Self-Gravitational Correction of the Vacuum Polari

arXiv:hep-th/9505191v121May1995hep-th/9505191Self-GravitationalCorrectionofthe“VacuumPolarization”FeynmanDiagramUsingFullEinsteinEquationPropagationoftheIntermediateVirtualGravitonsS.K.Kauffmann∗750GonzalezDrive,Apt.6D,SanFrancisco,CA94132-2208,U.S.A.AbstractThesecond-order“vacuumpolarization”radiativecorrectioninsertionFeynmandiagramisanultravioletdivergentvirtualobjectwhoseFouriertransformtoconfigurationspace–timehasthepropertiesofavirtualstress-energytensor.Thoughwell-defined,thisvirtualstress-energyisnon-integrablysingularonthelightcone—whichproducestheaforementionedul-travioletdivergenceofitsfour-momentumspaceFouriertransform.Toproperlycalculatetheself-gravitationallycorrectedversionofthissingularvirtualstress-energy,theusualgraviton-exchangeladdersumapproximationsareeschewedinfavoroffull,self-consistentEinsteinequa-tionpropagationoftheintermediatevirtualgravitons,whichtakesintoaccounttheirimportantnon-linearinteractionswitheachother.(Asaby-product,thesubsequentperturbativetreat-mentofthesenon-linearitiesisavoided,whicheliminatesthesourceoftheultravioletdivergencesofthesecond-quantizedgravitytheoryitself.)Theresultingcorrectedvirtualstress-energyisnon-singulareverywhereandFourier-transformsconvergentlytoafinitecorrectedversionofthediagram.Thiscorrecteddiagrammakesnocontributiontochargerenormalization(ascouldbeexpectedofadiagraminvolvingbutasingletransientvirtualpair),anditsdynamicalbehaviouraccordswiththestandardquantumelectrodynamicsresultexceptatinaccessiblyextremevaluesofthemomentumtransfer,|q2|∼(Ge2)−1.There,thestandardlogarithmicrisewithmomen-tumtransferwhichthisdiagramcontributestotheeffectivecouplingstrengthfallsaway,asthediagramproceedsinsteadtodecreasestronglytowardzero.Thesameself-gravitationalcorrectionismadetothecloselyrelatedquarticallydivergentsecond-ordervacuum-to-vacuumamplitudecorrectionFeynmandiagram,anditisfoundthattheresultvanishesidentically.PACSnumbers:04.20.-q11.10.-z11.10.Gh12.20.-m∗E-Mailaddress:[e-mail]:1(415)585-0678May19952IIntroductionSelf-gravitationalcorrectionasapossiblephysicalmechanismforresolvingultravioletdivergencesinquantumfieldtheorieshasbeendiscussedovertheyearsbyanumberofauthors.SomeoftheearliestpublishedspeculationsalongthislinewereputforthbyLandau[1],Klein[2,3,4],Pauli[5],andDeser[6].AdefinitivedeterminationofthefinitemassoftheclassicalpointchargewasworkedoutbyArnowitt,Deser,andMisner[7]inthecontextofanon-singularmetric(differentfromtheconventionalsingularReissner–Nordstr¨ommetricforthissituation).Thismassfortheclassicalpointchargehasrecentlybeenindependentlyrederivedbytheauthor[8].Ref.[7]emphasizesthat,forthishighlysingularsourceofgravitationalfield,thecountervailing(negative)effectiveenergydensityduetotheresultingverystronggravitationalfielditself,carefullytakenintoaccountinself-consistentfashion,iscriticaltoobtainingthecorrectfinitevalueforthepointcharge’smass.SubsequentworkbyDeWitt[9],Khriplovich[10],SalamandStrathdee[11],andIsham,Salam,andStrathdee[12]hasapproachedtheultravioletdivergencesofquantumelectrodynamicsthroughcorrectingthelowest-orderFeynmandiagramsinwhichtheyoccurbyvarioustypesofsummedgraviton-exchangeladders.Thesesimplegraviton-exchangeprocessesdonot,however,includethenon-linearfeaturethatthevirtualgravitonsinteractwitheachother,whichwasfoundinRef.[7]tobesocrucialtotheproperself-gravitationalresolutionoftheinfinitemassoftheclassicalpointcharge.Moreover,thegraviton-exchangeladdersumsaregenerallynotbythemselvesgravitation-allygauge-invariant[9,10],andcaninadditionintroducedifficultieswithelectromagneticgaugeinvariance[12].Khriplovichlookedintotheeffectsofmakingdifferentchoicesofgravitationalgaugeinhisapproach,andfoundthatsomeoftheseseemedunlikelytoresultinthesuppressionoftheultravioletdivergences[13].Furthermore,thequantizedelectronmasselectromagneticcorrectionobtainedinRef.[8](bystraightforwardextensionofthenon-singularmetricapproachthatwasusedfortheclassicalpointcharge)isfoundtoincorporateagravitationallyinducedeffectivecutoffradiuswhichdependsinaphysicallysensiblewayontheelectroncharge—incontrastwiththepuzzlingchargeindependenceofthateffectivecutoffradiusintheresultobtainedbymeansoftheladder-sumapproachesofRefs.[10]and[12].Theself-gravitationaltreatmentinthispaperoftheultraviolet-divergentsecond-order“vac-uumpolarization”radiativecorrectioninsertionFeynmandiagramofFig.1isessentiallythesameasthequantumfieldtheoreticapproachofRef.[12]inmostrespects.Sincethefocushereisonself-gravitationalsuppressionofthequantumelectrodynamicallyinducedultravioletdivergence,theRef.[12]restrictionthatgravityisonlycoupledtotheelectromagneticinteractiontermoftheactionintegralisobserved,asitisthisinteractiontermwhichgeneratestheFeynmandiagramelectromagneticvertices,withoutwhichtherecouldbenoelectromagneticallyinducedultravioletdivergences.ThusthediagramofFig.1ispermittedtocoupletointermediatevirtualgravitonswhichtravelbetweenitstwovertices—theseproducethepartofitsself-gravitationalcorrectionwhichcouldbecapableofimpingingonitsultravioletdivergence.Ref.[12]usesthe“