Dimensional Regulation & Axial Gauge

Dimensional Regulation & Axial Gauge Hooft &Veltman (1972) Leibbrandt (1975) Konetschny (1978) Frankel &Meuldermans (1976) Phil Lucht notes 1979 Hooft &Veltman (1972) a Q se CD.Ketser,Particlemuleplcities FallNuclearPhysics44(1972)189-213.NorthoandPubingCompany|19}11.Crameérin Mathematical methodsofstatistics(fincetonUniversityPress)1966. [10] GF. Chow andA.Pignotti, Phys, Rev. 176(1968) 2112. . [11] G.Y. Chow, Nuovo Cimento Letters3(1970)55:G.Y.Chowand5.Rix,Phys.Rev.D2 F (1970) 139. i112]F.Cervus,NuovoCimentoSupp.15(1960)402. '1131DSiRSnA,Anderson,Py,Rey,Latte23(1969)1044;D.8.Sit REGULARIZATION ANDRENORMALIZATION 114]Sanna elaboration fetaleerNaBohnite(1969) OFGAUGEFIELDS [15] 8.Nilsson, E.Breivik, T.Buran and H,T@fte, Nuovo Cimento 43A (1966) 716. [16]W.E.Elisetal.Dataquoted byR.Panvini, BNL14126 (1970). G."tHOOFT andM,VELTMAN [17]M,Bardadin, L,Michejda, S.Otwinowski, R,Sosnowski, Inst.Nucl. Research, Watsaw Institute forTheoreticol Physic *,University ofUtrecht report No$97/V1/PU; J.athe, Nuc. Phys. 82(1966) 673.[18]P.Fleury, G.Kayas, F,MullersndC.Pelletier, Proe.tnt,Conf,onhighenergy physics Received 21February 1972 (CERN, 1962) 597. U9]Ain.inDoneCERNCinmelsherg Warmclabortn,NilPh.13 sm Abstract:Anewregilarizationandrenormatizaionprocedureispresented.11fxpartculat 1201Fs,O;BsG,Coe,MSeanand,Sain,RensRomaloedePysgwe14(196%) wellsotheeaumenofagetheories.Theihoworfosmoethata .‘i Knowntoberenorimalizable awellsforYang-Mills typetheorles. Overlapping diver- (2JRatlam,A.D.Brody,G8.Chadwick,D.Fre,2.6:Guragosian.W.B,Zohoson, tracicesrentangled.Theprocedurerespectsunitaiy,eaveailyandallowssheof any ELAMNG,LetSLACTUD394O96) antcompla Ininrattonarises.Innoranomaouscees alsoWardleniesarsatisfiedatalla. 23]0.CapsewshlondFrRytiethCromeremand(910) lations {istransparentwhenanomialios,suchastheBell-Jackiw-Adler anomaly,mayoccur. 24}Alma-Ata-Budapest-Cracow-Dubua Moscow-Sofia-Tashkent-Ulan-Bator Collaboration. Phys.Letters 313 (1970) 237.[25]L.W-Jones,A.E.Busia,G.D.DeMeester,BW.Loo,D.E.LyonJ,P.V.Ramana-Musthy, 1.INTRODUCTIONILRothLG,beamed,FE.MilLDReeder,KNErcanandBConkPhy.Re.p61ArinelicrstteeaocherA.Stmude,©BarbonePDariusM,ankou, Oto, Recentlyithasbeonshown[1]thatiispossibletoformulaterenormatizableP.Palazzi,A.SantroniP.Siroin.K.Tittel,J.Pilcher,C.Rubbis,G.deZorn,M.Macti theories ofcharged massive vectorbosons. Thederived Feynman rulesinvalve shostG,Sette,C.Grosso-Pilcher, A.Fainberg andG.Maderni, Phys.Letters35B(1971)361. particles, andinorder(oestablish unitarity andcausality oftheSmatrix Wardiden-127]A.A.Logunew, Nguyen VanHou,M.A.Mestvirishvili andNguyen NgoeThuan,Proc titiesareneeded. ‘Thenecessary combinatorial techniques weregiveninref.[2],in Topical Coot. onhighenergy eolisions ofhadrons, CERN 68-7, vol.11(1968) 74, the(reatment ofmassless Yang-Mills fields. ICwasemphasized thatthese same (eel niques work alsointhecaseofmassive vector boson theories obtained from the massless theory hymeans oftheHiggs-Kibble [3]mechanism. Stated somewhat dit- ferently: themanifestly renormalizable set**ofFeynmanrulesinvolvingghostsmay—* betransformed intoasetofmanifestly unitary andcattsal Feynman rules bymeans : . ‘ofWardidentities. Actuallythesemanifestly unitaryandcausalFeynman rulesare‘quitemeaningless inviewoftheoccurring divergencies, andadirect proof ofunita- ryandcausality starting fromthemanifestly renormalizable rulesistobepreferred. ‘Thisisprecisely theprogram carried through inrefs.[1,21. . However, even with asetofmanifestly renormalizable rules onecannot besure thataconsistent theory results unless asuitable cut-off andsubtraction procedure hasbeendefined. Inparticular, sinceunitarity depends crucially onthevalidity of theWard indentities onemust have aprocedure thatrespects these Ward identities. Inref,[2]theexistenceofsuchaprocedure wasprovenfordiagramscontaining at *postal address: Mallesingst 23,Utrecht, theNetherlands. *¥i.e,renormalizable with respect topower counting. \ Og eZ a Seaeaektsae oe — ee . a Yvo-- ee - =Shewonwnbegah erH(ARp2 +My(ABa+DiaMelatehele(AS) SESS ©Bat (Alp=CRY(os)secanceoes a —_——~woe - . a WA SCAB) yo=2(AN)Spo+CAs)|lentes=TicksTht—2SoutaLoe ete .antl Sp] .a ae ve Cate yey -- a ee oaETAsee Sea2 F[nents Meccayin (SEY Se ~ + Sp) - Ee as)DayleSelly 2S (Lan=2enTEE “SELES es MaRS ne aw? Wea) One) khW288 FLMets) Mec)te {Eyer ns a ee ee ap - -- AFC(e-Am)_L Wye(183)=Urged) Gav SSa a a acs 2ElePanta jst aeaSWeeDee LPHIT_ANEEL SE bi SES ane | ©WSSede ia2 aE SaPGs) 12)sO nn *) a »eh Ny \-U+42yp..- Lo QSpxteat ><4ous)9=o. << a a Cp ——-Sagar (ny OOabitfountinEnapliesMAL,quant TOM gpaRan8ALyMM do OTe sucheDanang shagging ydatesSunday,ikullen ——agdedkioes seshask DavepaleoavinGa)!Siylnadanaflly Mane a—— -seMem2ee neUD tael TAICoreg whPio ¢)SaasstatingannabikdeaBegbogaiapesmusta on(8). ave OOCEOS eT:ae a”SS ae a -kthwnstpwdquadanbic =4comathmstubsJaptfooi. —- ——Gaoiele Babaat.WaVixadonaie pousseal £2;Doveobs @ YQee (inde geomet weeee .=.Sto aSora(ET TASVelma Wa (agt=ETNaTGenny re ~wedaki=pak 88 Soe QtyEGER xcs) ART es Cera OSE) we ee eee Se ASCRAs) we PTW RE ee ee a = Fiy* eye) aPGco CAME TRS GREE seyNy — On=WA 5Wana) CEEeee5a ce OMYEEey ae ee ee RE) Jee ee (ose Alpesbeya hte —mmyo teta(AS.ploy.=Alesco a fhBey OS yc=AR . ons AB)Mubu=SBpegy : - wy 7—oy | as lat Na Cz)eoeM — SE Keine)Coss) co AR RED a ee ee er wee a Pasty. ______.. we i .mata) a -——~ $$ -\&May(atTeyeee —_ _7PR) 9)Oey ce a eee EM teeAAS . Slo =ROS)SPT CARSBE a a AEBS aSRG) SS ee —. 1 an Tle. AR SRSBa“ieoO nia iyaeg —S Ree ccauoudaplGiie munimdygodanea kil. _aaeamaapiCs+Seite covensecu en Qo BO CPR rermr Te rN OeSegenepedevel tuaale Reta ainconti Vindlinadaonguiabe7 | -AaSNATR)Soshe,Ree256.LAWreatOFaysalintrr .. ee 2Nous oy : UY wenn renee 2.2 ne a .So,-Cocluinnts ThypastedoDewntnheaTR,LadbeWH.FaA=,—ypertet. Mat~Osanc(0,0),Gatesyetncoebraiat olfMateGet wolnmenanalQoGGagAW6,nee=2)oogphctennanchn staatyemereGnvreYesOs eon) VW. 3eeons) La se eeewoe . ee ee 2 Ok,Iamhappysiththis.justification. Thequestionaskedisthis:the“original._..expression_(1) diverges forn.GT.2;evenwhenyousmiteit_as(6),itstilldiverged ______..Also, ithasan IRdivergence thatsomehow isntreallythere,Youdosomederivatives ___te-lower_the (apparent) IR-divergence off.to theleftinthenaxis,sothat _____ -you-get-a-real_region-of convergence-between_som}_negative.n_andn=2. Then all... -youhaveto-dois-continue-off thisregion.to thé.neighborhood_of_n=l,. For the... . -—_ene-loop casethey-do-this-explicitly.—But.to.me, beingusedto analyticity, this .— ——_——-is- really noworry.—Once-I-have computed result.(2),I-can_see inmediately howto- -_——-eontinue thedamn-thing-to-nnear J.SoIdon!t-have to_worry about thedetails of ~_——_-the-tpartiel_p-operation_and-formila (12), - .—In-ether-words,-I-believe that—any integral -you-doinn-dimensions-will_be—an—— analytic-—functieon ef-n-and thus-you-can-take—a-limit-as_n_goes to 4.——______—. ——-—-Subtracting-out—the-pole—attel with itsreside ieequivalenttoadding acounterterm ---——-4o-the-Legrengien-with the-séme—pele-effeet.—Finey Iambeginning-to-like-this stuff. ——Tosw 3,Currecletye cmaiaeelcagermanie tM goQe a ll nee ee en ef ee OoFar oieTopdiagrams, youwanttoshowthat.theresidue oFthepolewhich(~*S-- -Yourare goingtosubtract OutI«polynomlal WIthvealCoetricientsinthe~ extemal momenta; Thepointnéreisthatyouwanttoavoldgetting anything ©~~~ ~"7" Gpaginary InThePesidie becansé thenwhenyousubtract, youwTchangethe- _———"thaginary partinthatorder;butthisisnotallowed because theImaginary part ~ “"“feuniquely determined through unitarity byloverorders andcannot bechanged "TT byasubtraction term” ©777 . 5 ee eas 7 "The"oneloopcaseiseasyandyoucanquickly convince yourself thatthere "5 simply realpolynomial residue. Bytheway,alogdivergence makéo"a pole at nekasdoesanydivergence. ©SO ~The twoLoopcaseismoreinteresting andbringsintheoverlap problem. “"""“Gonsdier atwo-loop diagram ,eg,equation (2)withgeneral exponents of(29)“—~" “ath obvious exponents. Thereiean,overall integrafion, endthereare three ~~~" “Gublntegeations. Tfanyofthesubsdiverge, youwillgetapoleatnah.Youmst “"" “subtract thisout.Soyoumakethreesubtractions asinfigure 5,seeequation (30). ~~" thieisal1quiteclear, TO a ~~ “tn additiontothesesubgraphsubtractions(yhichremovesubgraphdivergences), oO youalsohavetomakeanoverallsubtraction, ie,youmustremovea|polewhich correspondstothe"overall"integration divergence. Thisisshownin(31)—™ “the problem theyworryaboutisthis:whenyoudealwithasubgraph subtraction ~~ integral, you get apole times aresidual integral. Inprinciple, that subgraph "Sole times thefinite partoftheresidual integral cangeneral logs ofexternal “momenta intheobvious way.Suchlogsereunacceptible frompointofviewofbeing _renormalizeable (Idontknowwhyyet).Soyouhavetochowthattheoriginaltwo}ood “diagram exactly cancels anysuchlogs! a — —‘Theexample (29)showshowthishappens: (30)showsyouthe.potential log ~~generator fromasubtraction graph.Eq.(31)shawsyoutheoriginal graphafter ______ ithasbeen“partialled" forcontinuation. Youcanseethatthesecondtermhasapotential log-generator whichexactlycancels thesubtraction (30),so youareOK.Theothertwotermsin(31)willcontain termscancelling theothertwo subtraction potential log-generators, _______seherewehadthetoughexampleofoverlappingdivergences,yettheproot -—_-____0of renornalizability wentthrough OK,Torenormalizethisgraph,just’subtractout thesubgraph poles[proofshowsnologsleftover{].Theoverallsubtraction is Where does "overlap" come inhere? Inanon-overlapping situation, as here?_Inanon-overt Situal _. 6,youjustsubtract outthesubgraph poles,Inasubtraction _Ss dheselannine veranocantpronbee,«Fes}integral _____mitiplying yourpoleresides, soyouhavetowdry! ____ 5,ExtentiontoFermions Straightforward, aslongasnogamma-5 orepstensors. a- -4 Le a ——. .-&Limitations: Suppose youhaveQEDvithanaxialvertex, whichyourecall is__ __. —related totheAdlepBellJackivtriangle’ graph!anonoly, thoughTcantmemenber _.....uhat_thds anomoly is.Wardviolation? Anyway, whenyouhaveganma=§ youarescrewed _ _.... .“because itcanbeexpressed intermsofthéepstensor whichdoesnot,generalibe _______t.0._complex_n,_or atleast theydont.knowhow’to,doit. Nevertheless, they_go ___ _.....onto compute the"andmoly" andgettherightanswere 79 _-...App_Az__ dimensional integrals dietionary 01 oe --.-App.B: Theorem that.in_generalshows_cancellation_of potential logs. 2 OO oe |fin1.23.79 w - —~ - oH~ ——- —_—— ss Leibbrandt (1975) - Introduction tothe technique ofdimensional regularization* 7 George Leibbrandt" ° ‘Thepurpose ofthisreview article istoexplain andillustrate indetail the ve technique ofdimensional regularization, which isamajor mathematical toolin {therenormalization program ofgauge theories. The most important single feature ofthenew technique istheconcept ofanalytic continuation inthe number ofspace-time dimensions 2,where theregulating parameter ois complex ingeneral, and w=2corresponds tofour-dimensional space-time. The technique ofdimensional regularization preserves thelocal gauge symmetry of theunderlying Lagrangian and thereby permits aconsistent gauge-invariant treatment ofdivergent Feynman integrals toallorders inperturbation theory. . ‘The method canthus beapplicd—as demonstrated inthisarticle—not only to Abelian gauge models, butmore importantly tonon-Abelian theories such as ‘Yang-Mills fields and quantum gravity, towhich themajority ofconventional regularization procedures isinapplicable. Weillustrate both theadvantages and . thelimitation ofdimensional regularization, aswell asitsextension tomassless particles. CONTENTS ‘VI.Application toQuantum Gravity . 866 i ‘ArPurequantum gravity 866 TTntroduetion 849 1.Tntoduetion 366‘A.UltravioletinfitiesinquantumGeldtheory Fg 2:Fictitiousparticles 3671,Preliminary remarks 49 3.Slavnov-Taylor identities, ‘8672.Previous regularization techniques 850 4,Structure ofpoletermandcounterLagrangian 868a.Pauli-Villars regularization 850 B,Corrections tothegravitonpropagator 868 Db.Analyticegulatization 850 1Photonconection as €.Speers analytic renormalization andtheBPH 2.Neutrino correction 39approach 880 .Asecondlookatquantumgrav 4.Abelianandnon-Abelian gaugetheories 851 Oe ean sravity bd B.Conceptofdimensional regularization 852 2,Igquanturngravityrenormalicable? 8701.General idea 852 VI.Higher-Order Diagrams 3102.Usefulness 882 A.Multiple-loop integrals 870 C.Outline— . 882 1.Generalremarks 37 TL,TheTechniqueofDimensional Regulciation 853 2Eaponential parametrization a‘A.Mathematical tools, 833 B.Venterdings a31BThetechnique ofdimensional regularization 854 VIEL Gonloding Renerks an 1.Prescription . 854 Acknowledgments a42.Combinatorics andgauge invariance ass area a3.Infraredvs,ultravioletdivergences 855.Refecences a3 C.Other techniques 855 1.The *tHooft-Veltman approach 855 2.Otherone0°?techniques + 8561.INTRODUCTIONa."Hoots method 856 b.Brown's method 356 A.Uliraviolet infinities inquantum field theory . 2,Muyetons fanQantun Btdyeanis 851+.preliminary remarks Fintanttdivcionalregularization $55‘Thetreatmentofultravioletinfinitiescontinuestobeone 1.Canthecopystensorbegeneralized toarbitrary: ofthemostchallenging andtenacious problems inrelativisticdimensions? 860quantum fieldtheory.Onlyinquantumelectrodynamics has 2.Anoinalies . 861itbeenpossible toeliminate theseinfinities consistently and TV.ExtensionofDimensional Regularization toMassessFields862inaphysically meaningful mannerbyabsorbing theminto alednitionaf is« thebare,i.e.,unobserved, chargeandmassoftheelectron. B.Redefinition ofthegeneralized Gaussian integral 863 " Bi1.TheGaussian integeal 863 The result isthesuccessful renormalization program of 2:Thelowest-order tadpole integral 863 Dyson (1949a,b) and Salam (195ta,b), inwhich there- ch(0)terrevela 864normalizedquantities,massandcharge,correspondprecisely V.ApplicatontoYeoestePeds $64totheobservedmassandchargeoftheelectron.Although Anatroduction 86{ many questions remain tobeanswered foreitherthestrong, B.Themuon(g~-2)factorintheGeorgi-Glashow model865gravitational, orweakinteractions, thereareindications =. ,.Rowthatperhapsoneoreventwooftheseinteractions canaasepeentelinpartbytheNationalResearchCouncilofmightberenormalized inthenot-too-distant future.The {Alcander vonTlunioldt Felow,onsabiaiat kaveatDESY _feAsonsforthiscautious optimism are(ithereeentprogress andIt.Institut fr‘Theoretische Physik derUniversitit Hamburg, intheunification ofelectromagnetic andweak interactions, Hamburg, Germany. (ii)thebreaktirough intherenormalizability ofcertain non- Reviews ofModem Physies, Vol.47,No,4,October 1975 Copyright ©1975 Amorican Physical Society asa Sf 874 G.Leibbrandt: Introduction tothetechnique ofdimensional regularization 1974), weconclude onceagain thatthecontinuous dimen- anomalies cansometimes bemade tocancel byajudicious sionmethod yields thesame result asother moreconven- redefinition ofthefundamental fermion ficlds, itisneverthe- tionalregularizationprocedures. . lessdesirabletocontinuethesearchforca7*matrixvalid fe)inarbitrary dimensions, sothatthetechnique ofdimen- sional regularization maybeapplied unambiguously toan VIL CONCLUDING REMARKS even greater variety ofphysical models than hashitherto Inthis review wehave applied thetechnique ofdimen- been thecase. sional regularization todivergent Feynman integrals inthe context ofAbelian aswell asnon-Abelian gauge theories. ACKNOWLEDGMENTS “The technique has,inouropinion, three distinct advantages. ‘Inthefrstplace,itssimplerandmoreelegantthanother, Theauthorisgrateful toProfessor 7.D-Nestonand more conventional regularizationmethodssuchasthePauliDrs.D.A.Akyeampong,J.Strathdes,andG.Toot Villarsprescription. Secondly,dimensional regularization isserene ee Beitreiman forbisadreandeo.powerful enoughtohandleefficiently andonthesameGraftstoBrotissorS.©.Trinieoftheprojectandtofootingbothultraviolet andgenuineinfrareddivergences. fouragement VatheFilysfoeoesbeseitieameFinally, andthisisitsmostsignificant feature, thetech- Professor M.Veltman, forseveral constructive critics“que is emi - Sng at ofthefinal version ofthis review. Itisalso apleasure tonique iseminently well suited fordealing with gauge : ‘theories,sinceitpreservesthelocelgaugesymmetry ofthethankProfessorAbdusSalam,theIntemational Atomicunderlying Lagrangian. Thepreservation ofthissymmetry Energy Agency andUNESCO forhospitality attheInter‘ sti, national Centre forTheoretical Physics, Trieste, where the wasdemonstrated inSec.IIIforthevacuum polarization A peortensor inquantum clectrodynamics, andinSec.VIby™anuscript wascompleted. regulatizing non-Abelian massless spin-two quantum gravitytolowest orderinthegravitational coupling constant. ItAPPENDIX. USEFUL INTEGRATION FORMULAS wasspecifically shown’ there that thesumofthegraviton ‘Thefollowing listof2u-dimensional integrals isdivided andfictitious-particle contributions tothegraviton propa- ntotwocategories: theintegrals inthefirstcategory hold gator satisfies Slavnov-Taylor identities [Eqs. (6.17) and jormassive particles (mx0),whereas those incategory B (6.18)] andthatthefinite portion ofthissumcanbeex- arevalid forintegrals associated with massless fields tracted inamanner which isconsistent with these identities. (yp=0). : Insummary, dimensional regularization permits aconsist- centgauge-invariant treatment ofdivergent Feynman am- A,Massive Integrals litudes toall ordersinperturbationtheory. oe) ® ue fontheory: Formulas (A1)-(A6) belowaretakenfromAppendix A :‘Although 7 ie Jarization igoftHooftandVeltman(1972a).Intransferring themwecytevBoponceihenationofwealotccontraction izbave,forthesakeofconsistency, replacedthecomplexclearlyunderstood, themethod shouldnotbeapplied indis- variable »by2anddivided eachintegralby(25).These Giminatelytoanymodel ing.gaugesymmetry. integralsholdform*<0,warbitrary,andarepaticslacly Beforeembarking onanexplicitcalculation, itisbesttoUsefulinconnection withthediscussion inSecs.IITandV. ascertain first whether ornottheunderlying’ theory—be it Abelian ornon-Abelian—(i) ismassive, (fi)ismassless, or oq iii)contains, throughSlavnov-Taylor identities orother- tae kyu Gr =wise,factors of7%.Letusbriefly examine these three (a)(gt+2k-g+m2)*(4x)(mn?—HY possibilities.Ta—)Te-9 (at) Formassive theories theprescription of’tHooft and T@) Veltman (1972a), Bollini and Giamabiagi (1972), and Ash- more: (1972, 1973) works remarkably well, asdemonstrated : inSecs.III-AandV,andambiguities caneasilybeavoided. Pom ps Formassless theories theprescription given inSec.I1.B.1 On)a(g+2k-g +mis GxeGe =Berequires modification (Leibbrandt andCapper, 1974a)due Onin dha totheappearance ofinfrared divergences connected specifi T@-«) cally withmassless particles (seeSec.IV.B). Themodifica- ——(-h), (A2) tioninvolves basically aredefinition oftheoriginal 2s- T@) dimensional Gaussian integral which permits aconsistent treatment ofmassless tadpoles ‘aswellas§*(0)terms mae i 1(Sec.1V.C)andpreserves, moreover, thecrucialSlavnov— f 4 =Tasosdeniesassociatedwiththegravitonself-energy Onetddb-gtm"Gel—BPTOjoop(Sec.VI). X(T@—wk+Pe—1—w)o(m?—#)},(a3) fe)Finally, extreme care must beexercised whenever the theory contains anomalies characterized by‘apeor78.Since om j 1thelatter hasonlybeen generalized successfully toeven- ae - _— dimensional spaces (Sec.III.C), thecontinuous dimension Gn Gtth-g4 m= Gayl? —BI FOmethodcouldcreateambiguityproblemsiftheseanomalies Ox+hatwi)»persist inthefinal Slavnov-Taylor identities. Although the X(Tle~w)lgke+P@~1—a)lm—2},(AM) Rov. Mod. Phys., Vol. 47,No. 4,October 1975 \ 4 akin}Vachuiqut, ofDuinBag. aeeaLiebbromdt, Coseade eeoe »-2BBangegaya.Be - —___ @YadUnaag,conegutnaannyemmaasiat }aSabaryeyunaEararalSaas~~Vowels hretieomsvepaumtla A,geogagyeta cantonsfoam A nn. cemmmachasdo RUN ee Saget akRonen saleiegenea!Oochakgaban —lua:S'sputQkganDisoosanaWinganndarn pldaguatietGa divinZRTite tay —. “Bate dee _ SE. Qaadm Gat ee _EQweyygae ADinBeg ~WliMoaga,. FiloupAemam G01Gagea0quoponia YawQndada A AB=VuhSuman CeTE YW'SmSenslonpmatioh Armes conatved. onduingecrbe se a ea Wa POV OEOeOO80) “nee 9Sa eea i eee ——afAl"= Weg22(o-2)BA Busnotnadfonsgen LEGGED YoBone 0 te ReGp8gi) "Upesaypalt mapleSichNed) es regy24cGy TEAatyawlee RT Seacamatsod, seachaman Tegyye) To ae Sry - ce ange cominTES=FDBEwr Degc9 pedDeaiekWAaebptoe ue ee CSen 2)Quaninpadimonaoefiatdogk(amt)coPana)GET yk ——-—oasema Ral)>L.QaonQinWO Ro —Ts_on elu>Loree, encntoylen QuahioGeramanen War —anthadaa eae),fommiargse i a NURI ATS ee ee oe2OPMO) on-nay Sea a aar 5GOyoncaiBAYoa eat cee aOe ge EE Sores.oats=haei,OI ght] eg ETaeghatil Bays Ste)eo adhwadvr93hlaLesadn FeOhlvant chenWoe-ataggtt.. “Te,tet_cwnaler adulfaatach conbanenain d\Quavedere! a -Se\=Yondanin Schlaaati+Gogg==uk2speopaatioa. MleBuSofess~\ Oe\6 eee sake — TE_Geoe.\5.MeadWeel anee> joe CegaendWeabochwar,beMeocghemeQutrodussoae <fehGonnoth a ee a 55 ) SassSrascandiven” ygT=Ooaw->2froolsQueiaJponvevoibs [*)_.8, cee ee eee ee -@Waimgale Yeldia,on@uncdanwe boleanedWok ee SY ee afaa =OaWadad20.Gap a Onoii.saann)Beh2Se A Waygather ee "OK, 60Ihavereadthissection. Idontreallyknowwhattothinkebout,it.They_ _ .arereallysayingthatanypowerintegralgetsregularizedtozeroinaconsistent __ ———_fashien. Thebasiccutoff problem withtheseintegrals isthatyouhavetrouble _ ..... making theanalytic continuation bebause thereisnodomainofconvergence. Maybe . Iwilllookintotheirpaperstoseefurther coments-..1dontknowifthiswillhelpoubeffortmichbecause wealwaysare _e . dealing withn.kstuff. oo ___Thelistof"IRintegrals" attheendofthepaperdoesnotseemtohave____ _...thisIRproblem, ie,nodomain ofconvergence. Bg,theygiver ee SEE ae Seana :aero—-\8a,—!-—- woeGYTG) Bw)©GASP Carey CaP TeSopaGewandunsag—Cadardale gpI,iinMon6omodal. _SO(8)_._. -Ceesnandesea\ow,—Pavandis +QkewSROSejuuakMeathMaanrset —__—DBsatmerbowetong MsSte. Olanmfpopti OW . __- ~SEQuawrv GeewiyA)fnQuodGout plapecabgpanstm Jud.—— --Bay gF=apad” =OFgE 3 TR, Orrin BOK DLRulateoletogdacsdhbnrach.glaashes Saat bmnBabee lpamugy2bewaealy Saeeaa Ee Jee RaginBagsondatsam— Crags :oe. _UpuaBui=(at)Syuls®=ea)7 Sige Be Bee TeaBay=Qua Ayfasees, nudhepityQua)se Be TapaQien oor5onatanvoinmabe olDaunOoh |—B\Beusdnsee GuGudeaosccanulaianoontchiedeQaguauieaaameg on ded oe —S)Mew: conlesluacomasddoaasisalasLenedinagncssnoncdds: oe Qin Le ee . »The Otydove —N\Wutticteg balagate —yoncoathsoomenhpoofoads Deg.porregetecmairdasinn gousunnsove Drsenaoin fansrdre fora ——. ~ shaggalSalaspieDassLeaati—Pade a *) a Konetschny (1977) . cugsiint oduthlate 1cgahaa 2See* or weeNetseizeHieSsase,inee a 1 : . : FeaovSTADE H oa hoe ILNUOVOCIMENTO Vou.444,N.4 21Aprile1978 ea °sfor- . ° Besicular l . Berl zation : MassShellBehaviour ofQuantum Chromodynamics Ee sssical intheAxialGauge(’). pd vathe- t baa ample { W.Koxprscixy Seserote itt‘Theoretische PhysikIT,Technische Universitat Wien-Wien,Austric ee give 1 InsltutfiirTheorelisohe PhysikIT,Technische UniversitatWien-Wien,Austria Be . (ricevutoil22Dicembre1977). be : i : bat fess; Summary. —Itisshownthat,inthenoncovariaut axialgaugo,quark bedandgluonmassshellrenormalization constants(ifsuitablydefined) ‘ee _—. H obey theclassic Ward identity ofquantum clectrodynamics Z,=Z,. ig oe Similar relations hold forthepuro gluon sector. ‘Tho validity ofsuch Bd 1 ‘Thederivation intheaxialgaugeisconsiderably simpler thanincova- raH riant gauges. eh inger H 1.=Introduction. Beesteuisce . bad nopol , ‘Theinvestigation ofnon-Abelian Yang-Mills theoriesisbyfarnotaclosed Bd . subject. Whereas theultraviolet (UV)divergences havebeenshown tobe Setractable ((leading toarenormalizable theory), thosemecannot yetbesaid ati of.theinfra-red (IR)singularities. ‘Thereare,however, goodindications (*) pes : (‘)Tospeeduppublication, theauthor ofthispaperhasagreed tonotreceive the ae i proofsforcorrection. Rd HENNEa ()G.vHoorr:Nucl.Phys.,33B,173(1971);35B,167(1971);for#review,ef. Be epee.YG.CostaandM.Tosts:Iie,NuocoCimento,8,29°(1975). aa mecs! €)YoukPuseYao:Phys.Rev.Zelt.,36,653(1976);‘Tu.Avrerquisr, J.Canaz- ee greet . 208e,I.KuupERa-Srexs andM,Row:Phys.Rev.Lelt.,36,768(1976);L.'Frauks«t: Se mpi Phys. Ree. Lett., 37,319(1976); Nucl. Phys., 116B,201(1976); J.Prexwer, R.Mevr- = } penaaxs, I.Momastwap andJ,C.Tarton: Phys.Lelt., 64B,211(1970); Nucl. Phys. 9 ‘ 121B,83(1977); A.Gancta Anvawez: Nucl. Phye., 120B,355(1977); 't.Kisosiura, eta andA.Uxawa:Phys.Rev.D,13,1573(1076);15,1596(1977);G.Srenwax:Phys. Sea fe): Ree.D,14,2123(1976). eae ' | 4 j a lok 466 W.KovErScUNY os ~aM-thatinperturbationtheory(whichbyitselfmightnotbetheappropriateframe- 3bvade worktotackle thequestion ofTRbehaviour) thedivergences fromrealand :; SagVirtualsoftvectorquantamutually cancelforquantum chromodynamies . Aug(QCD, thetheory ofmassive quarks interacting withmassless giuons) inmuch4 Peerthesamewayasinquantumelectrodynamics (*)(QED)—at leastinthenoneKw +tforwarddirection(+).Theinfra-redfinitetransition ratesthusobtainedstill : d oeinvolvetheunrenormalized couplingconstant, andthenexttaskistounder- “894 standhowrenormalization affects thefinalresult, ; a Inthiscontext ithasbeenshownrecently byCvrraxovr6(*) thatin¢o- 1 . F¢ VariantgaugesofQCDthemassshellquarkrenormalization constants obey: 43 theclassicQEDWardidentityZ,=Z,,andsimilarlyforthegluonic2's. 4 £eg ‘Theremarkable featureoftheseWard‘identiesisthattheycallforaninter- yt4 play(*)ofUVandIRsingularities. ‘Thisrequirement ismetbythedimensional 4rg regularization 9scheme whereintegrals without anyinternal massscaleare. }“ne setequaltozero@. .. poe Covariant gauges ofnon-Abelian Yang-Mills theories necessitate theine f° g troduction offictitiousfieldsintotheLagrangian tomaintainunitarity. These i3 *Faddeev-Popoy ghosts»(*)complicate theSlavnov-‘Taylor @)identities which : “47. express thegaugeinvariance ofthetheory. Ghost-free gauges cieumventing 4FoASH thesecomplications canbeconstructed (2)andshowntoberenormalizable .6" aswellasunitary(19),Thepricetobepaidinthis«axialgauge»(4)is E ¥.Romustcn: TheLheoryofPhotonsandBlectrons(Reading, Mass.,1955). aot()Theforward direction ismorecomplicated thaniaQED:C.’T.Sacuraypa: Phys.. d Rev.D,V4,1072(1976); L.Marssoy andR.Mrvupemwaxs: Phys.Tell,70B,°309ne é1977). 4: _ v7) G'soorr andM.Venawax: Nucl.Phys., 44B, 189(1972). +day vot) G.Luwaanpr: Rev.Mod. Phys., 47,849(1973),rrr )L.D.Fapprev andV.N.Porov: Pliys.Lett.,258,20(1967). ;“ys ()A.Suavsov: Theor.Math.Phys.,10,99.(1972);J.C,Tavcon: Nuel.Phys.,33B,“ALog 436(1971),. we9 Yee)W.Kusamm: detaPhys,sustr,42,315(1975). . PayV(")J.Fresxen: Phys. Rev.D,13,2395(1976)..Y v(*)W.Koxerscusy andW.Kuster! Nucl.Piys.,100B,106(1975); 108.B,397Ys 1976). sod ‘UT-291 (August 1977).. 7 (eoRL.Ansowrrr ana8.I.Prexuan: Phys.Rev,127,1821(1962);W.Kup: waActsPhys.Austr.,14,149(1961);J.Seanwisern: Phys.Ree,130,402(1903);Y.P. “aYao:Journ,Math.Phys.,5,1319(1964);E.8.PRanxry andI.V,‘Trorix: Phys. oaksRee.D,2,2641(10970);A.Curaxnanantt audC.Danzixs: Phys.Rev.D,9,2484(1974); eeE,Tosnoutis: Phys,Rev.D,8,2736(1973);K,Dynwounao, A.Saax andJ.Srrarie, A Reypen:NuovoCimento,23A,237(1974);J.M.Conswaut: Phys.Rev.D,10,500(1974); SyRJ.Cenwrion: WeakandElectromagnetic Interactions atHighBneryies, editedby .v bask MM.Livyefal.(NowYork,N.Y.1976),p.345, . y A a a a ee seaksoreePE eeeeaea aT aay aa Konetschny 1978 . uo ° Section 2:Author writes amostgeneral kinematic formfortheelectron self-energy (axialgatigeQCD)andthenafterdoingmassrenorm goesontodefine aZp»Meanwhile, thequark-quark-glue vertex hasamostgeneral kinematic formshown in(2.6), andthe first function GAMMAI isusedtodefine 2.Thenapply theWardIéentity (called Leeidentity because this isQCD, butitlooks like usual QEDWard-Tak identity), and conclude that 2,=2) almost.. , . Aside: remember that SIGMA-SIGMA(sh}(the mass renormed ¢lectron self-energy) still hasdivergences which showupaspoles atneinthedimreg,ie,logdiv's. Inthedimereg,thislogdivergence appears asconst/pole (because ingeneral residue must bepolynomial inextmomenta andinlogcase that poly isconst, seetH-V). Thus, when youaddacounterterm like (1~Z,)prop owwhatever tocancel this logdivergence pole, thatcounterterm musthaveconst/pole also. Thus, Z)mustbeindependent ofp. This part ofZ,iscalled Z,,, anddiscussion here isontopofpage 470. So,the fact thet youconclude thet Z),=2, isenough toprove renormalizeablilty forthis sector ofthetheory, since itisthese Z,, which youputinthecounterterms. ©) section 3:Hereauthorexplicity verigies theLittleQaD-Like werdidentity tolowest order inQCDperturbation theory. Thegraphs arebbvious, theactual integrals are done inAppendix Busing rules ofAppendix A(Feynman rules). Mush useismade of “regularizing" power integrals tozero. Apower integral isalso anintegral without ascale, bytheway. Sothis section isOK,except Idont quite seehowadetail intheappendix Bworks, seetop ofpage 78. Notice, bytheway, that theWard identity isfully verified. Author does not simply show that both sides have thesame divergence (egmthat both sides have the samelogLAMBDA)(or bothsideshavethesamen=poleresidue. )Heshowscompletely theequailty, andavoids doing thelast pair ofintegrals. Section 4:Somuch forthequark-gluon QED-like Ward identity. Next, author considers. theWardwhich weareinterested in,relating the3-vertex tothegluepropagator. Irecall fromother papers thattheglue-Ward incovariant gauge says23/Z=23"/Z1," where: Z,relates tothegluepropagator,’ andZ,goeswiththevertex; andwhere theprimed Z'srefer tothesame objects fortheghosts. Asauthor notes, youmight expectthatintheaxialgaugewhereghostsdecouple (Z3'=Z' =1)youmightexpect O thet 23-24. tenow goes ontoshow that this isinfact true. How? Hewrites down ageneral form forthevertex dotted between twospecial fol polarization vectors (thus, heavoids writing dow amost-general tensor form forthevertex).ThentheWardbecomes(44.2).Healsowritesdownamostgeneraleoformfortheinverse propagator PIasin(4.4). Inthispropagator Z,isidentified, andinthevertex 2,isalsoidentified. Thewardthensaysthat23eZ,, nobjust for tle constant parts. | Whataboutvértying'this Ward? Fortermsonlyinvolving quark-loops, it iseasily verfied. Buttheyareunabléitodoitforthepureglueloopsbecause "there aretoomany terms". Wehave done this vertification only forthedivergent part, ie,weshowed Ward OKtologLAMBDA terms only, or,n=pole residue only. Wenever dealt withallthose lower terms. Sohereisperhaps anopenproblem that wecenutilize reduce todo. :e e Comments on Section 2. oO1,Intheaxialgauge,thepresenceofthevectorn"causesthegenerelformsforthe electron self energy and the electron-electron-photon vertex tobemore complicated. (By electron Imean quark, byphoton Imean gluon, this isQCD). Somehow itisstill pobbible todefine a2;andZ)inanalogy with QED, andyoususpect that youshould” getaWard result Z,=Z5. 2.Theauthor Konetchny goes ahead todefine Z,inanobvious wayandZ,inawaythat Ibelieve but which Ihave not really proved isright. With his Z's, the author finds that youdonotget simply Z)= Z)butthat there isanextra guage term as shown in(2.9). Itisthen argued that atrue Zused asacounterterm inrenorm theory cannot beafunction ofpbecause that means non-local. SotogettheZ's . youwould actually usetorenormalize, youaresupposed totake theconstant part ofZ,andZp,ie,pandnindependent. (first implies second). Butitisobvious that these constants are equal, soatleast wehave that much. 3.Interestingpointismadehere.Iamnowusedtop-depdendentZ'scertainly fe)‘through the ratio gamma; but renormalization Z's really have tobepindependent! : Having that extra vector n™really complicates thepie, butthereward issimple QHD-like Ward identities. | 4,Massrenormalization iseasytounderstand andconsitent withmyownnotes. You put inthe physical mass @m,then you add acounterterm dmsothat the mass does not get shifted when propagator dresses. Inote with someinterest that, because of the n.p thing, this mass counterterm isactually momentum dependent! You are then allowed tosubtract momentum dependent things from your self-energy inaxial gauge. Iawait with interest tosee how the gulon self-energy case works. Section 3: Byworking out afew feynman graphs, the ward identity (2.9) isverified . for the quark self-energy. The constant parts are thus also equal. This section involves appendices A,B, and C,inparticular, integrals are all done inthe dimreg method which Ishould now study alittle. Ithink the equality ofthe Z's isneeded for the proof ofrenormalizability and inthe axial quage this has been aquestion. oO Belowiat 4Ouni ee ee ©Daducten. Oiagagiesede gliaoulquakeimonahgatige GOEna —--Seowed adh(hrshoaaic, Ward 2,52, obtevue) botDeleumehtesovaniawk gauge.Gacegogaeillduugiisale, Maskscattiafacial gangs 2 aTossteak WE nsppt fy—we eeKEI O...Las. thrpgerys©Up]___agnsnaall {rnssiaotegrange(18)7 TRSeeReBac)oeconnnous OR$dataahonliaa Heston qeegiteGED,Masseueforod(Baonging arinABin) ee ES ea sa2.Oc) lamngy d e a nHengncGumafgh aescoon simonsoddena.Om AasimAt,A=Bdedsma Sen HE faonyown860dak @StonIshaRedeWertimarchWD.(hsaafpamtianA,waneiaQhakWDuWonape. okgent,Cede)bestgeesowdetarenaadsetywoakgantOle——. Ayesuanh[San =2Yesoe BienSEZ ren dae7emdbons Bo SoSmet fasthmedheFZ 18 Ba)==pw OCS) +6Tig om ZEwmo. Qenaquaecores galeteqi. yp SecuZe)Aeaedies QeTateJpatesQe,sonmattbeat.dv DegasTT @Neoware Aahdecaymunconte) gkDensadorenlh Reena YoRY —Yongrad)+diedaoutAOasisagpolaiie. vagsvitiaDeakyimDasoaiahgouges.—.—ysoslend tegh ee ae Ge)(asbg) aefom TamaOvinacoatDosSarmaaa2%onsts(gm) Qala,dailanalaeamngenuesit Tomaenedy(Vong akodtahSen : a ThowitefomMeee oe Pee. Ugshms tgey ee deere, Loaf foda AseVED,Conveonohgapguad byho JBMek 2SRategtTbeety aa - L OOICGMPT ew%_@ Psst. beh peat 2a —~\— awe(2a), ee ho Oe ee -2Qm\= adPgavo2Leg] ulesasave _ oeARR AK, tblstendetNort vir.Ind ce oeEPGnSIS ache Re Gal RZonia pag+deacba ee betne FO ceceee ~2B =2m=2ogo) +X(K-SE)teLd bssfomAG,==2k Apne =dm|xa —. —S_(-(hap) ==a1a — pe _- de =Wn {a2b] Tx)=Upmrmr6| _ebwier =heeoY4 __ZinA ¥GSUgemont. el gama a" Dj2by= am) Dfen) - — ee_beatae sede) a eee * 5ENE HeG)stag PS “Cea ae aCES a Ve, O—GEw=SU)=.Ze) eS WS pemado Bgew) =Gen) Sa gpaGeteeten)\ OS TS aig ESZEN)=at —— (GLSind)GLKcaemaneedHtguteawedonnedaepe TGS TeaGoeGtga _Jaue obGa)eg Pama tia yp _ VRS]DeaWeanDegdoGhQEDMewkdbe koGY, Te BEGETS 2s aso TS ONS eeee Z's AxBlpom) fm=Bien) _aE ee okfrm _ nn a= Bgobs ael a SoegeBO€Bfate)rene 2a7 ee =@,-4=we SO. oe hw —_—__pe se :See neeosCC 0 Gp) Gia) 2KYet) (dag2th*)) ~ we HEP pedpote KAM - @_fh Dal wig)eT “Bow+dolpheeofne 6 Ita.agSE NS Bgeae&(fran) ekoa) __ a AGEwa oneGfarm)aCd=e)meGem) Wea LLGs=eCgem) 984 £E)eeLAK oP oe =REew).— mA)(=S)FEO). — a Le Ge Gem) K$eKeg) d So2"=(a0)ng.Gove)=live)440d)bs+iGon @ Tee a fe S/Sfh8)adhops-3=~Pahang) =0%ee ngaccede Veal Gyn. (eealae +erLp) tasesew=BgTO eoSe ; _3- Paprearerions identDeGaal4 tt “()Saye GaReeheae my Oy “afayhenfedI.fgeoalfet} Faget) *SB&[Ge +ee(Sho) ®Gs) % SateevatSoswpea:SKgeo.Akoy ) ° 5aehd Wp. . =-8WAyMasex.weeINteedRKS-\ ertCurlLokYannoo(6)., fe} =Appadin Bsaiidsh LdmoehortgadSVond «DashSKdegksane. QueDob _ -..7Gre . Sompoiadnd faq adi =BWRL NR OM ~~ ok(N\. (6a)|) 2Dye PEEVE =taleoleae toy ao eS eee {7?erin(Qha td) =yp yy) Jenin) City pede)__& SMe wedsepey ema oe) =Ge Ge) 2 ie oo PS)Giemonid gogunsSusSotccmasilgangSaoosWER eaflgg. Wash SAR=0te -—@Baduher(Qe) ao NaginGanitee(suDaghends) eyesADE; -WDS80iaemominonch :puweselee Akgle,aad€psenao; qil=o} usbSPREE 2;qo, OQ apoemtnemel _. (2)lesdarter (else)_ oo 07)ootweaYimma: suataharogy, aaiel5naam..2's<ifembonhy OW)tattgoedfwJone ©Wi)adaQkZpoate) 5up20,1ameye§ a > — OQ ©GY)CRM Frankel &Meuldermans (1976) re EE EEEEEEEE | i 1 ee a : a Volume650,number1 PHIYSICS LETTERS 25October1976 Bs icfse i= INFRARED BEHAVIOUR OFSELF-ENERGY FUNCTIONS INTHEAXIAL GAUGE j ee . =e J,FRENKEL andR.MEULDERMANS** | =eUniversity ofOxford, Oxford, England =e |= Wednsinthaings,theinfraredsingulrtesofhestefnctinswhichdeterminetheexponen=a ‘alofleadinginfrareddiverggncies inmasslessYangMilstheories. = Recently, Cornwall andTiktopoulos [1]haveextensively discussed, intheFeynman gauge,theexponentiationat ofleadinginfrareddivergencies inmasslessYang-Mills theories. Furthermore, a8showninref.[2],innon-coveriantSEX ‘gaugesthesedivergencies occuronlyintheexternal lineself-energy. Thisfact,together withthegaugeinvarianceSyd oftheS-matrix,leadtoasimpleproofoftheexponentiation. Inordertoensuregaugeinvariance, usehasbeen aa‘made ofthedimensional regularization scheme [3]forbothultra-violet andinfrared divergences [4]. 4.Inthisnote,wewouldliketopresentintheaxialgauge,amoredetaileddiscussion oftheinfrareddivergences =whichoceurintheself-energy function. Inordertoisolatethesedivergences inasystematic wayweproceed as &follows.First,thedimensionnofspace-time ischosentobelessthan4inordertoregulatetheU.V.divergencies, E withtheexternalmomentum offmass-shell. Inthiscasenoinfraredsingularities arepresent.Bysubtracting off = theULV.divergencesweobtainafunctionwhichisregularatn=4,whichisthenanalytically continuedtom>4 ie toregulate theinfrared divergences. These singularities willmanifest themselves astheexternal momentum is finally putonshell,aspolesatn=4,theleading oneexhibiting adouble polestructure inlowest orderofpertur- bationtheory ®,Inordertoillustratethesefacts,weconsiderfieldtheoriesdescribedbytheLagrangian givenbelow,inwhich i ‘massless veetor particles interact with themselves andwith massless fermions: 4 12EFpFgy—VOYWLys @ie where 2=8429,At+gstbegb4c Bi Fy=8,45—2,As,+afPAPAS : (2a) . : isthegaugecovariant fieldstrength tensor and ‘ 2 Dy=2,—igr?Ag (2b) ‘ i isthecovariant derivative. ThetermL,specifies theaxialgauge[5]: f I_=tim1(A,a2 @) i BYgp 2a nn i Ay,beinganarbitraryfour-vector, withA?#0.Inthiscase,thevectorbosonpropagator becomes,inmomentum. iq space: i1 QA,+O,A, 070.0, i Dyy(Q,A)=>-(s,,-4“a toe ®|O2-ie @-ay i E wherethedenominators Q«Aand(Q-A)?aredefined bytheprincipal valueprescription. 41OneeofstancefomInstitutodeFie,UniversidadedeSaoPaul,Braz oq**Navorser L.K.W.(Belgium) andBritishCouncil iog : 1 64 } Frenkel and Meuldermans 1976 letter 9 ——_tquations onthefirst pagejustdefine thetheory: QODinaxialgauge. Notice the idea that you really dohave totake the paramter tozero toget axial gauge, as. Chiu noted. Second pageclaims theselfenergy isAg+Bf,although Idontseewhyyou dontalsohaveatermCyfforexample,asIfoundinthemassivecasewith wemyh Kon fiddlings. Inlowest order, theysetaboutcomputing thiselectron self-energy. The result is(10) where they usethedimregdeal. Fine, Bytheway, they usemassless electrons (quarks). Lookattheirresult (10).Thepolesatn=krepresent theUVdivergence, which hereislog(massrenorm 1salready performed, henceno1ternin(5).9I'ma little confused about howthemass renorm enters (they dont mention itbutKondid). Maybe inthis lowe order calc there isnothing tosubtract. Anyway, lookat(10).Youclearly seetheUWpolesatn=.Thisthingisbineg heldoffmass shell sothere arenoIRsinges. Ifyouletp*=0,youextra sings . because n.LT.4 will cause divide byzero. o Soholding offmass-she@l, justsubtract outtheUVpolesasin(11).Thenes youaresupposed togotothemass shell andobserve thesudden appearance ofsome ° newpoles atnek. These must betheIRsings since youalready removed theUT ings. Thissounds nice,butnotenough detail isgiventoletmefollow thecale. Inparticular, Idontthinkthat(12)canbeprovednfromwhatisgivenin(10), In(10),ifyoucontinue ton.GT.4 andletp®=0,thefirstterminSIBMA1dies, andfails tocancel the UVpoles subtracted off, that Isee. But Idont see wherethoseIRpolesareconging from,unlese itiein"f™and"finite" whichtheydontstate. 6Noticethat(12)reallygivesthebehavior oftheself-energy nearthemass shell,sincesecondtermisorderp*.ThisisIguesswhatyouneedtocompute 2a relatedtoNp and you see the presence ofdouble poles atnel which are associated with the IRsings. Sotheir point istoshow that there aredouble poles atn=, which arefretatedtotheIRsingofthemasslessquarkselfenergy. 7oa Next, they consider the gauge orgluon self energy inlowest order. The “leaf graph gets regularized away and you are left with just our "Born Loop" and the "Quark Loop" graphs. They then repeat the above analysis: keep off shell and find the UV poles, then subtract outthese poles. Then goonshell anddiscover that you now have some double-poles atn=4 which are connected tothe IRsings. Still the details elude me.IKdont even know ifthey aredoing Euclidean orMinkowski. Thebigresult isthen that 23hasdouble IRpoles. Soreally thispapershould betitled: discussion ofthedouble IRpoles inquarkandgluonselfenergy tdlowest orderperturbation theory .Thesignificance ofthese double-poles isthat Cornwall and Tiktop showed that these poles playsome critical role inproving thereal/virtual IRexpnnentiation/cancellation similar totheYFZ work inQED of1961. Ofcourse that assumes gluons arenot contained, ifthey can beemitted esreal particles.... hmmm.