QED and Radiation

: QED /Radiation | Bjorken &Drell(1964) Chapt 8:Higher Order Corrections "Insertions" and Ward Identity Magnetic Moment Triangle Anomaly Positronium Lamb Shift QED Processes Radiation Theory Recent QED Tests -Strauch Phil Luchtnotes 1976 a — Se mm TDS —_ Cn oN : i4 | \VeY re) "‘BjobkenandDrell,CHAPTER8,"HigherOrderCorrections "|Contents ofthesenotes:~ :orfaitComments” ee on oo om8.1 tenet scattering" 1~ Ba"VacuumPolarisation"~—~ aae —_— 7| charge renorm (Z3)- ¢|~Wehling See meI typed comments 10 1 “8.if "Self=Mass~of-Electron*--~ -----~ - EL = = ane 8.8 “Renorm, ofelectron prop." 12 Mass counterterms ~~ more charge renorm (Zp)-— - e+e -| typed comments wy + = ==-8.6-"The Vertex Correction” — - - -- _ Ward - =. 4 anomolous mag -moment, | more charge renorm (Z,) -— + - infrared problen —~ - ~ aief“TheLambShift" 22tte typedcomments -—— 27-8 -eee I | ie ee -|: . -a a 1 see ! ~ ee oe ee - mo ep attpee Oe ERE .| O_ ,Overall Comments onBD,chapter.8:"Higher OrderCorrections!:__. a oe "4, This wasmythird "shot" atthis chapter. Thefirst time Ilooked at . _itwasduring 230.Iabsorbed verylittle because Ifailed to.ignore... .~ the integration detaigs. Also, Iwas unhappy with the feynman rules atthetimeandsodid‘notbelievethevariousexpressions. The.— ~~~ second reading must have been during 230C. Again, Igot very littleout. ee - - -' Thistime, Igotthewholepicture. Thereason Isuppose isthat Ihave moregeneral knowledge ofwhatisgoingon,but.also,thigtime {Ytookcarefulnotes,sectionbysection,tokeepmyselfhonest.I flhade andecided effort tofind the "results" and toignore thedetails _ ' ofintegration. Maybe some other time Iwill look atall the tricks fordoingtheseintegrals. _ae -e - {o>Usually itisthejooftheauthortocleanly separate the“results” fromthemessy details, notthe job ofthereader. That isoneofthe | - tiajor faults ofBjreken andDrell, that you cannot gointhere and read the results without trampling over all calculational detail. _ - For example, Iwanted tolook upthe result for the Lamb shift. The dialogue eeenattasakteanresultwasinterstrewn withcal_2|_ culation,soIhadtoreadthewholedamnchaptertogettheresult. Inretrospect, ofcourse, itwas good tohave read it, but Iwould . . have preferred tomake that decision independently. ne “2.Meliy ideas arecontained inthis chapter, soIwill just list someoO keywords.Manyofthesewereonmy"list",sothis was 2profitable _ - “GigPéssions . 77 relative signsoffeynman graphs andgeneralized Pauli photon symmetrization cangenerate disconnected graphs | ‘ - the cutoff method ofdoing integralssubtractions oo. _gauge condition - 23and vac pol charge renormalization _. aan- ~~PROEGH waeBelBropegdter confection vac pol Uebling potential and -27 mc _ . soo ~ ->“order byonderniearity~~~ oseessgees Hiexternal wavefunction reriormelization fer photons _ _~ “lectPoh propagator incdifaction “*** TT 'mass shift din, mass renormalizationcial ' ~2gChargevenormalization ~~~ mm |external electron wavefunction renormalization hoe“mass“counterteting os 1Ward's Identity a H anomoleus magnetic SoNene en t the Infrared problem andBrem.Lambshiftand-vavitua’fluétuations =~~ ose aiid 3.‘Thischapter” should.haverdeen required redding beforeWewent”into ourrenormalization discussionin2300.Ittookmemaybe3days’to: fe)do this. soe whee | wae . ~ 2 we Gd - - - 1 On o™~ :,yd \ OC)—RigkogMilaGestion Bl"Eatshaskin, WMonun sO©Goumaites4case“gctalogiteBfaetheydid . 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Mo =BWA, augoxsomiot MaeLogeotWink,vasaendaber YP Mae,scion ubvavogaun: Maule 2Sent;HhRae=O,NaotsolesCeOCarp,Obrradinnn2h).ASee yw x Qa,. a -Browsastwalesom“oadsalayondeisdMey.rracrart NL =— . 10 .1 0 |Bjorke]Dreli.on"VacuumPolarization" ee _. |1sAnytimeyouhave anaelectron photon situation, itispossible that _1~"ghe’photoncan"spendstnieof{t's"tiiie”beingane+e=pair.WhenWe— correct the photon propagator Torthis possibility ofhaving the—-" =" photon ™polarize fheVabbtin”,twothingsresults a)theShares!getsrenormalized bya23¢1-(a/3PI)1n(M/m)?, ‘“whereWisreuters |~T |b)thepong spacepropagator hasaddedtoitasmallconstant - 1 pieté-(2/15PIN). Thiscondtant “piecein‘configuration spateappears ——L asaorigin point corfection toaCoulomb potential, astrengthening ae-of“thehegbtive potential: asaregult,. energyTevels-FOr L-0stoic —~statesarestabilized by{0-5ev.Porhydrogen, itis~27mé. 2.Thisatomic correction isonlyasmall partoftheLambshift which — ~ ~vis more~like-+1000metétal.Oursmallpiecefsknown‘astheUehling ~ correction andwasBound byhimin1935. . 3+Wecanalways consider an“éxternal photo’" tobe.just 4propagator 'very close tomass shéll; so-vdc polfias the same” effect. The self ' interactions ohthe propagator "increase" the size ofthe propagator ~~~ -|=-by-2ys Wecanassoctate @SQRT(Z3) witheachvend, and‘thensimply ~~' forget about the source énd. Hence, thé charge where anexternal ~~ ->4> photon-attaches~is-renormalized“in the satié-way asthe intérnal ~——~~ \changes. . fe)"hs IntheSZpicture weallow anexternal photon doallitsself )- —~ +=" interactions; just: asifitwere-any-other photons~then; ‘however; —— weamphtateitasifitwereanoninteracting protonpropagator. _- +> “Thus;—that~photon-hasdonatedfactorSeer(Z9)-to-ouramplitude—>~ ~~~ ofinterest whichwedonotwant,sointhe182ruleswedivide ~ -=+—~by-this-SQkTs~~In ether: words; we-should-completely remove the~ - full, slfinteracting ‘external line toget just ‘the Greens functions. . — 4 - ee ee Ee , + - 1 ae ee eee ' a Lococo cee eee weeee 1 -- . ~e eee - { eI —~ ‘ ul oO BiakonQo.Sahion8.4."Self-MassoftneCheatin! cee 8k DO"deenproperarilonnge parkfootnst"=ten ®VE ~Sak©Burtonaimee _ en @\S&level le] ee=5a0. .Henna 26)&aOisematerclojrdandAfunebane). wunad, Bout a ee : - xan a GR ERG)+iG)CieESO FeASOT ASEZOEQ)2aE Cm) @Ware uroshene,: .weactin compu,©PMeewatery |SegerotI Be=ENoeLemagi-as]TC}MN)|ergrtechrones a)@d- pistonvmneeCasqrtertn HS gh©SON) eaeonDeokBea.ondgococea yfZeya LoemaySL peplotseman a; - wee 6%OTaecubed.” .8s BEeu® —Goede — A - a - ed - wee SarFoeHora]dafaesetae)~—-* Awhe +xtci-a)— ptaq-2) a Se0_ ~.+amde athaBei)=Rowse) eg)(sa)oS(SSE))y* ©wartemd fp 2fogYn)‘ Bgy- == Seine -a Rp Lom) ' ~ |Rubstedeakflew’. -- ‘ : =soqplecrgan. ae a———————s a | . on io oO-—-BipeuDull,SectionBF”Rerown..oh SleckasPrapagater' _- - [==nwo.pe=petFrmCOO)Ae + -M04 nonaa wweDros, wadhowe oo. -. _ A+ARASARABA+-.=©wyA=ARC . ABA+ABABA +. -2=ABC =C=ARDC. _. . FLieG@has _. S: geLA" =GAA . 1-AB _ oe le)Tes(AL=2AeFR Ancagew) De BeCED) ceoeGAMECOEOS “SiCgame-20)" _ ‘Ss, ame ek] Bn .lL peal ° a ®Wore, weInoueoSasaday. ommntisbah ZG)__oo : =($8. RN~(gam) () 7 -oe =Brean =B14e@) oo. 2 DS#@=kwBetsey ‘4Gite C0N= Bape CE (OM.ty.Mm 0WedDikeC=O.Drew[&\+oiSaasAmyj] ~DSPforsee bo _.C Bomn-K0 Bde) 2a]SimeLEGA) SS Ee I - an _a- ‘ apRuy2mrR ece)Gow)._ we ee -skbe .~[,-QR ~{ a2 CBarX Bc) BME Lam -Em)(I+ 248) lowwonproud”OGZS,Dasa7CosaanSt)nelle © @wwe,Me=MAB5aahLemachateronary]: jWao SRY degree mrp. ~~oeae - -itaeey ae i a oe Ty35Pond,stageaclelaud10,sehuatysho . eo) epee ttNakaKacknoth owtdw0|Ae ae @Varrrast, Mh=OmMen :—"Boar stekoaahasl.” Ota eseK oe_. I . SN Gern eSee [RyAneAsoSepprenpiinn VasesOtleyekvestoonaedtLede RkeBKlorom‘oywayCiplacouda"voscoumbanbeam.” ~Bepanaiany [Remy =eK Sma LO “Spintecochin, oad fo) EAM™WalePreroubvoas.. —- we ~ Rplapiod. Pad . \\ fe) |___GommentsonElectronPropagatorRenormalization: -=-+-4. When we-add-the second-order~self energy part term tothe bare” — ~~ : electron pappagator, weofcourse get acorrected propagator, --- —- -! = jist-as-the-second order~photon~self-energypartcausedx tcorrection tothe photon propagator. -- : nanos eee eee |2.Thus,wefirsthavetheproblemofcomputing theselfenergypart,Ep) -eks“in-the-photon: case,we'dothis-vie-a-subtraction;and the-result |isgivenin8.40. in 3.Inthis self energy part calculation, you need both acutoff(subtraction) ' ~and aphoton mass (reguiator).-In éther words, -both-the-IR andWo =~ | catastrophes are.present,.)v |e So,what happens tothe electron propagator asaresult ofthis' correction? llboetannir —a)the mass isshifted bya(computable) amount. t b)thepropagator ismultiplied byZp 1 c)noother effects forp®very close tom@where C(p) =0. | 5.Whatabout ap2modification? Inthephoton casewehadaqchange -- b -inthe photen propagator which 3ed-to an-observable effect, namely ~ H vacuum polarization caused shifts inenergy levéls because the =.- ~-j-fstatdc~potentials- were-modified- (Uehling).-Of course "potentials" - 7Hinvolvedphotons,notelectrons, sewegetnocorresponding analysis e)—-% —here-- Still, Ifeel there -showld-be -seme-observable effect: due-to 4 thisC(p).We“ignoredthisinthecourse,BDignoreitintheirtext.”\6.WhataboutthescalefactorZz?Anelectronpropagater (aswiththewy a4 ~—photon) alwaysoccursbetween- two-"vertices".-Thusy we‘gould‘takd-. aSQRT(Zp) toeach vertex and renorm thecharge.. Then, wehave-wr SORT(Z3: so-farrenorming thecharge. Later-Z2's-effect,will-be--RWyeexactlycancelled viaWard'sidentity, butwaitonthat.“Gy 7»External electrons? Iretract myearlier paragraph onthis.In-- 4cs ~--computing-s-Greens. Function, you-do want-te include all-self —---Y ‘e interactinns onthose external lines, because thoseselfinteractionsresee are-part-of the -physical- external-particles.-Whenweapply-the— -— amputator operators, wedopickuptheZ,sitting oneachexternal.----electron-propagator.. = However, somehow this istoo much. Itisnot consistent, to-- += keep-the-entdre-Zp because. the.source" should get-half ofit.This isapoor argument, but there are other proofs that you only . want to-pick-up-a-SQRT(Z,) for.an external electron, not&full Zo.Inperturbation theory, then, wecaninclude all‘selfinteractions . byusing ‘bare.charge -and‘then gultiplying-by SQRT(Z2).-Or,-we ean -— just use the Zprenormed charge and donothing else. OR, inLSZ, we-.+4p++—willgain-a fullZpbytreating-the external-line asinternal. Then-onamputation wehave todivide bySQRT(Z2) tocorrect. * oO 8.Masscounterterms: ifinthefullfree.lagrangian youdecidetogo' with. thephysical.mass,then you.must correct with.dninteraction - - , mass counterterm. This isbecuse. lagrangians are’ reatly only -. = y+=supposed.tohave.barequantities inthem.Ofcourse,ifweneglect — t self energy graphs, wecan sometimes userenormalized charge, eg,—= ee- butthenyewouldnevergetthe,observable corrections! : - L ar a an . \r fe)[BinksDaal,Sestine5g)*THE‘iosCorrecnion”---=- DO"Yrarrsemd motormorlerepont’: - —Ke,pak - ayop pe x k asoehe \SydoSj&iePIGSMes) yy (eseoffMachargeck ya,Sooneorld cawgantconn NeSey, Yeteat . oe peBOthikmaceepphathonshellWorcdeo,2plop.BeSuaeae. ; BR-Leancppw(p)=Yuaaty=onastXTC YaWG), _. iOWDarioneSeooh mntng\- a Some Bl | 0~@Meocompubes ~el lg eles CE Se (a)[cy =24,Cdorn)]} yoo oe [inesachs _-ldaghe2%—Se =|GmGather). =Ye(—daw\Gem)e TRG tmLrn —=YGoiewt)a2on tm) _.Ce . eSUL. —~Y Quen,, t ary a: -_-e Sow f9 je |_+ -”i aeLeeRo —© Oo,Jajt. wpeyy ye Os late Hhalelisb we\Qua:J3p20)SFGQewfhPapel vs<M| NS eet ate \6 OC)DSS, Bakeexenpuhe AnCee)GromtheWordtindowubShZ, cot. =ees 32)2-3]yeE-\GERT.yt?oo 2phaotardsya SoeSour, =2)) WideUecompushed sorbsis, : BSL RAEN =Bs . . ~~Benaseled arkiaSuan" howYulayZp.Seytaney -/ - a A . -©Feceoaginwstoo ee fe) “s sel walevacaFES Xpee(FO)_%SOdfpone ahy} =HCOTemEV EH aeLOY 8mPaheneyOO‘ - weer gonnschin [ole keptoot|. w=.YowweSoussaad,Orahayprrerd omoakpeespagakt so—- sO a.Commenter eonsatade :_ae _.;—-FoeCBG pon a A«bhoantsa:ek—SuirEreCat - @Bo-Csh)auabehatte TE Rea Orod aittebeArwewodiy Maobeys by2,peed. _ Oe ONG vs -3- v oOSDSedesehomeanthcssit) pCi)abiyiwOeseas,Oak;eeeaoeStasagttoedaa_deathraileOrVs92.0:+Raqs0,wecenst useWardsous pansden usual: ,JPaswallgmafabs et~ ae 7 ~3at ,om.4 ra: ¢BAhehkGO=<4SESEles ww Kye Mike:woe , =Ms #\oeMeet SteadeSelestokhs . 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[oo 7 =A Rp. dtu 0... on | -._= eUTw ~ - - Bit,L=0BdwAs.BekWTdocespfDeeieLS ent -y —\- vied C)OMe cmog.oraansste abitSofood todSaida- >eee te eo be=£0s+)104)-4) --[aeaaS otal oo @Gaoct tv:— oo - hastParr : Loe -- 3 ar s LGR -om =VBSba SefvagoN ---=+teeOReGR B®Viera oe -- wee “a-te sEH <Bo-vR) ne =>Gra du. Gogxcesta) PV TO) - wo. Tae TS Shee RORY ST Wakva@sg =Srat| a com ON ad 72 ia . 1 . aahearromedlene oman, earecteiey QOREE Yacksft?Metuesfief? “= Gig +as-1) a+36 wee QeAYgic’—Mme,seangh-ce foOnceacinbdr-‘wsJumON ionos _ -- a —a - I ON wc 2) , | - weMy 1 = ro)-—-|coments: onBiorkenlandDiels__262)Np(e.dase). +TdYaw(e)+BFpole).aetnvertexCorrection. Inena’section wefirst-deal withtheDiracmatrix_ ‘element ofthevertex"at q=0. Weshow that when q-0, this matrix element _—~~mustbeproportional togammaso thisdefines constant 2+.____ ; ae ;2+However, fromtheWardIdentity. whichrelates thesecond ordervertex:part __ ‘atq0tothesecondorderelectron selfenergy,wecanshowthatZ;just Male)=2ate)_definedisequalto-2Z>.Theresult isthatthecharge renormalization ___.|.~dnduced bytheelectron self-energy isexactly cancelled bythecharge_. .—.-j}- -Fenormalization inducedbythevertexcorrection, ___:~--e |caer __ 3Specifically, BDaddupallthe-vertex conrections (6terms) including = .themass comterterms. Theresult ofadding allthese things (atq-0) _igthat onlytheSRT(Z3) charge renormalization survives. Thiswas _‘induced bythesecond order photon self energy (vacuum polarization). |4eNote_onmass_countertermsi tieclectron “prépagater* shouldréallyhave |"thebare masa. Butwe.like using théphysical mass in-this propagator. Inf|ordertobeabletousethephysicalmass,wehavetomodifytheDirac equationandtheLagrangian(fromwhence,thePropagator derives)with "amass coulterterme thesveavetoincludeallseperategraphswith|Counterterms: Ofcourse thiscounterterin "X*already isequaivalent . .totwoVertités in“order because aswecomputed, ifisproportional toaES }$.TheFeynmanFulforamaiscountertermis(di)stvertex,~~~~7 ie) |ewe'50begin"tdcompitethevertexcérrection £6FGlo.THISFequites-ondeH |” again both IRand UVcutoffs inthe photon miss position, but the small~ oePgil€dependsonlyontheInfrared‘ citorr.Thisrésult(page17)shows—three terms: : : “ a |" gyanvintrared divergent term~2«Cr/n) eeb)a-3/8termwhichisexactly likethevacpolwhichis-1/5,_- co>ts=*pot"Yetincliided inte"this"regult. “~~ \ ¢c)forthefirst time, theanomolous-magnetic moment termappears. .uf AITthiFee"teray’ Comefromthe“onefeymnahgréphi.inwhich“aphotonbridges ~ ‘the vertex. a a- Toy. te verte wee |7.Nowbegins«verylonganalysis ofthatinfrared divergent term.Thisterm -modifies elastic’ Coilomd scattering. “IfWeWereVOFemovethéintrarea ~~ | eutoff, this modification would beinfinite and Coulomb scattering would- |nottidkeSeRSE. aaeememmeneneataWeran into asimilar problem with Bremstraullung. There, unless we- ~“put”aniineFared™cutart onthepioton energies tiittable, theBreacrésy~~ section againbecameaninfinite modification, . |-Hereit4sarguedthatwhenthedifferential crosssectTon mbairicationsofthese two processes are added, the infrared cutoff disappears. a 2 os i[Yeor+APo>erate Leal 7[@) However,wecannotdirectlyaddourresultstoshowthiscancellation * because.the Bremwas.computed_using, massless. photons. and_excluded_energies —: uptoKmin- Onthe other hend, the vertex correction was done with massive|.photons. and.included alllower.energies. Wehaveto.getthese.onto thesame footing somehow. ic Bole ens - 1; ana - was ’ \ “\ _.8+ Inorder tegotothis“samefooting",BDineffectrecomputethevertex.__ Oo |correction beremoving photon phase space uptoKmin Theyassume thata ._;_-—knin is_much lerger thanthe_photon mass._So, aftercomputingthe |change inthevertex dueto.this photon phase space modification, they —-- _discover that they_also have:toincluded aneffective change_in the - '“Vertex caused byachange inthe electren self energy part, because it -===~-t90,hasaiphoton loop,Thesumofthesecorrections isillustrated on __‘top ofpage 176. —~s '9What. results From this photon phase space correction? -_—.- |a)theinfrared termwhichwas1n(m/,) in8.62ignowreplacedwith__ ant ‘In(m/2kmin) +Thus,wecanaddtotheBremresultinasecond.. '._.") a+5/6vae poltypetexmappearsoutofthinaire 1 10.Now:wecanaddtheBremcorrection totheVertex correction, ‘andalso ~~“adainthe zeroth order“elastic Scattering,Onpage2iweseethat the_sumisnolongerinfrared divergent, butdepends onlyonax-- : oT |“thisinternisdetermined byyourexperimental resolution. Although- thigequation doesnot:seemtosayso,Iwould-assume thatasyour =._' resolution“gets moreandmoreperfect, themodificatfon should wanish.——- 1: gNoteherethetweonlyaddedinthat infrared piece ofthevertexi ~Fromnowonwedonotsimply ignore thispiece. Wehaveexplained how we-experiuantelly aveldthe,inffared catastrophe, butthat“piece” is . still thereinthevertexcorrection andwillLeadto.still‘more a——-—- —.observable effects(Lamb). - ~ - ie) |i1.TheLambShift.Thevertexgraphistheprimarycontributor totheLamb -- “shift. Welearned ‘that,thevertex graphContributes thatkai,term. the=3/8term; andthe+5/6 term. It.turns outthat kpin hasa "==|natural Valuef”about“nineinverse BoiYFadiT.“Wecertainly expected —| it*to have something*to ‘dowith the scale ofthe problem (atomic radius). . | So,Tfthevacuiim polarization term(thé“1/5)ware-1;thenthe aa. k{nin)tera wouldbeabout.#35,andthe.othertwofractions‘contribute42.3. ~ee qoee eee MoweThus,whenthese.threeeffects areadded,Iget.+975 mcfortheenergy VoonbsSavee; |“shift. IhaveleftOuttteancnoTous magnetié momentCorrection, andmaybé~. some other .corrections Idont know about. “But the point isclear: the~\ *‘mainFedson TortheLambshiftisthefact“that theelectron inthehydrogenatomCartemitaphotonand.thencatchitagainontheoterhsideof. |AteColom itterdcticn vertex. Thisiscalléd“fluctiiations.” Tits~~~~" 7' "vertex" correction isabout 35.times, larger than the Uehling vatuum "| >potential Correction We"discussed eurliers |mm ~ ed T2. IqOf¥er toevaludte kqgai:BDgototheBetheMonrel-caleulation‘of ~~ theenergy shift for photons ofenergy below‘kwin» This isdone via -4stafiderd oldfashioned sétond oxde?Perturbation caltulation Sing ~~~as aperturbation the j+A, where-j-is the elctrors Dirac curréat}~and ~1Kisjustee*. Itisasifthe VACUUM caused “theelectron toemit’ -‘aphoton, andthentheelectron catches the-photon again. Thevacuum ~7 = tt 8Rels the electron and makesitdo~this;‘heticetheHaNe“vaeuun™ —-————— fluctuatioris"s Weknow that even the vacuus'-has acertain nonzero,-> ‘—VEVfor"stevtric ‘fivli, anditvisthtselectri¢-Tieldwhichisinducing” [@) |thistransition, loosély’ speaking, =~7 .m .Hl“Having“dorie-the ‘Bether"¢elCulatton;~BD additsrebultto-their~ -sone .former result, thek(min)'» cancel, andyougetaspecifié redult,(8,89). 1.: i - H a [nseenons +WardIdentity Nis.omneogtd'sinasn’Stow +Wod,Saortin ---= mol pent, loty -- —e- . : Dros. ieeatBQdwt :an _quantity. eat,caMmongaat . . ffLe&_wattanas x4.Avianoanalin: hanawant.Q9_Diao) Qualteolorgelu ves. re an 7 Lee. aAhem aeam gm | coo @NapSdhueddkinaMakLuecnaoy aasaseaeal Ye. OBWwamdry oesteely —fh—OestaanDae—--cpsnades occur Spimm ~Foseonght weaeL aa}! 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Magnehe Moment Qikvnandtrnornainsdcousbe | wryoreLacknae p__,_AmonolousMagneticMomentofElectron,\|0/7aM ‘e)1.The term "magnetic moment” seems tohava a“nominal meaning—|~~“Gndan“operational”meaning.Nominaliy\wewritedomen _—— operator forthemagnetite momentbyJustcribbing: fromvlassivel -|| BAMthisoperator iskacertainspatiel_infegral ofthecurrent—_____ J. Ifyou put inthe well-known Dirac electron current, you get ~~ inM=6/2My.‘Thisieexactly whatyouget,thereareno"corrections"~--}--whenyoudefine-magnetitc moment=reaydasterttperoe_.|.-dnsert fanciercurranta (withformfectors, ete),youegain find - . thenominsl magnetic moment forthat particle. - “25The“operational” magnetic moment is*themagnetic moment:youwoixtd=~~~~ ' operationally meanure. Theonlywayyou.can.measure itiswia.an. ____ ' interaction between theelectron andaclassical magnetic field. TT “| In“external potential fieldtheory"the“source” ofmagnetism =~ ween ee isrepresented by-an-"x";—This- "x"isexact:inthatif-you made~ —~-|afieldtheorymodelfortheexternalpotential(ie,youmakea__.__ fe)' proton goinacircle and have acurrent which generates avector '7"Sotential ete),youwouldhavetoinclude corréctions "ontheproton ~ |+= side"~to-all orders to-get-"x"=( In-passing, we-note that-this is~-~. 'why,therearenocorrectionstotheMottformulasyoucannottalk, i about "double Coulomb scattering" etc because the entire effect of oe :7thesource proton hasbeenincluded intheCoulomb Potential). ~~ - ee Howevery you-can-still do-eofrections-on-the “electron-side™ —-- _4oftheinteraction, Thismearisthattheoperational magnetic:moment canonlybeexactly computed fromaninfinite sumoffeynman c= >*”graphs. THelowest ordergraphisthenominel magnetic moment which~~~ — 4 you-cannet measure because-you-cannot-"turn-off"-the correctionse - |. -In_this picture, theoperatimnal magnetic moment isthatmagnetic : moment you would have touse with nocorrections toget the same Te Spatiace~ ~ Te —-- 3,Thefirst order correction ig1,16x10"9 =00116 relativeto1NaMELY, |a/2PI. Itwas first computed bySchwniger in1948. The second order oO 'cor¥ettion wasdénein1950byKarplus aliaKFOLLtoGe-2:973(e/Pr)®.| Theygoofed,-however,-and-in 1957-8.Somerfeld andPetermanm redid -|_thecaletoget,-.328(a/PI)°. Thisnewresult(quotedbyBDandGAS}. Asof1963, exper says .328 +.005, see BD). \ NS nT OC)-~Avensig MromatMaonniad ; © WW3deeque: VeTeRogeou(al),wee et Anaadados - a.wee fe _- = 122.0360 *.0002 _- .Arusha, w=,00729938 +i - a . ©@Onur,Oedyadkondegrain GE . : 7 ., —-3B=LOOiéttio tl. aeWeLssiost a a SateUmeacbadeaty daSEanae"asa pace” - erMiorh __. _7 . @Srovaninfusl +Paterna |153-8mdagendindl) at. . 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A)onegoomdaSavawetmapaobeke we ee- 1. ee ee ee Movenlulere, wekaveanikYoo aapagysowesa .o. a toe ~6— (o)—oovastnsdldocder,pashantoga! -a ee Ly gistceGRROR eSFED7 _[oweaanweeSERIEQEEP)boagh awe teak GRA - 2 . a) os-Ba,GQ=PoaEomywCorsega.Geb MWe raat 25 4 oeVeX*HS. =VeecqVand : .=_€qn_S Spe8xda expSoe3%Bo =ShFEY8=FRY LLL . _ 0.1STonia"fade,sateshaoceneingomanarwpiyte : Gry=aheON.[AREONvon] 7 ee ANeLE) -- QheaeStewwarm,C1,Dawn,C2CON?ond OO EE Oe PENSCLOLSL) Vay ——-SeyGayeuewarbans spacey.Git>iebasses DreeuofDeaanedk— SsdesGeLeZeLeyttacel=GeiesetTel . . _w=Vet mop -+ = ny“ -NYowo.cmauer vraamgiteunseya BamSiowledfoaduis: fo) aeSawag-|on — GR — . o> . - m——© - 2‘Ver,St=Vej=a|=a[4«|— te nnnneneen nen oOBeNada,coe,Wacom nomesomgull! 0ee - pak au2EEE. Je“tag yo“AIS ;...bentBRO=.re|f_-¢83] °tho}. . ..- “FT. O fowl. . ae Leatou 2, ~ Ko St ‘ a .-oa—_lanby=Eoie(otoraiul |° ~eeekigh”yype. ~oo ~.. _ _kits 2et[ctoole] =gett (otyi] 7 [suminaoshed[atMATa] re)Pryway EM =COGest) st=sedean ok ite - cewi S-didt _-- ‘_< Sa=ze Say=-(sat —- ~ ° é A de GU otot 1s mide SGatLedtotaia “=steinroe WoL —--- - - -Suggear adachen aapisagabing2:Chenwwe Co(SSN) =1 EIN onapiticmanast AMachos((austtoniss, cs 7 “=Balaguer oeeR.W4IOoy/geee=a=is C3Made:aspiateeestaanuthL=loldteratjack 2=Deam,eferearcattey )QuadasyAton,hence. Bolt Mograten. TrangleAnomoly ae 7 BYtis |3 pelBeA fe)PHYSICAL REVIEW VOLUME 177,NUMBER 5 25auany Toghy nae !-Axial-VectorVertexinSpinorElectrodynamics ¢fryeaslybecale :SrovmewL,Antex tosie InclituteforAdvancedStudy,Princeton,NewJersey08540 Re, (Received 24September 1968) ar Workingwithintheframework ofperturbation theory,weshowthattheaxial-vector vertexinspinor aegmBas.(3) jsper eranreece ihhtvymalrain olNaety ifield equations. Specifically, becauseofthepresenceofclored-loop “trianglediagrams,"thedivergence i . : fal-ector cirent ot‘theusa!expression calalted fromthefldequation,andthearrestor°SEER ~9/0,*(p, currentdoesaotsatisfytheusualWardidentity.Oneconsequenceisthat,evenaftertheexternaliine SSG | estan oration aehentia eeeealdeena Gadie) URRE order perturbation theory. Acorollaryisthattheradiativecorrectionstonfelasticscatteringin Breyocinth |acura thaydegenft(adhet),Aerines tntinasics"AyOuttacint i electrodynamics, despite thefact that ththeoryiavantundermstransformations, theaia-vecor YEE(9)whichw. eurentIsnotconserved. TnanAppendix wedemonstrate theuniqueness ofthetanglediagrams, and." ~*QHiBE"equations, act discussapossibleconnectionbetweenourresultsandthex"~»2yand»—+2ydecays.Inparticular, weaytfTothisend ; argue thatasresult oftriangle dagrams, theequations expressing partial conservation ofaxialvector MES io ' efined manner, whichcompletely altersthePCAC predictions forthe+andtheytwo-photondecays. ae. INTRODUCTION well-defined manner,whichcompletelyaltersthePCACP). iaa .ici ph a iT=axial-vectorverterinspinorelectrodynamics Predictionsforthe2?andtheytwo-photondecays.|BR“in i isofinterest because ofitsconnections (i)with UBYAwhere theveri 'radiative corrections tovdscatteringand(ii)withthe-vs I.AXIALCURRENT DIVERGENCE AND(igBijself-energy par invarianceofmassless electrodynamics. Wewillshow WARD IDENTITY VERAIEY, thefreepropa: :inthispaper,withintheframeworkofperturbation Weworkintheusualspinorelectrodynangt ewsthefrepro, ie)theory,thattheaxial-vector vertexhasanomalousdescribedbytheLagrangiandensity! eerenormalizatio : properties whichdisagreewiththosefoundbytheformal Bidand2,Eq.(5) imanipulation offieldequations, Inparticular, because £(x)=(x)(¢y-O—mo)(x)—4Fo)P"(2)Bedi3 {ofthepresenceofclosed-loop “triangle diagrams,” the sed 4te PMA, idivergence oftheaxial-vector currentisnottheusualex- sebre @Ane):,(OEtnordertodepressioncalculated fromthefieldequations, andthe aAg(@)a4,(2) a UBSceasing axial-vector currentdoesnotsatisfytheusualWardiden- F,(x)=—*———""" |).esqn—, NESinwhichthee tity.Oneconsequence isthat,evenafterexternal-line axaxe oat ResEtermionlinebwave-function renormalizations aremade,theaxial- . nary_SSIVandending vectorvertexisstilldivergent infourth-(andhigher-)or-Wedefinetheaxial-vector currentj,*(2)andai&andending | lerperturbationtheory.Acorollaryisthattheradiative PSeudoscalar densityj*(z)by aieSagams in+ cortectionstoviclastic scattering inthelocal current- iMeas : «eg at toan 4elastic ; 58@)= VePrarab(e): UAE? 10),respect. icurrent theorydiverge infourth(andhigher)order,A “W(e) - EL.)SNe tae :second consequenceisthat,inmasslessclectrodynamics, PE)=VO)@):5 ieiee. idespitethefactthatthetheoryisinvariantunderthecorresponding vertexpartsI',*(p,p')andTi(jip}ae .‘vstransformations, theaxial-vector currentisnotaredefinedby Bs.ITfos :conserved. oyna , Pi)ia AInSec.Iwederivetheusualformulasfortheaxial.S+’@)Ps*(0,0Se'(P) ™aeqvectordivergence andWardidentity,andthenshow Bi ihowtheyaremodifiedbythepresenceoftriangle==—fdeadlyoreXU)JOO, EE ' diagrams. InSec.ITwediscussvariousconsequencesof aberha i theadditional termfoundinSec.I.IntheAppendix 5,/(pyr6(p, 9.59" efBE7swherewehave| 'weshowthatitisnotpossibletoredefinethetriangle9?PF*(#2S9'(e aa&‘hexanvert. diagram inaphysically acceptable waysoastoelim- : dpeeeip! 7 “ee femainderinate theanomalousbehaviordiscussedinSecs.Tandif.=~JA¥4ye"-“TEHC);#OMON. adBfby(—p')anv WealsodiscussintheAppendix a,possibleconnection nMBybetweenourresultsandthex?—>2yand9—»2ydecays.UsingtheequationsofmotionwhichfollowfronEge>__(y fo)Inparticular, wearguethatasaresultoftriangle (1),thedivergence oftheaxial-vector currentm¢ian :diagrarhs,theequationsexpressingpartialconservation. wom i3 1ofaxial-vector current(PCAC)fortheneutralmembers 1,'¢ussthenotationandmetricconventions ofJ.DB LSoftheapial-vector curren octetmustbemodified ina3,NewVar/06SppIONocatee aD IT2426 a !ct tF “matsVectorVertexinSpinorBiectrodymamtes" ~~ 8PRI77; 2126 (1969) -- - —- 1.FirstrecallsomebasicQEDfects. ‘TheWardIdentity redlates thefull vertex tothefullspinor propagator. Analternative formrelates the— vertex corréction tothespinor proper sélf-anergy part. Inlowest order theWardidentiy maybetirtvatly-proven byUsingthéLittleDifaé~~—~ - spinor space identity. - - _- 2.Theaxial version ofthisidentity appears. in(2)of_Adlers_paper. There-are. — threetermsinstead oftwo_ontherightside.Thisleadsto_thethreetem,_ axial Ward identity shown in(4)andagain in(5). This Ward identity is _ formally "proved" intheunualwaybyconsidering allpossible "insertions" ofthe“axialphoton" ontotheaxialvertex blob.Unlike thevectorWard case,theclosed loopcontribttions donofvanish. However, whenoneadds “theloopantlinecontributions; theformal-result-(7) vbtaitis. “~~~ 4.--But-there.is acatch.Partoftheproofrequirestranslation-of-the-origin fe) atcinadivergent integral. Suchatranslation isonly_justifiable if. . theintegral isonlylogarithmically divergent. Itturnsoutthatthe triangle closedloopisactually linearly divergent’, asopposed tothehigher_loops, soyouarenotreally surethattheWardholds.Itturnsoutin factthatthenaiveWardiswrong, andthatthere isanextratermasshow in(G2). - - 5:-You can directly derive the Ward identityfrem eurrent conservation-and ~~ -~ - axiomatic_stuff (reduction_stuff). see.equations (3)and-(}.-The fact that . theactual perturbation theory Wardhasthat.extratermmeans_thatcurrent | conservation (axial) also hasanextra termasshownin(30). __ 6.Theeffect described aboveiscalled "thetriangle anomoly". Itisdirectly related toanother triangle effect whichisthis: according toWeinberg, ~ you can obtaifi the “asynptotic behavior" ihe certain sensé 6faféjrinan graphsbydoingacertain powercount,Foraxtalstuff,theaétiaTpower isonelowerthanthepowercount:gives-because theleading-power has 68vanishing coefficient. BUT, for the triangle graph, Weinbergs power~is : correct. oe 7-Nhat aresomeoftheimplications ofthis triangle anonoly? First, recall . _thetheoremprovedbyPreparate andWeisberger: ifthecurrentis_. _conserved spartfrom“usualméssterms,thenyoucanmultiplicatively [=}renormalize thevertex: ie,thétheory cenberenormalized. Adler ° 7 ”GiowsGhatWSektFateimiWtheCUFFENE covervation HHS HHS ~, -—tineprenmantcauses-thehth-order-vertex-correction tobe-divergent~evan a —efter-the-standard. wavefunction- renonnalizations. OUCH. - <-.. =. 8+Next,heshowsthatthe4thorderradiative corrections toeveeleastic |_ oe scattering diverge forthesamereason. (However, tounderstand thisyou have tounderstand that Fierz transformation which takes you tothe ~~~ "~"“Charge retention Ordering; 1donotknowaboutthis) ~ QuTieWHIRYibiuowal Pact—Ts-thatelthowga theaxialcwsrént isot soa ++ Mconserved'y the-tagrangian-stiti-has-perfect gamma-5 symmetry. . ... 10,-Pinally,on_page 238Adler_biveds sheformfora.coeficient inthe . ___ ._ broken current conservation. Ifyoucanmake your axial charges addup . .—Tight,youcanevadethisproblem altofether. Harariinanother paper ; _____@lains thattheconditionisactually‘thetthehadronicquark+lepton ry charges must edd upto0, This fact obtains inthe GIM theory with a fourth quark. so . ye ~ ee *] Positronium AboutPositroninm, [ee March26,1976. 1.reference cards:PondDicke,Deutschey-Green-Lee, OrePowell,—-——- - 8 other refs: Heitlor book, Segree book, oo. 2.General description. Positroniug hasthesamegeneral energylevelstructure ‘ashydrogen, axcept spacings are all scaled down byfactor 2due tosmaller reduced mass. Thus, ground state binding energy is~6ev.rolative-to-total - mass of about 1000 ev. ‘This iswhy potential convept still works. 3+Ground state hyperfine: due tothe obvious spin gitustion, youget asplitgroundstatewaethe triplet38,lyingaboutToSabovethesinglet 1S,Thesinglet (alse called "para” although spins are.nat."parallel") decays -into two phtons inXGkX 10°10 seconds. This isastandard QED calculation of annihilation, you have tomltiply bywavefunction atorigin etc. The triplet {also ealled- "ortho" moaning-straight inGreek, se-spins aligned)- state ismetastable inthat itcan only decay into three photons in10-7 sec. Ie,this triplet state islong lived. Ofcourse there wethreesuchstates. 7 Ihavenotseena“modern” calculation ofthree photon decay inQED,bubthe__ graphs are pretty clear, Old calaulation isOre Powell above. You would expect three photon decay tobedownbyorder-alpha. -Actually, factor--is- - more like 1000. 4.You can also radiatively gofrom triplet downtosinglet, buteasytoshow that only Mltransition can dothis. (lo parity change). The spontaneousW1lifetime isontheorder.ofamonth, sayoucanforget abou’this.But,when positronium isinaparamagnetic gas whose molecules like O»have magnetic moments, IgatherthatMi-radiabion-ftoats aroundand-can-stimutate: me) this triplet tosinglet transition. This effect iscalled quenching. Imfact, __thiscanhappen veryfastsothatquench tineis10"!seconds. Depends ongas pressure. - - —— | 5.Selection rules: youcanshowthatsinglet mustgointoevennumber ofphotons __nerely nycharge Conjugation arguments. Thecatch isthat(c1)'*S tellsyou ‘tho Cofaspectroscopic state. Similarly, this shows that triplet. cannot gointo ‘vo photons, Analysin isanalogous toPIO decgy into two photons.Bytheway,EXXXPIOhasnever~beon seen-going intothree photonst -{iitt-1970) - MW AR ABAR 7 ; Wo W= AZAR eee DeordyapaatayHLennarvend., \edhlemmaeonsskJapeana. @Didldoke WowP=+l=. _M=_AS-S .vee PH-h SL M= BSR. Ras GrokdusinrpiesWeddohepanded OadodQeAscendaoutJaloaned, aoWWoPHL a“ ©©Resdeoninas Sco, Yana.gerlaetivon JnSgon*S, . Que,yorseq bone Ze. gecameras omabesis |-. orpadroaGastLjoetf,saysSagipb, =- ©Qereaishs ayoveLeDeeaaaQueeens =+hiw dada. iwYai poss: y : 6dead,Se onoe ed eek————) ©RalpareASSAS Rave\hCoaWasa+Po!v%Q20._Sarvat! ~@hS” ~~©.d_DinkSighNassnamedeont. sr. oO.slums ook. -- BataloomaMeesamewrendorqeaiyohana‘So° --gostiunniuas. dacoyas AsgasdOO08 -anton fuakfeoA —- aroadeatim dealsonOST — Sone a* . no=&EarheyyEu zis)4Ex~hE\(EatErg)2.-wee SR Kel. - Soe ee ---=SEXETET god,annbeWales GSBeAeka Cay courak KLALRcee Gr RRLL ~~poss Loe © @Waorguena ooVedfrmonia Oia: wontdouate asdWW?vnWyATEN.QeieQrobowsalec*) Snow.MySQEL, dant 22)OuWy=0 _- RQe + RR= LL eo - . an _ . . am. -.QYowoun, Selatis=O siebidldlalewoWess)gre dbsoa—ahsa. Dae AndaoteAsysy +ems onda a ose - _ @doshoiwy: torr 4WSa‘So. Qdeah2S, \oscamae SS=1radraldiad aQuunavarn ©.one & conor owJpio Pradone Seefait,o\gosthontunnsoda ~\.©daycopaghinsayCi)=CA=e2S,consiqotubo\ Ania wesoar” ree Lamb Shift ABriefReviewoftheLanbShiftasDescribed In—BDvolume --—--Marob-a6, 1976_ @%=_1.First,letsgetUshling outoftheway.Evenweforethat,letssaywhatthe"-“LambSHIis:itisfieenergyGfTereicebetween the 25;and 2Plevels in. hydrogen. Inbasic hheory, thene are_degenarate._(ia,- Sehrodinger-theory}.Inthe relativistic Dirac theory ofhydrogen where you use the Dirac Hamiltonian _in the SE, you-get some --Yfine-structure"-namelys ~~~ — ~ ws "han "=/d,0co mba— ope Upinestucha =(0ocomte oT =os — - Gottfried_page 369 --- \ heFete28 =DEfovmHe. : -: ee : -- t foccet ---Bown. Resabunakee poyee eee ~ -rao Duac CanaleDitactheoty aitofatically {aéludee spineffects (perturbation theory calisahis spin-orbit. interestion) and.shus_yieldsthefinestruciure .BuiIthink - states with the same nand Jare mtill degenerate. For sure, the Dirac theory “of hydrogen does fot give the Lemb-shift.- ‘oureatiy-need QHD. Bythe way;theLamblineiswhatinterstellar hydrogen doesallthetime. 7 2.Now for Uehling. The Uehling corréctior accontsfor27mdofsha-Lavb_shifa about 1/40 thofthe full shift. Ibissimply the vaccuum polarization correction tothephotom-propagater. This-covrection-causes- a-constant-term—to-appear =- Ce)inthe renormalized (to lowest order) propagator. This constant term inthe mofieitim space photon propagutorappears a8a@deltdfunction termimthe . potentialinconfiguration space,asimplefourier fact,ThisistheUshling potential, also know as@contact potential. This potential clearly only‘affects S-waves whichhawpsomeamplitude atthe-origity Ittendsto-stabilize -S-orbitals, although asshown above the total Lamb shift isaraising ofthe S-state, For details onUshling, see pageS-of BDnoted ‘chapter6. Inthe messy formas which are obtainedinBDand.below, thisUehling term always appears asa-1/5 term inthevertex correction (vertex inthe non—proper" sense). Ie, vacuum_polarization-can -beconsidered-as-one-term in ->»full vertex correction. > 3,50,wheredoesthatbulk1000mcoftheLachshiftcomesfrom? To_get theanswer,‘Wecanthink interms oftheUshling effect described above. intheabove, weobtainedtheUehling corraction. as_a_cornection_te. thephotenpropagator. But youcouldalsothinkofthisasacorrection tothevertex. ThenthisUshlingcorrection -appearm in-this-way-t -qyrompop ———— -os= oe gt aS) BO hE oe ee) From this poimt ofView, youcanseetheg*factor which cancels theq@?of thephoton propagator coming fromyourxxexternal potential xtothiscorrected vertex. Thus,thesizeofthie-q@X,. -termin.tlievertex—coprectionWill alwoys determine the size of‘the contact potential, which inturn will determine theLaubshift. wo ae A Weshellnowshowthatthe(-i/5) isonlyasmsilpartofthetotal. . \jeQsNP . DreaOe (edaeconch ipoe tun . Soo — SON SESSeaasNe ToyimadineFEYvouehoateWeactual-testexcorbections Inlowektorder. -—-$O.AeRDshh,thismeansYou_add tdgether 6_dyaphs. Thieehavedimple -\.-photy Res,ond VP,andtwoaremadscounter}erm graphs whic! 2 ~~Fecal -erehalso-sX thie-dpders thenyou-add 421this aturtup;you .; oe get“thy me£0; Vertex cordectiom. Yalhaeleg @)theanomotous magneticmomenttermappearswithusuala/a.This UNGAR | Moennetvoneera thebuns“SuiTYhowever becuseHor@g?Yora. . eng d}-a-(~3/8)-verm appearslike-the-Ushling (1/5). : ae,~X)..aninfrared divergent ternln(u/\)-appears becauseyouhave-to -——- ond! S regulate thevertex loop integral. .w 5.Clarification omUehling versusvertex, Wearearguing thatthevertex sorrections givetertiswhichareeffectively liketheUeylimg term.BUT--+—-—¥ou_ptil]_have toimoludetheIehlingalongwiththevertexcorrection, Ushlingisacorzgetion totheghotompropagator, vertaxcorrection is ~vss>something elsen~ -Qe, —cod Keeareeghe doomolnig, Lawlawar.~~+64OK,so_now whab.do.you dowith thatinfinite tending infrared correction???This iswhere things getcomplicated. Thefirst thing that BDdois ++> —neke-a-vir¥ectionto“the-vertex correction so-thet looppiictons With ~~ ~ . below some K(inin)'are+not counted intheintegrations. When this ie _. ~ dons; thephoton massVanishes fronthenewcorrection andthenewlog.fernin_1a(o/fnin daAlso,tn theprocess ofthis.correction tothevertex -correction, a°term of(+5/6) appears, So,the"vertex correction without ~--Boftphotons-below-kgig is-now‘gievenby: as . we eye ~ ee ——= Agwd x a. A eBaeS ee e —Rotuntty FtgegAnctudes tee‘Uehhing-here-even thoughitis-nobreally apart oftheavertex correction itself. The (~3/8) came from the original “Vertex Calculation, the(45/6) comes whenyoumodify thevértex correction - -. . for soft photons. eee cee ee _ Te So, wehave tten ridofthephotenmass,butnowwehavethisarbitrary soft photog cutoff k(min) appearing. What dowedowith that?? This part.-.is-weny subtle, Thereis.stillanothervertex correctien "thatwehavenot yet done, and that isthe possiblity ofsoft physical photons beingradiated fromthevertex, (called Bremstraulhmg), Well,actually BDda=-not Work it:this way, Shey-entyshowtiat—z~ Theymakiclaimthatthat "~~"Tk(min) really ispresent inthevertex correction. I+mustbethereto_-. =-«aneal another kuinwhicheccurs intheBramstzaullung crosssection,Then when you calculate elestic scattering (Coulomb say) with Brem and with vertex--correctiom-containing- kmimy—the-tmo—kmins eancel-and--your. actual - Coulomb scattering isindependent ofk(min) aswell asphoton mass. 8.So,wostill havekminpresent inthevertex correction, Nowcomes astupendousgimpick. Thisisthe“hardphotons vertex correctiom. Weshowed abovethatifyou--inetude-the-soft—photons, ‘the vertex correction depends onphoton mans. However, somehow ifyou Handle the soft photons inaFermi nonrel “+ ealowletiontoger-thewortpltotowenergy“shift,againthekminscancel,. This extra calculation ineffect gives you avertex correction which cancels thekminterm.Notermslike(-1/5)are‘addedhere. ° 9. Total Lamb shift then becomes: .0. (Me) aS ‘ OE=Yule) \*Sn(BE)+E-2 Sr 1 : 2.E= Gaim Jn bday ~2- —-Hereistherelativesize.ofthesetémie?laSu|‘cone=BEons...= i ~ onA|vexook[SaZB)28-F-1| xcokLSET23-1) =BAK =WWDme. 10.Somenumbers. fromSchweber. IfItakeout-theUshling, my-actual” number: _.* 4s1002me,Schwever saysthisshould he_1010.. Now,theanomalous magnteci . moment increases theshiftby68mo,Ushling pulls itdomby27maThis” | takes youto1052. Experiment says 105748. Theremaining fewmzcome from higher order stuff plus finite size effects (Iread this es:proton or nucleus isnotreally apointbuthasarédius, Notclearhowthisohanges ~caloulation which wasbased onanexternal potential anyway. Perhaps finite proten mass also enters inthis catagory). i 11, Conclusion and comments: We have seen that the Lamb shift is adirect _consequence ofhigher order QEDcorrections. The vacuum polarization “correction isrelatively unimportant coming inat27mc. (However, the VPcorrection isproportional foelectron. mass sowould déabout 200 tities ~ zn larger formonic hydrogen. AIsay VPisproportional to-Z4 forhydrogenlikeoe) -system, Thisatleastsuggests thetVPismoresignificant-in heavyatoms, — .butIhavenotthought that.through.) Except_for bout5eVduetohigher order andfiniesiseeffects, theentire restofthe1057me Laubshiftcomesfrom thesimple lowestordervertex correctiom. Thisvertex correqtion generates among other things the anomolous magnetic moment term which accounté only for68mc+. Thereal problem ishowtohandle thesoft pheton problem inactually calculating the vertex correction graph. Somehow BDend up ~ -notusingQDforapartoftheircaluodtios WHIthissupposéd totestODT ~ Ithink abound state theory in=little-batter atdoing this Lamb thift: Roughly, the pictures you might dzaw for-Lamb shift are these: — a . . Roughly speaking, viriual_photoas fromthenucleus _allow . the elctron toemit and recapture photons. 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Probably thisisunrelated toatomic. radiations, butmaybe itwill 7 “berelevant. ‘ThebasicideaissimplytoSay:EandBfields maybecomputed f¥om 4,which inturn maybeobtained (atretarded time)fromthecurrent. ~~ density J...Bufthecurrent density ofasfriglé partielé istrivial. THEs; you * obbain the"trivial—tienard-itectart potentials on-page 4655“Theonly-faney~ ~ . -aspect isthe-littie-k factor which, causes-potential-fields to-cempress-into-—- -~ abeamin the direction o-f the particles_mation. woe —- ~ Now,given these LWpotentials, you_can calculate —and.B. Butit turns outthatB=nx8 soyouonly need calculate B. This isdone in1h. where.we getafastfalloff "velocity field” which weshall negelet, andthe standerd ratiihion fieldwhichisproportional toparticle acceleration, This isprobably the fundamental formula ofthe chapter. ~ a a Next, JDJ goes.to anoiiéél fFaine ‘and computed the radiaféd energy ~ ~ inte--solid angle. Yowgetthestandard result that dP/dOMEGA. =~alpha-x accel? xsin(angle)?.. This-then-integrates-over-solid-angle togivetheLarmour-formla, - , ‘This,Laxmour formila,is non.rel.but_is easily.relativissad to:give :6 result14.24or14.26. Themaindifferences fromthenonzelLarmour.are the presence ofaganma® factor and_asecond velbeity dependent term,JDJthengoes ontocalculate radiation fromlinear andcircular accelerators. Thisgives thefamous games”radiated powerdependence which30muchlimitsenergy ofeléctrons incircular accelérators: Ifisnotedthatradiation losses in— alinear accelerator-are’totally negléagable, littewhy“slacwas"BiiTE?* Next-comesthe-"angutar-distribution“of radtated power'~stuff:- You- -+ get the searchlight effect withawidthinangle-of-ganma’*, -Rrom-an-estimate - ofthe."sea_time" yoncanguessthatthefrequency spectrum_will runuptoganma? times thefundamental frequency oftherotating particle. Another ruleof thumb isthetitisalways the{transverse acceleration thatcauses radiation. Next, hementions that onesimple waytomake aparticle dcéelerate istoapplyaplanewave. ThisleadstotheThomson formula forbhecross section for radiation upon afree partitle (oreffectively free). Theresult istha-standard (11c0s®)- engutar-distribution;-with Thomson total-cross-sevtion-of SPI/3 times ro”,the-chassical -etectron-radiuss Atbigh“energy this-process iscalledComptonscattering..and.the-Iéin Nishima-formila-mst be-usedwhich le) shows.peaking intheforward firechion, ... .. - ~ ~ -——— ~Jackson-closes-out_the-chapter_wiht comments. onGerenkoy effect —-—- - Themainideaisthatwhenaparticle travels ataspeed greater thanthe oe __speedof Light inthe medium, the particle radiates only inaspecific direction . whoke cosine isthe, ratio c/v.Bymeasuring this radiation direction, you canfigureouttheexact speed oftheparticle. Ofcourseithastobegoing fasterthanthelocalspeedoflight oo a 2c"Havingreviewed thisisiness-on- radiation ofsingle’ particiss “in“wotion; wenOW ~~ -(Grr, eaenermorrmsinaie-Hekdar biopingthiswilhavemorerelevance - (QD) 460atone.and.-charmoiium transitions... e-are-still completely -calssical, no . quanfummechsnics yete2 ee : - 7 to. Westart onpage513 which shows that there aretwo ways tomake _ thewave equation describe electromagnetism intheabsence ofsources. Oneway hasBsatisfying thewave.equation. Bmustthenhavenodivergence (transversality ~"~" """“Gonditidon onB),andyougetB-from Bbytaking acurl. This isthe "electric" "7" “gdeTTHKeYorELtransition C6bsdescribed Tatey. ~~ ~But~once youknowthat=B- satisfies the-wave. equation (bytheway,the-+--+ —~bime-dependence-ts-always fixed-at~a-certain-frequency OMBGAy-Thus,{del? +k2)}----- isthe_wave_operator),.you-can_expand_+his.thing_onto. ppherical harmonics as . . 4n16.35... Ie.theangular stuffisdeseribed byspherical harmonics because x*]thesesolvetheangular equation. Theradialequation isthensolvedbysome version. of,spheéical Bessel functions. You usually choose hankels because these “"have distant planewavebehavior thatyouwantforradiation theory. Oféourse “inthisexpansion there are.cerfain undétérmined vector Coefficients, These ~WayDeparttattydetermined bycheervir:tratBomabs"eransverse" *.Herice youvend-up-with-B +-b-¥-whereI-tstheangudar momentum vector-operator andY-- -++-is-spherical-harionicThen-there,-is-only-a_scalar.cocfficient f(r) -See66.12. : - -Toese_fields By,=f)kr)LY),andEy,=curlBy,arethefamus | . "multipole fields". Ofcourse these are only half ofacomplete set ofexpansion .fields because: you also have toinclude the. "magnetic possibliities” asshonw in1bshhe: . 3.Sec TUSS: Jackson claimsthatangular mométum selection rulesaresimply Bivertyvector additions The-matbipslertieid carries offL,Ifyourinitial : state is-Js-finel- statevis Jt,—then-Jt aust be-contained inJxL-to get. action. Obviously this.is just-WignerUckart andthe multipole fieldisatensor operator. _,‘Theparity. selection ruleismoreinteresting. Theparity ofthe eo radiated "“miltipole" field isactually given bythepartay ofBymwhich is(-1)+ - -2- forelectrics, and(-aybrone formagnetigs. “ThiS,fOrexample, ivEItmmssrtion ~~~ (elevtire dipole), the field has purity<Ito-the-initizi-and-final-atomie-statés 6must have-opposite parity. Ifthe atomic-states-have same- parity,+then-El- is- forbidden-and you can then have only Mi‘orE2 whicharebathtwo.photon_operations .. Ithink. Ie, they are at‘the. same level ofweakness relative toBl._ Inthisdiscussion of‘stlection rulesthereis.mention onlyof parity and total J.Nomention oforbital LasinRussel Saunders stuff. Iwill have tolook elsewhere fordiscussionofmoredetailed atomicsdlection rulesandrules of thumb.5 ee as oa *OK,nowwetavesetupthe-multipote fiedd-anatysis.—The next-question - ishowdoyougenerate aparticular mitipole-field.-All fieldsare-generated-by —-movingcharges, soitallboilsdontolooking .at_the Jofthe.radiating. saurce.. Ifyou integrate your charge density against aspherical harmonic, you getthe ° electric multipole momentQm+Roughly; thisisthescAlar.coefficient intheyc radiationexpansion. Thus, just from the physical shape ‘ofthe radiating charge q/' Sisixibution youcanfigure outwhich electric miltipoles will go,Forexample, .alinear antenna hasag{1,m) onlywhenm-0and=1,3,5..... ACircular anterma 6probably hashoelectric moments. ItCeftainly haS-aStrongM1TotehtTHOME*Yougetthemagnetic moments My,byLooking-at M=-curkJ.=domotwant-to-make anyinferenees yet about therelatiin between-the- shape-of quantumwave--functions .andthe radiation, Wait until weincorporate quantum mechanics later.’ --Nowheresresomerulesofthumbforatomicandnuclear radiation without enyquantum mechanics. :thebelative contribution ofmultipoles is setbysizeof(ka)* where k=photon energy antia=characteristic distance. For ~ "atomics, a=Bohr like distance andk=1evmoreorless.Onegetsthaka=order ~ ofZxALPHAsogenerally smallexcept forheavyatoms. This,usually youkeedonly —dealwiththeleading multipole. Thos;youhave’a“highQuality" pertiepation—— - expansion. Bymalcing asimple niodel, you can‘show-that-M isdowm-frourE atthe - same1bysamematioof(2alpha)”, ThatiswhyElsits-by-itself, then-Mi and-E2 — - are_same strength, inatomic radiations... : — oeInnuclear stuffyou_take a=fewfermis, butkyariesoververylarge range from 10kev to10mev. Still, itisgenerally true that electrics are stronger thanmagnetics bymaybefactor25toMM.(ofthesameorder,ie,Elstronger thanHZMi,).Butwhataboutrelation between 2andMi?Inatomice thesewere 6”the_satie “size,Xbutyinnuclear M1tendstobe20timeslarger thenE2.~Sotheorder - would beEl, then M1, then £2, ahd soohifall allowed. — —s . L-Rirety-they-clanify thebusiness of“intrinsic magnetization. Recallhow- this_magnetizetiog founditswayinteJackgons multipole scalarcocffisients, e . Inthe weve equations youputasource. Wherever thecurrent Jappears, you _______alsowanttoinclude thebound current curl(M). Thus, thisMoccurs inthe varbous moments Qj,andMy,-Forexample, supposeyouhaveaneutron irradiated ““""""“byaplanewave. Since thereisnocharge density, therewillbenoelectric (77"HOIpeTeFacaeion. —(exdept Possibly IkHigherOEFdistoOy,>therewT “> “Be nagReeierantatror-homeverbevausee-you wilt have anvscLllating magnetic “+ +moment-Perhaps plane wave xx. Obviously, thie has something to-do with ~---spin-$-of-panbiclebecause spinisalwaystiedtointrinsic magnetic moment .-hence tointrinsic "magnetization" khichcauses. radiation. _ 2.Alsointhere pre-quantum comments. Blatt Weiskpf observe that theangular distribution ofradiation fororder(1,m)isthesameforeitherelectric "~~"“ormagnetic. See16,72ofJackson. . °=>“SUTINTirstpartof“trmnsition toQuBWnotethat-radtation actually occurs‘in quante-which-carry-hvs “This-is a"photon'-However, thisphoton-sam-carrkes angularnonentun_ofLshere Listhemultipole order.Roughly,this.mst eo bethe. tokel angular momentum carried off by,the photon, which isthevector . combination ofthephoton spin (1unit) with theorbital angular momentum, Thisidea comes, across intheHeitler typetheory below where theexiting photon isputinaplane wavee*’* which weJmowisasuperposition of 7~“erbital angular momentum states. “Thisalsosupports theideathataphoton ---~ -ganmotvurry-off “angular mometun-O-which” yourmight thinkpossible because: Sui-and-E=1-could- combine-bo-give~ O-ThereasonmustbethatMy=0isnot- possible fora.photon-due. to-masslessness. Thus,younevergetmulltipole ---Radiation with120,Alneysstartswithdipole. - :. _____ $0)Ewilgropmyerroneous ideathatmultipole radiation of order Linvolves and I-photon process. Infact, only one photon isinvolved inmultipole radiation. 4.Nowfinally conesthetransetisn wher'eyoumustsomehow replace theclassical The-idea isto-replace- charge density RHOwithug”w/Te,acharge deinsitybecomes-the-actual physical overlap.of twowavefunctions (orbitals). e $Similarly, thecurrent_goes over intoug pw/yhere p=momentum operator. —_— -~3- - The-reason thatthese.quantummechanical_analogues_appear isofcourse-that - : 6 inquantum mechanics, theradiation rate_(transition probability) isobtained _ fromperturbation matrix elements (ug/#'/u;) where H'=v.Aroughly, where A isthevector potential. Now,youmight alsoinclude inyourperturbation Hamiltonian somekindofspintermlikes.Bandthiswillthencausethe magnetization style terms toappear when youcalculate themultipole moments. BlattandWthengoontodisciss ‘someOftlienuéleer photo” ~ processes~in their chapter onnuclear physics~and-radiation. Iwill rewt-instesd ~ the abbreviated stuff inSegre. What bbout Schiff 7 a Hepresents some oftheHeitler style semiclassical theory ofradiation in whichtheatomicsystemisquantized buttheradiation fieldisclassical. Whenyoudothis,youfindamatrix element whichlookslikeintegral ofefK-r4UpPyeTheexponential Comes fromtheplane wave oftheoutgoing photon, theA.pcomes ™ frointheinteraction hamiltonian. Ifyoureplace theexponential byunityin ‘the long wavelengtir limit (asyouintegratge over the small atomic orbitals) ,- — you clearly seethat only electric.dipole radiation survives. Ie,-yow-can replaceODbhonatzix.elementofpwiththematrixelement_of_r_and this_you atonce. recognizeagdipoleelectric xxkimnradiation: integral( r¥up*uj)Yoyisgmneral Blectric multipole. . . “Ifyougoontoexpand theexponential, younextobtain B2andM1 radiation multipoles, andsooneIe,youexpand thephoton plane wavein spherical harmonics. ‘One also sees from this treatment that’ the treHSitidi rate is ~ 7 - ‘proportionat totheem-field present-{/a/2}. -This isusual stimutated-emission- - +» ideas The idea of—spentaneous. emission Ibelieve requires fullquantizatien : ofthe.radiation fields. (ie,“second quantization") Thisisthe,magicbusiness ofngoing ton+landground stateenergy (quantum fluctuation). Although you . really cannot explain this insemi classical, you canfake it. This iswhat Schiff doesinhischapter 11whichhasasection onspontemission. Thenwhenhedocs“second quantization attheendofthebool,hegets spont tobepartofthe . straight?dFward QED calculation. ~ - ~ Actually, Schiff isprobably egoodplacetbread“hoitorelatefheVYariolis 6levelsofradiation theory:classical,-semictassical, fullQup. Jackson's Chapter 17 - oe 6.1.Heopisbydiscussing thefactthatwehavebeenneglecting reactiveeffects sofar inthe book. when anelectron changes configuration inatime that is veryshort, ontheorder ofthecharacteristic time%=1074secforelectron, then youmust consider reactive effects. Forexample, anelectron thatis ~~‘accelerated byexternal fields radiates andthenrésponds foitsownradiated fielar - Se 7 ~ 7 7 7 2.Section 17.2 gives aquick Larmor-based derivation ofthe Abrsham-Lorentz equation “of motion, equatiicn 17.9. What isTewhere15S“ddiiping ters”whichisproportional -tod. Recall thatusualchassical damping termsareproportional justtov. - The effect ofsuch a#"term is-simtler-te- a-standard-damping term,-as shown later. 3.Inowskip; tosection 17.7 where theAbraham-Lornets equation istranscribed into amorenewtonian-looking form17.51. Byusingthisform,yougetridofundesired blow-up solutions oftheother equation. This"integrodifferential” equation" iscleatly acausal, asillustrated bythe preactéleration exaipe. But you could - never make observations onsuch asmall time-scale -anyway; and-@-uncertainty 6 wouldswampsuchaneffect anyway, s0-no-problem.- Referred to-as:-calesical-microscopica causality violation. oe . ~heSection 17.8 gives-the first application: linewidths radéating-atom has: a -- damping termgivenbytheALequation andyou-calculate auniverel linewidth -called[Algo,youfindashift,fromthenaturalfrequency whichismuchsmaller_ thathewidthinthismodel. However, whenyougooverintoQsD,yougetamuch larger shift then this classical model predicts due to photon loops and vertex corrections etc("vaccum fluctuations" so-called). ThismakestheLambshiftmuch larger. ~~ 5.Section 17.9 deals with atomic scattering oflight. Youassume enexternal electric field andputintheALdamping term into theequation ofmotion. Here,inaddition, Davéputsinastandard v-typé damping termtOAcéouHt forotherformsoFdamping” like thermal effects, vibrational decays; etes—You-sotve-for-the’electrons motion ~ x(t). From this (ie, from’) you-eakculate the-radiated field; at-the-same - frequency ofcourse. You then getthe-cross section for such-radiation.into AS andatenergy Q).Then17.67 shows thetotalradiative crosa section (elastic). Ifyou are above the resonance assumedinthemodelhere,butstillatreagonably_ fe) low energy, you recover the Thomson cross section result. (Remembef that athigh ehergyyouhezvetouseQEDtogetKlein-Nishina result)(ie, Compton scattering). 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Leeee ;=ponies,=x=Ps\Ssa®e-el QuininQoandlabed oyacdeasionkenmmrh, ©Astrobiaonde: dxVeaL vhsduo,eR poRE ots a48vt__NSa.e TR,Baangkcomandabondsbin, At-Je. —..esos gROAD.—Onaumns, Tvawit.Maan 8, ooTeka Saag aree ee _an Uw), .__| __SE Ee gee ee _TRY ~atysave “SET Wan) pe A Acapes dip TSS SEDs DEER YG. wet —\-~ R~&\8h ~-W(SS). F a Qravacdngotenbioh ia. ualteOsdipolemanna . Sanpaola URnaafin “0 eS, R= aH. eR” eeOS et=Nagaeatnfeaienctor BROdKKA - 5&Bale GFRV a ~-at? false Av.OS. WOQuaPlowveukdQua: . oeae at Eres eorneaaeeQHDSem HP. ite. ox\Cater!) maga=arabes Cay© -@Yoonrncansholea cage one Awecatte be- QiuCdnaakOardope. JE, SponoRTO Comments onthe spim-flip synchrotpon radiation . 6 lyFirst someremarks -omclassical theory. Ifyoutakesomecharge andmakeit7==kecelerste,, yougetradiation. Thore‘are_noveral wayo—to- analyze-this, One way_is tocompute the vector potential Adirectly fromthecurrent d,.then + get the fields BandBfromA,thengetradiated pewer spectrum-dI(w)/d(Qaw and soon. his ishow you compute radiation pattern from anantenna (see oe Jackson chapter 9)ofradigtion froma.charge (seeYackson chapter 14).This “might becalled "charge radiation, However, suppose youhave amagnetic 7 WoreR’ whichaccelerates, bityouhaveHo{Feecurreit assuch.Wellobviously ——+thetmagnetic moment~can besimlwted ‘by“acurrent -ucvording tod=-eurl fit}, -— and this current isbeing-accelereted- and hence-radiates. Thi: redietien will - - tendtobeAL,or"magnetic radiation" whereas.the-charge radiation tends.to : beEl.Incither case,yoncan_get thexelativintio. radiation. pattem bydeing - asimple timeintegral ofthepaththensquaring (thisisforaparticle) where ‘theparticle velocity appears (forchargeradiation). anditsmagneticmoment appearsformagneticradiation. Generallyspeaking,ifaparticlehasboth charge andmagnetic moment, thecharge radiation dominates. a : 2.Jackson reviews the notion efregular syachrobron radiation amspontaneous_. 8_.mission, asiftheaccelerator whereahugeatom.Youstartbypretending there isanexternal Afield andtake Ht=A,v e,Ie,asifAwere anextemal perturbation (time dependent) which, iscausing atrnasition, Buttheayoulet Abethepotential ofthesponitaneously emitted photon itself .Thisresults : inafPalidition rateandradiatioi spactrum shown 1a°(33), ie,theQHmatrixelement appears (2/€-Wwe"}9 inside atimeimtegral, alldevetAHeisenberg oo picture, Butthiscanbetaken-at-once to-classical limit end-@i-diseppears - -and.you are left with the-classical velocity (instead ofamatrix-element of a theHeisenberg velocity operator). ‘Thisisofcourse justified because the |__ __ seeelerater orbit hassuchahugen=10'7, thus, werecover viatheory of : spontaneous emission theclassical "charge radiation” ofaynchrotrons. The "“gpinoftheradiating electron isirrelevant tothisbusiness ofcharge . radiation. Notice how the Qimatrix element ofvelocity wasreplaced with ~~ "thé Classical velocity. ~ a 7 3.Jacksons text does onpage 481give aclassical formla forradiation by@ ~“fantmovingdipolemoment.However,thedagneticdipolemomentofthemoving , lok cliarg® Seeiistobealwayé ifaiCigenstate ofmagfonent. Ie,youseepas& . ~~>classical object-im theequation; In-QH-this isbike(up/y /up}-—~Theomg -- ~~ ~ -Ronent_is_always"up" because.classical physiondoes-not know-about—spim-flipss— -- - -4yPo-anrive-at-something-reasonable for-the magnetic moment radiation, you need~ —- something. withmoreQMinit, Jackson takesoverhispreviouw.spontanecus. -eo._emission argunent butgoeswithA"=gflg.3- BAsbefore, youfirst imagine that Bisanexternal timedependent perturbing magnetic field(like from aplane wave) whicl' isinducing transitions between unperturbed levels. Butthen youletB=field ofspontaneously radiated photon, Then indirect Vanalegy youcamcompute theradiation, pattern andlifetime ofthe“upper” 7 ”“Tever singportivbation tieory. “Yougetnearly therightaasier for - -- —Aifetime;-but predictst00f-polerisation-of course. ‘Thereason” in-that -the epin-is-act-at rest-really im your comiving frame and -one-mst include .—-.—Thomaa.pracassion terms. This.yields theeffective interaction. Hamiltonian giver irequation{(40}:— Note that-a=0 is-for-g= 2. Seali you need dois . -—grind—thpough-your-perturbation theory withthishamiltonian. -The-net _—effect is_shown in(46)where youreplace thevelocity operator ofcharge radiation with aspinoperator. Theelectric polarigation’é isofcourse _. noreorlessreplaced withthemagnetic polarization Xxe. Theresult is ° schematically given im(47) andcompletely stated in(55) with allThemas _-5aMow).this."spimradiation" hasbothspimflip andspim-nomflip pieces... 8 Theimpertant peiktisthatexlythespinflippieces canaffectthe.... beampolarization; non-flip transitions leavethebeamwhereitwas.So you simply compute the total rate of transitiom from up to down, and dow “7""“Youp (Jackson Gallsthis€),Theserateoarenotequal,asequation (57) ~“giows, andtheobvious result is2polarization. (59)/ Sincetherate oe 0=“5“GETRHeEe SPiNFACUStiGNe ISFERNEE low,JOUhavetowaltaWHLLOsdThat -—— ——-enough-ef these-transitions-ocour-to-getthe-beam toasteady state for~~~—— a+ +-up-wersus—dows transitions ---—-— -2+--— ceecee | an ee en nen °) Recent QEDTests Strauch (1973) OO i + LS Frou ee a N . va Se Oe eS - 7”os 7 —_— a annn 7 eee Qeds NW ro) meee —-- -_—- ZA \ -weelas7p re a — { ~ Shaudts Goon 1473 Lotte. The ‘2Quantumeleetrodynamic processes —-rset QEOlede. ~ Thethreereactions ofordera?forthestudyofthevalidityofQEDwith¢”e~colliding beamsare: 1 eTHe”setter, a)i !4 et+e”>ty, @ : : ete tte °) ForH annit TheQEDcalculations fortheseprocessesarebasedontheassumption that(1)leptonsbehavelike Allt- Point-like Diracparticles; (2)Maxwell's equations arevalid,thephoton propagator issimply given LAby1/¢*;G)diagramsoftheorderadominateinthecalculations, andcontributions fromhigherorder Bees diagramscanbetakencareofbytheusualmethods ofradiative corrections, Thevalidity oftheseas- spaesumptions hasbeentestedwithincreasing accuracyandoveranincreasingly largeenergyregionin 1 1experiments reportedatthisSymposium fromFrascatiandCEA:letmesayrightawaythatertaiesiay Fras 'siumrcimesecheersfomelt |willnotdiscuss reaction (3)sincenonewresults'have become available since 45 *lastyear’sChicago Conference. inf , - KK.Strauch, Recent experimental results 3 Reactions (2)and(3)contain noe*ore~inthefinal state andarethus “annihilation reactions”; they donotoccur ine~e~ collisions. Aswillbediscussed below, theBhabha scattering reaction (1) involvesanadditionalannihilation diagramnotpresentinthee~e~elasticscattering.Reaction(3) +istheonly one ofthethree touniquely involve atime-like photon. Interesting resultsonthevalidity ofQEDandonthesearchforheavyleptonsarereported atthis symposium from studies ofthea?order Bremsstrahlung reaction: ette” ete ty, @) Three QED reactions oforder «*have been studied ine*e™ colliding beam systems: etge” ettentyty, 6) ettemsettentet ten, G) ette ete tpt ty. () ‘Thedouble Bremsstrahlung reaction (5)wasfirststudied atNovosibirsk [5]andhassince served asa convenient luminosity monitor formedium luminosity experiments, New interesting results are sony, reported atthissymposium forreactions (6)and(7)from experiments atFrascati inwhich theinelas- ware tically scattered electron and positron aredetected. Inreactions (4)-(7) thefinal state contains theinitial state leptons plus one ormore particles or a photons. Thesereactions occurbothine*e™andine~e~collisions. Thename“electroproduction” ot istraditionally usedtodescribetheproduction ofparticlesbyelectrons(orpositrons) inelastically “hosescatteredfromatarget;eeesromrstetiet2arecxamplesnfielecteopRRTOEETPIn contrasttoreactions(2) erand (3)which areannihilation reactions and only occur ine7e” collisions. rable . 2.1 Bhabha scattering e*e~ +e*e~ Theelastic scattering process e*e” +e*e” isdescribed bytwoFeynman diagrams: wo Ne . gessm$as . @ @) Forexperiments which donotmeasure thecharge ofthelepton (theonly ones reported todate) the hkannihilation diagram (ontheright) plays itsmost significant role atthescattering angle of0=90 ike ‘AliaheeepertmentssirarenelectmusandonsitromsratteredinterasottamplereeTEAST iven AtFrascati theluminosity ismeasured usingsmallangleBhabha scattering; atCEAthedouble . derBremsstrahlung process isused forthispurpose. Both monitors detect events dominated bylowq?- as space-like processes where itisreasonable toassume thatQED isvalid [6]. nin . ‘Themost accurate experiment spanning theregion of//s=1.2-3.0 GeV hasbeen carried outat via~ Frascati bytheBCF group [7].Their results arebased on12,827 events scattered into, theregion ince 45°<0<135°anddistributed among 11values of\/s.Theauthors fittheir beautiful results shown infig.2with theexpression: