine ijusUiPasa op aeer Sa mannan a lavavppeard sinbe,nuplan,Taoleadamuseeatamalaecda) —J + HADRONDYNAMICS: "Wewlsouctierfnourwudierce mbasLedofsmall henry 9.3.Atmrtaved hierly wen tromSebnertndy, RevYore. Yeramyug National Accelerator laborw:cry .ue “ - 4eaieyBinet”BR posers. ' ASitat's Garde:ofGrass |tBS.YargotieandByclorfene asamct |Develommeats inthedynaaics ofhedcone entthelr Seterestione are unresuczion, meaosomty avo APcLcor : 4 presented sing thethresd oftherenormalization group tohold thefubeleeseisEhreadofthecevomsitsstion grouptobeléchefubrle ‘heaubsectoftheseLecturestescxgoneseltghifud$aparticle together. After anIntrotirtion totheideas andeguations oftherenomalt- iysics: thedynamics andstructure ofvisstrong decerectione. 1%Lsalso tationproup weSlacuis thesolution toondusefulness ofthese equations*<theseos f:fuedaverymichAna“openndorn'sex”wnatoftenaltoesogo Anwedvithehistoolveconsidertnwovedepthtwoseentagiycuaparateaspecte i ; * feyspsspest fon$8hope.Onthesrbarebonestherehesteen&creatdealofclever,thousht= ' ofbadron pnytice: (2)gaige Bheortes oftheatreng tntersctsone--in perticu:in . “* ios FulvorkwhiehatteaptetoaniversoaeoraZ-ofthequeettone: 7 tar Sutrviotes treedea'bas deep inelastic scattering, solels ofhadron strane A.Mae 18theortgin andmature cfthecvpuane structure, theReser tare)theuseendnecessity ofcherm andcolor, and 40onwill beAiseussed. ' fnchukoFvacuim singsiarity, whieh teresnccatbie feralnost constant total ‘ (2)pittrection settering andtheFosereschuk stzgulartty--in pacttculer, the tee ' “° : aa vous aecttose 500coleululy branchpointstntheJeplane,deccupiing *heorens,‘chancel FApane 2.fwcanlocalquantunfieldthecsybeconsistent withtheapproxt= | Aecontizuttyfomula forRggsont, tnd toonvill beaddressed. Sooe coments : savetolling ofdeepdnelastie seructure furstont? | VID bepresented ontherelation ofthese teJaporsant subject,: er“ 3,Arehadrons composites made outofconetizuents, generically called erowhout the Lectares Mere Asanempheeie onFecagoey rather thansony quarks?Howmaayandwhatquantunmuuberscarrychessquarks?Howisitpos- subtlety,Sdsas andelesentary emaplee areetzessed andtheresults ofe- thie thetthese pressed constituents have sever beesseentnthelaboratory? | tailed calculations, vhenpresented, arelifted outofthereferences like ‘ can vemake 8virtue out of this shyness of suarke? ; sagic. Titecourse ofaectires should serve asanSetroduction, then, we4.canwehaveaPoeronwifchgivesrivetoelnostconstant cross q ‘tothemainstream ofpresent ideas about hadrons andtothemoreadvanced ‘ sectons nd Limiting {nelusive eroes eccticns shows Zeetorization, reeulte So Lectures20begiven 4athe Topical Goaterence tofollow thie school. ™ ‘8triple Pomeron coupling, and yet does not decouple from particles? ’ : 5.Yoweas one inplenent unttarity athieh energies (direct chanael fod crossed channel unttarity)? Does thie provide the constraints tosecure se&solution tothehadron scattering probienst ub These are ascog the mejor tsnues ofthe éay. Tone ofthea baa cox pletely yielded todotstion; each ofSpun tasbeen vigsrousy andtnnglantively — stacked. She on thing th nach are the, eujees of thee stores, Be | ERT 8cae ite ontite Bonen, 29ty8 nga, |AginSant193. 39 2 at et | a sie ~Honeam epSachin OsRowe. Arg"PasirS19]Bysend OY(eBeSONS). 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First, youhavetofindawaytodefine a . _-Fenormalized field’theorywhichissortofcoordinated at@renormalization point fhe Te,this point isused aspart ofthedefinition oftherenormalized massandcoupling.Thenyouareabletoshowthattherenormalized greens functions f\")satisfy acertain _ homogeneity conéition, based only onknown dimensions ofagreens function, Theindependence _oftheunrenormalized greensfunctions onartificial pointJAthenleadstoacondition ontherenormalized f!,acondition known astherenormalization group equation forf. When you solve this equation, you learn acertain correlation between scalapg allthe momenta, andvalueoftheeffectivedcouplingconstant.Ifacertainfunction®@hase_ Linearzerowithnegative slope,thenasallp;in\'scaleofftoinfinity, youfind_that{canbecomputed intermsofanother f'whoseeffective coupling isgi,theplace __ __. where Phasazero[thisisanUWfixedpoint]. Ifg,=0,yourtheory isasymptotically _free,Thisisthought tobethecaseforQODbasedonparton datainUWregion, . 619aR Neoswins2\*GangeVenta.Pomegenot <@cmMIE) Orsoaarew Qneny: .7 .a)robctnda gpg.fepySahcupannnienomcEts2mVpwnyMODgmaryeHddeonBas :ay-8)rasaMawpuncntecalhgousionuncnndet Eamonfronds Bs. 8) wnaYralteandAanomnabryaedty seaBodnByes aA.weohrares0880spmocaineBSrarinenJackgpgyavanrouihNee eee BnCape QwOGoveg &abate EER OT QIN)Voss.Guaspade,QODDagangadnUecedel SoTasnstocaad Qk, PereBy geo 9Reasd ;VeeaoeonTice) gawkandOWieoarpengalssaldey fuse! ©usin aqugArarer wkD=\.blewet}aqua APY ~ WB-829), Ss rranroe al9;>&au.onwoosS08},yhure., atheAEE RoeOw“*hy- Y)inDaagineddatacodedoeWat) 0 2k 2)Dienasadobed do.Ce[BGO VIN). QksingWhimOpaintuotect vlan,Co=BOGE <piAip)xwe.2. ©QRFidhuay porcomesdureDaaETE=TG)9Gy,ena7 aadOutwight. PWeCRAE)—3 RS)\BFcandy aQedcellar AFM, CGE) —e+)fatesOokhoe.SLAPgureasoulcvedenasSa an (peers) Thasdiendiia cambiar, Iadamvoillsursak+o L- So,asfarastherenormalization group isconcerned ,thissecond part,ofAbarb's_paper.*: __does not_havemichtosay. .__. 3 a)QODtheory, people have computed BETA andthey find that theory isindeed __ - " _—- ..-asymptotheally freeinthe_renorm. groupsense.ThereisaZoxedpointatg,-0 a _. ..andconstant BETAhasthecorrect sign.Moreover, _ Lo. —_..-b) itisthought that_non-abelian gaugetheories aretheonlytheories whichshow - thisproperty. Abelain theories donot. =, . c)Renorm groupisutilized somehow intheWisonOperator Product. idea.toshow.that._ atheory whichisAPshowuld exhibit Bjscaling, justlikeafreetheory dbes,more. -oor less... See ee : ~——~— There.is:no. cnnmection made.between gauge.theory andcritical phenomena ormyprevious Imowledge of.therenormgroup.InfactIdon'tevenknowwhatcorresponds to-the. 9_.....- --— correlation length. 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DYQademinerSahe. ~ g [Phataays MRE) =oe(SDA) : i w. 8 X =Lslt \CAGA )vehquichem. _ 2 Ass: Sa 4 jl| : “focyAms) =&-&CAYG : a. =he he . - : -}own. - re)PreagtaumrrhQmamntle; . SgBye ghA=)baketdcoated, PO Ob,Aaerneasmorrelts .Ymages53vdmnleher. 1 .i Flexsye)= Fy, 28°) ; 7woud.JindkBichane~Que6,187 :a ‘_ - Dhonloame? a 8 SLACDecline wolue#/e-reft190She arnt. natrbrsle ovePOP 2 Dur_omsbtiouts geste :3 _—Shoo tanner Ao),contain prndisctdol Psadds willy. ; prwoIQ-— 12onoQs\e“4WeAdi Gop Quintaerat atl——MarpacedsNoam Or gecng Dra smGrater (met Cabdo - Seema? oat ———(beGe) btpron“ens ot ade “Q0 tem) Se oe Lee)2J 3 Gell-Mann &Low 1954 /1300 W.J.SPRY . -BtheresultsofthisexperimentR=cc/xb*=(1.0:0.2)totheenergyofPanofsky’sexperiment,butthatthisJ,tweentwo]10"sec~ifthelimitsPlacedon8,°—aj?aretakeninitialslopeis+(9.2°)q'.Theythenfitthisinitialslopechargesgseriously.ThisassumestheradiusofthemesonicBohrandthedataofthisexperimentwithasmoothcurve5aorbitbe22x10"cmandthatw%=8X10"cm/sec;forfay"tersus9.Weenshevaluesof8andafq)20° co(8x/9K)(24/0)(6\°—ay")*.Panofsky's experimentfromhigherenergiesareextrapolatedwiththisrestric. fer showsthatthiscapturerateshouldequalthecapturetionontheirdifference,itisdifficulttofitthedata rateforthecompetingprocess#(r~,7)# withoutassumingthatajvarieslessrapidlythany’and|Bemardini"” hasdiscussedthe‘crosssectionsforthat‘8svariesmorerapidlythan1!intheenergyregion. Plyx*)n,d(yxt)2n, andG(r,4~)2p forEybetween170between20and42Mev.Forthemostprobablefitand190Mevinthelaboratorysystem.Ifitisassumedundertheseassumptions ,°changessignbetween20|.thattheratioofx~tox*productionobtainedfromtheand30Mev.Thisenergydependence forAy?suggestsasecondreaction isthesameasthePhotoproduction Jastrow" potential forthisphaseshift. J - ratiobetween thefreeneutron andfreeProton then|.Herea=e,' theprincipleofdetailedbalanceandthisratiocanbe ACKNOWLEDGMENTS tureconst.usedtopredictthecorresponding crosssectionsforAnumberofpeopleaidedtheauthorinthisexperi- mrestmasso| P(r).Ifthesecrosssectionsareextrapolated tothement,‘ThanksavedueespeciallytoDr.A.Robertsand Tfrh, energyofPanofsky’sexperimentBernardiniobtainsaDr.J.Tinlotfortheirhelpandadvicethroughoutthef°form, captureratethatrequirestheinitialslopeof8—ay°toexperiment.TheauthoralsowishestothankDr.J.~a be=:(9.2°)y'incontrasttothevalueof—(16.5°)q’FrenchandDr.H.P.Noyesfortheirhelpfuldiscussions vont obtainedfromthisexperiment. oftheresults.Otherswhoassistedandwhocontributed 4nrl BetteandNoyes'*have givenanargumenttoexplain91ne768suggestionswereDr.E.Hafner,Dr.F.ff thisdiscrepancyintermsofMarshak’s"suggestion.InTenney,R.Santirocco,andD.Nelson.W.Coombs,briefthisargumentassumesthattheslopeofB’—aj) Enslein,L.Braun,H.VonThenen, andF.Palmer Jobtainedfromthisexperimentcannotbeextrapolated wereresponsibleforconstructionofpartsoftheequip- where721 ‘+H.BetheandH.P.Noyes, ProceedingsoftheFourthAnnual mentandforthereliableerationofthecyclotron. irie "'R,Marshak,Phys.Rev,8,1208(1959) WR,Jastrow,Phys,Rev.81,1165(1951). makinguse +invirtueof o. —_— tiesareintin| %,fenormalizaiPHYSICAL Review VOLUME95,NUMBERs Serremuer s,s9seJb[normalial~physical args QuantumElectrodynamics atSmallDistances* Atest bod ‘surrounding. .GztieManwt ano F.E.Low ¢ mur (ReceivedApril1,1954) Fsamesignas. 8g intheF| Therenormalized propagation functionsDreandSpeforphotonsandelectrons,respectively, arein- thetestbod;Yiglgatedformomentamuchgreaterthanthemassoftheelectron.Itsfoundthetfataregiontheindi- [*thebodyfroforma,AasthaPerturbationseriestoallordersinthecouplingconstanttakeonveyron seeaneffecti+ cernfont she entreseriesionlypartlysucelIt isfoundgtheceeee>izedcharge, Snitedrouts Stations byvstofthesenocmalianblty ofthethor.Iphotonselfenge mene,‘omittedfromtheseries,sothatDroDe,thenSpehastheasymptotic formACS/m*)[iy-p}", where Penetrate thA=A(6)andm=n(e;%).Whenalldiagramsareincluded,lessspecificresultsare.found.Oneconclusionis body,thecha tances anehtofthechargedistribution surroundingatitchargeinthevacadnsOneSe chargego,co fanciesrae, couplingconstantexceptthroughascalefactor,Thebehavioroftheennan clearthen? .Thscoordsmomentsirelstedtothemagnitudeoftherenormaisiton constantsinthetheory (1.2),muse Sanitshownthattheunrenormalized couplingconstantaf/éne,whichapheatgaa theory 12),must SenaTcesittherenormalized couplingconstantef/tehewithdivergentcooler eng Thus,using(itheoftwoways:(0)Kemayrellybeingniteasperturbation theoryindicate ©Reimay beaGitenumber ipa oeeo aaa{tt 1.INTRODUCTION vacuumpolarization. Theywerecalculatedtofrst wheretheiw 5#weltknown factthataccording toquantum orderinthecouplingconstant«bySerber'andUehling” Uithuicallyte electrodynamicstheelectrostatic potentialbetweenshortlyafterthefrstdiscussionofvacuumpolarization Toga in cracRastchargesinthevacuumisnotgivenbyDiracandHeisenbergWemaycapristhee1.sonnek 0exactlybyCoulomb'slaw.Thedeviationsareduetosultsbywritingaformuleforthepotentialenergybe ‘SuchdiversyahcreanfROBESageomeC8.OeofEaline”beeBay wheneverobs. {NowatDepartment ofPhysicsandinstituteforNuclear 385Pict,ProsCambridgePhilSoc.3,150(1938) °F.Schwinger Studies,University ofChicago “W.Heisenberg, Z.Physik90,209(19843. Summary oftheGell-Mann Low papers ‘ 8Byfiddling around with dimensional analysis, they come upwith certain "renorm| alization groupequations" whicharenotofthedifferential "CallanSymansik" form i but which instead ere the global Z=Z.Z form, where the cutoff papameter and momentum appears asarguements. Seeeg(3.9). Aleter section ofthis paper actually uses two separate cutoff parameters sothe multiplication equations are alittle messier. You cannot completely solve these renorm group equations, but you can learn something about the form ofthe solution, inthe sense ofKNO scaling say. You learn (asseems reasonable frommiltiplication idea) thet S(p) propagator goes asapower(P*)” butyoucannot compute n(later Ithink AFboys shown howyoumight compute n).Just how you solve the R-group equations toget this general power form isdiscussed inan appendix. Obviously, ifyouknow this "form ofsolution", youknow something about thelarge p® behavior ofthefunction Z(p). Since aZscales etoe,,youinprimciple arelearing about €,-Inother words, given thephysical charge e,bylearning about thelarge p* behavior ofZ(p) youcanlearn about e,.That wasthemain interest ofthis paper: just exactly what isthe "bare charge", the charge you see atsmall distances, high energy. o) The"scaleformofthesolutions" canbereformulated asanintegralstatement(5.9) .This statements contains functions PSI which islater tobecalled the BETA function. Here isthe important fact: ifthis function has azero for finite argument, sayX5ythenthelarge k*behavior isdescribed byafinite Z(called d),andhence the bare charge isfinite. Ifthere are nozeros ofthe function sSPI, then the bare charge isinifite. The authors are not able tostate which isinfact the case. Ithink even today 1978 the answer tothis question isunknown. So, tosummarize, this paper hwas two important contents: first, ithas golbal renorm group equations ofthe type you now see inBogo Shirkov and Bjorken Drell. Second, it shows the "solution" ofthese equations interms afaBETA function whose zeros have the significance that thehigh-energy charge "(bare charge) can befinite andnot inifite. Later welearn that in@NAGT you have asymptotic freedom which means that the high-energy charge isinfact zero. This paper does not contain differentiel R-group equations ofthe CZtype, nor does itsay anything about scale invariance orPCAC orconserve dcurrents etc. Their Gwin interest wassaithQEDandwhbther e,isorisnotafinite number. Afterall,thiswas written in 195). 8.23.78 re)Gellman-Low: "QEDatsmalldistances". 195k : Comment: This isthe paper which ley dormant for 15years and then led into asymptotic .freedom via the renorm group equations. Authors are ofcourse interested inQED only. ? : 1,Introduction. What isthe true pobentiel between two charges? Asweimow, itis the PTofthe photon propagator and will thus contain effects from the vacuum polarization graph. The finite parts ofthe vacpole effects are shown in1.3 as acertain 5/6 andsoon.There isadivergent piece also, ie, asyou take r=0 “ .*yougetalogdivergent piece. Thebracketed factor isthenjust25%andthisis its usual log divergence. Ineffect, (1.3) is2computation ofthe bare charge interms ofthe (Observed and finite) renormalized charge. But only toafinite order inperturbation theory. The authors are wondering this: just what isthe bare charge e,toellorders ofperturbation theory? Isitreally infinite? They have this nice way ofvisuelizing your measurement ofthe bare charge: you probe small. distances so you getthrough the polarization cloud. Asyou really: getinthere, you seethebare charge ey. Interesting howtheword "bare" ieused but,this picture notusually conjuredupbyme.* : fo) Asanaside, they quote that. QEDrenormalization was proved in1949-1951 by. Dyson, Salam and Ward. o 2.Representations forthePropagators. Basically thevarious Z'saredefined. Dp" Astheunrenormelized-but—full photon propagator, thething thatwillhaveZ,in its numerator’sttherenormalization pointwhichisp=0.ThenDggigwhatwewould call the renormalized propagator, see (2.2). Certain integral representations for 25-1and2571aregiven, Ithink these arecal@ed Kellen reps, nodoubt based on uniterity somehow. Acomment ismade about photon mass renormalization end how you have tothrow away the big quadratic divergent term. 3.QEDignoring photon propsgator renormalizatitn,eie, with 2,1. °Suppose you compute Zp You canuse aFeynmen cutoff onthephoton prop sothat electron self- energy perts erefinite inthecutoff. Bvaluting@(e) with renormed charge ey, ifyoulook near p=myoucangetZp,andifyoulook forlarge pyougetsome result asshown in(3.5). Again, just’ afewterms in’perturbation series for these quantities. Equation(3.9)showshowthenumerators oftheunrenormed andrenormedphuknxpelectron © props are related inacertain Limit: cutoff.MGT.p.MGP.m. You notice thet the renormalized propagators numerator Sgisafunction ofp/manddoes nothave problems involving thecutoff. This thing isfinite. Ofcourse 25isfunction ofX/m andtheunremormed prop isfunction ofYe,where mis neglected. This equation (3.9) isthekind ofrennormalization-group ratio-style equation Ihave senn in BSand BDand elsewhere. Roughly itcays that the UVdivergence ofthe unrenorn propisallcontained inZ.Asyoumight guess, theonlysolution thatequation (3.9) canhave isapower solution (ie,arepofthemultiplication group). thus, inthispartial theory youfindthetZ)isaconstant Atimes(S/#t)™ where n issome power. Ie,inperturbation theory you ereonly getting afewterms whéch arelogs, butthey seem toalladduptoapower (afamilar happening). Then you conclude that thefull electron propagator must behave as2power ofp2/m@, andtheb usingtheWardidentity youcanlearnthebehavior ofthevertex. Ithink basically this isjust dimensionality arguments atwork. Equation (3.9) arises merely because youknow what each function candepend on.So,using the renormalization group’argument, youcanin-effect compute theformofthefull propagetor ,although thé little power nisnot so-computed. That was probably the reason Gell-mann Low stfor awhile: noway tocompute n. “Notice, bythewey} that there arenodifferential renormalization group equations in’this paper. There seems asyettobenomention ofa"renormalization point" inthesense ofpofPagels review. Butthere isavague idea: (look at (3.9) that youcancompensate foracharnge inthecutoff ®bychanging 2p(bychanging thecharge...). Ie,thinkofsg=s/z9.Thisratiohastobeindependent e ofthe cutoff because itgives the finite, renormelizaed prop. Ithink-it isthis same idea which decomes the differential equation later on. ij.Ward's Method used asacutoff. Thevertex Dyson equation iswritten down (and Dyson given credit fordefining the1piobjects, Ithink). Then these equations ere "subtracted" inacertain wayh which causes theunrenormed SpaND Dppropagators tobenormalized (ie,unitresidue) stpoints f=\andkX. Thus, although they donot use the term "renormalization points", they are ineffect choosing independent renormalization points for the two basic propagators ofthe theory. This then quickly leads toequations like (4.7). This equation relates the originel divergent functicn Sp'(e,). totheZo(% X).andtheSp(d,%) +Ie,foranyparticular value ofthepairXs thisletter object isthefinite renormalized: propagator. There is sort ofadouble renormalization group here ,twofloating parameters. Now (4.7) isjust (4.7) with these parameters attheir usual values (ie, theusual renorm points used inQED). Nowifyoudivide (4.7) by(4.7'), then theoriginal infinite ordivergent function cancels out, andtheratio ofthetwoZp's is.called 25,thus yielding equation (4.7)". Thisequation tellsyouineffect howyoucandeform oo) from the"stendard” renormalized propagator Spy with itscharge e;toanother prop that isafinite distance awey, sotospeak. -2- One might refer tothis little triad ofequations astheRenormalizetion Group C)_emstions forGED,Mext,theygoontorewrite theseequations intermsofthenumerator functions only. Notice that25!asanumber isthenumerator ofthe standard finite function Spqevaluated attherenorm pointf=\.Thusyou endupwith thesetoffunctional equations (4.19) through (4.20). There seem to be4,functions and two parameters e;- 5.Asymptotic Behavior ofQEDPropagators. Using dimensional arguments, ourthree renorm group equations arecast into theform (5.3) through (5.5). NowIhave to golearn how the “solve "these things. This isappendix Bsowait. For now just look atthe solution: itislike KNO sceling: the function d,=photon prop numerator =F(z)where 2=(k2/m2) xsomeunknown function ofe;7. Asyoutakek?/m? toinf in(5.6), youareineffect computing thebare charge, asshown in(5.8). So,suppose F(2) =finite value aszgoes toinf.; then the thepty has afinite bare charge, which depends onthe function Fbut not atallonthe number ey. Onthe other hand perhaps F(z) =inf asz-inf, inwhich case there isaninfinite bare charge, this iscase (a), which wemight nowcall UVSlavery (coupling grows without limit as yougotohigh energy). Perhaps asaspecial case ofcase (b)youwill have eg fo} approaches 0.ThisisAsymptotic Freedom. WenowknowthatthisdeesnothappeninQED since itisabelian. Probably there isafinite UVcoupling constant for QED, which may befound innew HEphoton experiaents. See Pagels review page 161. In(5.9) they present analternate form of(5.6) solution totheRGequations. Here the unknown functions are qandA and itisclémed that you cam compute these inperturbation theory (whereas youcould notforFandJoftheother solution). Look now atthe form (5.9). Suppose the RHS diverges only asthe upper endpoint goes toinf. This might bethecase iffunction 4{x) never goes through zero. Inthis caseyouwould have that k*/m@-» ©ase9-700 sothisisourfriend Case (a). However, suppose (x) hasazero somewhere atsome finite x=x,. Then youare getting k2/n@ corresponding toe,=somefinite value, andthis iscase (b). Thus, (x) must betodays BETA function. Its zeros determine the positions ofthe UVvalues emight take. Somehow afuture author isgoing toredo all this stuff using the differentiel equation approach. Appendix A: Derives the Kallen-type representations for the Z's, not too relevent. Apoendix B: Shows howtosolve the renormalization group equations. Iwill look at 8 thisdiscussion leterifneedbe.Itisallsitting there.