Infrared Divergence - Eikonal Approximation

Infrared Divergence Yennie, Frautschi & Suura (1961) Grammar & Yennie (1973) Levy & Sucher (1969) Eikonal Appro ximation Phil Lucht notes (197 8) Yenne ,Fravtseh *Souya (461 PALucht niles 19173 70standard peewd, ! so ye i anthe ANNALS OFPitYstcs: 18:379-152 (1961) : 1spite :. skawaTheInfrared Divergence Phenomena andHigh-Energy i vial. Processes* !ofthe ."low. D.R.Yewnrep i +-pion +weak SchoolofPhysics,University ofMinnesota, Minneapolio, Minnesota ‘ “thou 8.C.Fraurscart cut 5 q ut5 DepartmenofPhy,UniversityofCalifor,Belen,Cooma AND =k H H.Suura- afi { PepartmentofPhysics,NihonUniversity,Tokyo,Japan wt 'i=4 7 Ageneraltreatmentoftheinfrareddivergenceproblemiuquantumelectro- onan 4 sRinamiesigiven,‘Themainfeatureofthistreatmentitheoperator ‘ Sw errareddivergences1smultiplicative factors,whicharotestededhe, £ ato).otperturbationtheory,andtheconversionoftheresidualrerioee £-_papsionintoonewhichhasnoinfrareddvergonee,andhenceaeeae, re acerCutoW,Iateinfraredfactors,whichareexponentialinfee.Gore : 5{rateddivergencesarisingfomherealandvirtualphotoes . ft ruaLacanThesefactorscanthonboexpressedsolelyintoosTepespaceinitial andfoalehargedpaxtiles andanintegraloverthoragor nes jweraorbtotheundotectedphotons;theydonotdependupsthea é ikeaoftheinteraction, Elestronseatterngromastategateetareees ; 6.478 {acousiderable deal,andseveralothe»examplesaednsee : “Tmpertantbyproductofthegeneraltreatmentieee eeinfra. ‘ ; #edcontributions areseparated inaparticular way.tdominate theradi- p# “ailvecorrectionsahigheneriegaadeteoheee hea= “magnetic terns” / Tongtteoatanionsorrectionseontogivealiheconto eeHonal-te In(H/a).Allofthese corrections canbeeasesa aaa «enaTr, fom&knosledgo oftheesternalmamonta oftheshoredet “tclthisthenprovidesaverypowerfulandaccuratewarotestimating 4k rae tdiativecorrections tohighsnoamypreeoeeSee;SupportedinpartbytheU.S.AtomieEnorgyCommission, ContractAT(ti-t)-60, i 1adeeycunoments woreaddedLothemananrpt (panels isRepos duringtheacademicyear1900-1961whilethisauthorwasaNationalScionceFoundation Seniorhain,sinktheUniversity ofPareeisgratefulioPeer aLévyforthehos. 3ySpeed BytheLaboratotodePhysioThoriquetHautesBeatOrsay. +Suppérted byNational Science Foundation Grant, . 379 aBi 4; Ke). a . Funes SV Introduction, AsIhoped, thie introduction makes much more sense now that Ihave readthepaperandthoughtaboutthings.Iwilltrytojustlistoffthethings ‘e)they have tosay. 1.Their propotype example iselectron scattering off apotential. First, by drawing the Purcell pictures, you can convince yourself that photons should tend tocome off inthe direction ofthe incident and scattered electron. This fact isthen verified bythemathematical form ofthefactor show in(1.1). This appears inJackson classical aswell asBDRQFT calevlation. Two facte: the higher theenergy Eandthelarger thesattering angle, themore radiation you aregoing toget. Beth rather obvious just bythinking about the“proper fields” fethe Purcell diagrams. Now,theenergy intheradiation iscenstant atsmell which implies adk/k photon distrikbtuion, the infrared blowup situation. 2.When yeucompute theeffects ofbeth real endvirtual phetens, youfind that tthedk/ksingularity iscancelled andyougetaresidual leg(4B)/B facter in your radiative corrections. Angular integration over thedivergent temmex anguler peaksgivesanetherlegtoradiativecorrections ofthefermleg(E/m)*A. (@) Thus, after cancellation eftheIRdivergence, yeur radiative cerrectéens are typically ofthis "deuble leg” ferm, tefirst erder inALPHA. Ie,itisreally anexponential butyoucancensider thefirsttermwhichisthenthatdeubleleg. The electren gase gives the typical result: the cross section with some definite resultion dBisgivenbyacorrection factor timesthecrose section youwould compute fortheprocess ignoring soft photons alltogether, andthe correction factor ismaybe 1.2 orso, ie, not very large. But finitel The explicit answer for electron scattering isgiven facing page 400. 3.History ofthis subject. Bloch and Nordsieck in1937 first noted the problem efsoft photons unavoidably accompanying your process; they first showed how therealandvirtual photondivergences cancelgid!hexyouseteFiniteradiative correction that isnetvery large. Then in1950's Lemon andfkeax friends did @semiclassical analysis which required anarbitrary hard/soft photen cuteff. Thenin1954JeuchandRerlich showedthevirtual/real cancellation toailorders ofALPHA (for electren scattering Iguess). This paper claims tebearefinement. Oo oftheJRarguement. Intro, con't heAcrucial peint inthe argunent isthat yeu only have tewerry abeut phetons fo) ”onexternallines.Intuitively thisisbecauseseftphotensarelargeanddon't much get inte the high energy small space guts efafeynnan graph. This isef course just anintuitive statemenet. You really have teshew it, and this Anvelves censidering “overlapping divergences" where one end efaphoten er beth land somewhere inside amessy diagram. InAppendix Athey deal with this preblem butnetcompletely andrigereusly, buttheyatleast censider itwhich isamimprevement ever Jaych-Rerlich. ‘This external lines thing isefcourse what lets yeuextract theIRfacters and get acalculable result. ‘Asecond improvement everJRisthattheygiveaseries expansion fer thenen-IR radiative cerrectien germs. Section 5.Icould only skim this. Relates te"Kallen's arguement abeut the incensistency ofQED" which Iamnet familair with. Basically they say that Kallens facters must bemedified byIRexponentials such asin(5.4) andmaybe this makes seme difference. Anether thing Ident knew abeut is"spurieus charge renermalizatien".° . fe) Someinteresting remarits,theughenpage131:thevertexandplectren prepagater are enly IRsingular when yeu are enthe mass shell. Sectien 6: Summary. There ore several ether types ef"corrections" besides theIRencs, butthey tend tebesingle legrather than deuble leg, sethat the infrared corrections tencd tebethe main cerrectiens, These corrections include: vacuum pelarizatien (ie,electren cerrectiens tepheten lines), "magnetic terms" whatever tha means. oO Section 2:Radiative Corrections toElectron Scattering. (oe) hegeneral scenario hereisthatofasingle‘electron comingalongarid scattering asmany times asyou like off apotential. Ie, wedont consider the heavy nucleus initself here. The blob shown infigure 1issupposed tobeasum ofall possible feynman graphs! The scatterin gcenter picks upnoenergy, so €=E-E! isthe energyb total ofany emitted physical photons. (a) Assume for the moment that there are nophysical photons, Ofcourse there always will bebecause adeflected particle must radiate. But here wewant toexamine only the internal photons. Arrange your infinite set ofFeynman graphs bytheir number nofinternal orvirtual photons. LetM,bethe sumofallgraphs having precisely nvirtual photons, noradiated photons. What question dowewant toask?!? First, byrecursion youcanderive (2.8) which shows theamplitude M,where allIRterms have been factored out. Virtual photons only cause IRsingularity when both ends are enexternal lines, aswas shown somehow inAppendix A, EG, the first term intheexpansion (2.8)forM,'sintegrand YqshowsS(kj).....5(k,) andisthus infrared singular inall the k's. But since these k's here are integration variables(whichisnotthecaseforexternalphtons),youarereallyaskingabout fe)singular portions ofthe mlti-integration .This sing gest isolated into the object called B. Look ahead to(2.23) where you get the IRsingular part ofB.Recall thatde=Wak,whereas theobjectS(k)(whichisthevirtial coupling ofthat photon toexternal lines) issingular inthesense thatitbehaves as1/k". So now you have kdk which isnot yet singular. But when you add the photon propagator, yougetanother 1?and50yougetlak which islogsingular. Sowhen you speak of"IR singular" you speak ofthe fact that Bhas singulerities inthevariable ),. Youdonotmean somuch that S(k) issingular. Butsince S(k) isreandcontributes tothesingularity, youmight refer tothisS(k)thing as being an"infrared" term, Sothe"infrared" piece pfM,isallintheobject B()). Notice that M,,involves all orders offield theory except there are only nvirtual photons. Thecomplete Misthefull sumofeverything, andagain theonly singularity in\ appears inBwhich isnow inanexponent. Sowhat makes the "virtual photons singularity"? Itis2combination ofthe photon propagators with the electron lines which are created byadding those photons. Hard tosayitisjust thephotons themselves, 0 So,youknowtheIRsingularity duetointegnal photons toallorders .interms ofanobject S(k) which isessentielly something offirst order! The "all orders'ness merely exponentiates the thing, something Iamused totrom Regge theory. (b)Sofarweknow the singularity inthephoton mass )ofthe complete electron 0 scatterin gamplitude dototheeffect ofallinternal photons. Nextweaskiwhatisthe effect ofexternal radiated photons which uptonow wehave ignored. Consider anamplitude which isradiating nphotons. Since this amplitude contains all possible internal photons, itcontains the exp(qB) factor just discussed. Let usnow compute directly the DCS for radiating nphotons ofenergy total =€=£'-£E, This isgiven in(2.13), showing thephase space andthe amplitude squared. Again youdoaniheration toexpress theamp”interms of IRfactors called M(k,)waichareLittle amp”factors formutkinxgxsh addingan external photon ofmomentum kj. This "add-a-photon" term appears and gives the entire \- singular part. Look, at2.24 toseethis. Notice that dk=k7dk, sothecombination ofk”*fron andthek7!fromtheenergy denominator causes Btobesinguler inthevariable at)=0. the8factor isessentially determined byafirst-order Feynman graph, actually thetwoorderg graphs shown inFig3(a)and(b). Itgets exponentiated inthe usual way, sothe big net result isthis: parkxsf DCS toradiate photons ofenergy€inelectronpotentialscattering fo)=expQg [B+B])times somthing non-singular in). Thus, the IRpart has been completely isolated and factorized off. And wehave done things toallorders inthefield theory! Allyouhave toshow isthat B+820, and this issomething you show infirst-order, sotospeak. Actually, you want to showthat thesingularity inBat \=0cancells thesingularity inBat\=O Then there isnoIRsingularity left. Conment": Experimentally, what are wesaying here? Ifyou compute upyour total feynman amplitude for radiating nophotons atall, just scatter the electron, you expect afinite andwer but you get aninfinite answer due toaninfrared singularity. Luckily, itisimpossible todothis experiment! Soyou dont care that itis singular. Inareal experiment, there will always besome photons coming off, no matter how small you make your resultution. What you measure isthe DCS for electron scatterin gplus unknown number ofphotons with energy less than something. Soyou have toadd the elastic rate tothe rate with photons. Inthis (incoherent )sum, theIRsingularitiescancelandnosingularityappearsinsomethingthatyoucando re)with anexperiment. (c)Herewehaveexplicit formulas forBand‘B.These arederived intheappendix A, butyoucanseewheretheycomefrom.Whyaretherenounknown"blobs"inthése oformulas? Because S(k) isamultiplicative factor, notthewhole thing. Youdo not include the blob inamultiplying factor. Howdoyoushowthat theX=Osingularities ofBandBcancel? well, you Just doitbybrute force. Inthelimit that pandp'arevery large (high energy) and€=small (forward scattering), you canwrite Band®inclosed formas (2.27) and(2.28). There youseeexplicitly the\=0singularity. Itislogarithomic inboth cases and cancels between the two terms. Sometimes the theory isreformulated interms ofalow-end momentum cutoff Kpininstead ofaphoton mass). Sameresult, samecancellation ofcourse. Itsjust that you want touse the same cutoff method inboth cases! (a)thenonIR terms: Ifyouknow B,theentire amplitude forelectron scattering isgiven bythevarious m. See2.3. The“non-infrared" terms aretheones that areleftafter thecanclleation oftheinfrared singer terms. Ttisthese non-IR terms that tell you"the answer", ie,theDCSforelectron scattering. ‘These terms are the"radiative corrections" that youoften compute around ablob. Notsurprising thatthethingswecomputedabovearesamethingspeoplecomputedforradiative le) corrections tocertain processes involving oneelecgron line. Actually, Ithink theterm radiative corrections" means only "virtual photon corrections". The corrections duetosoft external photons imconsidered inthenext section. (e)_the NonIRreal photon terms. Nowwewant toinclude theeffect ofsoft Brem andcompute explicitly theelectron scattering DCS. Earlier weonly wkamix that the IRsingular parts cancel. Here wewant toknow somw real numbers for what isleft over, The answer isgiven Ithink in2.57. (£)Nowsuppose wedotheexperiment where weforce E'~Etobeless than some smali number JE. Ie,wetryourbest tolook atelastic evernts withnophotons. Then 2.58 shows you what you aremeasuring. le) aden sacien (2.2). 6PDHP=ALgreghteonUalemokplates ‘\polental. ky 2 ae Gals ke)=omplindes Duce glecton moprgutes eeIRpoatait, feSRils ferlyFSC),hye: Woocomes«Gala Fa)=SEke)Gweeshon)«BaeFayhs) °SS wna Quace Mefous.badeauntKyewoah BRgasSCG 4A ,ae) seBa=SKK)Sean}Que*Slka)apn RakNaderdosymentioge Deaoe \ea" ail Qn(a=ek)=SCkm)SCKwn) SerCheeae) ‘5SCDBLCyMeee)Yeae ro)o +SGA)GosCeeee|~yNyvf +Jesh.)feibesby)+n(hakenot Ore ThcemamemeWoh,EABLOCK» baneKensby) QoCerkn)=Skee)Soma) Bare(leyKnee) © SK) BC keeKen)+SC) EethorKe)be ~*e-Clas-KaeejFeet,ke) Vuew .Nangshoraie' _—— Qn(eos)=BeBIE)|Se)Slk-s)Bae geete) =SESE) |SCs)Gua(leheen)+Bbite 4S(Ven)Bat(hogan) +S(K)pat(.co) e *3(lu)fait(leerbe)~~ocke. OK,oStee enubi guntetin. 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Quen regimedDaeree pops wh .AetnesCS art ~ cyaogsndtvon (28)Weareermpite Quer+| ()Emntintaswileg rya\ye‘de=\ntliyredaeSpstesinet . eee he COENEN ord3a. WE wewnails8(Gogegl Za)We) BakGayl=pede! . de=Vit!afiag,ake.”YM PK 2ELE onde2AE)Ole Wh crenS ao! : di4e=viktoS(e-Zt!)(Ge) “3} 'i) aas_(pias). sean). SaalgedntNa798} ie ° ba,w=Yehae Zeedonss VWrtud= mpd ot«Wy >-~ . —y Sree Mrsen ML=SEAM DhowsadugwihC2us). OVso:inShoredeanDOSine6!bethned,hencee=Zkbalkfired. @\WoATduebsductoed plese, e597,4halen, Dorrcemsinn &Bouse TRpork|Gi,semdan de(et)(20). nduneromeBerEli)sandglrrsynee &(2.17). Qfhe Yeawe(U0)ae: ~ c _2 \ die Sk ~~he Goa bo(19). yo GN foxy .as&=<ZzacomySte)Pre(kear--ks) 2a nw ak, -estar! Sk)Banelenuk).Aye €. Vel © in(e- 2b:*KSyeste)—$§ c . m ~i.Seri Ataki Aw dle: x WerquegeseMe|T,freTh,Nye le) Ee n= WeShQR). SouwaxCar) Wardtko@Swegetontayaw exposed. (9)24(B+8) cL » oege- < te e oe Added Comments onSection 2onElectron Scattering. fe) _Tamgettingmoreofanoverview nowastowhattheyaredoing.Insubsection (a)they computed.all virtual photon IRstuff andthey concluded that thecomplete electron scattering amplitude Mcanbewritten as(2.3). Thething Bcontains allthe potential IRsingularities, andthe"reast" ornon-IR.part ofthecomplete blob electron scattering amplitude isgiven interms ofthem,objects. Inturn, as shown in(2.12), these m,aregiven interms oftheQiwhich arecertain remainders which were obtained intheiteration. - :soe . Now,, wegotosection (b)where they compute theIReffect ofreal emitted photons. Nowthevarious objects (B,whichmakeupSq,arereallyfunctions.of - theoldm.Ie,inorder tocompute accurately the@,youneedtoknow themy, which istosay, you nedd toknow thenon-IR terms due tothevirtual phaton effects. Soletssayyouhaveworked oyt,thevirtual photon corrections andyouknow allthem.Thenyouknowthe, .Thus,fromformula (2.17) youknowexatly..in full details what theDCSwith energy loss €is. Thesumin(2.17) isover.the nuither ofreal_emitted photons. Ofcourse youknow&froma first order computation Everything isthemsummarized in(2.21) and(2.22). Notice thattheeffect ofvirtual photons iscontined both in8andinthevarious&. an 0 Soobviously, whatyouwanttodoisJomputewhat theDCSshould beforelectron scattering when there is€energy loss. Then gocompare ittoexperiment. Ofcourse this isjust aQEDproblem, butyouaresumming over allorders andyouhave todeal. with the IR"problem". Inparticular, athighenergy theinfrared exponent NxB+B isexplicitly given in(2.31). Soitremains only tofind the"non-IR" part ofthecross section, which iscalled d¥/ dé. Section (c)computes theexonent asIjust showed, itshows howtheIR singularities cancel, and ittells what the finite piece is! Section(d)thentriestocomputesomeofthem,,itonlygivesmy.Again, thepoint here istotrytocompute themsothatyoucanknowtheBy Section (e)thenconcentrates onthefirst fewdé/d€ terms involving zero and then one non~IR photon, which just means keeping first two terms inthe series offormula (2.21). Thetotal result keeping only these terms isstated then in(2.57). Again, you seethe IRexponent which was already computed, andyou seeafunction Fknownasin(2.45) asagammafunction, andyouseetheBeandCmfunctions ofcourse,thoughthesecdononeisrecodedasGy. fo) Finally insection (f)yougetananswer forthecross section whenyouhave unobserved photons ofenergy uptosomesmallDE.@%,isjusttheusual elastic cross section with nophtons, something youcould compute toanyorder. Inprinciple, thisisthecomplete answer ifyoucould compute themjhence the&hence G).(over) Now try tosummarize: jou want tocompute the observed electron ICS, but youknowtherearegoingtobesoftphotons whichyouwillaccidentally alsomeasure °50you have'to include these also, You know that you have toseparate outthe internal phtoon effects aswell asthe éxternal photon effectsx inorder tozget theIRsings tocancel. Yéudothings atonce toallorders; youfind that toall orders theIRsing cancdllation arises from something witch youcancompute in lowestorder. . But you want more than just showing that there are nonew IRsings left over. You want toknow exactly toell orders(in acertain sense) what the observed DCS will beifthere issome resultion soft photon loss. You can compiite the elastic DCSsoanyorder youwant, butyouwant toinclude soft photon effects toallorders, sort ofabybrid situation. ne Youdothis andthe result isanexplicit formula fortheDCS. Itinvolves certain functions which can becomputed byconsideringgkkax the effects toall orders ofboth internal andextémal soft photons. Soitallboils down tosorie brute force calculations. ~ : What dsoddhereisthatusually Ithink ofjustedding asmanyFeynman graphs asyou want toget ariacouarate ‘answer. But. here you really have toadd aninfinite numberofgraphstoget‘therightanswer. : fo) a e Section 3:Further Examples. (ae)(a)Consider electron scattering frompotential accompanied byonehardBremphoton. You still have tothink about the unknown number ofsoft photons emitted, sothis isjust anapplication ofthe preceeding stuff. The hot photon has k,the soft photons have EPSILON-k energy. Skimmed only,. (b) Inelastic (deep) electron scattering. Radiative corrections make radiative tail. You add correction photons tothe electron/photon blob. (c) Multiphoton production. Ie, just multiplie Bremstraullung, work done byGupta and repeated here using their new methods. (d)Recoil inelectron scattering. Here you doradiative corrections onboth sides ofthe carrier photon. Ie, there isanelectron blib tocorrect and aphoton blob to correct also. (e) Brem off nucleons innucleon-nucleus scattering. Gomments: Thetopichereisreallythatofradiativecorrections toscattering fe)processes infield theory. Nodoubt good references here ifIever want todelve into that subject; right now Iwant toconcentrate onQCD Dyson equation and Jim Ball's papers, soput this onthe back burner. . | AddednotesforSection 3._ Lastnoteswerenogoodbecause Ididntunderstnad thefunctions ofSections 2. 6(a)electron scattering with large energy loss. Inthe general formula, the integral over ,which involves iemitted non-IR photons isnowcalled G,andthe sumofalltheG,iscalled @.Since usuallydA41,theelectron DCSisgiven bymyequation (3.54) page 402. Now when you have alarge energy loss, Ithink the emission ofmore than onephton inthenon-IR issupporessed, soyouuseonly theG,term. Youcanthen goahead and compute this inthe Born approx [now weareback toadding afew Feynman graphs} andtheresult is(3.9) plotted inthefigure. This issomething you could compute without going through all this paper. The point here isthat we can compute the corrections tothis result. (b)Inelastic electron scattering. Assume nowthat the "target" can absorb some energy ,namely €!,andthere aresoftphotons with energy€4@.. Formla (3.14) gives the DCS inboth these energies: you see the usual IRexponent, and allnon-IR photons arenowbeirigignored soweonlydealwitheewhichhereis ae)»theinelasticexcitationcrosssection.Wewanttoknowsimply.d¢/de ro)integrating over the target, because this isthe thing you usually observe. This isgiven in(3.15)} ifthere were nophoton emission, the observed DSC would tell you the excitation function. But the presence ofthe photons causes aconvolution ofthis excitation thing with the photon emission, see (3.15). This isprecisely the theory ofthe radiative tail. Toget the actual excitation spectrum you have tosomehow de-convolute. (c)many photon emission. =have nonew comments here: they duplicate Guptas results with their methods. fe) Section 4:Infrared Analysis inthegeneral Case. 6 Sofarwehaveconsidered onlyelectron scattering offapotential. Nowconsider ageneral Feynman graphs with many incoming and exiting charged lines. Wenow want toconsider the IReffect ofany number ofvirtual photons onthese lines, and any number ofexternal soft emitted photons. All wewant todoisgeneralize the earlier casewhere there wasonlyonepairofexternal charged lines.” Nodoubt itwill turn out that you only have toconsider photons onexternal charged lines, sothings will becontrollable. The general reduction for the amplitude isgiven in(4.2) where the 8isthemultiplying IRfactor forphotons onexternal lines inallways, andwhere fy,issortoftheFeynman graph with iradiating photons, but including virtual photon corrections. When things are regrouped you get the same kind ofexponential factor out gxem front with a Banda$.These factors aregiven insubsection (b)andaregiven basically bysunming overline-pairs, hence theB,;objects. Itturns outthatnowBcan ‘becomplex soyou only useRe(B). TheIRsingularity cancels between theBandtheBasbefore. However, due tokinematic complications involving the way the soft photons come off, itistoo hardtowriteageneral,formulagivingthevariousradiativecorrections foran fe)arbitrary process. But they try togive some examples and general guidlines. IhaVE just skimmed this section. The main idea isthat you can ignore virtual and emitted photons from inside the blob and therefore the IRfactor can beisolated asamultiplicative factor which ends upinthe exponent. The virtual and radiated photon IRfactors have singularities which cancel. And there are left over nonsingular parts which are the radiative corrections. le) AppendixA:extractionoftheinfraredfactors... fe)Case a:Consider ageneral Feynman diagram, Pick out one electron line and trace it all the way through the diagram. Wewant toknow: suppose wetake anexternal photon with momentum kand sum over all the stick-on places for this line. Aswelet k gotozero, can weseparate out the piece ofthe result that isinfrared singular? Ifthere issome singularity, what causes it? Look at(A.3). When you stick this phtoon onto anexternal line, you create one extra internal electron propagator. Ie,youmake a1/(f -{-m).When this operator isputupagainst u(p), youseein(A.3) that a1/kterm isgenerated. That extra electron prop yields a 1/« singularity. When you sum over all insertions for anexternal photon inthis way, you can classify the insertions into three kinds: insertion onthe left end, insertion onthe right end, and insertion inside the mess ofthe diagram, Itturns out (this issomehow aresult ofWard like thing) that the inside insertions asinfigure 3(c) donotyield this infrared signularitiy. Sobasically, instead ofhaving to add aninfinite number ofthings, you just add graphs of(a) and (b). The infrared- singular piece ofthe blob with this addition isgiven in (A.11). Ofcourse for 6 atrueexternal photon youwilldotthisintoapolarization e,endyoucanthendropthek,terms bygauge invariance.” Also, setk*=0. Sowehave now figured out what isthe infrared-in-k piece ofthe amplitude when one extra photon isinserted everywhere onone line. Case b:More complicated. Now you want tosum over all ways ofadding avirtual photon toone line soboth ends ofthe photon are onthe line. Figure 2shows the6ways this might bedone, Itturns outwhen one orboth ends ofthe photon land inside the blob, there isnoIRsingular contribution. Thus you need toconsider only theoarts (a),(b),(c) ofthefigure. These yield theinfrared singular amplitude shown in(A.19) where youseethree terms after yousquare. Ithappens that this virtual photon thing istied upwith mass and charge renormalization inthe UVend ofthings, but Isee noreason toworry about that here. Wejust want the IRpiece. Comments: there must beamore elegant way todothis, something less brute force. Maybe intheir later paper they will explain anew method. For now this isOK. After all, this is1960, afairly pioneering paper inIRbehavior. Prior tothe idea of containment this was not such ahot subject. 6 Grammer +Yenme (9713 i PHYSICALREVIEWD VOLUME8,NUMBER12 18DECEMBER44,’ieJImprovedTreatmentfortheInfrared-Divergence ProbleminQuantumElectrodynamic PEyex\e) G.Grammer,Jr,andD.R.Yennie anes4 LaboratoryofNuclear.Studies,CornellUniversity, Mhaca,NewYork14850 Teee(Recetved9August1979) necesi ree} cimawceacebunculesassocatedwthinfcaveddivergencenquantumeleo~ wenn patagesincehavelongsincebeenresolved,aconvenienttechiesforWdenttyingtheappto= ReeteTheasebatofFeynmangrapishasbeenlacingSaskseePresented cones | Rese.(Thepolatization sumsforbothrealandvistaepee Tearranged Intotwoparte,tareporrany(inedthepolarization sun)resembles gaugeateeeswhosestruc. |parebermitsasimpledemonstration ofinfraredfactorization esexponentiation. Ittseasy ae 4Taleiedationthattheresidualfactors(vith6-polarnaton sereInfraredfree, fronpactsSteachsopnrategraph,inconttasttoearliernews contributions fromeosofgranhtadtobecoplacnelpeeseeeeaUlsstratedbyadetaitea me 1featmentoftheradiativecorrectionstolowastonlespantscattering,andgeneralleations erranttootherprocessesareindicated, eesteg at readin1.INTRODUCTION subjectevenifitisconceptually important, weed,seve. jnerefore, inthepresentpaper,weshallavoiy4wfF.oe Theinfrared-divergence probleminquantum theseconceptualissuesandsimplyassumethatoreineects electrodynamics hasaJonghistorywhichweshallthereissomeconvenientinfraredcutoff, a,eo iRotreviewhere.’Itisknownthattheinfrared- Guraimistopresentamoretransparentden.Mares | divergentcontributions toanobservableerese pustrationoftheinfraredseparationintodivernyusemtox {sectioncanceliuanygivenorderofperturbation factorsandinfrared-freo expressions. Thetech.fvractice.a, j pantieProvidedallappropriate real-andvistual- ‘nieaperewritethepolarization sumsfor)reyescoxa:Photoncorrections aretakenintoaccounttogether, bothrealandvirtualphotonsasasumoftwoORSUSSem }Asidefromtheleadingdivergenceineachorder, {aoaifiedpolarization sums.Oneofthesepolarin{weteve, |thecompleteproofofthiscancellation israthes tionsums(Ktype)willincludetheinfrared-diwer, “wceassc .complicated, Inpracticalcalculations, oneoften Senttermswhichveryconveniently factoroutae#4:etasvar:Ocean poeBaaletettoncuitatSxponentiatewiensummedoveraltampons”fcavern1intermediatestagesoftheworkandthendoesthePhotons.Theother(¢type)willleadtoinfrareé.§“+Howere: 1 infraredcancellation byhandattheveryend, freeexpressions whichnolongerrequireacut686caehave.Clearlyitwouldbedesirabletohaveaprocedure Theinfraredfactorsdependonlyonthemomeaa4#3,Fee:wherebyonlyinfrared-free expressionswould ofexternalchargedTarlciesantatealreadywetMérxcribe : havetobeevaluated, known.’WhenrealandvirtualcorrectionsaceSuestresF: i|Recenlyseveralauthorshavegivenreformula- suitablycombined,theyleadtoinfrared-finite }tionsofquantumelectrodynamics usingthecon results,asisalsowellknown, .iceptofcoherentphotonstates.Whilethesere.‘Thedifficultyofgivingeuchademonstration Mebeeseax:formulationsmayimprovethelogicalfoundations toshowthatitworkstoallorders,includingtr4torrens ‘Ofthesubject,theysofarseemtodoverylittleless-divergont termswhicharecontainedwithizAaaSe foractualpracticalproblems.Ineffect,they.proseigh-orderinfrareddivergence.Inoura_4‘reaisstan, :Seemtojustifytheuseofaminimumphotonmo-Broach,theinfrared factorization becomesasio74"a:vec, —Tentumingetualcalculations,‘Theconceptual pleprobleminalgebraandcombinatorics; Ittwo&beat Problemstreatedbytheseauthorsarisebecause Teratedfromtheactualanalysisofintegrals. “**roaserrn ~onewouldliketobeabletodefineasymptoticin‘Thedemonstration thattheresidualfactorisatexeandoutfields,IntherealWorld,anyexperiment infrared-freeisalsorathersimpleasitinvolret(fatten. iscarriedoutduringafinitetimeintervalsotheonlylookingat,butnotcarryingout,theintegr>CBeegenemissionofverysoftphotonsisnecessarilyin.ons,ThisistobecontrastedwiththecompliORToemaewaythiePathPhysicalresultissensitivetothecatedyalkiveninYFS(Ref,2)inwhichselSimame Waythishappens(preciselybecausethereisanintegralshadtobecombinedinordertoobta® LyMee . infraredcancellation), anditisnotinteresting to“heinfraredfactorization. Inthepresentdewmakeadetailedanalysisofthedependenceonthestration,setsofintegralsareagaincombined, 394bes .-timedurationofanexperiment,justasthede-butinasimplermannerbecausetheKztypepor(aeerie, . Pendenceofscatteringresultsonthodetailsorfatizationsumhasaresemblance toagauget"™ $2wares i thewavepackets hasnotbeenaverypractieal _formation, Risiohasa |oO. 8 4332\ ee cen sr Seg IEIEENGym!pareve11nnyth — Granmer andYennie, Cornel (1973) @) 1,Introduction. Thispaperiswrittenabout12yearsaftertheearlierone.They have found amuch better way todothings, asdescribed below. Although paperwas receitted inAugust 1973, paper isrestricted toQED and QCD isnot mentioned. Between the two papers the theory of“coherent states” appeared, but they dont. | think this has much use inpractice, sothey skip it. 2.Virtual photons. Asinthe YFS paper, the canonical example iselectron scattering, here where there isonly oneinteraction xith thepotential. Here ishow‘themethodworks: youwanttosumoverallvirtual photonsonthescattered electron line,skipelectron loopsfornow.‘hetrickistobreak eachvirtual photonpropagator intoaKandGterm,thenexpandthewholething.Usingfancycombinatorics andtheFeynmanidentity (whichmakestheWardidentity), you can show that the sum ofall insertions ofa photon onto agraph with. asetofGphotons causes sasimple multiplicative factor aBwherea=1/137. When youthen goahead andsumallgraphs including allmixtures ofboth kinds of photons,yougettheYFSresult:theKphotonssumtomaketheexonential,and re)thenthemyresidual IR-finite fumetions aréthefeynman graphswithonly@photons ineluded. Much nicer than the earlier proof. Sothefeynman identity isnowwhat tell syoutoignore inside insertions, somehow. Ihavenotreallyfollowed thedetaile butcouldifIhadto.-For elesed electron loops, youalways getzerointhelimitthatanattached photonissoft(Idon'tyetknowwhythisisso)soyoucanforgetthem.Thereisa detail aboutphotons whichareself-energy corrections ontheexternal lines, but not abig deal. . Inlast section ofthis part ofthe paper they show exactly how itisthat when youcompute asimple Feynman graph withonlyGphotons, there isnever anydivergences. Nor doyou seem toneed any kinds ofacutoff. 3.Real.photons. Howconsider thesituation where youhavenunobserved softphotons and maybe also some observed ones. The amplitude isasin(3.1) where you see the virtual exponential already exposed andyouseeallthepolarizations ofthe unobserved buys. Incomputin ght eDOS you havete sum over all these polarizations. Thesesumsmakepropagatorlikenumberatorsg.,+Asinthevirtualcase,you fe)break these down into aGand Ktype term. Somehow the Ktype terms cause the other exponential and you get the resul,t shown in(3.21). This isthe same as the old result, but itwas derived inamuch better way, hisGeneralizations. Gouptons scattering iseasysince wealready didthegeneral casewith observed photons. Youcandivide thep-leg fromthep'-leg ateither [o) hardphotonandgetequivalent results. Next the closed dfermion loop detail ismentioned and gives nothing. Finally, electron-electron scattering is“more complicated butcanbe done, ascan multiplie Coulomb scattering. length ofpaper: 26standard pages. 6 6 Levy ¢SucreR (964 8? sO pote) Sah RG : : io! ih ile PHYSICAL REVIEW VOLUME 186,NUMBER $ 25OCTOBER 196%h9B 186 ioni ‘ ae/ KikonalApproximation inQuantumFieldTheory* iydeeSARout, oO MaumceLévy}anoJoseruSucuent 4: CenterforTheoretical Pines, Department ofPhysics andAstronomy, University ofMorgland, 23S whe allege Pork, Maryland 20742 . ae (Received 5June1969) ny ‘Theckonaapprosimaton forhigh-energy colons Togfam inthetheoryofpotential scattering, aa{isconsidered fromtheviewpoint ofrelativistic quantum fieldtheory.Westudy,inparticular, theFeynman aaraplitade AG) describing thescattering oftwospin-O particle, oandb,interactingbytheexchangeof2a fpin-0mesons.”WeshowthatifMs),thecontribution toJf(f)atisingfromallwth-orderFeynman certGiagrams inwhichexactly mmesons areexchanged between aand8,iswritten inanappropriately sym< edimetrizedway,andifthetermsinanyoofbparticlepropagator whicharequadraticintheinternalmomenta a ForTrethendroped;theresultingexpresion,A,%*(),maybeevaluatedinclosedform,andthesumover, ey ‘whichdefines Mie(s,), maybecarried out.Therepresentation ofA!*(s) foundinthistrayinvolves the 3‘exponential ofafunction xofarelative space-time variable z=(ex)andtheexternal momenta; xisa Be 7{elativistic generalization oftheeikonalXjoefamiliarfromthetheoryofhigh-energy potential scattering, oy: 'Hie)inrbothcrossingsymmetricandtne-revereabiavariant, InthestaticHimit(w+«),xtends ||eto%pufortheappropriate Yukawapotentialand/**=—M=H/Brv/s hasalimitingformfra",which aeiforw{oefothe amtheCeoofpotentialseaterng;frsmallseateringangles,fotcoincides Mt Fey:Whithestandardreal.Theamplitadeforparticleantiparticlescatteringisstudiedinthesanemodel. ad|TtisshownthattheeikonalXs(s)associated withthecontribution ofallannihilation-type diagramshas& 5we 7logarithmae singularityatx0whosecoefficientisproportional toa()-+1,whereca()istheRegge-trajectory a the« functionobtained fromtheasymptotic behavior oftheladder-type diagrams alone.Anotherconnection Ua tribuswithReggebehaviorismadebyshowingthatthesummation ofacertaininfiniteclasofradiativecorrec- ERR fromTonstothelowest-order yeCompton amplitade givesrise,inoureikonal approximation, toaneikonal 2), —_—ex(a)whichhasasimilarlogarithmic singulasity withstrength1-+8(0);here(0)isthetrajectory function, eaeJntroduced lessdirectlyinearlierwork,whichreproduces themajorpartofthespectrum ofpositronium on Be Prop BitingBO)ctesmecL.Ageneralization ofasimplealgebraicidentityusedinthederivationoftheabove “he que! see Aeeedofaiintegral representation, permits theirextension tothecasewhereoneoFmore BH amp! forties areofthe massshelThisillustrated byacomputationofanelkonal-typeapproximation tothe YAform Green's function forarelativistic particle moving inanexternal scalarfieldandbythesummation ofan ae‘Green'sfunctionforSipationtotheverterfunctioninthemodelreferredtoabove.Thepossiblityof cs(eaApplying anoffshll eikonal approximation totheanalysis ofproduction processesisemphasized. )ne (eo)I.INTRODUCTION wherebisa2-component vectororthogonal to=p/Ip}s<)#t Piette eehasbeengreatinterestin24the“eikonal’"x isdefinedby ane high-energy approximations toscattering ampli- =m Reg, whertudeswhichexhibitanexponentialdependenceonsome x(b)=——[Vib+Bede. a.Ryvise‘ofthekinematical variables, especially inconnection [plJu. awiththerevivalofReggetheory.Simpleapproximations. MB x.‘ofthistypehavebeenknown foralongtimeintheItseemsworthwhile toasktowhatextentanalogous 4B ytheoryofnonrelativistic potential scattering: These pproximations fortwo-body: scattering amplitude 3i}aretheso-called eikonaltypeofapproximations,!-? maybeobtainedinquantumfieldtheory,Inthepresent.laa:Forexample,foraspinlespartieofmassmscattered -Papes.weshowthatthereisindeedanaturalrelativistic 3fhbyanexternal potential 7(a)thescattering amplitude -£enetalzationoftheeikonalspproximetien, re ae 'f(p',p)maybeapproximated, forlarge|p|andsmallHIGUESUteBYkeiersimpleandmaybeuselu!tor withsee eeeaetondersaiablerescicronson VitasCeevestigations oftheasymptotic behavior ofRetesttering suitableres onV5scattering amplitudes. HEH kept| ‘Theusualderivationsofequationssuchas(1.1)are}{#(1.4), reallfayeurvrv(eier—ty, (1.1)basedoncalculations whichstarwithexpresions forBeaksOmi ,1)thescatteringamplitudeandtheSchrodingerwaveSfMsHen. —_——function inposition space. InSec.IT,wereconsider the399? gene anguRaigfonrt bytheU.S.AirFores,underGrantNo,problem ofnonrelativistic potential scattering inYM also:. momentum space, starting with anexact expression for cronmneaveofabsencefromtheFacultédesSciencesoftheUni- Fi = eraeaveofsencefromtheFacultédesSinessofheUethewth-ondortermfaintheBornexpansionoff.WeEt‘Att $pinSosGuinn Plow showthatifintheenergydenominators appearinginthis2k the«)Yat, Naturforsch,2,199(1947asus,editedby¢XPFESSION,termsoftheformK*aredroppedrelative-Mwhich fuFenaBusanGiieyteedeSheytotemsoftheformp-K,whereKispartialsumoff portaak,1959)Vol.1,p35. internal momenta, theresulting approximation tofa J—behasani,F-Sehih.Phys,Rev.103,431950)soealoD.S.Saxonmaybeevaluatedinclosedform,withthehelpofanjiffy,— fo)See,Cg,Ref.2,pp.342-344. identity usedinearlier,closelyrelatedworkinquantiui# Ke on,‘ 1861656 et1 i . A| BallYovn*Zachartaser (1977) ‘Nuclear Physics B132 (1978) 509.-530 . ©North-Holland Publishing Company | NEARMASSSHELLSINGULARITIES INQUANTUM ELECTRODYNAMICS * JS. BALL** University ofUtah, SaltLake Cty, Utah 84112 D.HORN *andF.ZACHARIASEN . Galifornia institute ofTechnology, Pasadena, California 91125 Received 11July 1977: . (Rerised 30September 197) | WediscusthenearmassshelinfraredbehaviorofQEDbyperforming anexplicit sumoverall Feynman diagrams intheeikonal approximation, Wereview theinfiaredSingularities ofexclusive amplitudes inparticular limits((a)smallphotonmassordimen- son#4,(0)equal ofshellp?,(6largemomentumtransfers)asspecialcasesof«general ‘parametricformula.{ntheparametricrepresentation theinfraredsingularitiesalweys & exponentiate, Thsallows ustoderive simple differential equations forLaplace trans.formsofthescatteringamplitudes. Similardifferential equationshavebeenconjectured ‘toholdinQCDandwesummarize thepresentevidenceregerdingthisassumption, 1.Introduction tisfrequently suggestedthatconfinement ofcolorinnon-Abelian gaugetheories {saconsequence oftheirhighly singular infrared (IR)behavior1).Consequentlyit isofconsiderable interest tounderstand justwhatthisbehavior is, Non-Abelian gaugetheories aeanextension ofAbelian ones,andfortheAbeliancasetheIRbehavior iscompletely understood (2).Itisthereforenaturaltotryto buildthestudyofthenon-Abelian IRregime onthisbase,inordertoemphasize inexactly whatwaystheAbelian situation ismodified. Thismotivates ourdiscussion andreview ofIRsingularitiesinQEDinalanguageperhapsmoresuitabletothe Problematissueinthenon-AbelianQCDcase.Thenweoutlinethesimplestpossible. 'conjecture astohowQCDmodifies whathappensinQED.Wedonotclaimtoexhi- if contractEY76-C:03-0068 andbytheUS-IsralBinationalScienceFoundation underCom tractNo.28. **Supported inpartbytheNational Science Foundation. * Onleaveofabsence fromtheTel-Aviv University. 509 Questions: 6 1)whyisScalledunrenormalized ontopofpage514?ws 2)whathappened totheitstn(2.10)?vs :' vs 3)howare2.16 and2.17 computed ? .i 4)whycanexternal closedloopsbeignoredasinfigure17NS . 5)pointoutthatLevySucherismisquoted. ws—_ 6 ; , #8 Ball, HornandZach: "near-nass-shell sings inQED"(rec'd Sept. 1977, themonth =~ T. . IarrivedinUtth) io) 15Intoruduetion. “TheIRtheory ofQEDisunder control, references totheYennie Papers which Ihave nowstudied. They want tore-examine QEDinhopesofseeing howtogoabout attacking QCD. . Twomethods ofdoingregulation: first,youcanputinaphoton mass,or youcanvarythe’numberofdimensions; second, youcanmerelycompute. things which arenotIRsingular andmakenochange toyourtheory. Inparticular, off- stiell amplitudes tend tonothave theIRsingularities, . Thebasic question youareasking is-different inthetwotheories. In£39) youwanttoshowthattheoberved inclusive crosssections aretfinite. Youcan Nevermeasure anexclusive crosssection, sothefactthatthesecomeoutinfinite inQSDisnoproblem. . InQDhoweveryouareneversupposedtogetanygluonsout,softorotherwise. Thus,hereyouwanttoshowthatthecolorthresholds areabsent. Recall that gluon carries color, soyouwanttoseethatthere arenomulti-gluon branch points; ifthere are,then youcanproduce gluons through unitarity. Thusthereareno“inclusivecrosssections®withunobservedsoftgluonemission. fe) Thegoalisthis: consider off-she@l amplitudes (sinceyouthereby avoidtheIR singubarities) andseeifgluon thresholds areinhibited somehow. That.istheproblem. 2.Infrared behavior oftheIRpropagator. e Yevont.toknowwhatthefullpropagator lockslikenearp°=nafteryou dress onallthephotons. Apparaently youonlyhave toaddallgraphs oftheform offigure 4tothmgetthefullblob asfarastheIRsingular piece is concerned. Theyexplicitly addallthegraphs andgettheanswer given in(2.10). The characteristic exponentiation isobtained. Nearxe0, thebehavior oftheintegral in(2.10) iscontrolled bythebehavior ofH@s)atlarge s,hence theyfigure cut(2x8t) (2.12). Using amasscounterterm, thisallboils dowto(2.13). Wehaveofcourse anexplicit expression forG(s)asshown in(2.12). Bythe waywehave scalar electrons here sofar. Nowlets gofortheanswer: what doesthis renormalized ordressed propagator looklikenearpean? Heclaimsthatyoucanjustexamine G(s)forlargesand gettheresult (2.18): youseethepoleandacutduetothemulti-photons states. (Gntheother hand, youcanfirst putinaphoton cutoff mass andtake 5to infinity togettheform2.16." Ithink their point isthatthings areclearer ifyoustayawayfrom the exactpaintp°=n°asask:whatisthenatureofthesingularitytheredThen fo)youseethat cut. Notclear what yougetinthephoton-mass case. 8 dowe (25) eo) sees GO| Sy=Jae Wy dateOeSoongett? iCyrrenespow=e&swolll QeNaseeroe Wa|enact: a! . QhinWeeagets toy xe aro betomeWkdedae \ eons)wd. &ae SE yoetwhtwhe) big oon,geryvaigeh fedeme) byieneue): aan ANC Nsa\'=©>ers a)a SLim(gem)Sy=eh): 4(25) \Maomub hs Na ‘ist2k)on 6a=YSSSHos(es@)t zCat=@prk) & Qasim >OaaSe, bomen (Z/be), SBytewahseems aducheAMASflee. ‘ 8 Cevagahe Cateey Oued ve (em evry SE 1 \ oo (ebySenOYThapyoe) Cntyeonabetaunko. pk=Ek-pkessex=k(E- peosx)- soweBe peemt: Com (EEEkeZF~¢eased)(*'-¢leey) Pgece. ge“YEGOS a — QOAsage! ge=pee ; \ * GA@=_iNe® Qre®Nd. 2 te et? .Aerye_ KaeWP(k=292)reed) eas eek &. wl,As Nam aSvacSet ethxSco=Bed,ee LN gt ° SS . ee fe) Qracdk@ay) Powe AFAGRsay: OS G)* =Bec ~2"¥G,15x) YOu ewan: Qxt SeBinS(Eby =—C=Bax. Mons: S22(eo) fhkeik maha 6=SH(es)©~REM) H~eaaBae)° dammolee | fe) Appendix: Generalized Lewy Sucher Identity. ie) Thereisacertainamountofhistorybehindthisthing.First,supposeyou want tomake asum over all virtual photon insertions onascalar electron line. Assume there are noexternal photons. Ifyou are interested only inthe IR behavior ofthe result, you can make the approximation called Rytov orsuper- eikonal ofneglecting denominator terms which are quadratic inthe photon momenta. Younowwant tosum, inthis approximation, allgraphs oftheform shown infigure 4,ievirtual photons nomatter howtheyland, nexted orwhatever. This sumisaspecial case oftheGeneralized ISIdneity derived inthis index, ie,there areMinternal photons andnoexternal ones. Actually, weare summing over allpossible ways toinsert JustMinternal photons, wearenot yetsumming M.Theresult isgiven in(4.12) forthisspecial case, andthis then appears in(2.7) ofthetext. There they have summed over nphoton loops. Igather that no-one ever didthis simple calculation before????? ‘Theoriginal Levy-Sucher identity énswered thequestion: howdoyousumover allways ofinserting Nexternal photons onto anelectron line. That requires (intheRytoc) simply apermutation sumofterms like(A.2) andthesumis(A.3). ThisoriginalLevy-Sucher IhavenowderivedandIseehowitworks. fe) The ‘generalized question isthis: how doyou sum over all ways ofinserting Nexternal andMinternal photons?? Theanswer isgiven'in (A.7) asasingle integral ofaproduct oftwoproducts. This thing isderived byarecursion relation rather thanby‘somebruteforce proof. Alittle differential operator maybeused toconvert aphoton from external tointernal. 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