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Scalar Contribution to the Graviton Self-Energy during Inflation

Permanent Link: http://ufdc.ufl.edu/UFE0044637/00001

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Title: Scalar Contribution to the Graviton Self-Energy during Inflation
Physical Description: 1 online resource (103 p.)
Language: english
Creator: Park, Sohyun
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2012

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Subjects / Keywords: field -- inflation -- quantum
Physics -- Dissertations, Academic -- UF
Genre: Physics thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

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Abstract: We use dimensional regularization to evaluate the one loop contribution to the graviton self-energy from a massless, minimally coupled scalar on a locally de Sitter background. For noncoincident points our result agrees with the stress tensor correlators obtained recently by Perez-Nadal, Roura and Verdaguer. We absorb the ultraviolet divergences using the $R^2$ and $C^2$ counterterms first derived by 't Hooft and Veltman, and we take the $D=4$ limit of the finite remainder. The renormalized result is expressed as the sum of two transverse, 4th order differential operators acting on nonlocal, de Sitter invariant structure functions. In this form it can be used to quantum-correct the linearized Einstein equations so that one can study how the inflationary production of infrared scalars affects the propagation of dynamical gravitons and the force of gravity. We have seen that they have no effect on the propagation of dynamical gravitons. Our computation motivates a conjecture for the first correction to the vacuum state wave functional of gravitons. We comment as well on performing the same analysis for the more interesting contribution from inflationary gravitons, and on inferring one loop corrections to the force of gravity.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Sohyun Park.
Thesis: Thesis (Ph.D.)--University of Florida, 2012.
Local: Adviser: Woodard, Richard P.

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Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2012
System ID: UFE0044637:00001

Permanent Link: http://ufdc.ufl.edu/UFE0044637/00001

Material Information

Title: Scalar Contribution to the Graviton Self-Energy during Inflation
Physical Description: 1 online resource (103 p.)
Language: english
Creator: Park, Sohyun
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2012

Subjects

Subjects / Keywords: field -- inflation -- quantum
Physics -- Dissertations, Academic -- UF
Genre: Physics thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: We use dimensional regularization to evaluate the one loop contribution to the graviton self-energy from a massless, minimally coupled scalar on a locally de Sitter background. For noncoincident points our result agrees with the stress tensor correlators obtained recently by Perez-Nadal, Roura and Verdaguer. We absorb the ultraviolet divergences using the $R^2$ and $C^2$ counterterms first derived by 't Hooft and Veltman, and we take the $D=4$ limit of the finite remainder. The renormalized result is expressed as the sum of two transverse, 4th order differential operators acting on nonlocal, de Sitter invariant structure functions. In this form it can be used to quantum-correct the linearized Einstein equations so that one can study how the inflationary production of infrared scalars affects the propagation of dynamical gravitons and the force of gravity. We have seen that they have no effect on the propagation of dynamical gravitons. Our computation motivates a conjecture for the first correction to the vacuum state wave functional of gravitons. We comment as well on performing the same analysis for the more interesting contribution from inflationary gravitons, and on inferring one loop corrections to the force of gravity.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Sohyun Park.
Thesis: Thesis (Ph.D.)--University of Florida, 2012.
Local: Adviser: Woodard, Richard P.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2012
System ID: UFE0044637:00001


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SCALARCONTRIBUTIONTOTHEGRAVITONSELF-ENERGYDURINGINFLATIONBySOHYUNPARKADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOLOFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENTOFTHEREQUIREMENTSFORTHEDEGREEOFDOCTOROFPHILOSOPHYUNIVERSITYOFFLORIDA2012

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c2012SohyunPark 2

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Tomyparentsandmyance 3

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ACKNOWLEDGMENTS Foremost,IamdeeplyindebtedtomyadvisorProf.RichardWoodardforhisdedicatedguidance,patienceandencouragement.Itwasagreatpleasuretondsolutionsforseeminglyunsolvableproblemsafterlongdiscussionswithhim.Yearsofinteractionlater,talkingaboutphysicswithhimbecameoneofthemostenjoyableactivities.Iwouldliketothankmysupervisorycommitteemembers,Prof.StevenDetweiler,Prof.JamesFry,Prof.DavidGroisserandProf.PierreSikiviefortheirvaluablequestionsandsuggestions.IamverygratefultoDr.ScottDodelsonforsupervisingmyresearchatFermilab.Theinsightfulandchallengingquestionsheconstantlyofferedmegreatlybroadenedmyhorizonsincosmology.IamalsothankfultoDr.AndreasKronfeld,theFermilabFellowshipdirectorthatofferedmethegreatopportunityofworkingatFermilab.IalsowouldliketothankmyFermilabofcematesChrisKelso,FarinaldoQueirozandRitobanThakurformanyrichdiscussions.MythanksalsogotoOliviaandKristinforgreatlysimplifyingthepaperworkandhelpingmeintheprocessofmovingbetweenUFandFermilab.IwouldliketothankmyfriendsattheUFDepartmentofPhysicsfortheirfriendshipandforhelpingmeinsomanyways:JesusEscobar,HyoungjeenJeen,SungsuKim,InhaeKwak,KatieLeonard,PedroMora,ZahraNasrollahi,MyeonghunPark,FranciscoRojasandMinjunSon.FinallyIwouldlikegivemydeepestthankstomyparents.Theyhavealwayssupportedmewiththeirloveandcare.IwouldliketoalsothankmysistersandnephewsforcheeringmeupwhenIneededit.Duringmyrstyearsofgraduateschool,ImetmyancePablo.HiscompanyandloveencouragedmetoovercomeallthedifcultiesIhadduringmystudies.IwouldnothavebeenabletoachievewhatIhavewithouttheendlessinspirationfromthoseclosesttome. 4

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TABLEOFCONTENTS page ACKNOWLEDGMENTS .................................. 4 LISTOFTABLES ...................................... 7 LISTOFFIGURES ..................................... 8 ABSTRACT ......................................... 9 CHAPTER 1INTRODUCTION ................................... 10 1.1Ination ..................................... 10 1.2UnderstandingQuantumEffects ....................... 11 1.2.1UncertaintyPrincipleduringInation ................. 12 1.2.2ConformalInvariance .......................... 13 1.2.3ParticleProductionduringInation .................. 14 1.3UsingQuantumGravityasanEffectiveFieldTheory ............ 16 1.4Overview .................................... 19 2FEYNMANRULES .................................. 23 2.1InteractionVertices ............................... 23 2.2WorkingondeSitterSpace .......................... 25 2.3ScalarPropagatorondeSitter ........................ 27 3OneLoopGravitonSelf-energy ........................... 29 3.1Contributionfrom4-PointVertices ...................... 29 3.2Contributionfrom3-PointVertices ...................... 31 3.3CorrespondencewithFlatSpace ....................... 33 3.4CorrespondencewithStressTensorCorrelators ............... 34 4RENORMALIZATION ................................ 41 4.1OneLoopCounterterms ............................ 42 4.2RenormalizingtheFlatSpaceResult ..................... 45 4.3ThedeSitterStructureFunctions ....................... 48 4.4RenormalizingtheSpinZeroStructureFunction .............. 59 4.5RenormalizingtheSpinTwoStructureFunction ............... 64 5FLATSPACERESULT ................................ 69 5.1Schwinger-KeldyshEffectiveFieldEqns ................... 69 5.2SolvingforthePotentials ........................... 72 5.2.1AchievingAManifestlyRealandCausalForm ............ 72 5

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5.2.2SolvingtheEquationPerturbatively .................. 74 5.2.3CorrectiontoDynamicalGravitonsinFlatSpace .......... 75 5.2.4TheOneLoopSourceTerm ...................... 77 5.2.5TheOneLoopPotentials ........................ 79 6QUANTUMCORRECTIONSTODYNAMICALGRAVITONS ........... 81 6.1TheEffectiveFieldEquations ......................... 81 6.1.1TheSchwinger-KeldyshEffectiveFieldEquations .......... 81 6.1.2PerturbativeSolution .......................... 82 6.2ComputingtheOneLoopSource ....................... 82 6.2.1PartialIntegration ............................ 83 6.2.2ExtractingAnotherd'Alembertian ................... 84 6.2.3DerivativesoftheWeylTensor .................... 85 6.2.4TheFinalReduction .......................... 87 7CONCLUSION .................................... 90 REFERENCES ....................................... 96 BIOGRAPHICALSKETCH ................................ 103 6

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LISTOFTABLES Table page 2-13-pointverticesTIwhere gisthedeSitterbackgroundmetric,216Gandparenthesizedindicesaresymmetrized. ................... 24 2-24-pointverticesFIwhere gisthedeSitterbackgroundmetric,216Gandparenthesizedindicesaresymmetrized. ................ 24 4-1CoefcientofF2:eachtermismultipliedby1 16(D)]TJ /F7 7.97 Tf 6.59 0 Td[(2)(D)]TJ /F7 7.97 Tf 6.58 0 Td[(1) ............. 54 4-2CoefcientofF02:eachtermismultipliedby1 16(D)]TJ /F7 7.97 Tf 6.59 0 Td[(2)(D)]TJ /F7 7.97 Tf 6.58 0 Td[(1) ............. 54 4-3CoefcientofF002:eachtermismultipliedby1 16(D)]TJ /F7 7.97 Tf 6.58 0 Td[(2)(D)]TJ /F7 7.97 Tf 6.58 0 Td[(1) ............. 55 4-4CoefcientofF0002:eachtermismultipliedby1 16(D)]TJ /F7 7.97 Tf 6.59 0 Td[(2)(D)]TJ /F7 7.97 Tf 6.59 0 Td[(1) ............. 55 4-5CoefcientofF00002:eachtermismultipliedby1 16(D)]TJ /F7 7.97 Tf 6.59 0 Td[(2)(D)]TJ /F7 7.97 Tf 6.59 0 Td[(1) ............. 55 7

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LISTOFFIGURES Figure page 2-1Theoneloopgravitonself-energyfromMMCscalars. .............. 23 8

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AbstractofDissertationPresentedtotheGraduateSchooloftheUniversityofFloridainPartialFulllmentoftheRequirementsfortheDegreeofDoctorofPhilosophySCALARCONTRIBUTIONTOTHEGRAVITONSELF-ENERGYDURINGINFLATIONBySohyunParkAugust2012Chair:RichardP.WoodardMajor:PhysicsWeusedimensionalregularizationtoevaluatetheoneloopcontributiontothegravitonself-energyfromamassless,minimallycoupledscalaronalocallydeSitterbackground.FornoncoincidentpointsourresultagreeswiththestresstensorcorrelatorsobtainedrecentlybyPerez-Nadal,RouraandVerdaguer.WeabsorbtheultravioletdivergencesusingtheR2andC2countertermsrstderivedby'tHooftandVeltman,andwetaketheD=4limitoftheniteremainder.Therenormalizedresultisexpressedasthesumoftwotransverse,4thorderdifferentialoperatorsactingonnonlocal,deSitterinvariantstructurefunctions.Inthisformitcanbeusedtoquantum-correctthelinearizedEinsteinequationssothatonecanstudyhowtheinationaryproductionofinfraredscalarsaffectsthepropagationofdynamicalgravitonsandtheforceofgravity.Wehaveseenthattheyhavenoeffectonthepropagationofdynamicalgravitons.Ourcomputationmotivatesaconjecturefortherstcorrectiontothevacuumstatewavefunctionalofgravitons.Wecommentaswellonperformingthesameanalysisforthemoreinterestingcontributionfrominationarygravitons,andoninferringoneloopcorrectionstotheforceofgravity. 9

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CHAPTER1INTRODUCTIONMyresearchinvolvesquantumeffectsduringprimordialination.Primordialinationisaphaseofacceleratedexpansionduringtheveryearlyuniversewhichexplainswhythecurrentuniverseissohomogeneousandisotropiconlargescales,andwhyitissonearlyspatiallyat.Quantumeffectsarevastlystrengthenedduringinationbecausetherapidexpansionripsquantumuctuationsoutofthevacuumsothattheybecomerealparticles.Thisisthoughttobethesourceoftheobserveddensityperturbations.Myworkconcernshowtheensembleofscalarsproducedinthiswaywouldaffectthepropagationofgravitationalradiationandtheforceofgravity.Inthefollowingsections,wewillreviewwhatinationis,whyitenhancesquantumeffectsandhowonecanunderstandthisenhancementastheclassicalresponsetovirtualparticles.Wewillalsodiscusshowreliableinformationfromquantumgeneralrelativitycanbeobtainedinspiteofitsnonrenormalizability.Thischaptercloseswithanoverviewofmyproject. 1.1InationOnscalesabove100Mpcouruniverseisobservedtobehomogeneousandisotropic.Italsoseemstohavezerospatialcurvature.BasedonthesethreefeaturesouruniversecanbedescribedbytheFriedmann-Robertson-Walker(FRW)metric,withtheinvariantelement ds2=)]TJ /F3 11.955 Tf 9.3 0 Td[(dt2+a2(t)d~xd~x.(1)Herethecoordinatetisphysicaltimeandthea(t)iscalledthescalefactorbecauseitconvertscoordinatedistancek~x)]TJ /F6 11.955 Tf 11.53 .5 Td[(~ykintophysicaldistancea(t)k~x)]TJ /F6 11.955 Tf 11.54 .5 Td[(~yk. 10

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Therearethreeobservablecosmologicalquantitiesthatcanbeconstructedfromthescalefactor: RedShiftz(t)a(t0) a(t))]TJ /F8 11.955 Tf 11.96 0 Td[(1, (1) HubbleParameterH(t)_a a, (1) Decelerationparameterq(t))]TJ /F3 11.955 Tf 29.75 8.09 Td[(aa _a2=)]TJ /F8 11.955 Tf 9.3 0 Td[(1)]TJ /F8 11.955 Tf 18.57 10.75 Td[(_H H2=)]TJ /F8 11.955 Tf 9.3 0 Td[(1+(t) (1) wheret0isthecurrenttime.TheHubbleparameterH(t)tellsustheexpansionrateoftheuniverse.Thedecelerationparametermeasuresthefractionalaccelerationratea=ainunitsoftheHubbleparameter(_a=a)2.Inationisdenedasacceleratedexpansion,[ 1 2 ] H(t)>0and(q(t)<0orequivalently<1).(1)Theircurrentvaluesare:Hnow=(73.82.4)Km=s Mpc'2.410)]TJ /F7 7.97 Tf 6.59 0 Td[(18Hz'10)]TJ /F7 7.97 Tf 6.58 0 Td[(33eV[ 3 ]andnow'0.330.13[ 4 5 ].Soouruniverseiscurrentlyinating.Howevertheinationaryepochofrelevancetomyworkisprimordialination.BecausetheeffectsIstudyderivefromquantumgravity,theycontainpowersofGH2,andthecurrentHubbleparameterisjusttoosmallfortheseeffectstobeobservable.Incontrast,thelatestdata[ 5 6 ]plustheassumptionofsinglescalarinationimplyHI.1.71038Hz1013GeVwithI.0.011[ 7 ].ThisisonlyaboutsixordersofmagnitudebelowthePlanckscale,MPl1019GeVandthatmakestheseeffectssmall,butobservable.HereitisusefultocommentthatprimordialinationisveryclosetodeSitterwhichhasapositiveconstantforHandqexactly)]TJ /F8 11.955 Tf 9.3 0 Td[(1(or=0).ThisallowsustotakedeSitterspaceasaparadigmforprimordialination.AllmycalculationsconcerningquantumeffectsduringinationaredoneonSitterbackground. 1.2UnderstandingQuantumEffectsQuantumloopeffectscanbeunderstoodastheclassicalresponsetovirtualparticles.Forexample,considerthevacuumpolarizationofquantumelectrodynamics. 11

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Theenergy-timeuncertaintyprinciplesaysthatvirtualelectro-positronpairsarecreatedoutofthevacuumandexistforabriefperiodoftime.Ifwejustacceptthis,thevacuumpolarizationiscompletelyanalogoustothephenomenonofclassicalpolarizationinamediuumfullofchargedparticles.Thebottomlineisthatifwhateverincreasesthenumberofvirtualparticlesstrengthensquantumeffects.Inthefollowingsubsectionswewilldiscusshowinationdoesthis.Weconsidertwoaspects,thepersistencetimeandtheemergencerateofvirtualparticles. 1.2.1UncertaintyPrincipleduringInationTheenergy-timeuncertaintyprincipleofatspace Et>1.(1)saysthattoresolveanenergywithaccuracyEwehavetowaitatleastatimet.ToreslovetheproductionofavirtualpairofwavenumberkandmassmwehaveE=2p m2+k2.Wewouldnotnoticeaviolationofenergyconservationprovidedt<1 E.Thatis,wecantake1=Easthelifetimeofavirtualpair. t=1 E=1 2p m2+k2.(1)Howwouldthiseffectchangeduringination?Ifweconsiderthehomogeneous,isotropicandspatiallyatgeometrydescribedin( 1 ),fromitsspatialtranslationinvarianceonecanstilllabelparticlesbyconstantwavenumbers~k,justasinatspace.However,thisco-movingwavevector~kinvolvesaninverselengthandhenceonemustmultiplyitbythescalefactora(t)togetthephysicalwavevector~k=a(t).Thistimedependentwavenumberimpliestheexpressionfor( 1 )becomesanintegral. Zt+4ttdt02E(t0,~k)<1.(1)withE(t,~k)=p m2+k2=a2(t).Notethatspacetimeexpansionalwayslengthensthetimeavirtualpaircanexistbecausekphys=k=a(t)becomessmallerasa(t)grows.Just 12

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asinatspace,masslessparticlesofthesamewavenumberlivelongerthanmassiveones.Takingm=0andthedeSitterlimitofthescalefactor,a(t)=aIeHtwehave 2k Ha(t)(1)]TJ /F3 11.955 Tf 11.95 0 Td[(e)]TJ /F5 7.97 Tf 6.59 0 Td[(Ht)<1.(1)Thismeansthatmasslessvirtualparticlescanliveforeverduringinationiftheyemergewithk
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Thismeansthattheemergencerateofvirtualparticlespossessingconformalsymmetryissuppressedbyafactorof1=a.Thereforeanyconformal-invariant,masslessvirtualparticleswithk
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L=1 2a3Zd3k (2)3~_'(t,~k)~_'(t,~k))]TJ /F8 11.955 Tf 13.15 8.08 Td[(1 2aZd3k (2)3k2~'(t,~k)~'(t,~k).(1)Becausethereisnocouplingbetweendifferent~k's,letusconsideronemodewith~kandcallitq(t).ItsLagrangianis, L=1 2a3_q2)]TJ /F8 11.955 Tf 13.16 8.09 Td[(1 2k2aq2.(1)NowwenoticethisistheLagrangianofaharmonicoscillatorwithmassm(t)=a3(t)andfrequency!(t)=k=a(t).BecausemassandfrequencyaretimedependenttherearenostationarystatesbutwecanstillconstructtheHamiltonian H=_q@L @_q)]TJ /F3 11.955 Tf 11.95 0 Td[(L=1 2a3_q2+1 2k2aq2=1 2m(t)_q2+1 2m(t)!2(t)q2.(1)withtheequationofmotion q+3H_q+k2 a2q=0(1)Solvingthisequationforgenerala(t)isnoteasybutforthespecialcaseofdeSitter(a(t)=eHtand_H=0)whichisrelevantinourdiscussion,thegeneralsolutiontakestheform, q(t)=u(t)+u(t)y,u(t,k)=H p 2k31)]TJ /F3 11.955 Tf 23.25 8.09 Td[(ik Ha(t)eik Ha(t)(1)whereandyareoperatorswhichwecanonicallynormalizeasinnormalquantummechanics, [,y]=1.(1)WedeneBunch-Daviesvacuumjiasthestatewithminimumenergyinthedistantpast.TheBunch-Davisvacuumisj>isannihilatedby,j>=0.Tondthethenumberofparticleswhichemergewithwavenumberk,considertheexpectationvalueoftheenergyinthisstate =1 2a3(t)+1 2k2a(t)=k a1 2+Ha 2k2=k a1 2+\N(t)" (1) 15

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Werecognizethatthenumberofparticleswithwavenumberkgrowsintimeasthesquareofthescalefactor, N(k,t)=Ha(t) 2k2(1)Notethatthegrowthbecomessignicantforinfraredwavenumbers,kHa.Oneconsequenceofthisbeinganinfraredeffectisthatperturbativegeneralrelativitycanbeusedreliablyasaneffectiveeldtheory,eventhoughitisnotrenormalizable.Thisissuewillbediscussedinthenextsection. 1.3UsingQuantumGravityasanEffectiveFieldTheoryQuantumgravityisnotperturbativelyrenormalizable[ 9 ],however,ultravioletdivergencescanalwaysbeabsorbedinthesenseofBogoliubov,Parasiuk,HeppandZimmerman(BPHZ)[ 10 ].Awidespreadmisconceptionexiststhatnovalidquantumpredictionscanbeextractedfromsuchanexercise.Thisisfalse:whilenonrenormalizabilitydoesprecludebeingabletocomputeeverything,thatisnotthesamethingasbeingabletocomputenothing.TheproblemwithanonrenormalizabletheoryisthatnophysicalprinciplexesthenitepartsoftheescalatingseriesofBPHZcountertermsneededtoabsorbultravioletdivergences,order-by-orderinperturbationtheory.Henceanypredictionofthetheorythatcanbechangedbyadjustingthenitepartsofthesecountertermsisessentiallyarbitrary.However,loopsofmasslessparticlesmakenonlocalcontributionstotheeffectiveactionthatcanneverbeaffectedbylocalcounterterms.Thesenonlocalcontributionstypicallydominateintheinfrared.Further,theycannotbeaffectedbywhatevermodicationofultravioletphysicsultimatelyresultsinacompletelyconsistentformalism.Aslongastheeventualxintroducesnonewmasslessparticles,anddoesnotdisturbthelowenergycouplingsoftheexistingones,thefarinfraredpredictionsofaBPHZ-renormalizedquantumtheorywillagreewiththoseofitsfullyconsistentdescendant.Toseethisissuemorespecically,letusrstrecallthetheoremofBogoliubov,Parasiuk,HeppandZimmerman(BPHZ)whichconstructsthelocalcounterterms 16

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neededtoabsorbtheultravioletdivergencesofanyquantumeldtheoryatsomexedorderintheloopexpansion[ 10 ].Thisappliesaswellforquantumgeneralrelativityplusanymattertheory,andthetwooneloopcountertermsR2andC2requiredforscalarsinD=4spacetimedimensionshavelongbeenknown[ 9 ].TheproblemforquantumgeneralrelativityisthatthesecountertermsarenotpresentintheoriginalLagrangian.WecouldincludeR2;itwouldaddamassive,positiveenergyscalarparticlewhichposesnoessentialproblemforthetheory.However,incorporatingC2onanonperturbativelevelwouldaddanegativeenergy,spintwoparticlewhosepresencewouldcausetheuniversetodecayinstantly.Wemustthereforetreattheoneloopcountertermsperturbatively,andregardthemasproxiesforthestillunknownultravioletcompletionofthetheory.Theremainingproblemwiththeseperturbativecountertermsisthatwedon'tknowtheirniteparts.Theirdivergentpartsarexedbytheneedtosubtractofftheinnitiesoneencountersinloopcorrections,butnothingxestheniteparts,andthesenitepartsaffectphysicalresults,evenwhenweonlyusethemperturbatively.Ofcoursethisambiguityreectsthefactthatwedon'tknowtheultravioletcompletionofquantumgravity.Whatitmeansisthattheonlyreliablepredictionsarethoseforwhichthearbitrarynitepartsofthecountertermsareunimportant.Thattherearesuchpredictionsderivesfromtwothings: BPHZcountertermsareguaranteedtobelocal[ 10 ];and Masslessparticlesmakenonlocalcorrectionstotheeffectiveeldequations.Asanexample,consideroneloopcorrectionstothequantumgravitationaleffectiveactionwhicharequadraticinthegravitoneldh.Forsimplicity,letthebackgroundbeatspace,andletusagreenottoworryabouthowthevariousindicesarecontracted.Theoneloopcountertermscontributetotheeffectiveactionas, )]TJ /F7 7.97 Tf 6.77 5.44 Td[(1loopctermsZd4x@2h@2h+O(h3).(1) 17

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Asstated,theproblemwiththesesortsoftermsisthatwedon'tknowthenumericalcoefcientswhichmultiplythem.Incontrast,oneloopeffectsfrommasslessparticlescontributetermsoftheform, )]TJ /F7 7.97 Tf 6.77 5.44 Td[(1loopniteZd4x@2hln()]TJ /F6 11.955 Tf 9.3 0 Td[(@2)@2h+O(h3).(1)Perturbativequantumgeneralrelativitymakesanexactpredictionforthecoefcientsoftheseterms.Further,inthelargedistanceregimethenite,nonlocalcontributions( 1 )dominateoverthelocalcounterterms( 1 )owingtotheirenhancementbythefactorofln()]TJ /F6 11.955 Tf 9.29 0 Td[(@2),whichdivergesintheinfrared.Inmomentumspace@2!)]TJ /F3 11.955 Tf 26.05 0 Td[(p2,andthelongwavelengthregimeisp20.Thenthelocalcountertermgoeslikep4,andthenonlocalprimitiveeffectsgolikep4ln(p2).Forsmallenoughp2thenonlocaleffectsarelarger,nomatterhowbigthenitepartsofthecountertermsare.Itisworthwhiletoreviewthevastbodyofdistinguishedworkthathasemployedtoderivevalidquantumeffectsinthelongdistanceregime.TheoldestexampleisthesolutionoftheinfraredprobleminquantumelectrodynamicsbyBlochandNordsieck[ 13 ],longbeforethattheory'srenormalizabilitywassuspected.Weinberg[ 14 ]wasabletoachieveasimilarresolutionforquantumgravitywithzerocosmologicalconstant.ThesameprinciplewasatworkintheFermitheorycomputationofthelongrangeforceduetoloopsofmasslessneutrinosbyFeinbergandSucher[ 15 16 ].InpurequantumgravityDonoghueandothershasappliedtheprinciplesoflowenergyeffectiveeldtheorytocomputegravitoncorrectionstothelongrangegravitationalforce[ 17 23 ].Tosummarize,aslongasweconsiderthelowenergyregime,thenitequantumcorrectionsfromtheoriginalLagrangiancanbedistinguishedfromthoseoflocalcounterterms.IntheprevioussubsectionsIdiscussedthevirtualparticlesproducedduringination.Thetwokeyfactswere 1. Thenumberofvirtualparticlespresentduringaperiodofacceleratedexpansionisvastlylargerthanduringaphaseofdeceleration;and 18

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2. Theextravirtualparticleshavecosmologicalwavelengths.Therstfactmeansonecangetsignicantquantumeffects;thesecondpointmeansthattheseeffectscanbecomputedreliablywithoutknowingtheultravioletcompletionofquantumgravity. 1.4OverviewThelinearizedequationsforallknownforceeldsdotwothings: Theygivethelinearizedforceeldsinducedbysources;and Theydescribethepropagationofdynamicalparticleswhichcarrytheforcebutare,inprinciple,independentofanysource.Thisistheclassicdistinctionbetweentheconstrainedandunconstrainedpartsofaforceeld.InelectromagnetismitamountstotheCoulombpotentialversusphotons.IngravitythereistheNewtonianpotential,plusitsthreerelativisticpartners,versusgravitons.Quantumcorrectionstothelinearizedeldequationsderivefromhowthe0-pointuctuationsofvariouseldsinwhateverbackgroundisassumed,respondtothelinearizedforceelds.Thesequantumcorrectionsdonotchangethedichotomybetweenconstrainedandunconstrainedeldsbuttheycan,ofcourse,modifyclassicalresults.Aroundatspacebackgroundthereisnoeffect,afterrenormalization,onthepropagationofdynamicalphotonsorgravitonsbuttherearesmallcorrectionstotheCoulombandNewtonianpotentials.Asmightbeexpected,thelongdistanceeffectsaregreatestforthe0-pointuctuationsofmasslessparticlesandtheytaketheformrequiredbyperturbationtheoryanddimensionalanalysis[ 43 44 ], Coul.)]TJ /F3 11.955 Tf 24.02 8.09 Td[(e2 ~clnr r0, Newt.)]TJ /F19 11.955 Tf 26 8.09 Td[(~G c3r2,(1)whereristhedistancetothesource,r0isthepointatwhichtherenormalizedchargeisdened,andtheotherconstantshavetheirusualmeanings. 19

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Schrodingerwasthersttosuggestthattheexpansionofspacetimecanleadtoparticleproductionbyrippingthevirtualparticles(whichareimplicitin0-pointuctuations)outofthevacuum[ 45 ].FollowingearlyworkbyImamura[ 46 ],therstquantitativeresultswereobtainedbyParker[ 47 ].Hefoundthattheeffectismaximizedduringacceleratedexpansion,andformasslessparticleswhicharenotconformallyinvariant[ 48 ],suchasmassless,minimallycoupled(MMC)scalarsand(asnotedbyGrishchuk[ 8 ])gravitons.Thisresultwasreviewedintheprevioussections.ThedeSittergeometryisthemosthighlyacceleratedexpansionconsistentwithclassicalstability.FordeSitterbackgroundwithHubbleconstantHandscalefactora(t)=eHtwehaveshownthatthenumberofMMCscalars,oreitherpolarizationofgraviton,createdwithwavevector~kis[ 49 ], N(t,~k)=Ha(t) 2ck~kk2.(1)Itistheseparticleswhichcomprisethescalarandtensorperturbationsproducedbyination[ 50 ],thescalarcontributionofwhichhasbeenimaged[ 51 ].OfcoursethesameparticlesalsoenterloopdiagramstocauseanenormousstrengtheningofthequantumeffectscausedbyMMCscalarsandgravitons.AnumberofanalyticresultshavebeenobtainedforoneloopcorrectionstothewayvariousparticlespropagateondeSitterbackgroundandalsotohowlongrangeforcesact: InMMCscalarquantumelectrodynamics,infraredphotonsbehaveasiftheyhadanincreasingmass[ 52 ],andthechargescreeningveryquicklybecomesnonperturbativelystrong[ 53 ],butthereisnobigeffectonscalars[ 54 ]; ForaMMCscalarwhichisYukawa-coupledtoamasslessfermion,infraredfermionsbehaveasiftheyhadanincreasingmass[ 55 ]buttheassociatedscalarsexperiencenolargecorrection[ 56 ]; ForaMMCscalarwithaquarticself-interaction,infraredscalarsbehaveasiftheyhadanincreasingmass(whichpersiststotwolooporder)[ 57 ]; Forquantumgravityminimallycoupledtoamasslessfermion,thefermioneldstrengthgrowswithoutbound[ 58 ];and 20

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ForquantumgravityplusaMMCscalar,thescalarshowsnoseculareffectbutitseldstrengthmayacquireamomentum-dependentenhancement[ 59 ].Thegreatomissionfromthislistishowinationaryscalarsandgravitonsaffectgravity,bothasregardsthepropagationofdynamicalgravitonsandasregardstheforceofgravity.Myprojectrepresentsarststepincompletingthelist.Oneincludesquantumcorrectionstothelinearizedeldequationbysubtractingtheintegraloftheappropriateone-particle-irreducible(1PI)2-pointfunctionupagainstthelinearizedeld.Forexample,aMMCscalar'(x)inabackgroundmetricg(x)whose1PI2-pointfunctionis)]TJ /F3 11.955 Tf 9.3 0 Td[(iM2(x;x0),wouldhavethelinearizedeffectiveeldequation, @hp gg@'(x)i)]TJ /F12 11.955 Tf 11.96 16.27 Td[(Zd4x0M2(x;x0)'(x0)=0.(1)Toincludegravityonthelistwemustthereforecomputethegravitonself-energy,eitherfromMMCscalarsorfromgravitons,andthenuseittocorrectthelinearizedEinsteinequation.IntherstpartofmydissertationweevaluatethecontributionfromMMCscalarswhichisdescribedinchapters2through4;Inchapter2wegivethoseoftheFeynmanruleswhichareneededforthiscomputation,andwedescribethegeometryofourD-dimensional,locallydeSitterbackground.Chapter3derivestherelativelysimpleformfortheD-dimensionalgravitonself-energywithnoncoincidentpoints.Weshowthatthisversionoftheresultagreeswiththeatspacelimit[ 61 ]andwiththedeSitterstresstensorcorrelatorsrecentlyderivedbyPerez-Nadal,RouraandVerdaguer[ 99 ].Chapter4undertakesthevastlymoredifcultreorganizationwhichmustbedonetoisolatethelocaldivergencesforrenormalization.Attheendwesubtractoffthedivergenceswiththesamecountertermsoriginallycomputedforthismodelin1974by'tHooftandVeltman[ 9 ],andwetaketheunregulatedlimitofD=4.Thesecondpartofdissertationsolvesthelinearizedeffectiveeldequationstodeterminequantumcorrectionstothepropagationofgravitons.Inchapter5wecarry 21

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outthesamecalculationsforatspacewhichalsoservethecorrespondencelimitforthevastlymorecomplicateddeSittercase.Chapter6isdedicatedforthescalaroneloopcorrectiontodynamicalgravitons.Ourconclusioncompriseschapter7. 22

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CHAPTER2FEYNMANRULESInthischapterwederiveFeynmanrulesforthecomputation.Westartbyexpressingthefullmetricas g= g+h,(2)where gisthebackgroundmetric,histhegravitoneldwhoseindicesareraisedandloweredwiththebackgroundmetric,and216Gistheloopcountingparameterofquantumgravity.ExpandingtheMMCscalarLagrangianaroundthebackgroundmetricwegetinteractionverticesbetweenthescalaranddynamicalgravitons.WetaketheD-dimensionallocallydeSitterspaceasourbackgroundandintroducedeSitterinvariantbi-tensorswhichwillbeusedthroughoutthecalculation.WeclosethissectionbyprovidingtheMMCscalarpropagatoronthedeSitterbackground. 2.1InteractionVerticesTheLagrangianwhichdescribespuregravityandtheinteractionbetweengravitonsandtheMMCscalaris, L=1 16GhR)]TJ /F8 11.955 Tf 11.96 0 Td[((D)]TJ /F8 11.955 Tf 9.96 0 Td[(1)(D)]TJ /F8 11.955 Tf 9.96 0 Td[(2)H2ip )]TJ /F3 11.955 Tf 9.3 0 Td[(g)]TJ /F8 11.955 Tf 13.15 8.09 Td[(1 2@'@'gp )]TJ /F3 11.955 Tf 9.3 0 Td[(g.(2)whereRisRicciscalar,GisNewton'sconstantandHistheHubbleconstant.Computingtheoneloopscalarcontributionstothegravitonself-energyconsistsofsummingthe3FeynmandiagramsdepictedinFigure 2-1 .Thesumofthesethree Figure2-1. Theoneloopgravitonself-energyfromMMCscalars. 23

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diagramshasthefollowinganalyticform: )]TJ /F3 11.955 Tf 9.3 0 Td[(i[](x;x0)=1 22XI=1TI(x)2XJ=1TJ(x0)@@0i4(x;x0)@@0i4(x;x0)+1 24XI=1FI(x)@@0i4(x;x0)D(x)]TJ /F3 11.955 Tf 11.95 0 Td[(x0)+22XI=1CI(x)D(x)]TJ /F3 11.955 Tf 11.96 0 Td[(x0). (2) The3-pointand4-pointvertexfactorsTIandFIderivefromexpandingtheMMCscalarLagrangianusing( 2 ), )]TJ /F8 11.955 Tf 10.49 8.09 Td[(1 2@'@'gp )]TJ /F3 11.955 Tf 9.29 0 Td[(g (2) =)]TJ /F8 11.955 Tf 10.49 8.09 Td[(1 2@'@' gp )]TJ ET q .478 w 164.3 -277.15 m 171.06 -277.15 l S Q BT /F3 11.955 Tf 164.3 -284.47 Td[(g)]TJ /F6 11.955 Tf 13.15 8.09 Td[( 2@'@'1 2h g)]TJ /F3 11.955 Tf 11.96 0 Td[(hp )]TJ ET q .478 w 332.36 -277.15 m 339.12 -277.15 l S Q BT /F3 11.955 Tf 332.36 -284.47 Td[(g)]TJ /F6 11.955 Tf 10.5 8.09 Td[(2 2@'@'(h1 8h2)]TJ /F8 11.955 Tf 11.16 8.09 Td[(1 4hhi g)]TJ /F8 11.955 Tf 11.16 8.09 Td[(1 2hh+hh)p )]TJ ET q .478 w 356.94 -310.26 m 363.7 -310.26 l S Q BT /F3 11.955 Tf 356.94 -317.58 Td[(g+O(3). (2) Theresulting3-pointand4-pointvertexfactorsaregivenintheTables1and2,respectively.TheproceduretogetthecountertermvertexoperatorsCI(x)isgiveninsection4. Table2-1. 3-pointverticesTIwhere gisthedeSitterbackgroundmetric,216Gandparenthesizedindicesaresymmetrized. ITI 1)]TJ /F5 7.97 Tf 10.49 4.71 Td[(i 2p )]TJ ET q .478 w 72.78 -466.81 m 79.54 -466.81 l S Q BT /F3 11.955 Tf 72.78 -474.13 Td[(g g g2+ip )]TJ ET q .478 w 69.67 -482 m 76.43 -482 l S Q BT /F3 11.955 Tf 69.67 -489.32 Td[(g g( g) Table2-2. 4-pointverticesFIwhere gisthedeSitterbackgroundmetric,216Gandparenthesizedindicesaresymmetrized. IFI 1)]TJ /F5 7.97 Tf 10.5 4.71 Td[(i2 4p )]TJ ET q .478 w 113.23 -574.38 m 120 -574.38 l S Q BT /F3 11.955 Tf 113.23 -581.7 Td[(g g g g2+i2 2p )]TJ ET q .478 w 109.79 -590.15 m 116.55 -590.15 l S Q BT /F3 11.955 Tf 109.79 -597.47 Td[(g g( g) g3+i2 2p )]TJ ET q .478 w 66.91 -611.82 m 73.67 -611.82 l S Q BT /F3 11.955 Tf 66.91 -619.14 Td[(g g( g) g+ g g( g)4)]TJ /F8 11.955 Tf 9.3 0 Td[(2i2p )]TJ ET q .478 w 110.37 -634.04 m 117.13 -634.04 l S Q BT /F3 11.955 Tf 110.37 -641.36 Td[(g g( g)( g) 24

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Theseinteractionverticesarevalidforanybackgroundmetric g.InthenexttwosubsectionswespecializetoalocallydeSitterbackgroundandgivethescalarpropagatori4(x;x0)onit. 2.2WorkingondeSitterSpaceWespecifyourbackgroundgeometryastheopenconformalcoordinatesubmanifoldofD-dimensionaldeSitterspace.Aspacetimepointx=(,xi)takesvaluesintheranges )-222(1<<0and)-222(1
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Spatialspecialconformaltransformations-(D)]TJ /F8 11.955 Tf 11.96 0 Td[(1)transformations. 0= 1)]TJ /F8 11.955 Tf 11.96 0 Td[(2~~x+k~k2xx,x0=xi)]TJ /F6 11.955 Tf 11.96 0 Td[(ixx 1)]TJ /F8 11.955 Tf 11.96 0 Td[(2~~x+k~k2xx.(2)ItturnsoutthattheMMCscalarcontributiontothegravitonself-energyisdeSitterinvariant.ThissuggeststoexpressitintermsofthedeSitterlengthfunctiony(x;x0), y(x;x0)aa0H2"~x)]TJ /F6 11.955 Tf 10.84 2.74 Td[(~x02)]TJ /F12 11.955 Tf 11.96 13.27 Td[(j)]TJ /F6 11.955 Tf 9.96 0 Td[(0j)]TJ /F3 11.955 Tf 13.95 0 Td[(i2#.(2)Exceptforthefactorofi(whosepurposeistoenforceFeynmanboundaryconditions)thefunctiony(x;x0)iscloselyrelatedtotheinvariantlength`(x;x0)fromxtox0, y(x;x0)=4sin21 2H`(x;x0).(2)WiththisdeSitterinvariantquantityy(x;x0),wecanformaconvenientbasisofdeSitterinvariantbi-tensors.Notethatbecausey(x;x0)isdeSitterinvariant,sotooarecovariantderivativesofit.Withthemetrics g(x)and g(x0),therstthreederivativesofy(x;x0)furnishaconvenientbasisofdeSitterinvariantbi-tensors[ 54 ], @y(x;x0) @x=Hay0+2a0Hx, (2) @y(x;x0) @x0=Ha0y0)]TJ /F8 11.955 Tf 9.97 0 Td[(2aHx, (2) @2y(x;x0) @x@x0=H2aa0y00+2a0Hx0)]TJ /F8 11.955 Tf 9.96 0 Td[(2a0Hx)]TJ /F8 11.955 Tf 9.96 0 Td[(2. (2) Hereandsubsequentlyx(x)]TJ /F3 11.955 Tf 9.96 0 Td[(x0).Actingcovariantderivativesgeneratesmorebasistensors,forexample[ 54 ], D2y(x;x0) DxDx=H2(2)]TJ /F3 11.955 Tf 9.96 0 Td[(y) g(x), (2) D2y(x;x0) Dx0Dx0=H2(2)]TJ /F3 11.955 Tf 9.96 0 Td[(y) g(x0). (2) 26

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Thecontractionofanypairofthebasistensorsalsoproducesmorebasistensors[ 54 ], g(x)@y @x@y @x=H24y)]TJ /F3 11.955 Tf 11.95 0 Td[(y2= g(x0)@y @x0@y @x0, (2) g(x)@y @x@2y @x@x0=H2(2)]TJ /F3 11.955 Tf 11.95 0 Td[(y)@y @x0, (2) g(x0)@y @x0@2y @x@x0=H2(2)]TJ /F3 11.955 Tf 11.95 0 Td[(y)@y @x, (2) g(x)@2y @x@x0@2y @x@x0=4H4 g(x0))]TJ /F3 11.955 Tf 11.95 0 Td[(H2@y @x0@y @x0, (2) g(x0)@2y @x@x0@2y @x@x0=4H4 g(x))]TJ /F3 11.955 Tf 11.96 0 Td[(H2@y @x@y @x. (2) Ourbasistensorsarenaturallycovariant,buttheirindicescanofcourseberaisedusingthemetricattheappropriatepoint.Tosavespaceinwritingthisoutwedenethebasistensorswithraisedindicesasdifferentiationwithrespecttocovariantcoordinates, @y @x g(x)@y @x, (2) @y @x0 g(x0)@y @x0, (2) @2y @x@x0 g(x) g(x0)@2y @x@x0. (2) 2.3ScalarPropagatorondeSitterFromtheMMCscalarLagrangian( 2 )weseethatthepropagatorobeys @hp )]TJ ET q .478 w 121.91 -473.96 m 128.67 -473.96 l S Q BT /F3 11.955 Tf 121.91 -481.28 Td[(g g@ii4(x;x0)=p )]TJ ET q .478 w 249.08 -473.96 m 255.84 -473.96 l S Q BT /F3 11.955 Tf 249.08 -481.28 Td[(g i4(x;x0)=iD(x)]TJ /F3 11.955 Tf 11.95 0 Td[(x0)(2)AlthoughthisequationisdeSitterinvariant,thereisnodeSitterinvariantsolutionforthepropagator[ 62 ],hencesomeofthesymmetries( 2 )-( 2 )mustbebroken.Wechoosetopreservethehomogeneityandisotropyofcosmologyrelations( 2 )-( 2 )whichcorrespondstowhatisknownastheE3vacuum[ 63 ].ItcanberealizedintermsofplanewavemodesumsbymakingthespatialmanifoldTD)]TJ /F7 7.97 Tf 6.59 0 Td[(1,ratherthanRD)]TJ /F7 7.97 Tf 6.58 0 Td[(1,withcoordinateradiusH)]TJ /F7 7.97 Tf 6.59 0 Td[(1ineachdirection,andthenusingthe 27

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integralapproximationwiththelowerlimitcutoffatk=H[ 64 ].ThenalresultconsistsofadeSitterinvariantfunctionofy(x;x0)plusadeSitterbreakingpartwhichdependsuponthescalefactorsatthetwopoints[ 41 ], i4(x;x0)=Ay(x;x0)+kln(aa0).(2)Heretheconstantkisgivenas, kHD)]TJ /F7 7.97 Tf 6.58 0 Td[(2 (4)D 2\(D)]TJ /F8 11.955 Tf 9.96 0 Td[(1) \(D 2),(2)andthefunctionA(y)hastheexpansion, A(y)HD)]TJ /F7 7.97 Tf 6.59 0 Td[(2 (4)D 2(\(D 2) D 2)]TJ /F8 11.955 Tf 9.96 0 Td[(14 yD 2)]TJ /F7 7.97 Tf 6.59 0 Td[(1+\(D 2+1) D 2)]TJ /F8 11.955 Tf 9.96 0 Td[(24 yD 2)]TJ /F7 7.97 Tf 6.59 0 Td[(2)]TJ /F6 11.955 Tf 9.97 0 Td[(cotD 2\(D)]TJ /F8 11.955 Tf 9.96 0 Td[(1) \(D 2)+1Xn=1"1 n\(n+D)]TJ /F8 11.955 Tf 9.97 0 Td[(1) \(n+D 2)y 4n)]TJ /F8 11.955 Tf 31.66 8.09 Td[(1 n)]TJ /F5 7.97 Tf 11.16 4.7 Td[(D 2+2\(n+D 2+1) \(n+2)y 4n)]TJ /F16 5.978 Tf 7.78 3.26 Td[(D 2+2#). (2) TheinniteseriestermsofA(y)vanishforD=4,sotheyonlyneedtoberetainedwhenmultiplyingapotentiallydivergentquantity,andeventhenoneonlyneedstoincludeahandfulofthem.Thismakesloopcomputationsmanageable.WenotethattheMMCscalarpropagator( 2 )hasadeSitterbreakingterm,kln(aa0).However,theoneloopscalarcontributiontothegravitonself-energyonlyinvolvesthetermslike@@0i4(x;x0),whicharedeSitterinvariant, @@0i(x;x0)=@ @x(A0(y)@y @x0+Ha00)=A00(y)@y @x@y @x0+A0(y)@2y @x@x0.(2)Anotherusefulrelationfollowsfromthepropagatorequation, (4y)]TJ /F3 11.955 Tf 9.97 0 Td[(y2)A00(y)+D(2)]TJ /F3 11.955 Tf 9.96 0 Td[(y)A0(y)=(D)]TJ /F8 11.955 Tf 9.96 0 Td[(1)k.(2) 28

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CHAPTER3ONELOOPGRAVITONSELF-ENERGYInthischapterwecalculatethersttwo,primitive,diagramsofFigure1.Itturnsoutthatthecontributionfromthe4-pointvertex(themiddlediagram)vanishesinD=4dimensions.Thecontributionfromtwo3-pointvertices(theleftmostdiagram)isnonzero.Fornoncoincidentpointsitgivesarelativelysimpleformwhichagreeswiththeatspacelimit[ 61 ]andwiththedeSitterstresstensorcorrelatorrecentlyderivedbyPerez-Nadal,RouraandVerdaguer[ 99 ]. 3.1Contributionfrom4-PointVerticesThe4-pointcontributionfromthemiddlediagramofFigure1takestheform, )]TJ /F3 11.955 Tf 11.95 0 Td[(ihi4pt(x;x0)1 24XI=1FI(x)@@0i4(x;x0D(x)]TJ /F3 11.955 Tf 9.96 0 Td[(x0).(3)Recallthatthefour4-pointverticesFI(x)aregiveninTable 2-2 .Owingtothedeltafunction,weneedthecoincidencelimitofthedoublydifferentiatedpropagator( 2 ).Thecoincidencelimitsofthevarioustensorfactorsfollowfromsettinga0=a,x=0andy=0inrelations( 2 )-( 2 ), limx0!x@y(x;x0) @x=0=limx0!x@y(x;x0) @x0, (3) limx0!x@2y(x;x0) @x@x0=)]TJ /F8 11.955 Tf 9.3 0 Td[(2H2 g. (3) HencethecoincidencelimitofthedoublydifferentiatedpropagatorcanbeexpressedintermsofA0(y)evaluatedaty=0, limx0!x@@0i4(x;x0)=A00(0)0+A0(0)h)]TJ /F8 11.955 Tf 9.29 0 Td[(2H2 gi.(3) 29

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Fromthedenition( 2 )ofA(y),weseethatA0(y)is, A0(y)=1 4HD)]TJ /F7 7.97 Tf 6.58 0 Td[(2 (4)D 2()]TJ /F8 11.955 Tf 9.3 0 Td[(\(D 2)4 yD 2)]TJ /F8 11.955 Tf 9.96 0 Td[(\(D 2+1)4 yD 2)]TJ /F7 7.97 Tf 6.59 0 Td[(1+Xn=1"\(n+D)]TJ /F8 11.955 Tf 9.96 0 Td[(1) \(n+D 2)y 4n)]TJ /F7 7.97 Tf 6.58 0 Td[(1)]TJ /F8 11.955 Tf 13.15 8.79 Td[(\(n+D 2)]TJ /F8 11.955 Tf 9.96 0 Td[(1) \(n+2)y 4n)]TJ /F16 5.978 Tf 7.79 3.26 Td[(D 2+1#). (3) Nowwerecallthat,indimensionalregularization,anyD-dependentpowerofzerovanishes.Therefore,onlythen=1termoftheinniteseriesin( 3 )contributestothecoincidencelimit, A0(0)=1 4HD)]TJ /F7 7.97 Tf 6.59 0 Td[(2 (4)D 2\(D) \(D 2+1),(3)andwehave, limx0!x@@0i4(x;x0)=)]TJ /F8 11.955 Tf 10.49 8.09 Td[(1 2HD (4)D 2\(D) \(D 2+1) g. (3) Substituting( 3 ),andthe4-pointverticesfromTable 2-2 ,intoexpression( 3 )gives, )]TJ /F3 11.955 Tf 9.3 0 Td[(ihi4pt(x;x0)=)]TJ /F8 11.955 Tf 10.5 8.09 Td[(1 2HD (4)D 2\(D) \(D 2+1) gi2p )]TJ ET q .478 w 234.07 -381.28 m 240.84 -381.28 l S Q BT /F3 11.955 Tf 234.07 -388.6 Td[(g)]TJ /F8 11.955 Tf 13.15 8.09 Td[(1 4 g g g+1 2 g( g) g+1 2h g( g) g+ g g( g)i)]TJ /F8 11.955 Tf 11.95 0 Td[(2 g( g)( g)D(x)]TJ /F3 11.955 Tf 9.96 0 Td[(x0), (3) =D)]TJ /F8 11.955 Tf 9.97 0 Td[(4 4i2HD (4)D 2\(D) \(D 2+1)p )]TJ ET q .478 w 212.64 -449.07 m 219.4 -449.07 l S Q BT /F3 11.955 Tf 212.64 -456.38 Td[(g1 2 g g)]TJ ET q .478 w 287.4 -449.07 m 294.16 -449.07 l S Q BT /F3 11.955 Tf 287.4 -456.38 Td[(g( g)D(x)]TJ /F3 11.955 Tf 9.96 0 Td[(x0). (3) BecausetheGammafunctionsareniteforD=4dimensionssowecandispensewithdimensionalregularizationandsetD=4.Atthatpointthenetcontribution( 3 )vanishes. 30

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3.2Contributionfrom3-PointVerticesThecontributionfromtheleftmostdiagramofFigure1takestheform, )]TJ /F3 11.955 Tf 9.3 0 Td[(ihi3pt(x;x0)=1 22XI=1TI(x)2XJ=1TJ(x0)@@0i4(x;x0)@@0i4(x;x0). (3) Recallfromchapter2(section2.2)thatanydeSitterinvariantbitensorcanbeexpressedasalinearcombinationoffunctionsofy(x;x0)timesthevebasistensors, )]TJ /F3 11.955 Tf 9.3 0 Td[(ihi3pt(x;x0)=p )]TJ ET q .478 w 169.97 -196.73 m 176.73 -196.73 l S Q BT /F3 11.955 Tf 169.97 -204.05 Td[(gp )]TJ ET q .478 w 197.98 -196.73 m 204.74 -196.73 l S Q BT /F3 11.955 Tf 197.98 -204.05 Td[(g0(@2y @x@x0(@2y @x0)@x(y)+@y @x(@2y @x)@x0(@y @x0)(y)+@y @x@y @x@y @x0@y @x0(y)+ g g0H4(y)+h g@y @x0@y @x0+@y @x@y @x g0iH2(y)). (3) Bysubstitutingourresult( 2 )forthemixedsecondderivativeofthescalarpropagator,alongwiththeverticesfromTable 2-1 ,andthenmakinguseofthecontractionidentities( 2 )-( 2 ),itisstraightforwardtoobtainexpressionsforthevecoefcientfunctions, (y)=)]TJ /F8 11.955 Tf 10.5 8.09 Td[(1 22(A0)2, (3) (y)=)]TJ /F6 11.955 Tf 9.3 0 Td[(2A0A00, (3) (y)=)]TJ /F8 11.955 Tf 10.5 8.08 Td[(1 22(A00)2, (3) (y)=)]TJ /F8 11.955 Tf 10.5 8.09 Td[(1 82(A00)2(4y)]TJ /F3 11.955 Tf 11.95 0 Td[(y2)2+2A0A00(2)]TJ /F3 11.955 Tf 11.96 0 Td[(y)(4y)]TJ /F3 11.955 Tf 11.96 0 Td[(y2)+(A0)2h4(D)]TJ /F8 11.955 Tf 9.96 0 Td[(4))]TJ /F8 11.955 Tf 9.96 0 Td[((4y)]TJ /F3 11.955 Tf 9.96 0 Td[(y2)i, (3) (y)=1 42h(4y)]TJ /F3 11.955 Tf 11.96 0 Td[(y2)(A00)2+2(2)]TJ /F3 11.955 Tf 9.96 0 Td[(y)A0A00)]TJ /F8 11.955 Tf 11.96 0 Td[((A0)2i. (3) Expressions( 3 )-( 3 )forthecoefcientfunctionshavetheadvantageofbeingexactforanydimensionD,butthedisadvantagesofbeingneitherveryexplicitnorverysimplefunctionsofy(x;x0).Wecanobtainexpressionswhicharebothsimpleand 31

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explicit,andtotallyadequateforuseintheD=4effectiveeldequations,bynotingthateachpairoftermsintheinniteseriespartofA(y)( 2 )vanishesforD=4spacetimedimensions.Therefore,itisonlyneceesarytoretainthosepartsoftheinniteserieswhichcanpotentiallymultiplyadivergence.Forourcomputationthatturnsouttomeanonlythen=1terms,andwecanwritethetwoderivativesas, A0=\(D 2)HD)]TJ /F7 7.97 Tf 6.58 0 Td[(2 4(4)D 2()]TJ /F12 11.955 Tf 9.3 13.27 Td[(4 yD 2)]TJ /F3 11.955 Tf 13.15 8.09 Td[(D 24 yD 2)]TJ /F7 7.97 Tf 6.59 0 Td[(1)]TJ /F8 11.955 Tf 13.15 8.09 Td[(1 2D 2D 2+14 yD 2)]TJ /F7 7.97 Tf 6.59 0 Td[(2+\(D) \(D 2)\(D 2+1)+Irrelevant), (3) A00=\(D 2)HD)]TJ /F7 7.97 Tf 6.58 0 Td[(2 16(4)D 2(D 24 yD 2+1+D 2)]TJ /F8 11.955 Tf 9.96 0 Td[(1D 24 yD 2+1 2D 2)]TJ /F8 11.955 Tf 9.96 0 Td[(2)D 2D 2+14 yD 2)]TJ /F7 7.97 Tf 6.59 0 Td[(1+Irrelevant). (3) Substitutingtheseexpansionsin( 3 )-( 3 )gives, =K 25()]TJ /F12 11.955 Tf 9.3 13.27 Td[(4 yD)]TJ /F3 11.955 Tf 9.96 0 Td[(D4 yD)]TJ /F7 7.97 Tf 6.59 0 Td[(1)]TJ /F3 11.955 Tf 11.15 8.09 Td[(D(D+1) 24 yD)]TJ /F7 7.97 Tf 6.59 0 Td[(2+2\(D) \(D 2)\(D 2+1)4 yD 2+Irrelevant), (3) =K 27(D4 yD+1+(D)]TJ /F8 11.955 Tf 9.97 0 Td[(1)D4 yD+1 2(D)]TJ /F8 11.955 Tf 9.96 0 Td[(2)D(D+1)4 yD)]TJ /F7 7.97 Tf 5.18 0 Td[(1)]TJ /F3 11.955 Tf 26.7 8.09 Td[(D\(D) \(D 2)\(D 2+1)4 yD 2+1+Irrelevant), (3) =K 211()]TJ /F3 11.955 Tf 9.3 0 Td[(D24 yD+2)]TJ /F8 11.955 Tf 9.96 0 Td[((D)]TJ /F8 11.955 Tf 9.96 0 Td[(2)D24 yD+1)]TJ /F8 11.955 Tf 10.5 8.09 Td[(1 2(D2)]TJ /F8 11.955 Tf 9.96 0 Td[(3D)]TJ /F8 11.955 Tf 9.96 0 Td[(2)D24 yD+Irrelevant), (3) =K 25()]TJ /F8 11.955 Tf 9.3 0 Td[((D2)]TJ /F3 11.955 Tf 9.96 0 Td[(D)]TJ /F8 11.955 Tf 9.96 0 Td[(4)4 yD)]TJ /F8 11.955 Tf 11.96 0 Td[((D3)]TJ /F8 11.955 Tf 9.96 0 Td[(5D2+4D)]TJ /F8 11.955 Tf 9.97 0 Td[(4)4 yD)]TJ /F7 7.97 Tf 6.59 0 Td[(1)]TJ /F8 11.955 Tf 13.15 8.08 Td[(1 2D4)]TJ /F8 11.955 Tf 9.97 0 Td[(8D3+19D2)]TJ /F8 11.955 Tf 9.96 0 Td[(28D+84 yD)]TJ /F7 7.97 Tf 6.59 0 Td[(2)]TJ /F8 11.955 Tf 31.12 8.08 Td[(8\(D) \(D 2)\(D 2+1)4 yD 2+(Irrelevant)), (3) 32

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=K 28((D)]TJ /F8 11.955 Tf 9.97 0 Td[(2)D4 yD+1+(D3)]TJ /F8 11.955 Tf 9.96 0 Td[(5D2+6D)]TJ /F8 11.955 Tf 9.97 0 Td[(4)4 yD+1 2DD3)]TJ /F8 11.955 Tf 9.96 0 Td[(7D2+12D)]TJ /F8 11.955 Tf 9.97 0 Td[(124 yD)]TJ /F7 7.97 Tf 6.59 0 Td[(1+D\(D) \(D 2)\(D 2+1)4 yD 2+1+(Irrelevant)). (3) wheretheconstantKis, K2H2D)]TJ /F7 7.97 Tf 6.58 0 Td[(4)]TJ /F7 7.97 Tf 6.77 4.33 Td[(2(D 2) (4)D.(3) 3.3CorrespondencewithFlatSpaceAnimportantandilluminatingcorrespondencelimitcomesfromtakingtheHubbleconstanttozero,withtheconformaltimegoingtominusinnitysoastokeepthephysicaltimetxed, =)]TJ /F8 11.955 Tf 12.05 8.09 Td[(1 He)]TJ /F5 7.97 Tf 6.58 0 Td[(Ht=)]TJ /F8 11.955 Tf 12.06 8.09 Td[(1 H+t+O(H).(3)Whenthisisdonethebackgroundgeometrydegeneratestoatspaceandweshouldrecoverwell-knownresults[ 43 ].WewillalsoseeinthenextchapterthattheatspacelimitprovidescrucialguidanceinhowtoreorganizethedeSitterresultforrenormalization.Althougheachindependentconformaltimedivergesunder( 3 ),theconformalcoordinateseparationjustgoestotheusualtemporalseparationofatspace, x0)166(!t)]TJ /F3 11.955 Tf 11.96 0 Td[(t0.(3)Allscalefactorsapproachunity,andthedeSitterlengthfunctiongoestoH2timestheinvariantintervalofatspace, y(x;x0))166(!H2x2.(3)Intheatspacelimittheleadingbehaviorsofthevariousbasistensorsare, @y @x)166(!2H2x,@y @x0)166(!)]TJ /F8 11.955 Tf 31.88 0 Td[(2H2x,@y2 @x@x0)166(!)]TJ /F8 11.955 Tf 31.88 0 Td[(2H2.(3) 33

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AndtheleadingbehaviorsforderivativesofthefunctionA(y)are, H2A0(y))167(!)]TJ /F8 11.955 Tf 47.28 8.09 Td[(1 4D 2\(D 2) (x2)D 2)]TJ /F8 11.955 Tf 30.67 8.09 Td[(1 4D 2\(D 2) xD, (3) H4A00(y))167(!1 4D 2\(D 2+1) (x2)D 2+11 4D 2\(D 2+1) xD+2. (3) The4-pointcontribution( 3 )tothegravitonself-energyvanishesintheatspacelimit,evenforD6=4.Wecantaketheatspacelimitofthe3-pointcontribution( 3 )intwosteps.First,substitutetheleadingbehaviors( 3 )fory(x;x0)and( 3 )forthebasistensors.Thenuseexpressions( 3 )-( 3 )onthederivativesofA(y).Theresultis, )]TJ /F3 11.955 Tf 9.29 0 Td[(ihiat(x;x0)=limH!02(4H4())]TJ /F8 11.955 Tf 22.45 8.09 Td[(1 2(A0)2+8H6x()(x))]TJ /F3 11.955 Tf 21.26 0 Td[(A0A00+16H8xxxx)]TJ /F8 11.955 Tf 22.44 8.09 Td[(1 2(A00)2+H4)]TJ /F8 11.955 Tf 22.45 8.09 Td[(1 8h16H4x4(A00)2+16H2x2A0A00+4(D)]TJ /F8 11.955 Tf 9.96 0 Td[(4)(A0)2i+4H6hxx+xxi1 4h4H2x2(A00)2+4A0A00i), (3) =2)]TJ /F7 7.97 Tf 6.77 4.34 Td[(2(D 2) 16D(()h)]TJ /F8 11.955 Tf 20.77 8.08 Td[(2 x2Di+x()(x)h4D x2D+2i+xxxxh)]TJ /F8 11.955 Tf 18.92 8.09 Td[(2D2 x2D+4i+h)]TJ /F8 11.955 Tf 10.49 8.09 Td[(1 2(D2)]TJ /F3 11.955 Tf 9.96 0 Td[(D)]TJ /F8 11.955 Tf 9.96 0 Td[(4) x2Di+hxx+xxihD(D)]TJ /F8 11.955 Tf 9.96 0 Td[(2) x2D+2i). (3) Ourresult( 3 )agreeswithequation(26)of[ 61 ]. 3.4CorrespondencewithStressTensorCorrelatorsAlthoughtheatspacelimit( 3 )willproveausefulguidewhenwerenormalizeinthenextsection,itdoesnotcheckthepurelydeSitterpartsof( 3 ).AtruedeSittercheckisprovidedbythestresstensorcorrelatorsrecentlyderivedbyPerez-Nadal,RouraandVerdaguer[ 99 ].Toexploittheirresultwerstelucidatetherelationbetween 34

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thegraviton2-point1PIfunctionandcorrelatorsofthestresstensor.Thenweconverttheirnotationtoours.TheHeisenbergequationforthemetriceldoperatorcoupledtoamatterstresstensorTis, R)]TJ /F8 11.955 Tf 13.15 8.09 Td[(1 2gR+1 2(D)]TJ /F8 11.955 Tf 9.97 0 Td[(2)(D)]TJ /F8 11.955 Tf 9.97 0 Td[(1)H2g=1 22T.(3)Perturbationtheoryisimplementedbyexpressingthefullmetricg= g+hasthesumofavacuumsolution gplustimesthegravitoneldh.Expandingthelefthandsideof( 3 )inpowersofthegravitoneldgives, R)]TJ /F8 11.955 Tf 13.15 8.09 Td[(1 2gR+1 2(D)]TJ /F8 11.955 Tf 9.96 0 Td[(2)(D)]TJ /F8 11.955 Tf 9.96 0 Td[(1)H2g=Dh)]TJ /F8 11.955 Tf 13.15 8.09 Td[(1 22T,(3)wherethenonlineartermscomprisethegravitonpseudo-stresstensorT.TheLichnerowiczoperatorofthelineartermis, DD( g)(D))]TJ /F8 11.955 Tf 13.15 8.09 Td[(1 2h gDD+ gDDi+1 2h g g)]TJ ET q .478 w 168.13 -338.07 m 174.9 -338.07 l S Q BT /F3 11.955 Tf 168.13 -345.39 Td[(g( g)iD2+(D)]TJ /F8 11.955 Tf 9.97 0 Td[(1)h1 2 g g)]TJ ET q .478 w 338.95 -338.07 m 345.72 -338.07 l S Q BT /F3 11.955 Tf 338.95 -345.39 Td[(g( g)iH2, (3) whereDisthecovariantderivativeoperatorinthebackgroundgeometry.Substitutingtheseexpansionsin( 3 )andrearranginggives, Dh=1 2T+T1 2T.(3)Wearecomputingthe1PIgraviton2-pointfunction,whichcanbeobtainedfromthefullgraviton2-pointfunctionbyeliminatingtheoneparticlereduciblepartsandamputatingtheexternallegpropagators.Attheonelooporderweareworking,theoneparticlereduciblepartdropsoutifonecomputesthecorrelatoroftheeldminusitsexpectationvalue, h(x)h(x))]TJ /F12 11.955 Tf 11.96 13.27 Td[(Dh(x)E, (3) T(x)T(x))]TJ /F12 11.955 Tf 11.95 13.27 Td[(DT(x)E. (3) 35

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Toamputate,recallthatthegravitonpropagatorobeys, p )]TJ ET q .478 w 62.55 -40.5 m 69.31 -40.5 l S Q BT /F3 11.955 Tf 62.55 -47.82 Td[(g(x)Dihi(x;x0)=()iD(x)]TJ /F3 11.955 Tf 9.96 0 Td[(x0)+GaugeTerms,(3)whereGaugeTermsreferstotheextrapiecesneededtocompletetheprojectionoperatorontowhatevergaugeconditionisemployed.(Forexample,theprojectionoperatorfordeDondergaugeisgiveninequation(120)of[ 65 ].)Thismeansthatexternallegpropagatorsareamputatedby)]TJ /F3 11.955 Tf 9.3 0 Td[(ip )]TJ ET q .478 w 260.74 -148.09 m 267.51 -148.09 l S Q BT /F3 11.955 Tf 260.74 -155.4 Td[(gtimestheLichnerowiczoperator.Hencethedesiredrelationbetweenthe2-pointgraviton1PIfunctionanda2-pointcorrelatorofthestresstensoris, )]TJ /F3 11.955 Tf 9.3 0 Td[(ihi(x;x0)=D)]TJ /F3 11.955 Tf 9.3 0 Td[(ip )]TJ ET q .478 w 129.02 -258.66 m 135.78 -258.66 l S Q BT /F3 11.955 Tf 129.02 -265.98 Td[(gDh(x))]TJ /F3 11.955 Tf 9.3 0 Td[(ip )]TJ ET q .478 w 254.25 -258.66 m 261.01 -258.66 l S Q BT /F3 11.955 Tf 254.25 -265.98 Td[(gDh(x0)E+O(4), (3) =)]TJ /F8 11.955 Tf 10.49 8.09 Td[(1 42p )]TJ ET q .478 w 117.97 -285.56 m 124.73 -285.56 l S Q BT /F3 11.955 Tf 117.97 -292.88 Td[(g(x)p )]TJ ET q .478 w 164.39 -285.56 m 171.15 -285.56 l S Q BT /F3 11.955 Tf 164.39 -292.88 Td[(g(x0)DT(x)T(x0)E+O(4). (3) Theexpectationvalueontherighthandsideof( 3 )isthestresstensorcorrelatorFofPerez-Nadal,RouraandVerdaguer[ 99 ].Perez-Nadal,RouraandVerdagueractuallyderivedFforascalarwitharbitrarymass,butwecancompareourresult( 3 )forthemasslesscasewiththeirequation(28)[ 99 ] F=P()nnnn+Q()(nn g+nn g)+R()(nn g+nn g+nn g+nn g)+S()( g g+ g g)+T() g g. (3) Notethatheretheyexpressedthestresstensorcorrelatorintermsofvebasistensorswhicharedifferentfromoursgiveninequation( 3 ).EachofthesevebitensorsareformedasalinearcombinationofproductsofthedeSitterinvariantbitensors,na,na0, gab, ga0b0and gab0.Thevariableandbitensorsaredenedas[ 99 ]: 36

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(x,x0):thedistancealongtheshortestgeodesicjoiningxandx0,alsocalledthegeodesicdistance; naandna0:theunitvectorstangenttothegeodesicatthepointsxandx0respectively,pointingoutwardfromit; gab0:theparallelpropagatorwhichparallel-transportsavectorfromxtox0alongthegeodesic; gaband ga0b0:themetrictensorsatthepointsxandx0respectively.Thedistance(x,x0)(inournotation(x,x0)=H`(x;x0)whichisgiveninsection2)correspondstoourdeSitterinvariantfunctiony(x,x0)withtherelation, cos()Z=1)]TJ /F3 11.955 Tf 13.15 8.09 Td[(y 2.(3)Incomparingtheirresultswithoursitisalsousefultonotetherelationsbetweentheirbasistensorsandours, na=1 Hp y(4)]TJ /F3 11.955 Tf 9.96 0 Td[(y)@y @xa, (3) nb0=1 Hp y(4)]TJ /F3 11.955 Tf 9.96 0 Td[(y)@y @x0b0, (3) gab0=)]TJ /F8 11.955 Tf 17.66 8.09 Td[(1 2H2@2y @xa@x0b0+1 4)]TJ /F3 11.955 Tf 9.96 0 Td[(y@y @xa@y @x0b0. (3) Thusthevebasistensorsgivenin( 3 )areconvertedintoourbasistensorsas, nanbnc0nd0=1 H4(4y)]TJ /F3 11.955 Tf 11.96 0 Td[(y2)2@y @xa@y @xb@y @x0c0@y @x0d0, (3) nanb gc0d0+nc0nd0 gab=1 H2(4y)]TJ /F3 11.955 Tf 11.96 0 Td[(y2) gab@y @x0c0@y @x0d0+@y @xa@y @xb gc0d0, (3) 4n(a gb)(c0nd0)=)]TJ /F8 11.955 Tf 39.16 8.09 Td[(2 H4(4y)]TJ /F3 11.955 Tf 11.96 0 Td[(y2)@y @x(a@2y @xb)@x0(c0@y @x0d0))]TJ /F8 11.955 Tf 57.92 8.08 Td[(2 H4(4y)]TJ /F3 11.955 Tf 11.96 0 Td[(y2)(4)]TJ /F3 11.955 Tf 11.95 0 Td[(y)@y @xa@y @xb@y @x0c0@y @x0d0, (3) 2 ga(c0 gd0)b=1 2H4@2y @xa@x0(c0@2y @xd0)@x0b+1 H4(4)]TJ /F3 11.955 Tf 11.95 0 Td[(y)@y @x(a@2y @xb)@x0(c0@y @x0d0)+1 2H41 (4)]TJ /F3 11.955 Tf 11.95 0 Td[(y)2@y @xa@y @xb@y @x0c0@y @x0d0, (3) gab gc0d0= gab gc0d0. (3) 37

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(NotethatwehaverestoredthefactorofHwhichPerez-Nadal,RouraandVeraguersettounity.)Foramassless,minimallycoupledscalareld,the-dependentcoefcientsare[ 99 ], P=2G21,Q=)]TJ /F3 11.955 Tf 9.3 0 Td[(G21+2G1G2,R=G1G2,S=G22,T=1 2G21)]TJ /F3 11.955 Tf 11.95 0 Td[(G1G2+D)]TJ /F8 11.955 Tf 11.95 0 Td[(4 2G22. (3) HeretheG1andG2aredenedas G1()=G00())]TJ /F3 11.955 Tf 11.96 0 Td[(G0()csc(),G2()=)]TJ /F3 11.955 Tf 9.3 0 Td[(G0()csc(), (3) whereprimestandsforderivativewithrespectto.ThecomparisoncanbecompletedbynotingthattheWightmanfunctionG()becomesalmostthesameasourA(y)forthecaseofMMCscalar.Inthemasslesslimit,theirpropagatorhastheformalexpansion, G()=HD)]TJ /F7 7.97 Tf 6.59 0 Td[(2 (4)D=21Xn=0\(D)]TJ /F8 11.955 Tf 11.95 0 Td[(1+n)\(n) \(D 2+n)1 n!1+Z 2n. (3) (NotethatwehaverestoredthefactorofHD)]TJ /F7 7.97 Tf 6.58 0 Td[(2whichPerez-Nadal,RouraandVeraguersettounity.)Recallingthehypergeometricfunction, 2F1,;;z=1Xn=0\(+n) \()\(+n) \()\() \(+n)zn n!, (3) weseethatG(Z)canbewrittenas, G(y)=HD)]TJ /F7 7.97 Tf 6.59 0 Td[(2 (4)D=2\(D)]TJ /F8 11.955 Tf 9.96 0 Td[(1)\(0) \(D 2)2F1D)]TJ /F8 11.955 Tf 9.97 0 Td[(1,0;D 2;1)]TJ /F3 11.955 Tf 11.16 8.08 Td[(y 4. (3) 38

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Nowweuseoneofthetransformationformulaeforhypergeometricfunctions(Seeforexample,9.131of[ 66 ])toexpandGinpowersofy=4: G(y)=HD)]TJ /F7 7.97 Tf 6.58 0 Td[(2 (4)D 2(\(D 2) D 2)]TJ /F8 11.955 Tf 9.97 0 Td[(14 yD 2)]TJ /F7 7.97 Tf 6.58 0 Td[(1+\(D 2+1) D 2)]TJ /F8 11.955 Tf 9.96 0 Td[(24 yD 2)]TJ /F7 7.97 Tf 6.58 0 Td[(2)]TJ /F8 11.955 Tf 9.96 0 Td[(\(0)\(D)]TJ /F8 11.955 Tf 9.97 0 Td[(1) \(D 2)+1Xn=1"1 n\(n+D)]TJ /F8 11.955 Tf 9.97 0 Td[(1) \(n+D 2)y 4n)]TJ /F8 11.955 Tf 31.66 8.09 Td[(1 n)]TJ /F5 7.97 Tf 11.16 4.7 Td[(D 2+2\(n+D 2+1) \(n+2)y 4n)]TJ /F16 5.978 Tf 7.78 3.26 Td[(D 2+2#). (3) SoweseethatG(y)isthesameasthefunctionA(y)exceptforthereplacement, \(0)\(D)]TJ /F8 11.955 Tf 9.96 0 Td[(1) \(D 2))166(!cotD 2\(D)]TJ /F8 11.955 Tf 9.96 0 Td[(1) \(D 2). (3) ThismakesnodifferencebecauseG(y)onlyentersthestresstensorcorrelator( 3 )differentiated(Seeequations( 3 )-( 3 )).Thusforcomparison,wereplacethederivativesofGbytheonesofA: @G @=p 4y)]TJ /F3 11.955 Tf 11.96 0 Td[(y2G0p 4y)]TJ /F3 11.955 Tf 11.96 0 Td[(y2A0,@2G @2=(4y)]TJ /F3 11.955 Tf 11.95 0 Td[(y2)G00+(2)]TJ /F3 11.955 Tf 11.96 0 Td[(y)G0(4y)]TJ /F3 11.955 Tf 11.96 0 Td[(y2)A00+(2)]TJ /F3 11.955 Tf 11.96 0 Td[(y)A0. (3) Heretheprimestandforderivativewithrespecttoy.ThenthecoefcientsP,Q,R,SandTgiveninequation( 3 )arewrittenintermsofyas P=2(4y)]TJ /F3 11.955 Tf 11.96 0 Td[(y2)2(A00)2)]TJ /F8 11.955 Tf 11.96 0 Td[(4y(4y)]TJ /F3 11.955 Tf 11.95 0 Td[(y2)A00A0+2y2(A0)2,Q=)]TJ /F8 11.955 Tf 9.3 0 Td[((4y)]TJ /F3 11.955 Tf 11.95 0 Td[(y2)2(A00)2)]TJ /F8 11.955 Tf 11.96 0 Td[(2(2)]TJ /F3 11.955 Tf 11.96 0 Td[(y)(4y)]TJ /F3 11.955 Tf 11.95 0 Td[(y2)A00A0+(4y)]TJ /F3 11.955 Tf 11.95 0 Td[(y2)(A0)2.R=)]TJ /F8 11.955 Tf 9.3 0 Td[(2(4y)]TJ /F3 11.955 Tf 11.96 0 Td[(y2)A00A0+2y(A0)2,S=4(A0)2,T=1 2h(4y)]TJ /F3 11.955 Tf 11.96 0 Td[(y2)2(A00)2+2(2)]TJ /F3 11.955 Tf 11.96 0 Td[(y)(4y)]TJ /F3 11.955 Tf 11.96 0 Td[(y2)A00A0+f4(D)]TJ /F8 11.955 Tf 11.95 0 Td[(4))]TJ /F8 11.955 Tf 11.95 0 Td[((4y)]TJ /F3 11.955 Tf 11.95 0 Td[(y2)g(A0)2i. (3) 39

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Withthisequation( 3 )andtheconversionofbasisgiveninequations( 3 )-( 3 )wecanarrangeFfortheMMCscalarintermsofourbasistensors, F=)]TJ /F8 11.955 Tf 13.19 8.09 Td[(4 2(@2y @x@x0(@2y @x0)@x(y)+@y @x(@2y @x)@x0(@y @x0)(y)+@y @x@y @x@y @x0@y @x0(y)+ g g0H4(y)+h g@y @x0@y @x0+@y @x@y @x g0iH2(y)). (3) =)]TJ /F8 11.955 Tf 13.19 8.09 Td[(4 21 p )]TJ ET q .478 w 140.48 -180.34 m 147.24 -180.34 l S Q BT /F3 11.955 Tf 140.48 -187.66 Td[(g(x)p )]TJ ET q .478 w 186.89 -180.34 m 193.66 -180.34 l S Q BT /F3 11.955 Tf 186.89 -187.66 Td[(g(x0))]TJ /F3 11.955 Tf 21.25 0 Td[(ihi3pt(x;x0). (3) 40

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CHAPTER4RENORMALIZATIONOurresult( 3 )isvalidaslongasx06=x,eitherwiththeexactcoefcientfunctions( 3 )-( 3 )orwiththerelevantexpansions( 3 )-( 3 )forD=4.However,itisnotimmediatelyusableinthequantum-corrected,linearizedEinsteinequationsbecausetheyinvolveanintegrationoverx0, p )]TJ ET q .478 w 80.43 -150.58 m 87.19 -150.58 l S Q BT /F3 11.955 Tf 80.43 -157.9 Td[(gDh(x))]TJ /F12 11.955 Tf 11.96 16.28 Td[(Zd4x0hiren(x;x0)h(x0)=1 2p )]TJ ET q .478 w 356.32 -150.58 m 363.09 -150.58 l S Q BT /F3 11.955 Tf 356.32 -157.9 Td[(gTlin(x).(4)Toobtainausableformwemustexpress( 3 )asaproductofuptosixdifferentialoperatorsactinguponafunctionofy(x;x0)whichisintegrableinD=4spacetimedimensions.Thederivativeswithrespecttoxcanbepulledoutsidetheintegral,andthosewithrespecttox0canbepartiallyintegratedtoactbackontheh(x0),1leavinganexpressionforwhichtheD=4limitcouldbetakenwereitnotforsomefactorsof1=(D)]TJ /F8 11.955 Tf 11.96 0 Td[(4).Atthisstageoneaddszerointheformofidentitiessuchas, )]TJ /F3 11.955 Tf 13.15 8.09 Td[(D 2D 2)]TJ /F8 11.955 Tf 9.96 0 Td[(1H2#4 yD 2)]TJ /F7 7.97 Tf 6.59 0 Td[(1)]TJ /F8 11.955 Tf 18.15 8.09 Td[((4)D 2iD(x)]TJ /F3 11.955 Tf 9.97 0 Td[(x0) \(D 2)]TJ /F8 11.955 Tf 9.97 0 Td[(1)HD)]TJ /F7 7.97 Tf 6.59 0 Td[(2p )]TJ ET q .478 w 343.73 -351.77 m 350.49 -351.77 l S Q BT /F3 11.955 Tf 343.73 -359.09 Td[(g=0.(4)Wecombine( 4 )withtermswhicharisefromextractingderivativestosegregatethedivergencesonlocal,deltafunctionterms,forexample, 1 D)]TJ /F8 11.955 Tf 9.96 0 Td[(4" )]TJ /F3 11.955 Tf 13.15 8.08 Td[(D 2D 2)]TJ /F8 11.955 Tf 9.96 0 Td[(1H2#4 yD)]TJ /F7 7.97 Tf 6.59 0 Td[(3=" )]TJ /F3 11.955 Tf 11.16 8.09 Td[(D 2D 2)]TJ /F8 11.955 Tf 9.96 0 Td[(1H2#((4 y)D)]TJ /F7 7.97 Tf 6.59 0 Td[(3)]TJ /F8 11.955 Tf 11.96 0 Td[((4 y)D 2)]TJ /F7 7.97 Tf 6.58 0 Td[(1 D)]TJ /F8 11.955 Tf 9.96 0 Td[(4)+(4)D 2iD(x)]TJ /F3 11.955 Tf 9.96 0 Td[(x0)=p )]TJ ET q .478 w 390.55 -471.25 m 397.32 -471.25 l S Q BT /F3 11.955 Tf 390.55 -478.57 Td[(g (D)]TJ /F8 11.955 Tf 9.96 0 Td[(4)\(D 2)]TJ /F8 11.955 Tf 9.96 0 Td[(1)HD)]TJ /F7 7.97 Tf 6.58 0 Td[(2, (4) =)]TJ /F8 11.955 Tf 10.49 8.09 Td[(1 2h )]TJ /F8 11.955 Tf 9.96 0 Td[(2H2i(4 ylny 4)+O(D)]TJ /F8 11.955 Tf 9.96 0 Td[(4)+(4)D 2iD(x)]TJ /F3 11.955 Tf 9.96 0 Td[(x0)=p )]TJ ET q .478 w 377.1 -511.1 m 383.87 -511.1 l S Q BT /F3 11.955 Tf 377.1 -518.42 Td[(g (D)]TJ /F8 11.955 Tf 9.96 0 Td[(4)\(D 2)]TJ /F8 11.955 Tf 9.96 0 Td[(1)HD)]TJ /F7 7.97 Tf 6.58 0 Td[(2. (4) Renormalizationconsistsofsubtractingoffthedivergentdeltafunctionswithcounterterms.Insection4.1weexhibittheoneloopcountertermsforquantumgravity.Wereviewhow 1Theresultingsurfacetermscanbeabsorbedbycorrectingtheinitialstate[ 67 ]. 41

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torenormalizetheatspacelimit( 3 )insection4.2.Thatsuggestsaconvenientwayoforganizingthetensoralgebraintotwotransverse,4thorderdifferentialoperators,onewithspinzeroandtheotherwithspintwo.Insection4.3weimplementthisfordeSitter.Thespinzeropartisrenormalizedinsection4.4,andthespintwopartinsection4.5. 4.1OneLoopCountertermsGravity+ScalarisnotrenormalizableinD=4dimensions[ 9 ].However,thetheoremofBogoliubov,Parasiuk,HeppandZimmerman(BPHZ)showsushowtoconstructlocalcountertermswhichabsorbtheultravioletdivergencesofanyquantumeldtheorytoanyxedorderintheloopexpansion[ 10 ].ForquantumgravityatonelooporderthenecessarycountertermscanbetakentobethesquaresoftheRicciscalarandtheWeyltensor[ 9 ].TheproblemofquantumgravityisthattheWeylcountertermwoulddestabilizetheuniverseifitwereregardedasafundamental,nonperturbativeinteraction[ 68 ].Weshallthereforeconsideritonlyperturbatively,inthesenseofeffectiveeldtheory,asaproxyfortheyetunknownultravioletcompletionofquantumgravity.ThequantumeffectsweseektostudyderivefrominfraredvirtualscalarswithwavelengthsontheorderoftheHubbleradius,andtheywillmanifestasnonlocalandultravioletnitecontributionstothegravitonself-energywhicharenotaffectedbyhownatureresolvestheultravioletproblemofquantumgravity.BecausethebackgroundRicciscalarisnonzeroitisusefultoreorganizeR2intoapartwhichisquadraticinthegravitoneld, R2=hR)]TJ /F3 11.955 Tf 11.96 0 Td[(D(D)]TJ /F8 11.955 Tf 9.96 0 Td[(1)H2i2+2D(D)]TJ /F8 11.955 Tf 9.96 0 Td[(1)H2R)]TJ /F3 11.955 Tf 11.95 0 Td[(D2(D)]TJ /F8 11.955 Tf 9.96 0 Td[(1)2H4.(4)Sowewillemployfourcounterterms, L1c1hR)]TJ /F3 11.955 Tf 11.95 0 Td[(D(D)]TJ /F8 11.955 Tf 9.97 0 Td[(1)H2i2p )]TJ /F3 11.955 Tf 9.3 0 Td[(g, (4) L2c2CCp )]TJ /F3 11.955 Tf 9.3 0 Td[(g, (4) 42

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L3c3H2hR)]TJ /F8 11.955 Tf 11.95 0 Td[((D)]TJ /F8 11.955 Tf 9.96 0 Td[(1)(D)]TJ /F8 11.955 Tf 9.96 0 Td[(2)H2ip )]TJ /F3 11.955 Tf 9.3 0 Td[(g, (4) L4c4H4p )]TJ /F3 11.955 Tf 9.3 0 Td[(g. (4) OfcoursethedivergencescanreallybeeliminatedwithjustL2andtheparticularlinearcombinationofL1,L3andL4whichisproportionaltojustR2p )]TJ /F3 11.955 Tf 9.3 0 Td[(g.Itmustthereforebethattwolinearcombinationsofthecoefcientsarenite, limD!4h)]TJ /F8 11.955 Tf 9.29 0 Td[(2D(D)]TJ /F8 11.955 Tf 9.96 0 Td[(1)c1+c3i=Finite, (4) limD!4hD2(D)]TJ /F8 11.955 Tf 9.96 0 Td[(1)2c1)]TJ /F8 11.955 Tf 11.95 0 Td[((D)]TJ /F8 11.955 Tf 9.97 0 Td[(1)(D)]TJ /F8 11.955 Tf 9.96 0 Td[(2)c3+c4i=Finite. (4) Andthedivergentpartsofc1andc2mustagreewiththevaluesobtainedlongagoby`tHooftandVeltman[ 9 ].Atthispointwedigresstodenetwo2ndorderdifferentialoperatorsofgreatimportancetooursubsequentanalysis.TheycomefromexpandingthescalarandWeylcurvaturesarounddeSitterbackground, R)]TJ /F3 11.955 Tf 11.96 0 Td[(D(D)]TJ /F8 11.955 Tf 9.96 0 Td[(1)H2Ph+O(2h2), (4) CPh+O(2h2). (4) From( 4 )wehave, P=DD)]TJ ET q .478 w 216.61 -445.74 m 223.37 -445.74 l S Q BT /F3 11.955 Tf 216.61 -453.06 Td[(ghD2+(D)]TJ /F8 11.955 Tf 9.96 0 Td[(1)H2i,(4)whereDisthecovariantderivativeoperatorindeSitterbackground.ThemoredifcultexpansionoftheWeyltensorgives, P=D+1 D)]TJ /F8 11.955 Tf 9.96 0 Td[(2h gD)]TJ ET q .478 w 210.61 -541.38 m 217.37 -541.38 l S Q BT /F3 11.955 Tf 210.61 -548.7 Td[(gD)]TJ ET q .478 w 257.7 -541.38 m 264.46 -541.38 l S Q BT /F3 11.955 Tf 257.7 -548.7 Td[(gD+ gDi+1 (D)]TJ /F8 11.955 Tf 9.97 0 Td[(1)(D)]TJ /F8 11.955 Tf 9.97 0 Td[(2)h g g)]TJ ET q .478 w 337.18 -570.49 m 343.94 -570.49 l S Q BT /F3 11.955 Tf 337.18 -577.8 Td[(g giD, (4) 43

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wherewedene, D1 2h()DD)]TJ /F6 11.955 Tf 9.96 0 Td[(()DD)]TJ /F6 11.955 Tf 9.97 0 Td[(()DD+()DDi, (4) D gD=1 2h(D)D)]TJ /F6 11.955 Tf 9.97 0 Td[(()D2)]TJ ET q .478 w 287.69 -68.61 m 294.46 -68.61 l S Q BT /F3 11.955 Tf 287.69 -75.93 Td[(gDD+(DD)i, (4) D g gD=D(D))]TJ ET q .478 w 229.13 -95.51 m 235.89 -95.51 l S Q BT /F3 11.955 Tf 229.13 -102.83 Td[(gD2. (4) Oneobtainsthecountertermverticesbyfunctionallydifferentiatingitimeseachcountertermactiontwice,andthensettingthegravitoneldtozero.Theyare, iS1 h(x)h(x0)h=0=2c12p )]TJ ET q .478 w 209.17 -191.14 m 215.94 -191.14 l S Q BT /F3 11.955 Tf 209.17 -198.46 Td[(gPPiD(x)]TJ /F3 11.955 Tf 9.96 0 Td[(x0), (4) iS2 h(x)h(x0)h=0=2c22p )]TJ ET q .478 w 209.17 -231.49 m 215.94 -231.49 l S Q BT /F3 11.955 Tf 209.17 -238.81 Td[(g g g g gPPiD(x)]TJ /F3 11.955 Tf 9.96 0 Td[(x0), (4) iS3 h(x)h(x0)h=0=)]TJ /F3 11.955 Tf 9.3 0 Td[(c32H2p )]TJ ET q .478 w 226.53 -271.84 m 233.29 -271.84 l S Q BT /F3 11.955 Tf 226.53 -279.16 Td[(gDiD(x)]TJ /F3 11.955 Tf 9.97 0 Td[(x0), (4) iS4 h(x)h(x0)h=0=c42H4p )]TJ ET q .478 w 217.23 -312.19 m 223.99 -312.19 l S Q BT /F3 11.955 Tf 217.23 -319.51 Td[(gh1 4 g g)]TJ /F8 11.955 Tf 11.16 8.09 Td[(1 2 g( g)iiD(x)]TJ /F3 11.955 Tf 9.97 0 Td[(x0). (4) RecallthattheLichnerowiczoperatorinexpression( 4 )wasdenedinexpression( 3 ).Alsonotetheatspacelimits, iS1 h(x)h(x0)h=0)166(!2c12iD(x)]TJ /F3 11.955 Tf 9.97 0 Td[(x0), (4) iS2 h(x)h(x0)h=0)166(!2c22D)]TJ /F8 11.955 Tf 9.96 0 Td[(3 D)]TJ /F8 11.955 Tf 9.96 0 Td[(2h())]TJ /F8 11.955 Tf 11.16 8.08 Td[( D)]TJ /F8 11.955 Tf 9.96 0 Td[(1iiD(x)]TJ /F3 11.955 Tf 9.96 0 Td[(x0), (4) iS3 h(x)h(x0)h=0)166(!0, (4) iS4 h(x)h(x0)h=0)166(!0, (4) wherewedene, @@)]TJ /F6 11.955 Tf 11.95 0 Td[(@2.(4) 44

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4.2RenormalizingtheFlatSpaceResultRenormalizingtheatspaceresult( 3 )providesanexcellentguideforthevastlymorecomplicatedreductionrequiredondeSitterbackground.Webeginbyextractinga4thorderdifferentialoperatorfromeachtermusingtheidentities, 1 x2D=@4 4(D)]TJ /F8 11.955 Tf 9.96 0 Td[(2)2(D)]TJ /F8 11.955 Tf 9.97 0 Td[(1)D1 x2D)]TJ /F7 7.97 Tf 6.59 0 Td[(4, (4) xx x2D+2=1 8(D)]TJ /F8 11.955 Tf 9.96 0 Td[(2)2(D)]TJ /F8 11.955 Tf 9.97 0 Td[(1)D(@@@2+@4 D)1 x2D)]TJ /F7 7.97 Tf 6.59 0 Td[(4, (4) xxxx x2D+4=1 16(D)]TJ /F8 11.955 Tf 9.96 0 Td[(2)(D)]TJ /F8 11.955 Tf 9.96 0 Td[(1)D(D+1)(@@@@+6 D)]TJ /F8 11.955 Tf 9.96 0 Td[(2(@@)@2+3 (D)]TJ /F8 11.955 Tf 9.96 0 Td[(2)D()@4)1 x2D)]TJ /F7 7.97 Tf 6.59 0 Td[(4. (4) Substitutingtheserelationsinto( 3 ),andthenorganizingthevariousderivativesintofactorsofthetransverseoperatorofexpression( 4 ),givesamanifestlytransverseform, )]TJ /F3 11.955 Tf 9.3 0 Td[(ihiat(x;x0)=2)]TJ /F7 7.97 Tf 6.78 4.33 Td[(2(D 2) 16D()]TJ /F8 11.955 Tf 15.75 8.09 Td[( 8(D)]TJ /F8 11.955 Tf 9.96 0 Td[(1)2)]TJ /F8 11.955 Tf 13.85 9.62 Td[([())]TJ /F7 7.97 Tf 17.9 4.7 Td[(1 D)]TJ /F7 7.97 Tf 6.59 0 Td[(1] 4(D)]TJ /F8 11.955 Tf 9.96 0 Td[(2)2(D)]TJ /F8 11.955 Tf 9.97 0 Td[(1)(D+1))1 x2D)]TJ /F7 7.97 Tf 6.59 0 Td[(4. (4) Letuspauseatthispointtonotethatwecouldhaveguessedmostoftheformofexpression( 4 ).Gaugeinvarianceimpliestransversality.WealsohavePoincareinvariance,symmetryundertheinterchanges$and$,andsymmetryunderinterchangeoftheprimedandunprimedcoordinatesandindices.Allthisimpliestheform, )]TJ /F3 11.955 Tf 11.95 0 Td[(ihiat(x;x0)=F1(x2)+h())]TJ /F8 11.955 Tf 11.16 8.09 Td[( D)]TJ /F8 11.955 Tf 9.96 0 Td[(1iF2(x2).(4) 45

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Takingthetraceofthisandourresult( 3 )againstgivesanequationforthespinzerostructurefunctionF1(x2), )]TJ /F3 11.955 Tf 21.25 0 Td[(ihiat=(D)]TJ /F8 11.955 Tf 9.96 0 Td[(1)2@4F1(x2)=2)]TJ /F7 7.97 Tf 6.78 4.34 Td[(2(D 2) 16D)]TJ /F8 11.955 Tf 22.45 8.09 Td[((D)]TJ /F8 11.955 Tf 9.96 0 Td[(2)2(D)]TJ /F8 11.955 Tf 9.96 0 Td[(1)D 2x2D.(4)Ofcoursethesolutionisjustwhatwefoundin( 4 )bydirectcomputation, F1(x2)=2)]TJ /F7 7.97 Tf 6.78 4.34 Td[(2(D 2) 16D)]TJ /F8 11.955 Tf 43.15 8.09 Td[(1 8(D)]TJ /F8 11.955 Tf 9.96 0 Td[(1)21 x2D)]TJ /F7 7.97 Tf 6.58 0 Td[(2.(4)DeterminingthespintwostructurefunctionF2(x2)isdonebyrstactingthederivativesonthespinzerostructurefunction, F1=()8F001+x()(x)32F0001+xxxx16F00001+h4(D2)]TJ /F8 11.955 Tf 9.97 0 Td[(3)F001+16(D+1)x2F0001+16x4F00001i+hxx+xxih)]TJ /F8 11.955 Tf 9.3 0 Td[(8(D+3)F0001)]TJ /F8 11.955 Tf 11.96 0 Td[(16x2F00001i. (4) Wesubtractthesefromeachtensorfactorin( 3 )andthenactthespintwooperator[())]TJ /F7 7.97 Tf 20.5 4.71 Td[(1 D)]TJ /F7 7.97 Tf 6.59 0 Td[(1]onF2(x2)toreadoffanequationforeachofthevetensorfactors, ())4(D)]TJ /F8 11.955 Tf 9.96 0 Td[(2)D(D+1) D)]TJ /F8 11.955 Tf 9.96 0 Td[(1F002+16(D+1)x2F0002+16x4F00002=2)]TJ /F7 7.97 Tf 6.77 4.34 Td[(2(D 2) 16D()]TJ /F3 11.955 Tf 18.95 8.08 Td[(D D)]TJ /F8 11.955 Tf 9.96 0 Td[(11 x2D), (4) x()(x)))]TJ /F8 11.955 Tf 25.77 8.09 Td[(16D(D+1) D)]TJ /F8 11.955 Tf 9.96 0 Td[(1F0002)]TJ /F8 11.955 Tf 11.96 0 Td[(32x2F00002=2)]TJ /F7 7.97 Tf 6.77 4.34 Td[(2(D 2) 16D(4D D)]TJ /F8 11.955 Tf 9.96 0 Td[(11 x2D), (4) xxxx)16D)]TJ /F8 11.955 Tf 9.96 0 Td[(2 D)]TJ /F8 11.955 Tf 9.96 0 Td[(1F00002=2)]TJ /F7 7.97 Tf 6.77 4.34 Td[(2(D 2) 16D()]TJ /F8 11.955 Tf 15.81 8.09 Td[(4D D)]TJ /F8 11.955 Tf 9.97 0 Td[(11 x2D), (4) 46

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))]TJ /F8 11.955 Tf 35.98 8.09 Td[(4 D)]TJ /F8 11.955 Tf 9.97 0 Td[(1h(D)]TJ /F8 11.955 Tf 9.97 0 Td[(2)(D+1)F002+4(D+1)x2F0002+4x4F00002i=2)]TJ /F7 7.97 Tf 6.77 4.33 Td[(2(D 2) 16D(1 D)]TJ /F8 11.955 Tf 9.96 0 Td[(11 x2D), (4) hxx+xxi)16 D)]TJ /F8 11.955 Tf 9.96 0 Td[(1h(D+1)F0002+x2F00002i=0. (4) Eachoftheseequationshasthesamesolution,whichofcourseagreeswith( 4 ), F2(x2)=2)]TJ /F7 7.97 Tf 6.77 4.33 Td[(2(D 2) 16D)]TJ /F8 11.955 Tf 79.62 8.09 Td[(1 4(D)]TJ /F8 11.955 Tf 9.96 0 Td[(2)2(D)]TJ /F8 11.955 Tf 9.96 0 Td[(1)(D+1)1 x2D)]TJ /F7 7.97 Tf 6.58 0 Td[(2.(4)Wenoteforfuturereferencethataparticularlinearcombinationoftheverelations( 4 )-( 4 )givesasecondorderequationforF2(x2), (4)]TJ /F8 11.955 Tf 9.96 0 Td[(39)+x2(4)]TJ /F8 11.955 Tf 9.96 0 Td[(40)=)]TJ /F8 11.955 Tf 20.71 8.08 Td[(4 D)]TJ /F8 11.955 Tf 9.97 0 Td[(1(D)]TJ /F8 11.955 Tf 9.96 0 Td[(2)(D+1)F002=2)]TJ /F7 7.97 Tf 6.78 4.34 Td[(2(D 2) 16D(1 D)]TJ /F8 11.955 Tf 9.96 0 Td[(11 x2D).(4)Evenafterextractingthe4thorderdifferentialoperatorsfromtheintegrationof( 4 ),thefactorof1=x2D)]TJ /F7 7.97 Tf 6.59 0 Td[(4islogarithmicallydivergent.Wemustthereforeextractonemored'Alembertian, 1 x2D)]TJ /F7 7.97 Tf 6.59 0 Td[(2=@2 2(D)]TJ /F8 11.955 Tf 9.97 0 Td[(3)(D)]TJ /F8 11.955 Tf 9.97 0 Td[(4)1 x2D)]TJ /F7 7.97 Tf 6.59 0 Td[(3.(4)Afterthisnalderivativeisextractedtheintegrandconverges,however,westillcannottaketheD=4limitowingtothefactorof1=(D)]TJ /F8 11.955 Tf 12.34 0 Td[(4).Thesolutionistoaddzerointheformoftheidentity, @21 x2D 2)]TJ /F7 7.97 Tf 6.59 0 Td[(1)]TJ /F8 11.955 Tf 13.15 8.09 Td[(4D 2iD(x)]TJ /F3 11.955 Tf 9.97 0 Td[(x0) \(D 2)]TJ /F8 11.955 Tf 9.96 0 Td[(1)=0.(4)Tomakethisdimensionallyconsistentwith( 4 )wemustmultiplybythedimensionalregualrizationmassscaleraisedtothe(D)]TJ /F8 11.955 Tf 11.95 0 Td[(4)power, 1 x2D)]TJ /F7 7.97 Tf 6.59 0 Td[(2=@2 2(D)]TJ /F8 11.955 Tf 9.96 0 Td[(3)(D)]TJ /F8 11.955 Tf 9.96 0 Td[(4)(1 x2D)]TJ /F7 7.97 Tf 6.59 0 Td[(6)]TJ /F6 11.955 Tf 17.14 8.09 Td[(D)]TJ /F7 7.97 Tf 6.58 0 Td[(4 xD)]TJ /F7 7.97 Tf 6.59 0 Td[(2)+4D 2D)]TJ /F7 7.97 Tf 6.59 0 Td[(4iD(x)]TJ /F3 11.955 Tf 9.96 0 Td[(x0) 2(D)]TJ /F8 11.955 Tf 9.96 0 Td[(3)(D)]TJ /F8 11.955 Tf 9.96 0 Td[(4)\(D 2)]TJ /F8 11.955 Tf 9.97 0 Td[(1),=)]TJ /F8 11.955 Tf 10.49 8.09 Td[(1 4@2(ln(2x2) x2+O(D)]TJ /F8 11.955 Tf 9.96 0 Td[(4))+4D 2D)]TJ /F7 7.97 Tf 6.59 0 Td[(4iD(x)]TJ /F3 11.955 Tf 9.96 0 Td[(x0) 2(D)]TJ /F8 11.955 Tf 9.96 0 Td[(3)(D)]TJ /F8 11.955 Tf 9.96 0 Td[(4)\(D 2)]TJ /F8 11.955 Tf 9.96 0 Td[(1). (4) 47

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Thedivergenceshavenowbeensegregatedondeltafunctiontermswhichcanberemovedwithlocalcounterterms.Fromexpressions( 4 )-( 4 )weseethatthecountertermsmakethefollowingcontributiontothegravitonself-energy, )]TJ /F3 11.955 Tf 9.29 0 Td[(ihiat(x;x0)=(2c12iD(x)]TJ /F3 11.955 Tf 9.96 0 Td[(x0))+h())]TJ /F8 11.955 Tf 13.15 8.08 Td[( D)]TJ /F8 11.955 Tf 9.96 0 Td[(1i(2D)]TJ /F8 11.955 Tf 9.96 0 Td[(3 D)]TJ /F8 11.955 Tf 9.96 0 Td[(2c22iD(x)]TJ /F3 11.955 Tf 9.97 0 Td[(x0)). (4) Thedeltafunctiontermswillbeentirelyabsorbedbychoosingtheconstantsc1andc2as, c1=D)]TJ /F7 7.97 Tf 6.59 0 Td[(4\(D 2) 28D 2(D)]TJ /F8 11.955 Tf 9.97 0 Td[(2) (D)]TJ /F8 11.955 Tf 9.96 0 Td[(1)2(D)]TJ /F8 11.955 Tf 9.96 0 Td[(3)(D)]TJ /F8 11.955 Tf 9.96 0 Td[(4), (4) c2=D)]TJ /F7 7.97 Tf 6.59 0 Td[(4\(D 2) 28D 22 (D+1)(D)]TJ /F8 11.955 Tf 9.96 0 Td[(1)(D)]TJ /F8 11.955 Tf 9.96 0 Td[(3)2(D)]TJ /F8 11.955 Tf 9.96 0 Td[(4). (4) Ofcoursethedivergentpartsagreewiththeresultsobtainedlongagoby`tHooftandVeltman[ 9 ],withthearbitrarynitepartsrepresentedby.Thefullyrenormalizedgravitonself-energy(foratspacebackground)is, )]TJ /F3 11.955 Tf 9.3 0 Td[(ihirenat=limD!4()]TJ /F3 11.955 Tf 9.3 0 Td[(ihiat(x;x0))]TJ /F3 11.955 Tf 11.96 0 Td[(ihiat(x;x0)), (4) =@2(2 29324ln(2x2) x2)+h())]TJ /F8 11.955 Tf 13.15 8.09 Td[(1 3i@2(2 21031514ln(2x2) x2). (4) 4.3ThedeSitterStructureFunctionsWemustnowextendtheatspaceansatz( 4 )todeSitteranddeterminetheresultingstructurefunctionsbycomparisonwiththeexplicitresult( 3 )ofsection3.Asbefore,gaugeinvarianceimpliestransversality,whichsuggeststhatwemakeuseofthedifferentialoperatorsPandPwhichweredenedinexpressions( 4 )and( 4 ),respectively.InplaceofPoincareinvariancewenowhavedeSitter 48

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invariance.Wealsohavesymmetryundertheinterchanges$and$,andunderinterchangeoftheprimedandunprimedcoordinatesandindices.Asimplegeneralizationis, )]TJ /F3 11.955 Tf 9.3 0 Td[(ihi(x;x0)=p )]TJ ET q .478 w 155.64 -88.31 m 162.4 -88.31 l S Q BT /F3 11.955 Tf 155.64 -95.63 Td[(g(x)P(x)p )]TJ ET q .478 w 238.04 -88.31 m 244.8 -88.31 l S Q BT /F3 11.955 Tf 238.04 -95.63 Td[(g(x0)P(x0)nF1(y)o+p )]TJ ET q .478 w 69.64 -120.99 m 76.41 -120.99 l S Q BT /F3 11.955 Tf 69.64 -128.31 Td[(g(x)P(x)p )]TJ ET q .478 w 160.76 -120.99 m 167.52 -120.99 l S Q BT /F3 11.955 Tf 160.76 -128.31 Td[(g(x0)P(x0)(TTTTD)]TJ /F8 11.955 Tf 9.96 0 Td[(2 D)]TJ /F8 11.955 Tf 9.96 0 Td[(3F2(y)), (4) wherethebitensorTis,2 T(x;x0))]TJ /F8 11.955 Tf 30.28 8.08 Td[(1 2H2@2y(x;x0) @x@x0.(4)Asinatspace,thesecondtermistraceless.Notetheatspacelimitsofthebitensorandthetwostructurefunctions, limH!0T=,limH!0F1(y)=F1(x2),limH!0F2(y)=F2(x2).(4)Theselimitsmeanonecanimmediatelyreadoffthemostsingularpartsoftheexpansionsforeachstructurefunctionfromthecorrespondingatspaceresult, F1(y)=2H2D)]TJ /F7 7.97 Tf 6.59 0 Td[(4)]TJ /F7 7.97 Tf 6.77 4.34 Td[(2(D 2) (4)D()]TJ /F8 11.955 Tf 9.3 0 Td[(1 8(D)]TJ /F8 11.955 Tf 9.96 0 Td[(1)24 yD)]TJ /F7 7.97 Tf 6.58 0 Td[(2+...), (4) F2(y)=2H2D)]TJ /F7 7.97 Tf 6.59 0 Td[(4)]TJ /F7 7.97 Tf 6.77 4.34 Td[(2(D 2) (4)D()]TJ /F8 11.955 Tf 9.29 0 Td[(1 4(D)]TJ /F8 11.955 Tf 9.96 0 Td[(3)(D)]TJ /F8 11.955 Tf 9.96 0 Td[(2)(D)]TJ /F8 11.955 Tf 9.96 0 Td[(1)(D+1)4 yD)]TJ /F7 7.97 Tf 6.58 0 Td[(2+...). (4) TheinterestingdeSitterphysicsweseektoelucidatederivesfromthesubdominantterms.Justasfortheatspacelimit,wecanobtainanequationforthespinzerostructurefunctionbytracing( 4 )andthencomparingwiththetraceoftheexplicitcomputation 2Onecouldactuallyemployanybitensorforexample,theparalleltransportmatrix( 3 )whichreducestointheatspacelimit.DifferentchoicesforT(x;x0)makecorrespondingchangesinthesubdominantpartsofthespintwostructurefunctionF2(y).Wehavenottroubledtodeterminethesimplestchoice. 49

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( 3 ).Tracingtheansatzgives, g(x) p )]TJ ET q .478 w 30.32 -50.53 m 37.08 -50.53 l S Q BT /F3 11.955 Tf 30.32 -57.85 Td[(g(x) g(x0) p )]TJ ET q .478 w 91.75 -50.53 m 98.51 -50.53 l S Q BT /F3 11.955 Tf 91.75 -57.85 Td[(g(x0))]TJ /F3 11.955 Tf 21.25 0 Td[(ihi(x;x0)=(D)]TJ /F8 11.955 Tf 9.96 0 Td[(1)2h +DH2ih 0+DH2iF1(y).(4)Tracingtheexplicitresult( 3 ),substituting( 3 )-( 3 ),andthenmakinguseof( 2 )gives, g(x) p )]TJ ET q .478 w 57.3 -146.16 m 64.06 -146.16 l S Q BT /F3 11.955 Tf 57.3 -153.48 Td[(g(x) g(x0) p )]TJ ET q .478 w 118.72 -146.16 m 125.48 -146.16 l S Q BT /F3 11.955 Tf 118.72 -153.48 Td[(g(x0))]TJ /F3 11.955 Tf 21.25 0 Td[(ihi3pt(x;x0)=H4(h4D)]TJ /F8 11.955 Tf 9.97 0 Td[((4y)]TJ /F3 11.955 Tf 9.97 0 Td[(y2)i+(2)]TJ /F3 11.955 Tf 9.96 0 Td[(y)(4y)]TJ /F3 11.955 Tf 9.96 0 Td[(y2)+(4y)]TJ /F3 11.955 Tf 9.96 0 Td[(y2)2+D2+2D(4y)]TJ /F3 11.955 Tf 9.96 0 Td[(y2)), (4) =1 8(D)]TJ /F8 11.955 Tf 9.96 0 Td[(2)22H4(h(4y)]TJ /F3 11.955 Tf 9.96 0 Td[(y2))]TJ /F8 11.955 Tf 11.96 0 Td[(4Di(A0)2)]TJ /F8 11.955 Tf 9.3 0 Td[(2(2)]TJ /F3 11.955 Tf 9.97 0 Td[(y)(4y)]TJ /F3 11.955 Tf 9.96 0 Td[(y2)A0A00)]TJ /F8 11.955 Tf 11.96 0 Td[((4y)]TJ /F3 11.955 Tf 9.96 0 Td[(y2)2(A00)2), (4) =)]TJ /F8 11.955 Tf 10.49 8.09 Td[(1 8(D)]TJ /F8 11.955 Tf 9.96 0 Td[(1)2(D)]TJ /F8 11.955 Tf 9.96 0 Td[(2)22H4(4 D)]TJ /F8 11.955 Tf 9.96 0 Td[(1(A0)2+h(2)]TJ /F3 11.955 Tf 9.97 0 Td[(y)A0)]TJ /F3 11.955 Tf 11.96 0 Td[(ki2). (4) Nownotethattheprimedandunprimedscalard'Alembertian'sagreewhenactingonanyfunctionofonlyy(x;x0).Equating( 4 )and( 4 )andexpandingimplies, h H2+Di2F1(y)=)]TJ /F8 11.955 Tf 10.49 8.09 Td[(1 8(D)]TJ /F8 11.955 Tf 9.97 0 Td[(2)22(4 D)]TJ /F8 11.955 Tf 9.96 0 Td[(1(A0)2+h(2)]TJ /F3 11.955 Tf 9.97 0 Td[(y)A0)]TJ /F3 11.955 Tf 11.96 0 Td[(ki2). (4) =)]TJ /F3 11.955 Tf 11.86 8.09 Td[(K 32(D)]TJ /F8 11.955 Tf 9.96 0 Td[(2)2 (D)]TJ /F8 11.955 Tf 9.96 0 Td[(1)(D4 yD+(D)]TJ /F8 11.955 Tf 9.96 0 Td[(2)24 yD)]TJ /F7 7.97 Tf 6.59 0 Td[(1+1 2(D3)]TJ /F8 11.955 Tf 9.96 0 Td[(7D2+16D)]TJ /F8 11.955 Tf 9.96 0 Td[(8)4 yD)]TJ /F7 7.97 Tf 6.59 0 Td[(2+Irrelevant), (4) wheretheconstantKwasdenedin( 3 )andIrrelevantmeanstermswhicharebothintegrableatcoincidence,andwhichvanishinD=4dimensions.LetusrstnotethatwecanndaGreen'sfunctionforthedifferentialoperator[ =H2+D].Toseethis,acttheoperatoronsomefunctionf(y)whichisfreeoftheuniquepoweryD 2)]TJ /F7 7.97 Tf 6.58 0 Td[(1whichproducesadeltafunction, h H2+Dif(y)=(4y)]TJ /F3 11.955 Tf 9.97 0 Td[(y2)f00+D(2)]TJ /F3 11.955 Tf 9.97 0 Td[(y)f0+Df.(4) 50

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Nownotethatf1(y)=(2)]TJ /F3 11.955 Tf 12.02 0 Td[(y)isahomogeneoussolution,whichmeanswecanfactortoobtainarstorderequation(andhencesolvable)forthesecondsolution, f1(y)=(2)]TJ /F3 11.955 Tf 9.97 0 Td[(y)=)f2(y)f1(y)g(y)withg0(y)=1 (4y)]TJ /F3 11.955 Tf 9.96 0 Td[(y2)D 2f21(y).(4)Withthetwo,linearlyindependentsolutionsonecanconstructaGreen'sfunction, G1(y;y0)=(y)]TJ /F3 11.955 Tf 9.96 0 Td[(y0)hf2(y)f1(y0))]TJ /F3 11.955 Tf 9.96 0 Td[(f1(y)f2(y0)i(4y0)]TJ /F3 11.955 Tf 9.96 0 Td[(y02)D 2)]TJ /F7 7.97 Tf 6.59 0 Td[(1.(4)Hencewecansolve( 4 )toobtainonintegralepxressionforthespinzerostructurefunction, F1(y)="1 H2+D#2(Righthandsideof(4)]TJ /F8 11.955 Tf 9.97 0 Td[(61))(4)AlthoughwewilleventuallymakeuseoftheGreen'sfunction( 4 ),itisbesttodelaythisuntilthepointatwhichonecansetD=4.Forthemoresingulartermsthebeststrategyistoexploitthefactthatthesourcetermsontherighthandsideof( 4 )uponwhichwewishtoacttheinverseof[ =H2+D]2arejustpowersofy.Consideractingtheoperatoruponapowerp)]TJ /F8 11.955 Tf 12.27 0 Td[(26=D 2)]TJ /F8 11.955 Tf 12.27 0 Td[(1orD 2)]TJ /F8 11.955 Tf 12.27 0 Td[(2(thosepowersproducedeltafunctions), h H2+Di24 yp)]TJ /F7 7.97 Tf 6.59 0 Td[(2=(p)]TJ /F8 11.955 Tf 9.96 0 Td[(2)(p)]TJ /F8 11.955 Tf 9.97 0 Td[(1)(p)]TJ /F8 11.955 Tf 9.96 0 Td[(1)]TJ /F3 11.955 Tf 11.16 8.08 Td[(D 2)(p)]TJ /F3 11.955 Tf 11.16 8.08 Td[(D 2)4 yp+(p)]TJ /F8 11.955 Tf 9.96 0 Td[(2)(p)]TJ /F8 11.955 Tf 9.96 0 Td[(1)]TJ /F3 11.955 Tf 11.15 8.08 Td[(D 2)hD(2p)]TJ /F8 11.955 Tf 9.96 0 Td[(1))]TJ /F8 11.955 Tf 9.96 0 Td[(2(p)]TJ /F8 11.955 Tf 9.96 0 Td[(1)2i4 yp)]TJ /F7 7.97 Tf 6.58 0 Td[(1+(p)]TJ /F8 11.955 Tf 9.96 0 Td[(1)2(D)]TJ /F3 11.955 Tf 9.97 0 Td[(p+2)24 yp)]TJ /F7 7.97 Tf 6.58 0 Td[(2. (4) Wecanthereforedeveloparecursiveprocedureforreducingthepowerofthesource, "1 H2+D#24 yp=1 (p)]TJ /F8 11.955 Tf 9.96 0 Td[(2)(p)]TJ /F8 11.955 Tf 9.97 0 Td[(1)(p)]TJ /F8 11.955 Tf 9.96 0 Td[(1)]TJ /F5 7.97 Tf 11.16 4.71 Td[(D 2)(p)]TJ /F5 7.97 Tf 11.16 4.71 Td[(D 2)4 yp)]TJ /F7 7.97 Tf 6.58 0 Td[(2)]TJ /F12 11.955 Tf 11.96 20.44 Td[("1 H2+D#2([D(2p)]TJ /F8 11.955 Tf 9.96 0 Td[(1))]TJ /F8 11.955 Tf 9.97 0 Td[(2(p)]TJ /F8 11.955 Tf 9.96 0 Td[(1)2] (p)]TJ /F8 11.955 Tf 9.97 0 Td[(1)(p)]TJ /F5 7.97 Tf 11.15 4.71 Td[(D 2)4 yp)]TJ /F7 7.97 Tf 6.59 0 Td[(1+(p)]TJ /F8 11.955 Tf 9.96 0 Td[(1)(D+2)]TJ /F3 11.955 Tf 9.96 0 Td[(p)2 (p)]TJ /F8 11.955 Tf 9.96 0 Td[(2)(p)]TJ /F8 11.955 Tf 9.97 0 Td[(1)]TJ /F5 7.97 Tf 11.16 4.71 Td[(D 2)(p)]TJ /F5 7.97 Tf 11.16 4.71 Td[(D 2)4 yp)]TJ /F7 7.97 Tf 6.59 0 Td[(2). (4) Thestrategyistoapplythisuntilthesourceisintegrable,atwhichpointthedimensioncanbesettoD=4(unlesstherearefactorsof1=(D)]TJ /F8 11.955 Tf 12.72 0 Td[(4))andtheD=4Green'sfunctioncanbeusedtoobtainthefullsolutionforF1(y). 51

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Itisusefultoexaminethesortsoftermsgeneratedwhenthisrecursiveprocedureisappliedtothesourcetermsontherighthandsideof( 4 ).Themostsingulartermintroducesnofactorsof1=(D)]TJ /F8 11.955 Tf 12.37 0 Td[(4),nordoesitproduceremaindertermsdifferentfromthoseintheoriginalsourceterm( 4 ), "1 H2+D#24 yD=4 (D)]TJ /F8 11.955 Tf 9.96 0 Td[(2)D(D)]TJ /F8 11.955 Tf 9.96 0 Td[(2)(D)]TJ /F8 11.955 Tf 9.96 0 Td[(1)4 yD)]TJ /F7 7.97 Tf 6.58 0 Td[(2)]TJ /F12 11.955 Tf 9.3 20.45 Td[("1 H2+D#2(2(3D)]TJ /F8 11.955 Tf 9.97 0 Td[(2) D(D)]TJ /F8 11.955 Tf 9.96 0 Td[(1)4 yD)]TJ /F7 7.97 Tf 6.59 0 Td[(1+16(D)]TJ /F8 11.955 Tf 9.96 0 Td[(1) (D)]TJ /F8 11.955 Tf 9.96 0 Td[(2)D(D)]TJ /F8 11.955 Tf 9.96 0 Td[(2)4 yD)]TJ /F7 7.97 Tf 6.59 0 Td[(2). (4) Neitherstatementistruefortheremainingtwosourceterms, "1 H2+D#24 yD)]TJ /F7 7.97 Tf 6.59 0 Td[(1=4 (D)]TJ /F8 11.955 Tf 9.96 0 Td[(4)(D)]TJ /F8 11.955 Tf 9.96 0 Td[(2)(D)]TJ /F8 11.955 Tf 9.96 0 Td[(3)(D)]TJ /F8 11.955 Tf 9.97 0 Td[(2)4 yD)]TJ /F7 7.97 Tf 6.59 0 Td[(3)]TJ /F12 11.955 Tf 9.3 20.44 Td[("1 H2+D#2(2(5D)]TJ /F8 11.955 Tf 9.96 0 Td[(8) (D)]TJ /F8 11.955 Tf 9.96 0 Td[(2)(D)]TJ /F8 11.955 Tf 9.96 0 Td[(2)4 yD)]TJ /F7 7.97 Tf 6.59 0 Td[(2+36(D)]TJ /F8 11.955 Tf 9.96 0 Td[(2) (D)]TJ /F8 11.955 Tf 9.96 0 Td[(4)(D)]TJ /F8 11.955 Tf 9.96 0 Td[(2)(D)]TJ /F8 11.955 Tf 9.96 0 Td[(3)4 yD)]TJ /F7 7.97 Tf 6.58 0 Td[(3), (4) "1 H2+D#24 yD)]TJ /F7 7.97 Tf 6.59 0 Td[(2=4 (D)]TJ /F8 11.955 Tf 9.96 0 Td[(6)(D)]TJ /F8 11.955 Tf 9.96 0 Td[(4)(D)]TJ /F8 11.955 Tf 9.96 0 Td[(4)(D)]TJ /F8 11.955 Tf 9.97 0 Td[(3)4 yD)]TJ /F7 7.97 Tf 6.59 0 Td[(4)]TJ /F12 11.955 Tf 9.3 20.45 Td[("1 H2+D#2(2(7D)]TJ /F8 11.955 Tf 9.96 0 Td[(18) (D)]TJ /F8 11.955 Tf 9.96 0 Td[(4)(D)]TJ /F8 11.955 Tf 9.96 0 Td[(3)4 yD)]TJ /F7 7.97 Tf 6.59 0 Td[(3+64(D)]TJ /F8 11.955 Tf 9.96 0 Td[(3) (D)]TJ /F8 11.955 Tf 9.96 0 Td[(6)(D)]TJ /F8 11.955 Tf 9.96 0 Td[(4)(D)]TJ /F8 11.955 Tf 9.96 0 Td[(4)4 yD)]TJ /F7 7.97 Tf 6.58 0 Td[(4). (4) Theserelationsallowthethespinzerostructurefunctiontobeexpressedasaquotientandaremainderoftheform, F1(y)=Q1(y)+h1 H2+Di2R1(y), (4) Q1(y)=)]TJ /F3 11.955 Tf 9.3 0 Td[(K(f1a4 yD)]TJ /F7 7.97 Tf 6.58 0 Td[(2+f1b D)]TJ /F8 11.955 Tf 9.96 0 Td[(44 yD)]TJ /F7 7.97 Tf 6.58 0 Td[(3+f1c (D)]TJ /F8 11.955 Tf 9.96 0 Td[(4)24 yD)]TJ /F7 7.97 Tf 6.59 0 Td[(4), (4) R1(y)=)]TJ /F3 11.955 Tf 9.3 0 Td[(K(f1d D)]TJ /F8 11.955 Tf 9.96 0 Td[(44 yD)]TJ /F7 7.97 Tf 6.59 0 Td[(3+f1e (D)]TJ /F8 11.955 Tf 9.96 0 Td[(4)24 yD)]TJ /F7 7.97 Tf 6.59 0 Td[(4+Irrelevant), (4) wherethecoefcientsare, f1a=1 8(D)]TJ /F8 11.955 Tf 9.96 0 Td[(1)2, (4) 52

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f1b=D(D2)]TJ /F8 11.955 Tf 9.96 0 Td[(5D+2) 8(D)]TJ /F8 11.955 Tf 9.96 0 Td[(3)(D)]TJ /F8 11.955 Tf 9.96 0 Td[(1)2, (4) f1c=D2(D4)]TJ /F8 11.955 Tf 9.96 0 Td[(12D3+39D2)]TJ /F8 11.955 Tf 9.96 0 Td[(16D)]TJ /F8 11.955 Tf 9.96 0 Td[(36) 16(D)]TJ /F8 11.955 Tf 9.96 0 Td[(6)(D)]TJ /F8 11.955 Tf 9.96 0 Td[(3)(D)]TJ /F8 11.955 Tf 9.96 0 Td[(1)2, (4) f1d=)]TJ /F8 11.955 Tf 10.49 8.08 Td[(8 3+79 9(D)]TJ /F8 11.955 Tf 9.96 0 Td[(4)+O(D)]TJ /F8 11.955 Tf 9.96 0 Td[(4)2, (4) f1e=32 3)]TJ /F8 11.955 Tf 13.15 8.08 Td[(64 9(D)]TJ /F8 11.955 Tf 9.96 0 Td[(4))]TJ /F8 11.955 Tf 13.16 8.08 Td[(274 9(D)]TJ /F8 11.955 Tf 9.96 0 Td[(4)2+O(D)]TJ /F8 11.955 Tf 9.97 0 Td[(4)3. (4) AlthoughthepowersyD)]TJ /F7 7.97 Tf 6.58 0 Td[(3andyD)]TJ /F7 7.97 Tf 6.59 0 Td[(4intheremaindertermof( 4 )areintegrable,thefactorsof1=(D)]TJ /F8 11.955 Tf 12.67 0 Td[(4)theycarryprecludeussettingD=4andthenobtaininganexplicitformusingtheD=4Green'sfunction.Inthenextsectionwewillseehowtoaddzerosoastolocalizethedivergences,andthenabsorbthemintocounterterms.Fornow,letusassumeF1(y)hasbeenderivedandexplaintheprocedureforcomputingthespintwostructurefunctionF2(y).Thespinzeropartofthegravitonself-energycanbeexpressedasasumofthevedeSitterinvariantbitensorstimesfunctionsofy, P(x)P(x0)F1(y)=@2y @x@x0(@2y @x0)@x1(y)+@y @x(@2y @x)@x0(@y @x0)1(y)+@y @x@y @x@y @x0@y @x01(y)+H4 g(x) g(x0)1(y)+H2h g(x)@y @x0@y @x0+@y @x@y @x g(x0)i1(y), (4) Herethespinzerocoefcientfunctionsare, 1=2F001, (4) 1=4F0001, (4) 1=F00001, (4) 1=(4y)]TJ /F3 11.955 Tf 9.97 0 Td[(y2)2F00001+2(D+1)(2)]TJ /F3 11.955 Tf 9.96 0 Td[(y)(4y)]TJ /F3 11.955 Tf 9.97 0 Td[(y2)F0001)]TJ /F8 11.955 Tf 11.95 0 Td[(4(4y)]TJ /F3 11.955 Tf 9.96 0 Td[(y2)F001+(D2)]TJ /F8 11.955 Tf 9.96 0 Td[(3)(2)]TJ /F3 11.955 Tf 9.96 0 Td[(y)2F001+(D)]TJ /F8 11.955 Tf 9.96 0 Td[(1)2(2)]TJ /F3 11.955 Tf 9.96 0 Td[(y)F01+(D)]TJ /F8 11.955 Tf 9.96 0 Td[(1)2F1, (4) 1=)]TJ /F8 11.955 Tf 9.3 0 Td[((4y)]TJ /F3 11.955 Tf 9.96 0 Td[(y2)F00001)]TJ /F8 11.955 Tf 11.96 0 Td[((D+3)(2)]TJ /F3 11.955 Tf 9.96 0 Td[(y)F0001+(D+1)F001. (4) 53

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Ofcoursethespintwocontributioncanbereducedtothesameform, P(x)P(x0)(TTTTD)]TJ /F8 11.955 Tf 9.96 0 Td[(2 D)]TJ /F8 11.955 Tf 9.96 0 Td[(3F2(y))=@2y @x@x0(@2y @x0)@x2(y)+@y @x(@2y @x)@x0(@y @x0)2(y)+@y @x@y @x@y @x0@y @x02(y)+H4 g(x) g(x0)2(y)+H2h g(x)@y @x0@y @x0+@y @x@y @x g(x0)i2(y), (4) Determiningthecoefcientfunctionsisanextremelytediousexercisethatwasdonebycomputer.Theresultsforeachcoefcientfunctionareexpressedasanexpansioninpowersofderivativesofthespintwostructurefunction,forexample, 2=4Xk=02kdkF2 dyk.(4)Thevariouscoefcients,whicharefunctionsofDandy,arereportedinTables 4-1 4-5 Table4-1. CoefcientofF2:eachtermismultipliedby1 16(D)]TJ /F7 7.97 Tf 6.58 0 Td[(2)(D)]TJ /F7 7.97 Tf 6.59 0 Td[(1) CoefcientofF2 20)]TJ /F8 11.955 Tf 9.3 0 Td[((D)]TJ /F8 11.955 Tf 9.96 0 Td[(3)D2(D+1)2h)]TJ /F8 11.955 Tf 9.3 0 Td[(4(D)]TJ /F8 11.955 Tf 9.96 0 Td[(2)+(D)]TJ /F8 11.955 Tf 9.96 0 Td[(1)(4y)]TJ /F3 11.955 Tf 9.97 0 Td[(y2)i202(D)]TJ /F8 11.955 Tf 9.96 0 Td[(3)(D)]TJ /F8 11.955 Tf 9.96 0 Td[(1)D2(D+1)2(2)]TJ /F3 11.955 Tf 9.97 0 Td[(y)20(D)]TJ /F8 11.955 Tf 9.96 0 Td[(3)(D)]TJ /F8 11.955 Tf 9.96 0 Td[(1)D2(D+1)2204(D)]TJ /F8 11.955 Tf 9.96 0 Td[(3)D(D+1)2h)]TJ /F8 11.955 Tf 9.3 0 Td[(4(D)]TJ /F8 11.955 Tf 9.97 0 Td[(2)+D(4y)]TJ /F3 11.955 Tf 9.97 0 Td[(y2)i20)]TJ /F8 11.955 Tf 9.29 0 Td[(4(D)]TJ /F8 11.955 Tf 9.96 0 Td[(3)D2(D+1)2 Table4-2. CoefcientofF02:eachtermismultipliedby1 16(D)]TJ /F7 7.97 Tf 6.58 0 Td[(2)(D)]TJ /F7 7.97 Tf 6.59 0 Td[(1) CoefcientofF2 214(D)]TJ /F8 11.955 Tf 9.96 0 Td[(3)(D+1)2(2)]TJ /F3 11.955 Tf 9.97 0 Td[(y)h)]TJ /F8 11.955 Tf 9.29 0 Td[(2(D)]TJ /F8 11.955 Tf 9.96 0 Td[(2)D+(D)]TJ /F8 11.955 Tf 9.97 0 Td[(1)(D+1)(4y)]TJ /F3 11.955 Tf 9.96 0 Td[(y2)i218(D)]TJ /F8 11.955 Tf 9.96 0 Td[(3)(D+1)2h)]TJ /F8 11.955 Tf 9.29 0 Td[(3D2+(D)]TJ /F8 11.955 Tf 9.96 0 Td[(1)(D+1)(4y)]TJ /F3 11.955 Tf 9.96 0 Td[(y2)i21)]TJ /F8 11.955 Tf 9.3 0 Td[(4(D)]TJ /F8 11.955 Tf 9.96 0 Td[(3)(D)]TJ /F8 11.955 Tf 9.96 0 Td[(1)(D+1)3(2)]TJ /F3 11.955 Tf 9.97 0 Td[(y)21)]TJ /F8 11.955 Tf 9.3 0 Td[(16(D)]TJ /F8 11.955 Tf 9.97 0 Td[(3)(D+1)2(2)]TJ /F3 11.955 Tf 9.96 0 Td[(y)h)]TJ /F8 11.955 Tf 9.3 0 Td[(2(D)]TJ /F8 11.955 Tf 9.97 0 Td[(2)+(D+1)(4y)]TJ /F3 11.955 Tf 9.96 0 Td[(y2)i2116(D)]TJ /F8 11.955 Tf 9.97 0 Td[(3)(D+1)3(2)]TJ /F3 11.955 Tf 9.96 0 Td[(y) Nowrecallthesecondorderequation( 4 )wewereabletondfortheatspacestructurefunctionF2(x2)byaddingandx2.Afterlongcontemplationof 54

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Table4-3. CoefcientofF002:eachtermismultipliedby1 16(D)]TJ /F7 7.97 Tf 6.58 0 Td[(2)(D)]TJ /F7 7.97 Tf 6.59 0 Td[(1) CoefcientofF002 222h8(D)]TJ /F8 11.955 Tf 9.96 0 Td[(2)2D(D+1))]TJ /F8 11.955 Tf 11.96 0 Td[(4(D+1)(3D3)]TJ /F8 11.955 Tf 9.96 0 Td[(8D2)]TJ /F8 11.955 Tf 9.96 0 Td[(6D+12)(4y)]TJ /F3 11.955 Tf 9.96 0 Td[(y2)+(D)]TJ /F8 11.955 Tf 9.97 0 Td[(3)(D)]TJ /F8 11.955 Tf 9.97 0 Td[(1)(3D2+9D+7)(4y)]TJ /F3 11.955 Tf 11.95 0 Td[(y2)2i22)]TJ /F8 11.955 Tf 9.29 0 Td[(4(2)]TJ /F3 11.955 Tf 9.97 0 Td[(y)h)]TJ /F8 11.955 Tf 9.3 0 Td[(2D(D+1)(3D2)]TJ /F8 11.955 Tf 9.96 0 Td[(5D)]TJ /F8 11.955 Tf 9.96 0 Td[(10)+(D)]TJ /F8 11.955 Tf 9.96 0 Td[(3)(D)]TJ /F8 11.955 Tf 9.96 0 Td[(1)(3D2+9D+7)(4y)]TJ /F3 11.955 Tf 9.96 0 Td[(y2)i22)]TJ /F8 11.955 Tf 9.3 0 Td[(2h)]TJ /F8 11.955 Tf 9.3 0 Td[(12(D4)]TJ /F3 11.955 Tf 9.97 0 Td[(D3)]TJ /F8 11.955 Tf 9.97 0 Td[(7D2+D+10)+(D)]TJ /F8 11.955 Tf 9.97 0 Td[(3)(D)]TJ /F8 11.955 Tf 9.97 0 Td[(1)(3D2+9D+72)(4y)]TJ /F3 11.955 Tf 9.96 0 Td[(y2)i22)]TJ /F8 11.955 Tf 9.29 0 Td[(8h8(D)]TJ /F8 11.955 Tf 9.96 0 Td[(2)2(D+1))]TJ /F8 11.955 Tf 11.96 0 Td[(2(D+1)(6D2)]TJ /F8 11.955 Tf 9.96 0 Td[(11D)]TJ /F8 11.955 Tf 9.96 0 Td[(18)(4y)]TJ /F3 11.955 Tf 9.97 0 Td[(y2)+(D)]TJ /F8 11.955 Tf 9.96 0 Td[(3)(3D2+9D+7)(4y)]TJ /F3 11.955 Tf 9.96 0 Td[(y2)2i228h)]TJ /F8 11.955 Tf 9.3 0 Td[(2(D+1)(5D2)]TJ /F8 11.955 Tf 9.96 0 Td[(6D)]TJ /F8 11.955 Tf 9.96 0 Td[(24)+(D)]TJ /F8 11.955 Tf 9.96 0 Td[(3)(3D2+9D+7)(4y)]TJ /F3 11.955 Tf 9.96 0 Td[(y2)i Table4-4. CoefcientofF0002:eachtermismultipliedby1 16(D)]TJ /F7 7.97 Tf 6.58 0 Td[(2)(D)]TJ /F7 7.97 Tf 6.58 0 Td[(1) CoefcientofF0002 23)]TJ /F8 11.955 Tf 9.3 0 Td[(4(D)]TJ /F8 11.955 Tf 9.96 0 Td[(1)(2)]TJ /F3 11.955 Tf 9.96 0 Td[(y)(4y)]TJ /F3 11.955 Tf 9.97 0 Td[(y2)h)]TJ /F8 11.955 Tf 9.3 0 Td[(2(D)]TJ /F8 11.955 Tf 9.96 0 Td[(2)(D+1)+(D)]TJ /F8 11.955 Tf 9.96 0 Td[(3)(D+2)(4y)]TJ /F3 11.955 Tf 9.96 0 Td[(y2)i23)]TJ /F8 11.955 Tf 9.3 0 Td[(8h4(D)]TJ /F8 11.955 Tf 9.96 0 Td[(2)D(D+1))]TJ /F8 11.955 Tf 11.95 0 Td[((5D3)]TJ /F8 11.955 Tf 9.96 0 Td[(8D2)]TJ /F8 11.955 Tf 9.96 0 Td[(23D+22)(4y)]TJ /F3 11.955 Tf 9.97 0 Td[(y2)+(D)]TJ /F8 11.955 Tf 9.96 0 Td[(3)(D)]TJ /F8 11.955 Tf 9.96 0 Td[(1)(D+2)(4y)]TJ /F3 11.955 Tf 9.96 0 Td[(y2)2i234(2)]TJ /F3 11.955 Tf 9.96 0 Td[(y)h)]TJ /F8 11.955 Tf 9.3 0 Td[(4(D)]TJ /F8 11.955 Tf 9.97 0 Td[(2)(D2)]TJ /F8 11.955 Tf 9.97 0 Td[(5)+(D)]TJ /F8 11.955 Tf 9.96 0 Td[(3)(D)]TJ /F8 11.955 Tf 9.96 0 Td[(1)(D+2)(4y)]TJ /F3 11.955 Tf 9.96 0 Td[(y2)i2316(2)]TJ /F3 11.955 Tf 9.96 0 Td[(y)(4y)]TJ /F3 11.955 Tf 9.96 0 Td[(y2)h)]TJ /F8 11.955 Tf 9.29 0 Td[(2(D)]TJ /F8 11.955 Tf 9.96 0 Td[(2)(D+1)+(D)]TJ /F8 11.955 Tf 9.96 0 Td[(3)(D+2)(4y)]TJ /F3 11.955 Tf 9.96 0 Td[(y2)i23)]TJ /F8 11.955 Tf 9.3 0 Td[(16(2)]TJ /F3 11.955 Tf 9.96 0 Td[(y)h)]TJ /F8 11.955 Tf 9.3 0 Td[(2(D)]TJ /F8 11.955 Tf 9.96 0 Td[(2)(D+1)+(D)]TJ /F8 11.955 Tf 9.96 0 Td[(3)(D+2)(4y)]TJ /F3 11.955 Tf 9.96 0 Td[(y2)i Table4-5. CoefcientofF00002:eachtermismultipliedby1 16(D)]TJ /F7 7.97 Tf 6.59 0 Td[(2)(D)]TJ /F7 7.97 Tf 6.59 0 Td[(1) CoefcientofF00002 24)]TJ /F8 11.955 Tf 9.3 0 Td[((D)]TJ /F8 11.955 Tf 9.96 0 Td[(1)(4y)]TJ /F3 11.955 Tf 9.96 0 Td[(y2)2h)]TJ /F8 11.955 Tf 9.3 0 Td[(4(D)]TJ /F8 11.955 Tf 9.96 0 Td[(2)+(D)]TJ /F8 11.955 Tf 9.96 0 Td[(3)(4y)]TJ /F3 11.955 Tf 9.96 0 Td[(y2)i242(D)]TJ /F8 11.955 Tf 9.96 0 Td[(1)(2)]TJ /F3 11.955 Tf 9.96 0 Td[(y)(4y)]TJ /F3 11.955 Tf 9.97 0 Td[(y2)h)]TJ /F8 11.955 Tf 9.3 0 Td[(4(D)]TJ /F8 11.955 Tf 9.96 0 Td[(2)+(D)]TJ /F8 11.955 Tf 9.97 0 Td[(3)(4y)]TJ /F3 11.955 Tf 9.96 0 Td[(y2)i24h4(D)]TJ /F8 11.955 Tf 9.97 0 Td[(2))]TJ /F8 11.955 Tf 11.96 0 Td[((D)]TJ /F8 11.955 Tf 9.96 0 Td[(3)(4y)]TJ /F3 11.955 Tf 9.96 0 Td[(y2)ih4(D)]TJ /F8 11.955 Tf 9.96 0 Td[(2))]TJ /F8 11.955 Tf 11.95 0 Td[((D)]TJ /F8 11.955 Tf 9.96 0 Td[(1)(4y)]TJ /F3 11.955 Tf 9.96 0 Td[(y2)i244(4y)]TJ /F3 11.955 Tf 9.96 0 Td[(y2)2h)]TJ /F8 11.955 Tf 9.3 0 Td[(4(D)]TJ /F8 11.955 Tf 9.96 0 Td[(2)+(D)]TJ /F8 11.955 Tf 9.97 0 Td[(3)(4y)]TJ /F3 11.955 Tf 9.96 0 Td[(y2)i24)]TJ /F8 11.955 Tf 9.29 0 Td[(4(4y)]TJ /F3 11.955 Tf 9.96 0 Td[(y2)h)]TJ /F8 11.955 Tf 9.3 0 Td[(4(D)]TJ /F8 11.955 Tf 9.96 0 Td[(2)+(D)]TJ /F8 11.955 Tf 9.96 0 Td[(3)(4y)]TJ /F3 11.955 Tf 9.97 0 Td[(y2)i thebewilderingdatainTables 4-1 4-5 itbecomesapparentthatasimilarsecondorder 55

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equationforF2(y)derivesfromthecombination, 2(y)+(4y)]TJ /F3 11.955 Tf 9.97 0 Td[(y2)2(y)=h(y))]TJ /F6 11.955 Tf 9.96 0 Td[(1(y)i+(4y)]TJ /F3 11.955 Tf 9.96 0 Td[(y2)h(y))]TJ /F6 11.955 Tf 9.96 0 Td[(1(y)i, (4) =)]TJ /F12 11.955 Tf 9.29 13.27 Td[(D+1 D)]TJ /F8 11.955 Tf 9.96 0 Td[(1((D)]TJ /F8 11.955 Tf 9.96 0 Td[(2)F002)]TJ /F8 11.955 Tf 11.96 0 Td[((D)]TJ /F8 11.955 Tf 9.96 0 Td[(3)"(4y)]TJ /F3 11.955 Tf 9.96 0 Td[(y2)F002+2(D+1)(2)]TJ /F3 11.955 Tf 9.96 0 Td[(y)F02)]TJ /F3 11.955 Tf 11.95 0 Td[(D(D+1)F2). (4) HencewecanexpresstheequationforF2(y)as, DF2=)]TJ /F12 11.955 Tf 9.3 13.27 Td[(D)]TJ /F8 11.955 Tf 9.96 0 Td[(1 D+1(h(y))]TJ /F6 11.955 Tf 9.96 0 Td[(1(y)i+(4y)]TJ /F3 11.955 Tf 9.96 0 Td[(y2)h(y))]TJ /F6 11.955 Tf 9.96 0 Td[(1(y)i),(4)wherethesecondorderoperatorDis, D4(D)]TJ /F8 11.955 Tf 9.96 0 Td[(2)d dy2)]TJ /F8 11.955 Tf 9.3 0 Td[((D)]TJ /F8 11.955 Tf 9.97 0 Td[(3)"(4y)]TJ /F3 11.955 Tf 9.96 0 Td[(y2)d dy2+2(D+1)(2)]TJ /F3 11.955 Tf 9.96 0 Td[(y)d dy)]TJ /F3 11.955 Tf 11.96 0 Td[(D(D+1)#, (4) =4d dy2+(D)]TJ /F8 11.955 Tf 9.97 0 Td[(3)"(2)]TJ /F3 11.955 Tf 9.96 0 Td[(y)2d dy2)]TJ /F8 11.955 Tf 11.95 0 Td[(2(D+1)(2)]TJ /F3 11.955 Tf 9.97 0 Td[(y)d dy+D(D+1)#. (4) Thesourcetermontherighthandsideof( 4 )hastheform, )]TJ /F12 11.955 Tf 9.3 13.27 Td[(D)]TJ /F8 11.955 Tf 9.97 0 Td[(1 D+1(h(y))]TJ /F6 11.955 Tf 9.96 0 Td[(1(y)i+(4y)]TJ /F3 11.955 Tf 9.96 0 Td[(y2)h(y))]TJ /F6 11.955 Tf 9.96 0 Td[(1(y)i)=K(sa4 yD+sb D)]TJ /F8 11.955 Tf 9.97 0 Td[(44 yD)]TJ /F7 7.97 Tf 6.59 0 Td[(1+sc D)]TJ /F8 11.955 Tf 9.96 0 Td[(44 yD)]TJ /F7 7.97 Tf 6.58 0 Td[(2+sc04 yD 2+sd D)]TJ /F8 11.955 Tf 9.96 0 Td[(44 yD)]TJ /F7 7.97 Tf 6.59 0 Td[(3+se (D)]TJ /F8 11.955 Tf 9.96 0 Td[(4)24 yD)]TJ /F7 7.97 Tf 6.59 0 Td[(4+Irrelevant)+R, (4) wheretheremaindertermRderivesfromtheremainderR1ofF1, R=D)]TJ /F8 11.955 Tf 9.96 0 Td[(1 D+1((D)]TJ /F8 11.955 Tf 9.97 0 Td[(1)(2)]TJ /F3 11.955 Tf 9.96 0 Td[(y)(4y)]TJ /F3 11.955 Tf 9.96 0 Td[(y2)@ @y3)]TJ /F3 11.955 Tf 11.96 0 Td[(D(D)]TJ /F8 11.955 Tf 9.96 0 Td[(1)(4y)]TJ /F3 11.955 Tf 9.96 0 Td[(y2)@ @y2+4(D2)]TJ /F8 11.955 Tf 9.96 0 Td[(3)@ @y2+(D)]TJ /F8 11.955 Tf 9.97 0 Td[(1)2(2)]TJ /F3 11.955 Tf 9.96 0 Td[(y)@ @y+(D)]TJ /F8 11.955 Tf 9.96 0 Td[(1)2)"1 H2+D#2R1. (4) 56

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Thecoefcientsin( 4 )are, sa=)]TJ /F8 11.955 Tf 31.86 8.09 Td[(1 16(D+1), (4) sb=)]TJ /F8 11.955 Tf 11.87 8.08 Td[((D)]TJ /F8 11.955 Tf 9.97 0 Td[(2)D 16(D)]TJ /F8 11.955 Tf 9.97 0 Td[(1), (4) sc=)]TJ /F8 11.955 Tf 10.49 8.09 Td[((D)]TJ /F8 11.955 Tf 9.96 0 Td[(4)(D)]TJ /F8 11.955 Tf 9.96 0 Td[(2)D(D+3) 32(D)]TJ /F8 11.955 Tf 9.97 0 Td[(6)(D)]TJ /F8 11.955 Tf 9.96 0 Td[(1), (4) sc0=)]TJ /F8 11.955 Tf 19.64 8.09 Td[((D)]TJ /F8 11.955 Tf 9.96 0 Td[(4)(D)]TJ /F8 11.955 Tf 9.96 0 Td[(1)\(D) 16(D+1)\(D 2)\(D 2+1), (4) sd=)]TJ /F8 11.955 Tf 10.49 8.09 Td[(7 5+263 100(D)]TJ /F8 11.955 Tf 9.97 0 Td[(4)+O(D)]TJ /F8 11.955 Tf 9.96 0 Td[(4)2, (4) se=18 5)]TJ /F8 11.955 Tf 13.15 8.09 Td[(18 25(D)]TJ /F8 11.955 Tf 9.96 0 Td[(4))]TJ /F8 11.955 Tf 13.15 8.09 Td[(11331 1000(D)]TJ /F8 11.955 Tf 9.96 0 Td[(4)2+O(D)]TJ /F8 11.955 Tf 9.97 0 Td[(4)3. (4) Justasforthedifferentialoperator( H2+D),itisstraightforwardtoconstructaGreen'sfunctiontoinvertD.Therststepistochangevariablesinthesecondform( 4 ), wr D)]TJ /F8 11.955 Tf 9.96 0 Td[(3 4(2)]TJ /F3 11.955 Tf 8.68 0 Td[(y)=)D=(D)]TJ /F8 11.955 Tf 8.67 0 Td[(3)h(1+w2)d dw2+2(D+1)wd dw+D(D+1)i.(4)ThehomogeneousequationDf(w)=0givesrisetoasimple,2-termrecursionrelationwhichgeneratesevenandoddsolutions.TheseseriessolutionscanbeexpressedashypergeometricfunctionsthatreducetoelementaryfunctionsforD=4, fe(w)=2F1D 2,D+1 2;1 2;w2)166(!(1)]TJ /F8 11.955 Tf 9.96 0 Td[(6w2+w4) (1+w2)4, (4) fo(w)=w2F1D+1 2,D+2 2;3 2;w2)166(!(w)]TJ /F3 11.955 Tf 9.96 0 Td[(w3) (1+w2)4. (4) BecauseweagainhavebothhomogeneoussolutionsitissimpletowritedownaGreen'sfunction, G2(w;w0)=(w)]TJ /F3 11.955 Tf 9.96 0 Td[(w0) D)]TJ /F8 11.955 Tf 9.96 0 Td[(3hfo(w)fe(w0))]TJ /F3 11.955 Tf 9.96 0 Td[(fe(w)fo(w0)i(1+w02)D.(4)Aswasthecaseforitspinzerocousin( 4 ),thespintwoGreen'sfunction( 4 )isnotsimpletouseforarbitraryD.Wethereforeadoptthesamestrategywe 57

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usedforF1,ofrecursivelyextractingpowersuntiltheremainderisintegrableandtheD=4formscanbeemployed.ActingDonapowergives, D4 yp)]TJ /F7 7.97 Tf 6.58 0 Td[(2=1 4(D)]TJ /F8 11.955 Tf 9.97 0 Td[(2)(p)]TJ /F8 11.955 Tf 9.96 0 Td[(2)(p)]TJ /F8 11.955 Tf 9.96 0 Td[(1)4 yp+(D)]TJ /F8 11.955 Tf 9.97 0 Td[(3)(p)]TJ /F8 11.955 Tf 9.96 0 Td[(2)(D+2)]TJ /F3 11.955 Tf 9.96 0 Td[(p)4 yp)]TJ /F7 7.97 Tf 6.59 0 Td[(1+(D)]TJ /F8 11.955 Tf 9.96 0 Td[(3)(D+2)]TJ /F3 11.955 Tf 9.97 0 Td[(p)(D+3)]TJ /F3 11.955 Tf 9.96 0 Td[(p)4 yp)]TJ /F7 7.97 Tf 6.59 0 Td[(2. (4) Henceweconclude, 1 D4 yp=4 (D)]TJ /F8 11.955 Tf 9.96 0 Td[(2)(p)]TJ /F8 11.955 Tf 9.96 0 Td[(2)(p)]TJ /F8 11.955 Tf 9.96 0 Td[(1)4 yp)]TJ /F7 7.97 Tf 6.58 0 Td[(2)]TJ /F8 11.955 Tf 12.13 8.09 Td[(4 D((D)]TJ /F8 11.955 Tf 9.96 0 Td[(3)(D+2)]TJ /F3 11.955 Tf 9.97 0 Td[(p) (D)]TJ /F8 11.955 Tf 9.96 0 Td[(2)(p)]TJ /F8 11.955 Tf 9.96 0 Td[(1)4 yp)]TJ /F7 7.97 Tf 6.58 0 Td[(1+(D)]TJ /F8 11.955 Tf 9.97 0 Td[(3)(D+2)]TJ /F3 11.955 Tf 9.96 0 Td[(p)(D+3)]TJ /F3 11.955 Tf 9.96 0 Td[(p) (D)]TJ /F8 11.955 Tf 9.97 0 Td[(2)(p)]TJ /F8 11.955 Tf 9.96 0 Td[(2)(p)]TJ /F8 11.955 Tf 9.96 0 Td[(1)4 yp)]TJ /F7 7.97 Tf 6.59 0 Td[(2). (4) Forthefourpowersofrelevanceexpression( 4 )gives, 1 D4 yD=4 (D)]TJ /F8 11.955 Tf 9.96 0 Td[(2)2(D)]TJ /F8 11.955 Tf 9.96 0 Td[(1)4 yD)]TJ /F7 7.97 Tf 6.58 0 Td[(2)]TJ /F8 11.955 Tf 12.13 8.09 Td[(1 D(8(D)]TJ /F8 11.955 Tf 9.97 0 Td[(3) (D)]TJ /F8 11.955 Tf 9.96 0 Td[(2)(D)]TJ /F8 11.955 Tf 9.96 0 Td[(1)4 yD)]TJ /F7 7.97 Tf 6.58 0 Td[(1+24(D)]TJ /F8 11.955 Tf 9.96 0 Td[(3) (D)]TJ /F8 11.955 Tf 9.96 0 Td[(2)2(D)]TJ /F8 11.955 Tf 9.96 0 Td[(1)4 yD)]TJ /F7 7.97 Tf 6.58 0 Td[(2), (4) 1 D4 yD)]TJ /F7 7.97 Tf 6.59 0 Td[(1=4 (D)]TJ /F8 11.955 Tf 9.96 0 Td[(3)(D)]TJ /F8 11.955 Tf 9.96 0 Td[(2)24 yD)]TJ /F7 7.97 Tf 6.58 0 Td[(3)]TJ /F8 11.955 Tf 12.13 8.09 Td[(1 D(12(D)]TJ /F8 11.955 Tf 9.96 0 Td[(3) (D)]TJ /F8 11.955 Tf 9.96 0 Td[(2)24 yD)]TJ /F7 7.97 Tf 6.58 0 Td[(2+48 (D)]TJ /F8 11.955 Tf 9.96 0 Td[(2)24 yD)]TJ /F7 7.97 Tf 6.58 0 Td[(3), (4) 1 D4 yD)]TJ /F7 7.97 Tf 6.59 0 Td[(2=4 (D)]TJ /F8 11.955 Tf 9.96 0 Td[(4)(D)]TJ /F8 11.955 Tf 9.96 0 Td[(3)(D)]TJ /F8 11.955 Tf 9.96 0 Td[(2)4 yD)]TJ /F7 7.97 Tf 6.58 0 Td[(4)]TJ /F8 11.955 Tf 12.13 8.08 Td[(1 D(16 (D)]TJ /F8 11.955 Tf 9.96 0 Td[(2)4 yD)]TJ /F7 7.97 Tf 6.59 0 Td[(3+80 (D)]TJ /F8 11.955 Tf 9.96 0 Td[(4)(D)]TJ /F8 11.955 Tf 9.96 0 Td[(2)4 yD)]TJ /F7 7.97 Tf 6.59 0 Td[(4), (4) 1 D4 yD 2=16 (D)]TJ /F8 11.955 Tf 9.96 0 Td[(4)(D)]TJ /F8 11.955 Tf 9.96 0 Td[(2)24 yD 2)]TJ /F7 7.97 Tf 6.59 0 Td[(2)]TJ /F8 11.955 Tf 12.13 8.09 Td[(4 D((D)]TJ /F8 11.955 Tf 9.97 0 Td[(3)(D+4) (D)]TJ /F8 11.955 Tf 9.97 0 Td[(2)24 yD 2)]TJ /F7 7.97 Tf 6.58 0 Td[(1+(D)]TJ /F8 11.955 Tf 9.96 0 Td[(3)(D+4)(D+6) (D)]TJ /F8 11.955 Tf 9.96 0 Td[(4)(D)]TJ /F8 11.955 Tf 9.96 0 Td[(2)24 yD 2)]TJ /F7 7.97 Tf 6.59 0 Td[(2). (4) 58

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Theserelationsallowthespintwostructurefunctiontobeexpressedasaquotientandremainderoftheform, F2=Q2(y)+1 DR2(y), (4) Q2=)]TJ /F3 11.955 Tf 9.3 0 Td[(K(f2a4 yD)]TJ /F7 7.97 Tf 6.59 0 Td[(2+f2b D)]TJ /F8 11.955 Tf 9.96 0 Td[(44 yD)]TJ /F7 7.97 Tf 6.59 0 Td[(3+f2c (D)]TJ /F8 11.955 Tf 9.96 0 Td[(4)24 yD)]TJ /F7 7.97 Tf 6.58 0 Td[(4+f2c0 D)]TJ /F8 11.955 Tf 9.96 0 Td[(44 yD 2)]TJ /F7 7.97 Tf 6.59 0 Td[(2), (4) R2=)]TJ /F3 11.955 Tf 9.3 0 Td[(K(f2d D)]TJ /F8 11.955 Tf 9.96 .01 Td[(44 yD)]TJ /F7 7.97 Tf 6.58 0 Td[(3+f2e (D)]TJ /F8 11.955 Tf 9.96 -.01 Td[(4)24 yD)]TJ /F7 7.97 Tf 6.59 0 Td[(4+Irrelevant)+R, (4) wherethecoefcientsare, f2a=1 4(D)]TJ /F8 11.955 Tf 11.95 0 Td[(2)2(D)]TJ /F8 11.955 Tf 11.96 0 Td[(1)(D+1), (4) f2b=D4)]TJ /F8 11.955 Tf 9.96 0 Td[(3D3)]TJ /F8 11.955 Tf 9.96 0 Td[(8D2+60D)]TJ /F8 11.955 Tf 9.96 0 Td[(96 4(D)]TJ /F8 11.955 Tf 9.96 0 Td[(3)(D)]TJ /F8 11.955 Tf 9.96 0 Td[(2)3(D)]TJ /F8 11.955 Tf 9.96 0 Td[(1)(D+1), (4) f2c=D8)]TJ /F8 11.955 Tf 9.96 0 Td[(8D7)]TJ /F8 11.955 Tf 9.97 0 Td[(13D6+348D5)]TJ /F8 11.955 Tf 9.96 0 Td[(1136D4)]TJ /F8 11.955 Tf 9.97 0 Td[(210D3+15056D2)]TJ /F8 11.955 Tf 9.96 0 Td[(38208D+34560 8(D)]TJ /F8 11.955 Tf 9.96 0 Td[(6)(D)]TJ /F8 11.955 Tf 9.96 0 Td[(3)(D)]TJ /F8 11.955 Tf 9.96 0 Td[(2)4(D)]TJ /F8 11.955 Tf 9.96 0 Td[(1)(D+1), (4) f2c0=(D)]TJ /F8 11.955 Tf 9.96 0 Td[(4)(D)]TJ /F8 11.955 Tf 9.96 0 Td[(1)\(D) (D)]TJ /F8 11.955 Tf 9.96 0 Td[(2)2(D+1)\(D 2)\(D 2+1), (4) f2d=17 5+161 300(D)]TJ /F8 11.955 Tf 9.96 0 Td[(4)+O(D)]TJ /F8 11.955 Tf 9.97 0 Td[(4)2,f2e=82 5+243 25(D)]TJ /F8 11.955 Tf 9.97 0 Td[(4)+13343 3000(D)]TJ /F8 11.955 Tf 9.96 0 Td[(4)2+O(D)]TJ /F8 11.955 Tf 9.97 0 Td[(4)3. (4) 4.4RenormalizingtheSpinZeroStructureFunctionRecalltheform( 4 )weobtainedforthespinzerostruncturefunctionfromtakingthetraceofthegravitonself-energy, F1(y)=Q1(y)+h1 H2+Di2R1(y).(4)RecallalsothatthequotientQ1(y)andtheremainderR1(y)aregiveninrelations( 4 )-( 4 ).Fromtheseexpressionsweperceivethreesortsofultravioletdivergences: Thefactorof(4 y)D)]TJ /F7 7.97 Tf 6.58 0 Td[(2inQ1,whichhasanitecoefcientbutisstillnotintegrableinD=4dimensions; 59

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Thefactorsof1 D)]TJ /F7 7.97 Tf 6.59 0 Td[(4(4 y)D)]TJ /F7 7.97 Tf 6.59 0 Td[(3inQ1andR1whichareintegrableinD=4dimensionsbuthavedivergentcoefcientsthatprecludetakingtheunregulatedlimits;and Thefactorsof(1 D)]TJ /F7 7.97 Tf 6.58 0 Td[(4)2(4 y)D)]TJ /F7 7.97 Tf 6.58 0 Td[(4inQ1andR1whichareintegrableinD=4dimensionsbuthaveevenmoredivergentcoefcients.Inthissectionwewillexplainhowtolocalizeallthreedivergencesontodeltafunctiontermswhichcanbeabsorbedbythecounterterms( 4 ),( 4 )and( 4 ).Wewillalsotaketheunregulatedlimitsoftheremaining,niteparts,andusetheD=4Green'sfunction( 4 )toobtainanexplicitresultfortherenormalizedstructurefunction.Indealingwiththefactorof(4 y)D)]TJ /F7 7.97 Tf 6.59 0 Td[(2inQ1,therststepistoextractad'Alembertian, 4 yD)]TJ /F7 7.97 Tf 6.59 0 Td[(2=2 (D)]TJ /F8 11.955 Tf 9.97 0 Td[(4)(D)]TJ /F8 11.955 Tf 9.97 0 Td[(3)" H24 yD)]TJ /F7 7.97 Tf 6.58 0 Td[(3)]TJ /F8 11.955 Tf 11.96 0 Td[(2(D)]TJ /F8 11.955 Tf 9.97 0 Td[(3)4 yD)]TJ /F7 7.97 Tf 6.59 0 Td[(3#.(4)Theresultingfactorsof(4 y)D)]TJ /F7 7.97 Tf 6.59 0 Td[(3areintegrableinD=4dimensions,atwhichpointwecouldtaketheunregulatedlimitexceptforthefactorof1=(D)]TJ /F8 11.955 Tf 12.52 0 Td[(4)in( 4 ).Wecanlocalizethedivergenceonadeltafunctionbyaddingzerointheformoftheidentity( 4 ), 4 yD)]TJ /F7 7.97 Tf 6.59 0 Td[(2=2 (D)]TJ /F8 11.955 Tf 9.96 0 Td[(4)(D)]TJ /F8 11.955 Tf 9.96 0 Td[(3)( H2"4 yD)]TJ /F7 7.97 Tf 6.59 0 Td[(3)]TJ /F12 11.955 Tf 11.95 13.27 Td[(4 yD 2)]TJ /F7 7.97 Tf 6.59 0 Td[(1#)]TJ /F8 11.955 Tf 9.3 0 Td[(2(D)]TJ /F8 11.955 Tf 9.96 0 Td[(3)4 yD)]TJ /F7 7.97 Tf 6.58 0 Td[(3+D 2D 2)]TJ /F8 11.955 Tf 9.96 0 Td[(14 yD 2)]TJ /F7 7.97 Tf 6.58 0 Td[(1+(4)D 2 \(D 2)]TJ /F8 11.955 Tf 9.96 0 Td[(1)iD(x)]TJ /F3 11.955 Tf 9.96 0 Td[(x0) HDp )]TJ ET q .478 w 378.37 -412.76 m 385.13 -412.76 l S Q BT /F3 11.955 Tf 378.37 -420.08 Td[(g), (4) =)]TJ /F12 11.955 Tf 9.29 13.27 Td[(h H2)]TJ /F8 11.955 Tf 9.96 0 Td[(2in4 ylny 4o)]TJ /F8 11.955 Tf 13.44 8.09 Td[(4 y+O(D)]TJ /F8 11.955 Tf 9.96 0 Td[(4)+2(4)D 2iD(x)]TJ /F3 11.955 Tf 9.97 0 Td[(x0)=p )]TJ ET q .478 w 388.22 -434.81 m 394.98 -434.81 l S Q BT /F3 11.955 Tf 388.22 -442.13 Td[(g (D)]TJ /F8 11.955 Tf 9.96 0 Td[(4)(D)]TJ /F8 11.955 Tf 9.96 0 Td[(3)\(D 2)]TJ /F8 11.955 Tf 9.96 0 Td[(1)HD. (4) Weturnnowtothefactorsof1 D)]TJ /F7 7.97 Tf 6.59 0 Td[(4(4 y)D)]TJ /F7 7.97 Tf 6.59 0 Td[(3and(1 D)]TJ /F7 7.97 Tf 6.58 0 Td[(4)2(4 y)D)]TJ /F7 7.97 Tf 6.59 0 Td[(4inQ1andR1.Thekeyrelationsforresolvingthesetermsfollowfrom( 4 ), h H2+Di24 yD 2)]TJ /F7 7.97 Tf 6.59 0 Td[(1=1 16D2(D+2)24 yD 2)]TJ /F7 7.97 Tf 6.58 0 Td[(1+(4)D 2 \(D 2)]TJ /F8 11.955 Tf 9.96 0 Td[(1)HDp )]TJ ET q .478 w 198.41 -582.56 m 205.17 -582.56 l S Q BT /F3 11.955 Tf 198.41 -589.88 Td[(g" H2+D+1 4D(D+2)#iD(x)]TJ /F3 11.955 Tf 9.96 0 Td[(x0), (4) 60

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h H2+Di24 yD 2)]TJ /F7 7.97 Tf 6.59 0 Td[(2=)]TJ /F8 11.955 Tf 10.49 8.08 Td[(1 4(D)]TJ /F8 11.955 Tf 9.97 0 Td[(4)(D2+2D)]TJ /F8 11.955 Tf 9.96 0 Td[(4)4 yD 2)]TJ /F7 7.97 Tf 6.58 0 Td[(1+1 16(D)]TJ /F8 11.955 Tf 9.96 0 Td[(2)2(D+4)24 yD 2)]TJ /F7 7.97 Tf 6.59 0 Td[(2)]TJ /F8 11.955 Tf 25.25 8.09 Td[((D)]TJ /F8 11.955 Tf 9.96 0 Td[(4)(4)D 2 2\(D 2)]TJ /F8 11.955 Tf 9.96 0 Td[(1)HDp )]TJ ET q .478 w 336.37 -53.7 m 343.13 -53.7 l S Q BT /F3 11.955 Tf 336.37 -61.02 Td[(giD(x)]TJ /F3 11.955 Tf 9.96 0 Td[(x0), (4) h H2+Di21=D2. (4) OneaddszerousingtheserelationssoastoresolvetheproblematictermsinQ1,andtheremainderautomaticallyresolvestheproblematictermsinR1, f1b D)]TJ /F8 11.955 Tf 9.96 0 Td[(44 yD)]TJ /F7 7.97 Tf 6.59 0 Td[(3+f1c (D)]TJ /F8 11.955 Tf 9.96 0 Td[(4)24 yD)]TJ /F7 7.97 Tf 6.59 0 Td[(4+"1 H2+D#2(f1d D)]TJ /F8 11.955 Tf 9.96 0 Td[(44 yD)]TJ /F7 7.97 Tf 6.58 0 Td[(3+f1e (D)]TJ /F8 11.955 Tf 9.96 0 Td[(4)24 yD)]TJ /F7 7.97 Tf 6.58 0 Td[(4)=f1b D)]TJ /F8 11.955 Tf 9.96 -.01 Td[(4(4 yD)]TJ /F7 7.97 Tf 6.58 0 Td[(3)]TJ /F12 11.955 Tf 11.95 13.27 Td[(4 yD 2)]TJ /F7 7.97 Tf 6.59 0 Td[(1)+f1c (D)]TJ /F8 11.955 Tf 9.96 .01 Td[(4)2(4 yD)]TJ /F7 7.97 Tf 6.59 0 Td[(4)]TJ /F8 11.955 Tf 11.95 0 Td[(24 yD 2)]TJ /F7 7.97 Tf 6.58 0 Td[(2+1)+"1 H2+D#2(f1d D)]TJ /F8 11.955 Tf 9.97 0 Td[(44 yD)]TJ /F7 7.97 Tf 6.58 0 Td[(3+[D2(D+2)2f1b)]TJ /F8 11.955 Tf 9.97 0 Td[(8(D2+2D)]TJ /F8 11.955 Tf 9.96 0 Td[(4)f1c] 16(D)]TJ /F8 11.955 Tf 9.97 0 Td[(4)4 yD 2)]TJ /F7 7.97 Tf 6.59 0 Td[(1+f1e (D)]TJ /F8 11.955 Tf 9.96 0 Td[(4)24 yD)]TJ /F7 7.97 Tf 6.59 0 Td[(4+(D)]TJ /F8 11.955 Tf 9.96 0 Td[(2)2(D+4)2f1c 8(D)]TJ /F8 11.955 Tf 9.97 0 Td[(4)24 yD 2)]TJ /F7 7.97 Tf 6.58 0 Td[(2)]TJ /F3 11.955 Tf 19.78 8.08 Td[(D2f1c (D)]TJ /F8 11.955 Tf 9.97 0 Td[(4)2+(4)D 2=p )]TJ ET q .478 w 126.24 -323.01 m 133 -323.01 l S Q BT /F3 11.955 Tf 126.24 -330.33 Td[(g \(D 2)]TJ /F8 11.955 Tf 9.97 0 Td[(1)HD"f1b D)]TJ /F8 11.955 Tf 9.97 0 Td[(4h H2+Di+D(D+2)f1b)]TJ /F8 11.955 Tf 9.96 0 Td[(4f1c 4(D)]TJ /F8 11.955 Tf 9.96 0 Td[(4)#iD(x)]TJ /F3 11.955 Tf 9.96 0 Td[(x0)), (4) =1 184 ylny 4)]TJ /F8 11.955 Tf 11.16 8.09 Td[(1 6ln2y 4+O(D)]TJ /F8 11.955 Tf 9.97 0 Td[(4)+"1 H2+4#2(4 34 ylny 4+8 34 y+8 3ln2y 4)]TJ /F8 11.955 Tf 9.96 0 Td[(8lny 4+1 3)+"1 H2+D#2((4)D 2=p )]TJ ET q .478 w 404.67 -407.21 m 411.43 -407.21 l S Q BT /F3 11.955 Tf 404.67 -414.53 Td[(g \(D 2)]TJ /F8 11.955 Tf 9.96 0 Td[(1)HD"f1b D)]TJ /F8 11.955 Tf 9.97 0 Td[(4h H2+Di+D(D+2)f1b)]TJ /F8 11.955 Tf 9.96 0 Td[(4f1c 4(D)]TJ /F8 11.955 Tf 9.96 0 Td[(4)#iD(x)]TJ /F3 11.955 Tf 9.96 0 Td[(x0)). (4) Employingexpressions( 4 )and( 4 )in( 4 )allowsustoseparatethespinzerostructurefunctionintoanitepartandadivergentpart, F1=F1R+O(D)]TJ /F8 11.955 Tf 9.96 0 Td[(4)+F1.(4) 61

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Thenitepartconsistsoftherenormalizedspinzerostructurefunction, F1R=2H4 (4)4( H2"1 724 ylny 4#)]TJ /F8 11.955 Tf 14.3 8.09 Td[(1 124 ylny 4+1 724 y+1 6ln2y 4)+2H4 (4)4"1 H2+4#2()]TJ /F8 11.955 Tf 10.49 8.08 Td[(4 34 ylny 4)]TJ /F8 11.955 Tf 11.16 8.08 Td[(8 34 y)]TJ /F8 11.955 Tf 11.16 8.08 Td[(8 3ln2y 4+8lny 4)]TJ /F8 11.955 Tf 11.16 8.08 Td[(1 3). (4) Thedivergentpartconsistsof[ H2+D])]TJ /F7 7.97 Tf 6.59 0 Td[(2actingonasumofthreelocalterms, F1=2HD)]TJ /F7 7.97 Tf 6.58 0 Td[(4(D 2)]TJ /F8 11.955 Tf 9.96 0 Td[(1)\(D 2) (4)D 2"1 H2+D#2()]TJ /F8 11.955 Tf 9.3 0 Td[(2f1a (D)]TJ /F8 11.955 Tf 9.97 0 Td[(4)(D)]TJ /F8 11.955 Tf 9.97 0 Td[(3)" H2+D#2iD(x)]TJ /F3 11.955 Tf 9.96 0 Td[(x0) p )]TJ ET q .478 w 404.6 -171.91 m 411.37 -171.91 l S Q BT /F3 11.955 Tf 404.6 -179.23 Td[(g)]TJ /F3 11.955 Tf 17.18 8.08 Td[(f1b D)]TJ /F8 11.955 Tf 9.96 0 Td[(4" H2+D#iD(x)]TJ /F3 11.955 Tf 9.96 0 Td[(x0) p )]TJ ET q .478 w 195.21 -212.1 m 201.97 -212.1 l S Q BT /F3 11.955 Tf 195.21 -219.42 Td[(g)]TJ /F12 11.955 Tf 11.96 20.44 Td[("D(D+2)f1b)]TJ /F8 11.955 Tf 9.97 0 Td[(4f1c 4(D)]TJ /F8 11.955 Tf 9.97 0 Td[(4)#iD(x)]TJ /F3 11.955 Tf 9.97 0 Td[(x0) p )]TJ ET q .478 w 371.86 -212.1 m 378.62 -212.1 l S Q BT /F3 11.955 Tf 371.86 -219.42 Td[(g). (4) OfcourseonecancelsF1withcounterterms.Fromexpressions( 4 )-( 4 )weseethatthefourcountertermscontributetothegravitonself-energyas, )]TJ /F3 11.955 Tf 9.3 0 Td[(ihi(x;x0)=p )]TJ ET q .478 w 167.12 -299.04 m 173.89 -299.04 l S Q BT /F3 11.955 Tf 167.12 -306.36 Td[(g"2c12PP+2c22 g g g gPP)]TJ /F3 11.955 Tf 9.3 0 Td[(c32H2D+c42H4p )]TJ ET q .478 w 218.04 -338.89 m 224.8 -338.89 l S Q BT /F3 11.955 Tf 218.04 -346.21 Td[(gh1 4 g g)]TJ /F8 11.955 Tf 11.16 8.09 Td[(1 2 g( g)i#iD(x)]TJ /F3 11.955 Tf 9.97 0 Td[(x0). (4) Tracingaswedidin( 4 )gives, g(x) p )]TJ ET q .478 w 57.79 -420.64 m 64.55 -420.64 l S Q BT /F3 11.955 Tf 57.79 -427.96 Td[(g(x) g(x0) p )]TJ ET q .478 w 115.23 -420.64 m 121.99 -420.64 l S Q BT /F3 11.955 Tf 115.23 -427.96 Td[(g(x0))]TJ /F3 11.955 Tf 19.26 0 Td[(ihi(x;x0)=(D)]TJ /F8 11.955 Tf 9.97 0 Td[(1)2H4"2c12h H2+Di2+0)]TJ /F8 11.955 Tf 11.16 8.08 Td[(1 2D)]TJ /F8 11.955 Tf 9.96 0 Td[(2 D)]TJ /F8 11.955 Tf 9.96 0 Td[(1c32h H2+Di+D(D)]TJ /F8 11.955 Tf 9.96 0 Td[(2) 4(D)]TJ /F8 11.955 Tf 9.96 0 Td[(1)2c42#iD(x)]TJ /F3 11.955 Tf 9.97 0 Td[(x0) p )]TJ ET q .478 w 380.3 -459.17 m 387.06 -459.17 l S Q BT /F3 11.955 Tf 380.3 -466.49 Td[(g. (4) WecanentirelyabsorbF1bymakingthechoices, c1=HD)]TJ /F7 7.97 Tf 6.59 0 Td[(4(D 2)]TJ /F8 11.955 Tf 9.97 0 Td[(1)\(D 2) (4)D 2f1a (D)]TJ /F8 11.955 Tf 9.96 0 Td[(4)(D)]TJ /F8 11.955 Tf 9.96 0 Td[(3)=HD)]TJ /F7 7.97 Tf 6.59 0 Td[(4\(D 2) 16(4)D 2(D)]TJ /F8 11.955 Tf 9.96 0 Td[(2) (D)]TJ /F8 11.955 Tf 9.97 0 Td[(4)(D)]TJ /F8 11.955 Tf 9.97 0 Td[(3)(D)]TJ /F8 11.955 Tf 9.97 0 Td[(1)2, (4) 62

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c3=HD)]TJ /F7 7.97 Tf 6.59 0 Td[(4(D 2)]TJ /F8 11.955 Tf 9.96 0 Td[(1)\(D 2) (4)D 2)]TJ /F8 11.955 Tf 21.25 0 Td[(2D)]TJ /F8 11.955 Tf 9.96 0 Td[(1 D)]TJ /F8 11.955 Tf 9.96 0 Td[(2f1b D)]TJ /F8 11.955 Tf 9.96 0 Td[(4=HD)]TJ /F7 7.97 Tf 6.58 0 Td[(4\(D 2) 16(4)D 2)]TJ /F8 11.955 Tf 35.06 8.09 Td[(2D(D2)]TJ /F8 11.955 Tf 9.96 0 Td[(5D+2) (D)]TJ /F8 11.955 Tf 9.96 0 Td[(4)(D)]TJ /F8 11.955 Tf 9.96 0 Td[(3)(D)]TJ /F8 11.955 Tf 9.96 0 Td[(1), (4) c4=HD)]TJ /F7 7.97 Tf 6.59 0 Td[(4(D 2)]TJ /F8 11.955 Tf 9.96 0 Td[(1)\(D 2) (4)D 24(D)]TJ /F8 11.955 Tf 9.97 0 Td[(1)2 D(D)]TJ /F8 11.955 Tf 9.97 0 Td[(2)"D(D+2)f1b)]TJ /F8 11.955 Tf 9.96 0 Td[(4f1c 4(D)]TJ /F8 11.955 Tf 9.96 0 Td[(4)#=HD)]TJ /F7 7.97 Tf 6.58 0 Td[(4\(D 2) 16(4)D 2)]TJ /F3 11.955 Tf 22.45 8.09 Td[(D(D3)]TJ /F8 11.955 Tf 9.97 0 Td[(11D2+24D+12) (D)]TJ /F8 11.955 Tf 9.97 0 Td[(6)(D)]TJ /F8 11.955 Tf 9.97 0 Td[(3)(D)]TJ /F8 11.955 Tf 9.97 0 Td[(2). (4) Thelinearcombinations( 4 )and( 4 )arenite, )]TJ /F8 11.955 Tf 9.3 0 Td[(2(D)]TJ /F8 11.955 Tf 9.96 0 Td[(1)Dc1+c3=HD)]TJ /F7 7.97 Tf 6.59 0 Td[(4\(D 2) 16(4)D 2)]TJ /F8 11.955 Tf 9.3 0 Td[(2D2 (D)]TJ /F8 11.955 Tf 9.96 0 Td[(3)(D)]TJ /F8 11.955 Tf 9.96 0 Td[(1), (4) (D)]TJ /F8 11.955 Tf 9.96 0 Td[(1)2D2c1)]TJ /F8 11.955 Tf 11.95 0 Td[((D)]TJ /F8 11.955 Tf 9.97 0 Td[(2)(D)]TJ /F8 11.955 Tf 9.97 0 Td[(1)c3+c4=HD)]TJ /F7 7.97 Tf 6.59 0 Td[(4\(D 2) 16(4)D 2D(D3)]TJ /F8 11.955 Tf 9.96 0 Td[(6D2+8D)]TJ /F8 11.955 Tf 9.97 0 Td[(24) (D)]TJ /F8 11.955 Tf 9.96 0 Td[(6)(D)]TJ /F8 11.955 Tf 9.96 0 Td[(3). (4) ThereforeneithertheNewtonconstantnorthecosmologicalconstantrequiresadivergentrenormalization,althoughwearefreetocontinuemakingtheniterenormalizationsoftheseconstantswhichareimpliedbyequations( 4 )-( 4 ).ItremainstoacttheD=4Green'sfunction( 4 )twiceontherenormalizedremainderterminexpression( 4 ).Theresultis, "1 H2+4#2()]TJ /F8 11.955 Tf 10.49 8.08 Td[(4 34 ylny 4)]TJ /F8 11.955 Tf 11.16 8.08 Td[(8 34 y)]TJ /F8 11.955 Tf 11.16 8.08 Td[(8 3ln2y 4+8lny 4)]TJ /F8 11.955 Tf 11.16 8.08 Td[(1 3)=)]TJ /F8 11.955 Tf 10.49 8.09 Td[(1 3y 4ln2y 4+1 3y 4lny 4)]TJ /F8 11.955 Tf 16.77 8.09 Td[(7 540(122+265)y 4+842)]TJ /F8 11.955 Tf 11.95 0 Td[(131 1080+1 9y 4ln(y 4))]TJ /F8 11.955 Tf 16.29 8.09 Td[(1 45ln(y 4)+1 454 4)]TJ /F3 11.955 Tf 11.95 0 Td[(yln(y 4))]TJ /F8 11.955 Tf 13.63 8.09 Td[(1 30(2)]TJ /F3 11.955 Tf 11.95 0 Td[(y)7Li2(1)]TJ /F3 11.955 Tf 13.15 8.09 Td[(y 4))]TJ /F8 11.955 Tf 11.95 0 Td[(2Li2(y 4)+5ln(1)]TJ /F3 11.955 Tf 13.15 8.09 Td[(y 4)ln(y 4)+43 2164 4)]TJ /F3 11.955 Tf 11.95 0 Td[(y)]TJ /F8 11.955 Tf 13.15 8.08 Td[(5 6y 4ln(1)]TJ /F3 11.955 Tf 13.15 8.08 Td[(y 4))]TJ /F8 11.955 Tf 16.28 8.08 Td[(1 20ln(1)]TJ /F3 11.955 Tf 13.15 8.08 Td[(y 4)+7 904 yln(1)]TJ /F3 11.955 Tf 13.15 8.08 Td[(y 4). (4) 63

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HereLi2(z)isthedilogarithmfunction, Li2(z))]TJ /F12 11.955 Tf 23.91 16.27 Td[(Zz0dtln(1)]TJ /F3 11.955 Tf 9.96 0 Td[(t) t=1Xk=1zk k2.(4)Henceournalresultfortherenormalizedspinzerostructurefunctionis, F1R=2H4 (4)4( H2"1 724 ylny 4#)]TJ /F8 11.955 Tf 14.3 8.08 Td[(1 124 ylny 4+1 724 y+1 6ln2y 4+1 454 4)]TJ /F3 11.955 Tf 11.96 0 Td[(yln(y 4))]TJ /F8 11.955 Tf 16.29 8.09 Td[(1 45ln(y 4)+43 2164 4)]TJ /F3 11.955 Tf 11.95 0 Td[(y)]TJ /F8 11.955 Tf 13.15 8.09 Td[(5 6y 4ln(1)]TJ /F3 11.955 Tf 13.15 8.09 Td[(y 4)+7 904 yln(1)]TJ /F3 11.955 Tf 13.15 8.08 Td[(y 4))]TJ /F8 11.955 Tf 16.29 8.08 Td[(1 20ln(1)]TJ /F3 11.955 Tf 13.15 8.08 Td[(y 4))]TJ /F8 11.955 Tf 13.15 8.08 Td[(7(122+265) 540y 4+842)]TJ /F8 11.955 Tf 11.95 0 Td[(131 1080)]TJ /F8 11.955 Tf 13.15 8.09 Td[(1 3y 4ln2y 4+4 9y 4lny 4)]TJ /F8 11.955 Tf 13.63 8.09 Td[(1 30(2)]TJ /F3 11.955 Tf 11.95 0 Td[(y)7Li2(1)]TJ /F3 11.955 Tf 13.15 8.09 Td[(y 4))]TJ /F8 11.955 Tf 11.96 0 Td[(2Li2(y 4)+5ln(1)]TJ /F3 11.955 Tf 13.15 8.09 Td[(y 4)ln(y 4)). (4) 4.5RenormalizingtheSpinTwoStructureFunctionRecalltheform( 4 )weobtainedforthespintwostructurefunction, F2(y)=Q2(y)+1 DR2(y),(4)wherethesecondorderdifferentialoperatorDwasdenedin( 4 ).RecallalsothatthequotientQ2(y)andtheremainderR2(y)aregiveninrelations( 4 )-( 4 ).TheseexpressionimplythatF2harborsthesamesortofultravioletdivergencesasF1: Thefactorof(4 y)D)]TJ /F7 7.97 Tf 6.58 0 Td[(2inQ2,whichhasanitecoefcientbutisstillnotintegrableinD=4dimensions; Thefactorsof1 D)]TJ /F7 7.97 Tf 6.59 0 Td[(4(4 y)D)]TJ /F7 7.97 Tf 6.59 0 Td[(3inQ2andR2whichareintegrableinD=4dimensionsbuthavedivergentcoefcientsthatprecludetakingtheunregulatedlimits;and Thefactorsof(1 D)]TJ /F7 7.97 Tf 6.58 0 Td[(4)2(4 y)D)]TJ /F7 7.97 Tf 6.58 0 Td[(4inQ2andR2whichareintegrableinD=4dimensionsbuthaveevenmoredivergentcoefcients.Onlytheleadingdivergencerequiresanewcounterterm.Itishandledbyrstextractinganotherderivativeandthenaddingzerointheform( 4 ),justaswedidin 64

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equations( 4 )and( 4 ).Thenalresultis, )]TJ /F3 11.955 Tf 9.3 0 Td[(Kf2a4 yD)]TJ /F7 7.97 Tf 6.58 0 Td[(2=2H4 (4)4( H2"1 2404 ylny 4#)]TJ /F8 11.955 Tf 17.44 8.09 Td[(1 1204 ylny 4+1 2404 y)+O(D)]TJ /F8 11.955 Tf 9.96 0 Td[(4))]TJ /F6 11.955 Tf 13.15 8.78 Td[(2HD)]TJ /F7 7.97 Tf 6.58 0 Td[(4\(D 2) 16(4)D 24iD(x)]TJ /F3 11.955 Tf 9.97 0 Td[(x0)=p )]TJ ET q .478 w 343.09 -70.37 m 349.85 -70.37 l S Q BT /F3 11.955 Tf 343.09 -77.69 Td[(g (D)]TJ /F8 11.955 Tf 9.96 0 Td[(4)(D)]TJ /F8 11.955 Tf 9.96 0 Td[(3)(D)]TJ /F8 11.955 Tf 9.96 0 Td[(2)(D)]TJ /F8 11.955 Tf 9.96 0 Td[(1)(D+1). (4) Comparingexpressions( 4 )and( 4 )impliesthatthedivergentpartcanbeentirelyabsorbedbychoosingthecoefcientc2oftheWeylcounterterm( 4 )tobe, c2=HD)]TJ /F7 7.97 Tf 6.59 0 Td[(4\(D 2) 16(4)D 22 (D)]TJ /F8 11.955 Tf 9.96 0 Td[(4)(D)]TJ /F8 11.955 Tf 9.96 0 Td[(3)2(D)]TJ /F8 11.955 Tf 9.96 0 Td[(1)(D+1).(4)Ofcoursethedivergentpartagreeswith[ 9 ].ItturnsoutthatthelowerdivergencesofF2arecanceledbythethreefactorsweaddedtoQ1tocancelitslowerdivergences, Q1=K(f1b D)]TJ /F8 11.955 Tf 9.97 0 Td[(44 yD 2)]TJ /F7 7.97 Tf 6.59 0 Td[(1+2f1c (D)]TJ /F8 11.955 Tf 9.96 0 Td[(4)24 yD 2)]TJ /F7 7.97 Tf 6.59 0 Td[(2)]TJ /F3 11.955 Tf 27.14 8.09 Td[(f1c (D)]TJ /F8 11.955 Tf 9.96 0 Td[(4)2).(4)ThesechangesinQ1inducechangesinthesourcetermuponwhichweactD)]TJ /F7 7.97 Tf 6.58 0 Td[(1togetF2, SD)]TJ /F8 11.955 Tf 9.96 0 Td[(1 D+1((D)]TJ /F8 11.955 Tf 9.96 0 Td[(1)(2)]TJ /F3 11.955 Tf 9.96 0 Td[(y)(4y)]TJ /F3 11.955 Tf 9.97 0 Td[(y2)Q0001)]TJ /F3 11.955 Tf 11.96 0 Td[(D(D)]TJ /F8 11.955 Tf 9.96 0 Td[(1)(4y)]TJ /F3 11.955 Tf 9.96 0 Td[(y2)Q001+4(D2)]TJ /F8 11.955 Tf 9.96 0 Td[(3)Q001+(D)]TJ /F8 11.955 Tf 9.96 0 Td[(1)2(2)]TJ /F3 11.955 Tf 9.96 0 Td[(y)Q01+(D)]TJ /F8 11.955 Tf 9.97 0 Td[(1)2Q1), (4) =K(sb D)]TJ /F8 11.955 Tf 9.97 0 Td[(44 yD 2+1+sc D)]TJ /F8 11.955 Tf 9.97 0 Td[(44 yD 2+sd D)]TJ /F8 11.955 Tf 9.97 0 Td[(44 yD 2)]TJ /F7 7.97 Tf 6.58 0 Td[(1+se (D)]TJ /F8 11.955 Tf 9.96 0 Td[(4)24 yD 2)]TJ /F7 7.97 Tf 6.59 0 Td[(2+se0 (D)]TJ /F8 11.955 Tf 9.96 0 Td[(4)2). (4) Herethecoefcientsare, sb=)]TJ /F8 11.955 Tf 13.64 8.08 Td[(1 16(D)]TJ /F8 11.955 Tf 9.97 0 Td[(2)(D)]TJ /F8 11.955 Tf 9.97 0 Td[(1)Df1b, (4) sc=(D)]TJ /F8 11.955 Tf 9.96 0 Td[(2)(D)]TJ /F8 11.955 Tf 9.96 0 Td[(1) 16(D+1)h)]TJ /F8 11.955 Tf 9.3 0 Td[((D)]TJ /F8 11.955 Tf 9.96 0 Td[(1)(D2)]TJ /F8 11.955 Tf 9.96 0 Td[(2D)]TJ /F8 11.955 Tf 9.97 0 Td[(4)f1b+2(D)]TJ /F8 11.955 Tf 9.96 0 Td[(3)f1ci, (4) sd=(D)]TJ /F8 11.955 Tf 9.96 0 Td[(1)2 8(D+1)hD3f1b)]TJ /F8 11.955 Tf 11.96 0 Td[((D2+2D)]TJ /F8 11.955 Tf 9.96 0 Td[(4)f1ci, (4) 65

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se=(D)]TJ /F8 11.955 Tf 11.96 0 Td[(2)2(D)]TJ /F8 11.955 Tf 9.96 0 Td[(1)2(D+2) 4(D+1)f1c, (4) se0=)]TJ /F8 11.955 Tf 10.49 8.09 Td[((D)]TJ /F8 11.955 Tf 9.96 0 Td[(1)3 (D+1)f1c. (4) ToinferthecorrespondingchangesinthespintwoquotientandremainderweneedtoinvertDon(4 y)D 2+1,(4 y)D 2and1.Thesecondonewasgivenin( 4 ).Fromexpression( 4 )wend, 1 D4 yD 2+1=16 (D)]TJ /F8 11.955 Tf 9.96 0 Td[(2)2D4 yD 2)]TJ /F7 7.97 Tf 6.58 0 Td[(1)]TJ /F8 11.955 Tf 12.13 8.09 Td[(4 D((D)]TJ /F8 11.955 Tf 9.96 0 Td[(3)(D+2) (D)]TJ /F8 11.955 Tf 9.97 0 Td[(2)D4 yD 2+(D)]TJ /F8 11.955 Tf 9.97 0 Td[(3)(D+2)(D+4) (D)]TJ /F8 11.955 Tf 9.96 0 Td[(2)2D4 yD 2)]TJ /F7 7.97 Tf 6.58 0 Td[(1), (4) 1 D1=1 (D)]TJ /F8 11.955 Tf 9.96 0 Td[(3)D(D+1). (4) Althoughwewanttomoveallthe(4 y)D 2+1and(4 y)D 2termsfromtheremaindertothequotient,wemustallowforanarbitraryamountf2c0ofthe1term.Hencethechangesinthequotientandtheremaindertaketheform, Q2=K(f2b D)]TJ /F8 11.955 Tf 9.97 0 Td[(44 yD 2)]TJ /F7 7.97 Tf 6.59 0 Td[(1+f2c (D)]TJ /F8 11.955 Tf 9.96 0 Td[(4)24 yD 2)]TJ /F7 7.97 Tf 6.59 0 Td[(2+f2c0 (D)]TJ /F8 11.955 Tf 9.96 0 Td[(4)2), (4) R2=K(f2d D)]TJ /F8 11.955 Tf 9.97 0 Td[(44 yD 2)]TJ /F7 7.97 Tf 6.59 0 Td[(1+f2e (D)]TJ /F8 11.955 Tf 9.96 0 Td[(4)24 yD 2)]TJ /F7 7.97 Tf 6.59 0 Td[(2+fe0 (D)]TJ /F8 11.955 Tf 9.96 0 Td[(4)2). (4) Thevariouscoefcientsare, f2b=16 (D)]TJ /F8 11.955 Tf 9.96 0 Td[(2)2Dsb, (4) f2c=)]TJ /F8 11.955 Tf 10.5 8.09 Td[(64(D)]TJ /F8 11.955 Tf 9.96 0 Td[(3)(D+2) (D)]TJ /F8 11.955 Tf 9.96 0 Td[(2)3Dsb+16 (D)]TJ /F8 11.955 Tf 9.97 0 Td[(2)2sc, (4) f2d=4(D)]TJ /F8 11.955 Tf 9.97 0 Td[(3)(D+2)(D+4)(3D)]TJ /F8 11.955 Tf 9.96 0 Td[(10) (D)]TJ /F8 11.955 Tf 9.96 0 Td[(2)3Dsb)]TJ /F8 11.955 Tf 10.49 8.08 Td[(4(D)]TJ /F8 11.955 Tf 9.96 0 Td[(3)(D+4) (D)]TJ /F8 11.955 Tf 9.96 0 Td[(2)2sc+sd, (4) 66

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f2e=16(D)]TJ /F8 11.955 Tf 9.96 0 Td[(3)2(D+2)(D+4)(D+6) (D)]TJ /F8 11.955 Tf 9.96 0 Td[(2)3Dsb)]TJ /F8 11.955 Tf 10.49 8.08 Td[(4(D)]TJ /F8 11.955 Tf 9.96 0 Td[(3)(D+4)(D+6) (D)]TJ /F8 11.955 Tf 9.97 0 Td[(2)2sc+se, (4) f2e0=)]TJ /F8 11.955 Tf 9.3 0 Td[((D)]TJ /F8 11.955 Tf 9.96 0 Td[(3)D(D+1)f2c0+se0. (4) ItispossibletomakethecombinationQ2+Q2possessaniteunregulatedlimitbychoosing, f2c0=1)]TJ /F8 11.955 Tf 13.15 8.09 Td[(271 60(D)]TJ /F8 11.955 Tf 9.97 0 Td[(4)+11057 3600(D)]TJ /F8 11.955 Tf 9.96 0 Td[(4)2.(4)Withthischoicetherenormalizedspintwoquotientis, Q2R=2H4 (4)4( H2"1 2404 ylny 4#)]TJ /F8 11.955 Tf 17.43 8.08 Td[(1 1204 ylny 4+1 2404 y+1 124 ylny 4)]TJ /F8 11.955 Tf 16.29 8.09 Td[(7 304 y+1 4ln2y 4)]TJ /F8 11.955 Tf 13.15 8.09 Td[(119 60lny 4). (4) Choosing( 4 )alsoproducesaniteresultforthespintworemainderterm, R2R=2H4 (4)4(17 104 ylny 4)]TJ /F8 11.955 Tf 13.15 8.09 Td[(149 304 y)]TJ /F8 11.955 Tf 13.15 8.09 Td[(41 10ln2y 4+193 6lny 4+359 20+32 15(4)]TJ /F3 11.955 Tf 11.96 0 Td[(y)390y 44)]TJ /F8 11.955 Tf 11.96 0 Td[(291y 43+333y 42)]TJ /F8 11.955 Tf 11.95 0 Td[(152y 4+214 yln(y 4)+4 45(4)]TJ /F3 11.955 Tf 11.95 0 Td[(y)3432y 43)]TJ /F8 11.955 Tf 11.95 0 Td[(792y 42)]TJ /F8 11.955 Tf 11.96 0 Td[(288y 4+991)]TJ /F8 11.955 Tf 11.96 0 Td[(4744 y)]TJ /F8 11.955 Tf 11.95 0 Td[(844 y2)]TJ /F8 11.955 Tf 16.29 8.09 Td[(7 60(4 y)3ln(1)]TJ /F3 11.955 Tf 13.15 8.09 Td[(y 4))]TJ /F8 11.955 Tf 16.29 8.09 Td[(9 10ln2(y 4)). (4) ActingtheD=4Green'sfunction( 4 )ontheremainderandaddingtheresulttothequotientgivesournalresultfortherenormalizedspintwostructurefunction(recallthedenition( 4 )ofthedilogarithmfunction), 67

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F2R=2H4 (4)4( H2"1 2404 yln(y 4#+3 404 ylny 4)]TJ /F8 11.955 Tf 11.16 8.09 Td[(11 484 y+1 4ln2y 4)]TJ /F8 11.955 Tf 10.49 8.09 Td[(119 60lny 4+4096 (4y)]TJ /F3 11.955 Tf 11.95 0 Td[(y2)]TJ /F8 11.955 Tf 11.96 0 Td[(8)4")]TJ /F8 11.955 Tf 10.49 8.09 Td[(47 15y 48+141 10y 47)]TJ /F8 11.955 Tf 10.49 8.08 Td[(2471 90y 46+34523 720y 45)]TJ /F8 11.955 Tf 13.15 8.08 Td[(132749 1440y 44+38927 320y 43)]TJ /F8 11.955 Tf 10.49 8.08 Td[(10607 120y 42+22399 720y 4)]TJ /F8 11.955 Tf 13.15 8.08 Td[(3779 9604 4)]TJ /F3 11.955 Tf 11.96 0 Td[(y+193 30y 44)]TJ /F8 11.955 Tf 13.15 8.08 Td[(131 10y 43+7 20y 42+379 60y 4)]TJ /F8 11.955 Tf 13.15 8.09 Td[(193 120ln(2)]TJ /F3 11.955 Tf 13.15 8.09 Td[(y 2)+)]TJ /F8 11.955 Tf 10.5 8.09 Td[(14 15y 45)]TJ /F8 11.955 Tf 13.15 8.09 Td[(1 5y 44+19 2y 43)]TJ /F8 11.955 Tf 13.15 8.09 Td[(889 60y 42+143 20y 4)]TJ /F8 11.955 Tf 13.15 8.09 Td[(13 20)]TJ /F8 11.955 Tf 16.29 8.09 Td[(7 604 yln(1)]TJ /F3 11.955 Tf 13.15 8.09 Td[(y 4)+)]TJ /F8 11.955 Tf 10.49 8.09 Td[(476 15y 49+160y 48)]TJ /F8 11.955 Tf 13.15 8.09 Td[(5812 15y 47+8794 15y 46)]TJ /F8 11.955 Tf 10.5 8.09 Td[(18271 30y 45+54499 120y 44)]TJ /F8 11.955 Tf 13.15 8.09 Td[(59219 240y 43+1917 20y 42)]TJ /F8 11.955 Tf 10.5 8.09 Td[(1951 80y 4+367 1204 4)]TJ /F3 11.955 Tf 11.96 0 Td[(yln(y 4)+4y 47)]TJ /F8 11.955 Tf 11.96 0 Td[(12y 46+20y 45)]TJ /F8 11.955 Tf 9.3 0 Td[(20y 44+15y 43)]TJ /F8 11.955 Tf 11.95 0 Td[(7y 42+y 44)]TJ /F3 11.955 Tf 11.95 0 Td[(y 4ln2(y 4)+367 30y 44)]TJ /F8 11.955 Tf 13.15 8.09 Td[(4121 120y 43+237 16y 42+1751 240y 4)]TJ /F8 11.955 Tf 13.15 8.09 Td[(367 120ln(y 2)+1 64(y2)]TJ /F8 11.955 Tf 11.96 0 Td[(8)h4(2)]TJ /F3 11.955 Tf 11.96 0 Td[(y))]TJ /F8 11.955 Tf 11.96 0 Td[((4y)]TJ /F3 11.955 Tf 11.95 0 Td[(y2)i1 5Li2(1)]TJ /F3 11.955 Tf 13.15 8.09 Td[(y 4)+7 10Li2(y 4)#). (4) 68

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CHAPTER5FLATSPACERESULT 5.1Schwinger-KeldyshEffectiveFieldEqnsThegravitonself-energy)]TJ /F3 11.955 Tf 9.29 0 Td[(i[](x;x0)istheone-particle-irreducible(1PI)2-pointfunctionforthegravitoneldh(t,~x).ItservestoquantumcorrectthelinearizedEinsteinequation. Dh(x)+Zd4x0hi(x;x0)h(x0)=8G c2M003(~x).(5)However,thisequationsuffersfromtwoembarrassments: Itisn'tcausalbecausethein-outself-energyisnonzeroforpointsx0whicharespacelikeseparatedfromx,orlietoitsfuture;and Itdoesn'tproducerealpotentialshbecausethein-outself-energyhasanimaginarypart.Onecangettherightresultforastaticpotentialbysimplyignoringtheimaginarypart[ 76 77 79 ],butcircumventingthelimitationsofthein-outformalismbecomesmoreandmoredifcultastimedependentsourcesandhigherordercorrectionsareincluded,andthesetechniquesbreakdownentirelyforthecaseofcosmologyinwhichtheremaynotevenbeasymptoticvacua.Itisnotthatthein-outself-energyissomehowwrong.Infact,itisexactlytherightthingtocorrecttheFeynmanpropagatorforasymptoticscatteringcomputationsinatspace.Thepointisratherthatequation( 5 )doesn'tprovidethegeneralizationweseekoftheclassicaleldequation.ThebettertechniqueisknownastheSchwinger-Keldyshformalism[ 84 ].ItprovidesawayofcomputingtrueexpectationvaluesthatisalmostassimpleastheFeynmandiagramswhichproducein-outmatrixelements.TheSchwinger-Keldyshrulesarebeststatedinthecontextofascalareld'(x)whoseLagrangian(thespaceintegralofitsLagrangiandensity)attimetisL['(t)].SupposewearegivenaHeisenbergstatejiwhosewavefunctionalintermsoftheoperatoreigenketsattimet0is['(t0)],andwewishtotaketheexpectationvalue,inthepresenceofthisstate,ofaproductoftwo 69

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functionalsoftheeldoperator:A['],whichisanti-time-ordered,andB['],whichistime-ordered.TheSchwinger-Keldyshfunctionalintegralforthisis[ 61 ], DA[']B[']E=%&[d'+][d')]TJ /F8 11.955 Tf 6.26 1.79 Td[(]h')]TJ /F8 11.955 Tf 6.25 1.79 Td[((t1))]TJ /F6 11.955 Tf 9.97 0 Td[('+(t1)iA[')]TJ /F8 11.955 Tf 6.25 1.79 Td[(]B['+][')]TJ /F8 11.955 Tf 6.25 1.79 Td[((t0)]eiRt1t0dtnL['+(t)])]TJ /F5 7.97 Tf 6.59 0 Td[(L[')]TJ /F7 7.97 Tf 6.25 1.08 Td[((t)]o['+(t0)]. (5) Thetimet1>t0isarbitraryaslongasitislaterthanthelatestoperatorwhichiscontainedineitherA[']orB['].TheSchwinger-Keldyshrulescanbereadofffromitsfunctionalrepresentation( 5 ).Becausethesameeldoperatorisrepresentedbytwodifferentdummyfunctionalvariables,'(x),theendpointsoflinescarryapolarity.Externallinesassociatedwiththeanti-time-orderedoperatorA[']havethe)]TJ /F1 11.955 Tf 9.08 0 Td[(polaritywhereasthoseassociatedwiththetime-orderedoperatorB[']havethe+polarity.Interactionverticesareeitherall+orall)]TJ /F1 11.955 Tf 5.76 0 Td[(.Verticeswith+polarityarethesameasintheusualFeynmanruleswhereasverticeswiththe)]TJ /F1 11.955 Tf 9.08 0 Td[(polarityhaveanadditionalminussign.Ifthestatejiissomethingotherthanfreevacuumthenitcontributesadditionalinteractionverticesontheinitialvaluesurface[ 67 ].Propagatorscanbe++,+)]TJ /F1 11.955 Tf 5.76 0 Td[(,)]TJ /F17 5.978 Tf 5.75 0 Td[(+,or\000.Allfourpolarityvariationscanbereadofffromthefundamentalrelation( 5 )whenthefreeLagrangianissubstitutedforthefullone.Itisusefultodenotecanonicalexpectationvaluesinthefreetheorywithasubscript0.Withthisconventionweseethatthe++propagatorisjusttheordinaryFeynmanpropagator, i++(x;x0)=DT'(x)'(x0)E0=i(x;x0),(5)whereTstandsfortime-orderingand Tdenotesanti-time-ordering.TheotherpolarityvariationsaresimpletoreadoffandtorelatetotheFeynmanpropagator, 70

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i)]TJ /F17 5.978 Tf 5.75 0 Td[(+(x;x0)=D'(x)'(x0)E0=(t)]TJ /F3 11.955 Tf 9.96 0 Td[(t0)i(x;x0)+(t0)]TJ /F3 11.955 Tf 9.96 0 Td[(t)hi(x;x0)i, (5) i+)]TJ /F8 11.955 Tf 6.25 1.8 Td[((x;x0)=D'(x0)'(x)E0=(t)]TJ /F3 11.955 Tf 9.96 0 Td[(t0)hi(x;x0)i+(t0)]TJ /F3 11.955 Tf 9.96 0 Td[(t)i(x;x0), (5) i\000(x;x0)=D T'(x)'(x0)E0=hi(x;x0)i. (5) ThereforewecangetthefourpropagatorsoftheSchwinger-KeldyshformalismfromtheFeynmanpropagatoroncethatisknown.Becauseexternallinescanbeeither+or)]TJ /F1 11.955 Tf 9.08 0 Td[(intheSchwinger-Keldyshformalism,every1PIN-pointfunctionofthein-outformalismgivesriseto2N1PIN-pointfunctionsintheSchwinger-Keldyshformalism.Foreveryclassicaleld(x)ofanin-outeffectiveaction,thecoorrespondingSchwinger-Keldysheffectiveactionmustdependupontwoeldscallthem+(x)and)]TJ /F8 11.955 Tf 6.25 1.79 Td[((x)inordertoaccesstheappropriate1PIfunction[ 85 ].Forthescalarparadigmwehavebeenconsideringthiseffectiveactiontakestheform, [+,)]TJ /F8 11.955 Tf 6.26 1.79 Td[(]=S[+])]TJ /F3 11.955 Tf 11.96 0 Td[(S[)]TJ /F8 11.955 Tf 6.25 1.79 Td[(])]TJ /F8 11.955 Tf 13.15 8.09 Td[(1 2Zd4xZd4x08><>:+(x)M2++(x;x0)+(x0)++(x)M2+)]TJ /F8 11.955 Tf 4.27 2.96 Td[((x;x0))]TJ /F8 11.955 Tf 6.25 1.8 Td[((x0)+)]TJ /F8 11.955 Tf 6.26 1.79 Td[((x)M2)]TJ /F17 5.978 Tf 5.76 0 Td[(+(x;x0)+(x0)+)]TJ /F8 11.955 Tf 6.26 1.79 Td[((x)M2\000(x;x0))]TJ /F8 11.955 Tf 6.25 1.79 Td[((x0)9>=>;+O(3), (5) whereSistheclassicalaction.Theeffectiveeldequationsareobtainedbyvaryingwithrespectto+andthensettingbotheldsequal[ 85 ], [+,)]TJ /F8 11.955 Tf 6.25 1.79 Td[(] +(x)==h@2)]TJ /F3 11.955 Tf 11.96 0 Td[(m2i(x))]TJ /F12 11.955 Tf 9.97 16.27 Td[(Zd4x0hM2++(x;x0)+M2+)]TJ /F8 11.955 Tf 4.26 2.95 Td[((x;x0)i(x0)+O(2).(5)Thetwo1PI2-pointfunctionswewouldneedtoquantumcorrectthelinearizedscalareldequationareM2++(x;x0)andM2+)]TJ /F8 11.955 Tf 4.26 2.95 Td[((x;x0).Theirsumin( 5 )giveseffectiveeldequationswhicharecausalinthesensethatthetwo1PIfunctionscancelunlessx0liesonorwithinthepastlight-coneofx.Theirsumisalsoreal,whichneither1PIfunctionisseparately. 71

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Asmentionedbefore,thepointofthepresentpaperistolaythegroundworkforacomputationoftheoneloopcorrectiontotheforceofgravityondeSitterbackground.Thegravitoncontributiontotheself-energywascomputedsomeyearsago[ 86 ]buthasneverbeenusedintheeffectiveeldequations.Acomputationofthescalarcontributionisunderway.Althoughthecurrentcomputationwillbethersttoexplorecorrectionstoaforcelaw,thelinearizedeffectiveeldequationshavebeenstudiedondeSitterbackgroundformanysimplermodels.Inscalarquantumelectrodynamicstheoneloopvacuumpolarizationwascomputedandusedtocorrectforthepropagationofdynamicalphotons[ 49 52 ],butnotyetfortheCoulombforce.Theoneloopscalarself-mass-squaredhasalsobeenusedtocorrectforthepropagationofcharged,massless,minimallycoupledscalars[ 87 ].Boththefermion[ 55 ]andscalar[ 88 ]1PI2-pointfunctionsofYukawatheoryhavebeencomputedandusedtocorrectthemodefunctions.Theoneandtwoloopscalarself-mass-squaredof'4theoryhasbeencomputedandusedtocorrectforthepropagationofmassless,minimallycoupledscalars[ 57 ].InEinstein+Diractheoneloopfermionself-energyhasbeencomputedandusedtocorrectthefermionmodefunction[ 58 ].AndthesamethinghasbeendoneforscalarsinScalar+Einstein[ 59 ]. 5.2SolvingforthePotentialsWebeginbyexpressingthelinearizedeffectiveeldequationsinaformwhichisbothmanifestlyrealandcausal.Wethenexplainhowtheseequationscanbesolvedperturbatively.Thehardeststepisintegratingtheoneloopself-energyagainstthetreeordersolution.Thesectionclosesbyworkingoutthetwoonelooppotentials. 5.2.1AchievingAManifestlyRealandCausalFormThebasisforourworkisapositionspaceresultfortheoneloopcontributiontothe1PIgraviton2-pointfunctionfromaloopofmassless,minimallycoupledscalars,usingdimensionalregularizationandaminimalchoiceforthehigherderivativecounterterms[ 61 ].(PreviousSchwinger-Keldyshcomputationsofthisquantityhadbeengivenin 72

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momentumspace[ 89 ]whichisnotasusefulforus.)Allfourpolarizationvariationstaketheform, hi(x;x0)=D(x;x0),(5)wherethe4th-order,tensor-differentialoperatoris, Dh@2)]TJ /F6 11.955 Tf 9.96 0 Td[(@@ih@2)]TJ /F6 11.955 Tf 9.96 0 Td[(@@i+1 3h()@4)]TJ /F8 11.955 Tf 9.96 0 Td[(2@()(@)+@@@@i.(5)Thefourbi-scalarsare, (x;x0)=()()i2@2 51204"ln(2x2) x2#,(5)where2istheusualscaleofdimensionalregularization,216~G=c3istheloop-countingparameterofquantumgravityandthefourSchwinger-Keldyshlengthfunctionsare, x2++(x;x0)~x)]TJ /F6 11.955 Tf 10.83 2.74 Td[(~x02)]TJ /F3 11.955 Tf 11.95 0 Td[(c2jt)]TJ /F3 11.955 Tf 9.97 0 Td[(t0j)]TJ /F3 11.955 Tf 13.94 0 Td[(i2, (5) x2+)]TJ /F8 11.955 Tf 6.26 2.96 Td[((x;x0)~x)]TJ /F6 11.955 Tf 10.83 2.73 Td[(~x02)]TJ /F3 11.955 Tf 11.95 0 Td[(c2t)]TJ /F3 11.955 Tf 9.97 0 Td[(t0+i2, (5) x2)]TJ /F17 5.978 Tf 5.76 0 Td[(+(x;x0)~x)]TJ /F6 11.955 Tf 10.83 2.73 Td[(~x02)]TJ /F3 11.955 Tf 11.95 0 Td[(c2t)]TJ /F3 11.955 Tf 9.97 0 Td[(t0)]TJ /F3 11.955 Tf 9.96 0 Td[(i2, (5) x2\000(x;x0)~x)]TJ /F6 11.955 Tf 10.83 2.74 Td[(~x02)]TJ /F3 11.955 Tf 11.95 0 Td[(c2jt)]TJ /F3 11.955 Tf 9.97 0 Td[(t0j+i2. (5) Althoughthedivergentpartsof( 5 )havebeensubtractedoff[ 61 ],itshouldbenotedthattheyagreeexactlywiththoseoriginallyfoundby'tHooftandVeltman[ 9 ].Wecanachieveasignicantsimplicationbyrstextractinganotherd'Alembertianfrom( 5 ), (x;x0)=()()i2@4 409604hln2(2x2))]TJ /F8 11.955 Tf 11.95 0 Td[(2ln(2x2)i.(5)Nowdenethepositionandtemporalseparations,andtheassociatedinvariantlength-squared, rk~x)]TJ /F6 11.955 Tf 10.84 2.73 Td[(~x0k,tt)]TJ /F3 11.955 Tf 9.97 0 Td[(t0,x2r2)]TJ /F3 11.955 Tf 11.96 0 Td[(c2t2.(5) 73

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The++and+)]TJ /F1 11.955 Tf 9.08 0 Td[(logarithmscanbeexpandedintermsoftheirrealandimaginaryparts, ln(2x2++)=ln(2jx2j)+i()]TJ /F8 11.955 Tf 9.3 0 Td[(x2), (5) ln(2x2+)]TJ /F8 11.955 Tf 6.26 2.96 Td[()=ln(2jx2j))]TJ /F3 11.955 Tf 11.96 0 Td[(isgn(t)()]TJ /F8 11.955 Tf 9.3 0 Td[(x2). (5) The++and+)]TJ /F1 11.955 Tf 9.08 0 Td[(logarithmsagreeforspacelikeseparation(x2>0),andfort0>t,whereastheyarecomplexconjugatesofoneanotherforx0=(ct0,~x0)inthepastlight-coneofx=(ct,~x).Hencethesumof++(x;x0)and+)]TJ /F8 11.955 Tf 6.26 1.8 Td[((x;x0)isbothcausalandreal, ++(x;x0)++)]TJ /F8 11.955 Tf 6.25 1.8 Td[((x;x0)=)]TJ /F6 11.955 Tf 20.47 8.08 Td[(2@4 102403(ct)]TJ /F8 11.955 Tf 9.96 0 Td[(r)hln()]TJ /F6 11.955 Tf 9.29 0 Td[(2x2))]TJ /F8 11.955 Tf 9.96 0 Td[(1i.(5)Letusassumethatthestateisreleasedinfreevacuumattimet=0.Ournalresultforthelinearized,oneloopeffectiveeldequationsis, Dh(t,~x))]TJ /F6 11.955 Tf 13.15 8.09 Td[(2D@4 102403Zt0dct0Zd3x0(ct)]TJ /F8 11.955 Tf 9.96 0 Td[(r)hln()]TJ /F6 11.955 Tf 9.3 0 Td[(2x2))]TJ /F8 11.955 Tf 9.96 0 Td[(1ih(t0,~x0)=8GM c2003(~x). (5) Recallthat216~G=c3istheloopcountingparameterofquantumgravity,theLichnerowitzoperatorDwasgivenin( 3 )andthe4thorderdifferentialoperatorDwasgivenin( 5 ). 5.2.2SolvingtheEquationPerturbativelyThereisnopointintryingtosolveequation( 5 )exactlybecauseitonlyincludestheoneloopgravitonself-energy.Abetterapproachistoseekaperturbativesolutioninpowersoftheloopcountingparameter2, h(t,~x)=1X`=02`h(`)(t,~x).(5) 74

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Ofcoursethe`=0termobeysthelinearizedEinsteinequationwhosesolutioninSchwarzschildcoordinatesis, Dh(0)(t,~x)=8GM c2003(~x)=)h(0)00=2GM c2r,h(0)ij=2GM c2rbribrj.(5)Theoneloopcorrectionh(1)obeystheequation, Dh(1)(t,~x)=D@4 102403Zt0dt0Zd3x0(t)]TJ /F8 11.955 Tf 9.97 0 Td[(r)hln()]TJ /F6 11.955 Tf 9.3 0 Td[(2x2))]TJ /F8 11.955 Tf 9.97 0 Td[(1ih(0)(t0,~x0).(5)Findingthetwoloopcorrectionh(2)wouldrequirethetwoloopself-energy,whichwedonothave,soh(1)isashighaswecango. 5.2.3CorrectiontoDynamicalGravitonsinFlatSpaceTheoneloopcontributiontothegravitonself-energyfromMMCscalarsinaatbackgroundwasrstcomputedby`tHooftandVeltmanin1974[ 9 ].WhenrenormalizedandexpressedinpositionspaceusingtheSchwinger-Keldyshformalismtheresulttakestheform[ 61 ],(thisresultwasreviewedinsection4.2) hati(x;x0)=F0(x2)+h())]TJ /F8 11.955 Tf 11.16 8.09 Td[(1 3iF2(x2).(5)Here@@)]TJ /F6 11.955 Tf 11.95 0 Td[(@2andthetwostructurefunctionsare, F0(x2)=i2 (4)4@2 9"ln(2x2++) x2++)]TJ /F8 11.955 Tf 13.15 8.45 Td[(ln(2x2+)]TJ /F8 11.955 Tf 6.26 2.95 Td[() x2+)]TJ /F12 11.955 Tf 22.76 31.6 Td[(#, (5) F2(x2)=i2 (4)4@2 60"ln(2x2++) x2++)]TJ /F8 11.955 Tf 13.15 8.45 Td[(ln(2x2+)]TJ /F8 11.955 Tf 6.25 2.95 Td[() x2+)]TJ /F12 11.955 Tf 22.76 31.6 Td[(# (5) Thetwocoordinateintervalsare, x2++~x)]TJ /F6 11.955 Tf 10.84 2.73 Td[(~x02)]TJ /F12 11.955 Tf 11.95 13.27 Td[(jx0)]TJ /F3 11.955 Tf 9.96 0 Td[(x00j)]TJ /F3 11.955 Tf 13.94 0 Td[(i2, (5) x2+)]TJ /F2 11.955 Tf 16.22 2.96 Td[(~x)]TJ /F6 11.955 Tf 10.84 2.73 Td[(~x02)]TJ /F12 11.955 Tf 11.95 13.27 Td[(x0)]TJ /F3 11.955 Tf 9.96 0 Td[(x00+i2. (5) OfcoursethissameformfollowsfromtakingtheatspacelimitofthedeSitterresultsummarizedintheprevioussection. 75

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Inatspace,themodefunctionforaplanewavegravitonwithwavevector~kis, hat(x)=(~k)1 p 2ke)]TJ /F5 7.97 Tf 6.58 0 Td[(ikx0+i~k~x.(5)Theoneloopcorrectiontothis(fromMMCscalars)issourcedby, Source(x)=Zdx4x0hati(x;x0)hat(x0).(5)Itmightseemnaturaltoextractthevariousderivativeswithrespecttoxfromtheintegration,forexample, Zd4x0F0(x2)hat(x0)=i2 (4)4@2 9Zd4x0"ln(2x2++) x2++)]TJ /F8 11.955 Tf 13.15 8.45 Td[(ln(2x2+)]TJ /F8 11.955 Tf 6.26 2.96 Td[() x2+)]TJ /F12 11.955 Tf 22.76 31.6 Td[(#hat(x0). (5) Thatwouldreducethesource( 5 )toatedioussetofintegrations,followedbysomeequallytediousdifferentiations.Thepointofthissub-sectionisthatamoreefcientstrategyistorstconvertallthexderivativestox0derivativeswhichcanbedonebecausetheyactonfunctionsofx2.Thenignoresurfacetermsandpartiallyintegratethex0derivativestoactuponhat(x0).Forexample,doingthisforthespinzerocontribution( 5 )gives, Zd4x0F0(x2)hat(x0))166(!i2 (4)4Zd4x0"ln(2x2++) x2++)]TJ /F8 11.955 Tf 13.15 8.45 Td[(ln(2x2+)]TJ /F8 11.955 Tf 6.25 2.95 Td[() x2+)]TJ /F12 11.955 Tf 22.75 31.6 Td[(#@02 900hat(x0). (5) Becausethegravitonmodefunctionisbothtransverseandtraceless,wehave0hat(x0)=0.Thespintwocontributionisonlyalittlemorecomplicated, Zd4x0h())]TJ /F8 11.955 Tf 13.15 8.09 Td[(1 3iF2(x2)hat(x0))166(!i2 (4)4Zd4x0"ln(2x2++) x2++)]TJ /F8 11.955 Tf 13.15 8.45 Td[(ln(2x2+)]TJ /F8 11.955 Tf 6.25 2.96 Td[() x2+)]TJ /F12 11.955 Tf 22.75 31.6 Td[(#@06 60hat(x0). (5) Thisalsovanishesbecause@02hat(x0)=0. 76

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Inexpressions( 5 )and( 5 )wehaveemployedarightarrow,ratherthananequalssign,becausethesurfacetermsproducebypartialintegrationwereignored.TherearenosurfacetermsatspatialinnityintheSchwinger-Keldyshformalismbecausethe++and+)]TJ /F1 11.955 Tf 12.62 0 Td[(termscancelforspacelikeseparation.The++and+)]TJ /F1 11.955 Tf -418.31 -23.91 Td[(contributionsalsocancelwhenx00>x0,sotherearenofuturesurfaceterms.However,therearenonzerocontributionsfromtheinitialvaluesurface.1Weassumethatallsuchcontributionsareabsorbedintoperturbativecorrectionstotheinitialstate,suchashasrecentlybeenworkedoutforaMMCscalarwithquarticself-interaction[ 102 ]. 5.2.4TheOneLoopSourceTermInthissubsectionweevaluatetherighthandsideofequation( 5 ),whichsourcestheoneloopcorrectionh(1)(t,~x).Thisisdoneinthreesteps:werstperformtheintegral,thenactthe@4,andnallyacttheD.Fromtheformofthetreeorderpotentials( 5 )itisapparentthatweneedtwointegrals.Therstcomesfromh(0)00, Zt0dt0Zd3x0(t)]TJ /F8 11.955 Tf 9.96 0 Td[(r)hln()]TJ /F6 11.955 Tf 9.3 0 Td[(2x2))]TJ /F8 11.955 Tf 9.96 0 Td[(1i1 k~x0kF(t,r).(5)Thesecondintegralderivesfromtheothernonzeropotentialh(0)ij.ItstracepartisobviouslythesameasF(t,r),andweshallcallitstracelesspartG(t,r), Zt0dt0Zd3x0(t)]TJ /F8 11.955 Tf 9.96 0 Td[(r)hln()]TJ /F6 11.955 Tf 9.3 0 Td[(2x2))]TJ /F8 11.955 Tf 9.96 0 Td[(1ibr0ibr0j k~x0k1 2hij)]TJ /F12 11.955 Tf 9.4 .5 Td[(bribrjiF(t,r))]TJ /F8 11.955 Tf 13.15 8.09 Td[(1 2hij)]TJ /F8 11.955 Tf 9.96 0 Td[(3bribrjiG(t,r), (5) =1 3ijF(t,r)+1 2h3bribrj)]TJ /F6 11.955 Tf 9.96 0 Td[(ijihG(t,r))]TJ /F8 11.955 Tf 11.16 8.08 Td[(1 3F(t,r)i. (5) 1Foratwoloopexample,see[ 101 ]. 77

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TheintegralsaretediousbutstraightforwardandgivethefollowingresultsforF(t,r)andthecombinationG(t,r))]TJ /F7 7.97 Tf 13.15 4.7 Td[(1 3F(t,r), F(t,r)=4 r(r4 6ln(2r))]TJ /F8 11.955 Tf 11.16 8.09 Td[(25 72r4+11 18r3ct)]TJ /F8 11.955 Tf 11.16 8.09 Td[(11 18rc3t3+h1 12(ct+r)4)]TJ /F3 11.955 Tf 10.88 8.09 Td[(r 6(r+ct)3ilnh(ct+r)i)]TJ /F12 11.955 Tf 11.95 13.27 Td[(h1 12(ct)]TJ /F3 11.955 Tf 9.96 0 Td[(r)4+r 6(ct)]TJ /F3 11.955 Tf 9.96 0 Td[(r)3ilnh(ct)]TJ /F3 11.955 Tf 9.96 0 Td[(r)i), (5) G(t,r))]TJ /F3 11.955 Tf 11.16 8.09 Td[(F(t,r) 3=4 r()]TJ /F3 11.955 Tf 10.49 8.09 Td[(r4 9ln(2r)+23 108r4)]TJ /F8 11.955 Tf 11.16 8.09 Td[(199 675r3ct)]TJ /F8 11.955 Tf 14.3 8.09 Td[(13 135rc3t3+2c5t5 45r+h)]TJ /F8 11.955 Tf 10.5 8.09 Td[((ct+r)6 45r2+2 15(ct+r)5 r)]TJ /F8 11.955 Tf 14.29 8.09 Td[(5 18(ct+r)4+2 9r(ct+r)3ilnh(ct+r)i+h(ct)]TJ /F3 11.955 Tf 9.96 0 Td[(r)6 45r2+2 15(ct)]TJ /F3 11.955 Tf 9.97 0 Td[(r)5 r+5 18(ct)]TJ /F3 11.955 Tf 9.97 0 Td[(r)4+2 9r(ct)]TJ /F3 11.955 Tf 9.96 0 Td[(r)3ilnh(ct)]TJ /F3 11.955 Tf 9.96 0 Td[(r)i). (5) Thenextstepisactingthetwod'Alembertians.Thispurgesallthetimedependentterms, @4F(t,r)=4 r4ln(2r), (5) @4(1 3ijF(t,r)+1 2h3bribrj)]TJ /F6 11.955 Tf 11.96 0 Td[(ijihG(t,r))]TJ /F8 11.955 Tf 11.15 8.09 Td[(1 3F(t,r)i)=4 r(4 3ijln(2r)+h3bribrj)]TJ /F6 11.955 Tf 9.96 0 Td[(ijih4 3ln(2r))]TJ /F8 11.955 Tf 9.97 0 Td[(2i). (5) Atthisstagethelinearized,oneloopeffectiveeldequations( 5 )taketheform, Dh(1)(t,~x)=GM 12802c2Df(~x),(5)wherethenonzerocomponentsofthetensorf(~x)are, f00(~x)=4 rln(2r), (5) fij(~x)=ij4 3ln(2r) r+h3bribrj)]TJ /F6 11.955 Tf 9.96 0 Td[(ijih4 3ln(2r) r)]TJ /F8 11.955 Tf 11.16 8.09 Td[(2 ri. (5) 78

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ItremainsonlytoacttheoperatorDonf(~x).Thersttwoderivativesgive, @@f=4 r3,@jfij=bri[4ln(2r))]TJ /F8 11.955 Tf 9.97 0 Td[(4] r2, (5) @2f00=)]TJ /F8 11.955 Tf 12.57 8.09 Td[(4 r3,@2fij=ij[8ln(2r))]TJ /F8 11.955 Tf 9.96 0 Td[(12] r3)]TJ /F12 11.955 Tf 11.39 .5 Td[(bribrj[24ln(2r))]TJ /F8 11.955 Tf 9.96 0 Td[(32] r3, (5) @i@jf00=@k@ifjk,@i@kfjk=ij[4ln(2r))]TJ /F8 11.955 Tf 9.96 0 Td[(4] r3)]TJ /F12 11.955 Tf 11.39 .5 Td[(bribrj[12ln(2r))]TJ /F8 11.955 Tf 9.96 0 Td[(16] r3. (5) Thesourcetermcanthenbeexpressedintermsoftwomorederivativesofthequantitiesg(~x)r2f(~x)andg(~x)@i@jfij(~x), Dh(1)(t,~x)=GM 12802c2n)]TJ /F6 11.955 Tf 9.3 0 Td[(r2g+4 3@@g+1 3r2g)]TJ /F8 11.955 Tf 13.15 8.09 Td[(2 3@@(g)o.(5)Thenalreductionemploystheidentities, r2g=24 r5,@i@jg=)]TJ /F8 11.955 Tf 10.5 8.09 Td[(12 r5ij+60 r5bribrj, (5) r2g00=)]TJ /F8 11.955 Tf 10.49 8.09 Td[(24 r5,r2gij=)]TJ /F8 11.955 Tf 10.49 8.09 Td[(48 r5ij+120 r5bribrj, (5) @kgjk=)]TJ /F8 11.955 Tf 10.5 8.09 Td[(12 r4brj,@i@kgjk=)]TJ /F8 11.955 Tf 10.5 8.09 Td[(12 r5ij+60 r5bribrj. (5) Thenontrivialcomponentsoftheeffectiveeldequationsare, D00h(1)(t,~x)=GM 802c21 r5, (5) Dijh(1)(t,~x)=GM 802c2n)]TJ /F8 11.955 Tf 10.49 8.09 Td[(3ij r5+5bribrj r5o. (5) 5.2.5TheOneLoopPotentialsWewishtoexpresstheonelooppotentialsinSchwarzschildcoordinatessotheirnonzerocomponentstaketheform, h(1)00(~x)=a(r),h(1)ij(~x)=bribrjb(r).(5) 79

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ActingtheLichnerowitzoperator( 3 )onthesegives, D00h(1)=b0 r+b r2, (5) Dijh(1)=ijh)]TJ /F3 11.955 Tf 10.49 8.09 Td[(a00 2)]TJ /F3 11.955 Tf 14.61 8.09 Td[(a0 2r)]TJ /F3 11.955 Tf 14.42 8.09 Td[(b0 2ri+bribrjha00 2)]TJ /F3 11.955 Tf 14.6 8.09 Td[(a0 2r+b0 2r)]TJ /F3 11.955 Tf 15.15 8.09 Td[(b r2i. (5) Comparing( 5 )with( 5 )implies, b(r)=GM 1602c2)]TJ /F8 11.955 Tf 24.54 8.08 Td[(1 r3.(5)Substitutingthisin( 5 )andcomparingwith( 5 )implies, a(r)=GM 1602c21 r3.(5)Combiningtheclassicalandquantumcorrectionsgivesthefollowingtotalresultsforthepotentials, h00(~x)=2GM c2r(1+~G 20c3r2+O4 r4), (5) hij(~x)=2GM c2r(1)]TJ /F19 11.955 Tf 25.85 8.08 Td[(~G 20c3r2+O4 r4)bribrj. (5) Expression( 5 )agreeswithequation(3.9)ofHamberandLiu[ 20 ],andalsowithequation(32)of[ 90 ].WhentransformedtodeDondergaugeourresults( 5 )-( 5 )givethesametraceobtainedinequation(59)byDalvitandMazzitelli[ 91 ]. 80

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CHAPTER6QUANTUMCORRECTIONSTODYNAMICALGRAVITONS 6.1TheEffectiveFieldEquationsThepurposeofthissectionistopresenttheeffectiveeldequationwhichwesolveinthenextsection.Webeginbyreviewingsomeusefulfactsaboutthebackgroundgeometry.WethengiveourrecentlyderivedresultfortheoneloopMMCscalarcontributiontothegravitonself-energy[ 98 ].ThesectioncloseswithadiscussionoftheSchwinger-Keldysheffectiveeldequationsandhowonesolvesthemperturbatively. 6.1.1TheSchwinger-KeldyshEffectiveFieldEquationsBecausethegravitonself-energyisthe1PIgraviton2-pointfunction,itgivesthequantumcorrectiontothelinearizedEinsteinequation, p )]TJ /F3 11.955 Tf 9.3 .01 Td[(gDh(x))]TJ /F12 11.955 Tf 11.96 16.27 Td[(Zd4x0hi(x;x0)h(x0)=1 2p )]TJ /F3 11.955 Tf 9.29 .01 Td[(gTlin(x),(6)HereDistheLichnerowiczoperator,( 3 )specializedtodeSitterbackground DD(g)(D))]TJ /F8 11.955 Tf 13.15 8.09 Td[(1 2hgDD+gDDi+1 2hgg)]TJ /F3 11.955 Tf 9.96 0 Td[(g(g)iD2+(D)]TJ /F8 11.955 Tf 9.97 0 Td[(1)h1 2gg)]TJ /F3 11.955 Tf 9.96 0 Td[(g(g)iH2, (6) andDisthecovariantderivativeoperatorinthebackgroundgeometry.ThepointoftheSchwinger-KeldyshformalismisexplainedinSec5.1.HerewegivetheexpressionforthedeSittercase.Attheonelooporderweareworking[]++(x;x0)agreesexactlywiththein-outresultgivenintheprevioussub-section.Toget[]+)]TJ /F8 11.955 Tf 6.25 1.79 Td[((x;x0),atthisorder,onesimplyaddsaminussignandreplacesthedeSitterlengthfunctiony(x;x0)everywherewith, y(x;x0))166(!y+)]TJ /F8 11.955 Tf 4.26 1.79 Td[((x;x0)H2a()a(0)hk~x)]TJ /F6 11.955 Tf 10.83 2.74 Td[(~x0k2)]TJ /F8 11.955 Tf 11.95 0 Td[(()]TJ /F6 11.955 Tf 9.96 0 Td[(0+i)2i.(6)Itwillbeseenthatthe++and+)]TJ /F1 11.955 Tf 12.62 0 Td[(self-energiescancelunlessthepointx0isonorinsidethepastlight-coneofx.Thatmakestheeffectiveeldequation( 6 )causal. 81

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Whenx0isonorinsidethepastlight-coneofxthe+)]TJ /F1 11.955 Tf 12.62 0 Td[(self-energyisthecomplexconjugateofthe++one,whichmakestheeffectiveeldequation( 6 )real.Thisalsoeffectsagreatsimplicationinthestructurefunctionsbecauseonlythosetermswithbranchcutsinycanmakenonzerocontributions,forexample, ln(y++))]TJ /F8 11.955 Tf 11.96 0 Td[(ln(y+)]TJ /F8 11.955 Tf 6.25 1.79 Td[()=2i)]TJ /F6 11.955 Tf 9.96 0 Td[(0)-221(k~x)]TJ /F6 11.955 Tf 10.83 2.73 Td[(~x0k.(6) 6.1.2PerturbativeSolutionBecauseweonlyknowtheself-energyatonelooporder,allwecandoistosolve( 6 )perturbativelybyexpandingthegravitoneldandtheself-energyinpowersof2, h(x)=h(0)(x)+2h(1)(x)+O(4).(6)Ofcourseh(0)(x)obeystheclassical,linearizedEinsteinequation.Giventhissolution,thecorrespondingoneloopcorrectionisdenedbytheequation, p )]TJ /F3 11.955 Tf 9.3 0 Td[(g(x)D2h(1)(x)=Zd4x0hi(x;x0)h(0)(x0).(6)Theclassicalsolutionforadynamicalgravitonofwavevector~kis[ 70 ], h(0)(x)=(~k)u(,k)ei~k~x,(6)wherethetreeordermodefunctionis, u(,k)=H p 2k3h1)]TJ /F3 11.955 Tf 15.64 8.08 Td[(ik Haiexphik Hai,(6)andthepolarizationtensorobeysallthesamerelationsasinatspace, 0=0=kiij=jjandijij=1.(6) 6.2ComputingtheOneLoopSourceThepointofthissectionistoevaluatetheoneloopsourcetermontherighthandsideofequation( 6 )foradynamicalgraviton( 6 )-( 6 ).Webeginbydrawing 82

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inspirationfromwhathappensintheatspacelimit.OurdeSitteranalysiscommencesbypartiallyintegratingtheprojectors.Thisresultsinconsiderablesimplicationbuttheplethoraofindicesisstillproblematic.Toeffectfurthersimplicationweextractandpartiallyintegrateanotherd'Alembertian,whereuponthexprojectorcanbeactedontheresidualstructurefunctiontoeliminatefourcontractions.AtthispointwedigresstoderivesomeimportantidentitiesconcerningcovariantderivativesoftheWeyltensor.Thenalreductionrevealszeronetresult. 6.2.1PartialIntegrationWenowstarttoevaluatetheoneloopsourceterm( 6 )foradynamicalgraviton, Zd4x0hi(x;x0)h(0)(x0)=iZd4x0p )]TJ /F3 11.955 Tf 7.31 0 Td[(g(x)P(x)p )]TJ /F3 11.955 Tf 7.3 0 Td[(g(x0)P(x0)nF0oh(0)(x0)+2iZd4x0p )]TJ /F3 11.955 Tf 7.3 0 Td[(g(x)P(x)p )]TJ /F3 11.955 Tf 7.31 0 Td[(g(x0)P(x0)(TTTTF2)h(0)(x0). (6) InthisexpressionandhenceforthwesimplywriteF0andF2tostandforthefullSchwinger-Keldyshexpressions, F0F0(y++))-222(F0(y+)]TJ /F8 11.955 Tf 6.25 1.8 Td[(),F2F2(y++))-222(F2(y+)]TJ /F8 11.955 Tf 6.26 1.8 Td[().(6)Theintegral( 6 )canbesimpliedintwosteps.First,theprojectorsP(x)andP(x),whichactonafunctionofx,canbepulledoutsidetheintegrationoverx0.Second,theprojectorsP(x0)andP(x0),whichactonx0,canbepartiallyintegratedtoactonthegravitonwavefunctionh(0)(x0).Afterthesetwosteps,theintegral( 6 )becomes, Zd4x0hi(x;x0)h(0)(x0)=ip )]TJ /F3 11.955 Tf 7.31 0 Td[(g(x)P(x)Zd4x0p )]TJ /F3 11.955 Tf 7.31 0 Td[(g(x0)F0nP(x0)h(0)(x0)o+2ip )]TJ /F3 11.955 Tf 7.3 0 Td[(g(x)P(x)Zd4x0p )]TJ /F3 11.955 Tf 7.31 0 Td[(g(x0)TTTTF2(P(x0)h(0)(x0)). (6) 83

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Notethatthespinzerotermdropsoutduetothetranversalityandtracelessnessofthedynamicalgraviton,h(0): Ph(0)=nDD)]TJ /F12 11.955 Tf 11.95 13.27 Td[(hD2+(D)]TJ /F8 11.955 Tf 9.97 0 Td[(1)H2igoh(0)=0.(6)Thusweonlyhavethespintwoterm,whichgivesthelinearizedWeyltensor, P(x0)h(0)(x0)=C(x0).(6)Theoneloopsourcetermthenreducestotheintegral, Zd4x0hi(x;x0)h(0)(x0)=2ip )]TJ /F3 11.955 Tf 7.31 0 Td[(g(x)P(x)Zd4x0p )]TJ /F3 11.955 Tf 7.3 0 Td[(g(x0)TTTTF2C(x0). (6) 6.2.2ExtractingAnotherd'AlembertianAchallengetoevaluatingexpression( 6 )isthecomplicatedtensorstructureoftheexternalprojectorP(x)actingontheinternalfactorsofTF2.Recallfromtheatspacelimitthatallofthiswasconvertedtoderivativeswithrespecttox0andthenpartiallintegratedontothegravitonwavefunctiontogivezero.TofollowthisondeSitterwemustmakethestructurefunctionmoreconvergentbyextractingafactorof 0andthenpartiallyintegratingitontothegravitonwavefunction.Afterthistheexternalprojectorcanbeacted,whicheliminatesfourindices,andanalfurtherpartialintegrationcanbeperformed.Therststepisextractingtheextrad'Alembertian, F2= 0 H2bF2.(6) 84

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Wenextcommutethe 0throughthefactorofTTTT: TTTT 0 H2bF2= 0 H2+4TTTTbF2)]TJ /F8 11.955 Tf 14.52 8.09 Td[(1 H2cF02(@y @x@y @x0TTT++TTT@y @x@y @x0))]TJ /F8 11.955 Tf 17.66 8.08 Td[(1 2H2cF2(g@y @x0@y @x0TT+g@y @x0@y @x0TT+g@y @x0@y @x0TT+g@y @x0@y @x0TT+g@y @x0@y @x0TT+g@y @x0@y @x0TT). (6) ExploitingthetracelessnessoftheWeyltensoronanytwoindices,anditsantisymmetryonthersttwoandlasttwoindices,gives, PTTTT 0 H2bF2C=P 0 H2bF2TTTTC=P(4bF2TTTT)]TJ /F8 11.955 Tf 17.18 8.09 Td[(4 H2cF02@y @x@y @x0TTT)C. (6) Forthersttermof( 6 )wecanpartiallyintegratethe 0ontothelinearizedWeyltensor.Thentheoneloopsourcetermbecomes Zd4x0hi(x;x0)h(0)(x0)=2ip )]TJ /F3 11.955 Tf 7.3 0 Td[(g(x)P(x)Zd4x0p )]TJ /F3 11.955 Tf 7.3 0 Td[(g(x0)(TTTTbF2 0 H2C(x0)+4bF2TTTT)]TJ /F8 11.955 Tf 17.18 8.08 Td[(4 H2cF02@y @x@y @x0TTTC(x0)). (6) Thissetsthestageforactingtheouterprojector. 6.2.3DerivativesoftheWeylTensorAtthispointitisusefultomakeashortdigressiononthecovariantderivativesoftheWeyltensor.Inthissub-sectionweusegforthefullmetric,notthedeSitterbackground.Allcurvaturesaresimilarlyforthefullmetric. 85

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TheBianchiidentitytellsus, DR+DR+DR=0.(6)Ifthestress-energyvanishes,allsolutionstotheEinsteinequationobey, R)]TJ /F8 11.955 Tf 13.16 8.08 Td[(1 2gR=)]TJ /F8 11.955 Tf 9.29 0 Td[(3H2g=)R=3H2g.(6)InD=3+1theWeyltensorcanbeexpressedintermsoftheothercurvaturesas, C=R)]TJ /F8 11.955 Tf 12.81 8.09 Td[(1 2gR)]TJ /F3 11.955 Tf 11.61 0 Td[(gR+gR)]TJ /F3 11.955 Tf 11.61 0 Td[(gR+1 6gg)]TJ /F3 11.955 Tf 11.61 0 Td[(ggR.(6)Nownotethatthecovariantderivativeofthemetricvanishes.Substituting( 6 )in( 6 )implies, DC=DR.(6)Combiningthisrelationinto( 6 )gives, DC+DC+DC=0.(6)Ourrstkeyidentityderivesfromcontractinginto,andexploitingthetracelessnessoftheWeyltensor, DC=0.(6)OursecondidentityderivesfromcontractingDintorelation( 6 ),commutingderivativesandthenusingrelation( 6 ), C=)]TJ /F3 11.955 Tf 9.3 0 Td[(DDC+DDC, (6) =6H2C)]TJ /F3 11.955 Tf 11.96 0 Td[(RC+RC)]TJ /F3 11.955 Tf 9.3 0 Td[(RC+RC)]TJ /F3 11.955 Tf 11.95 0 Td[(RC. (6) Relations( 6 )and( 6 )hold,toallordersinthegravitoneld,foranysolutiontothesource-freeEinsteinequations.Takingtherstorderinthegravitoneldamounts 86

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tojustreplacingthefullWeyltensorbythelinearizedWeylCwehavebeenusing,replacingthefullcovariantderivativeoperatorsbythecovariantderivativesindeSitterbackgroundandreplacingthefullRiemanntensorbyitsdeSitterlimit.Whenthesethingsaredonethetwoidentitiesbecome, DC=0+O(h2), (6) C=6H2C+O(h2). (6) Notealsothatifthestress-energyhadbeennonzerotherighthandsidesofrelations( 6 )and( 6 )wouldhavecontainedsimplecombinationsofderivativesofthestresstensor. 6.2.4TheFinalReductionWearenowreadytoacttheouterprojectorontheremainingterms, Zd4x0hi(x;x0)h(0)(x0)=2ip )]TJ /F3 11.955 Tf 7.31 0 Td[(g(x)Zd4x0p )]TJ /F3 11.955 Tf 7.31 0 Td[(g(x0)C(x0)(P(x)10bF2TTTT)]TJ /F8 11.955 Tf 17.18 8.09 Td[(4 H2cF02@y @x@y @x0TTT). (6) Thesecondlineofthisexpressionisquitecomplicatedbyitself,butitisgreatlysimpliedwhencontractedintothelinearizedWeyltensor, C(x0)P(x)10bF2TTTT)]TJ /F8 11.955 Tf 17.18 8.09 Td[(4 H2cF02@y @x@y @x0TTT=C(x0)(@y @x0@y @x0T(T)f1(y)+@y @x0@y @x0T(T)f2(y)+@y @x0@y @x0T(T)f3(y)+@y @x0@y @x0T(T)f4(y)). (6) 87

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Herethefunctionsfi(y)are, f1=)]TJ /F8 11.955 Tf 9.3 0 Td[(125bF2+115(2)]TJ /F3 11.955 Tf 9.96 0 Td[(y)cF02)]TJ /F8 11.955 Tf 9.96 0 Td[((68)]TJ /F8 11.955 Tf 11.96 0 Td[(116y+29y2)cF002)]TJ /F8 11.955 Tf 9.96 0 Td[(2(2)]TJ /F3 11.955 Tf 9.96 0 Td[(y)(4y)]TJ /F3 11.955 Tf 9.97 0 Td[(y2)cF0002f2=)]TJ /F8 11.955 Tf 10.49 8.08 Td[(75 2bF2+69 2(2)]TJ /F3 11.955 Tf 9.96 0 Td[(y)cF02)]TJ /F8 11.955 Tf 9.96 0 Td[((28)]TJ /F8 11.955 Tf 11.96 0 Td[(44y+11y2)cF002)]TJ /F8 11.955 Tf 9.96 0 Td[((2)]TJ /F3 11.955 Tf 9.96 0 Td[(y)(4y)]TJ /F3 11.955 Tf 9.96 0 Td[(y2)cF0002f3=)]TJ /F8 11.955 Tf 10.49 8.09 Td[(85 2bF2+15 2(2)]TJ /F3 11.955 Tf 9.96 0 Td[(y)cF02f4=)]TJ /F8 11.955 Tf 9.3 0 Td[(5bF2)]TJ /F8 11.955 Tf 9.96 0 Td[(13(2)]TJ /F3 11.955 Tf 9.96 0 Td[(y)cF02)]TJ /F8 11.955 Tf 11.16 8.09 Td[(5 2(4y)]TJ /F3 11.955 Tf 9.97 0 Td[(y2)cF002 (6) Changingthedummyindicesin( 6 )gives, C(x0)P(x)10bF2TTTT)]TJ /F8 11.955 Tf 17.18 8.09 Td[(4 H2cF02@y @x@y @x0TTT=@y @x0@y @x0T(T)f(y)C(x0). (6) Herethefunctionf(y)is, f(y)=)]TJ /F8 11.955 Tf 9.29 0 Td[(50bF2+60(2)]TJ /F3 11.955 Tf 9.97 0 Td[(y)cF02)]TJ /F8 11.955 Tf 9.96 0 Td[((40)]TJ /F8 11.955 Tf 11.95 0 Td[(62y+31 2y2)cF002)]TJ /F8 11.955 Tf 9.96 0 Td[((2)]TJ /F3 11.955 Tf 9.96 0 Td[(y)(4y)]TJ /F3 11.955 Tf 9.96 0 Td[(y2)cF0002. (6) Thenalreductionisaccomplishedbyonemorepartialintegration.LetusdenetheintegralI[f]ofafunctionf(y)bytherelations, @y @x0f(y)@ @x0I[f](y)suchthat@I[f] @y=f(y).(6)Thentheoneloopsourcebecomes, Zd4x0hi(x;x0)h(0)(x0)=2ip )]TJ /F3 11.955 Tf 7.3 0 Td[(g(x)Zd4x0p )]TJ /F3 11.955 Tf 7.3 0 Td[(g(x0)@y @x0f(y)@y @x0T(T)C(x0) (6) =)]TJ /F8 11.955 Tf 9.3 0 Td[(2ip )]TJ /F3 11.955 Tf 7.3 0 Td[(g(x)Zd4x0p )]TJ /F3 11.955 Tf 7.3 0 Td[(g(x0)I[f](D2y Dx0Dx0T(T)C(x0)+DT(T) Dx0@y @x0C(x0)+@y @x0T(T)DC(x0)). (6) 88

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Therstandsecondtermsincludethemetric, D2y Dx0Dx0=H2(2)]TJ /F3 11.955 Tf 11.96 0 Td[(y)g(x0),DT(T) Dx0=1 2@y @x(T)(g)(x0),(6)sotheygivezerowhencontractedintothelinearizedWeyltensor.ThethirdtermvanishesbythetransversalityofthelinearizedWeyltensor(fordynamicalgravitonsonly)whichweshowedin( 6 ).Hencetheoneloopsourcetermforadynamicalgravitoniszero: Zd4x0hi(x;x0)h(0)(x0)=0. (6) Beforeconcludingweshouldcommentonthevalidityofourresult( 6 ),inviewoftheenormousdifferencebetweendeSitterandtheactualexpansionhistoryoftheuniverse.Ofcourseequation( 6 )iscorrectforanygeometry,butweonlyknowthegravitonself-energyfordeSitterbackground.Thisdoesnotmakeanydifferenceforcosmologicallyobservabletensorperturbationsfortworeasons: Asexplainedsection1.1,deSitterisanexcellentapproximationtoprimordialinationupuntilcosmologicallyobservableperturbationsexperiencersthorizoncrossing.AfterthistimethedeSitterapproximationbreaksdown,butthoseperturbationsarealmostconstant. Ourresult( 6 )isvalidforanygeometry,andthelinearizedWeyltensorvanishesforconstantperturbations.Sothereisnocontribtuionfromtheportionoftheintegrationwhichderivesfromtimesaftertheendofination.Toseethesecondpoint,notethatgeneralcoordinateinvariancerequiresmattercontributionstothegravitonself-energytotaketheform( 4 ),providedoneusesexpressions( 4 )-( 4 )todenetheprojectorsforageneralmetric,andprovidedthegeneralformofexpression( 4 )isrelatedtothegeodeticlengthfunctionthrough( 2 ).Thatformisallwerequiredtoderiveequation( 6 ). 89

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CHAPTER7CONCLUSIONWehavecomputedtheoneloopcontributiontothegravitonself-energyfromamassless,minimallycoupledscalaronalocallydeSitterbackground.Weusedittosolvetheoneloop-corrected,linearizedEinsteineldequationstostudytheeffectofinationaryscalarsondynamicalgravitons.ThecomputationwasdoneusingdimensionalregularizationandrenormalizedbyabsorbingthedivergenceswithBPHZcounterterms.Thegravitonself-energyhasbeengivenintwoforms.Therstform( 3 )isfullydimensionallyregulated,withtheultraviolatedivergencesneitherlocalizednorsubtractedoffwithcounterterms.ThisversionoftheresultagreeswiththestresstensorcorrelatorrecentlycomputedbyPerez-Nadal,RouraandVerdaguer[ 99 ].Oursecondformisfullyrenormalized,withtheunregulatedlimittaken, )]TJ /F3 11.955 Tf 9.3 0 Td[(ihreni(x;x0)=p )]TJ ET q .478 w 161.95 -317.93 m 168.72 -317.93 l S Q BT /F3 11.955 Tf 161.95 -325.25 Td[(g(x)P(x)p )]TJ ET q .478 w 244.35 -317.93 m 251.11 -317.93 l S Q BT /F3 11.955 Tf 244.35 -325.25 Td[(g(x0)P(x0)hF1R(y)i+2p )]TJ ET q .478 w 108.8 -344.83 m 115.56 -344.83 l S Q BT /F3 11.955 Tf 108.8 -352.15 Td[(g(x)P(x)p )]TJ ET q .478 w 199.92 -344.83 m 206.68 -344.83 l S Q BT /F3 11.955 Tf 199.92 -352.15 Td[(g(x0)P(x0)hTTTTF2R(y)i. (7) InthisexpressionthespinzerooperatorPwasdenedin( 4 ),thespintwooperatorPwasdenedin( 4 ),andthebitensorTwasgivenin( 4 ).Ourresultsfortherenormalizedspinzeroandspintwostructurefunctionsareexpressions( 4 )and( 4 ),respectively.Aninterestingapplicationofthisworkisthetransverse-tracelessprojector( 4 ),whichplayedacrucialroleintherecentsolutionforthegravitonpropagatorindeDondergauge[ 103 104 ].Itshouldbenotedthatequations( 4 )and( 4 )aretherst(andsofaronly)fullyrenormalizedresultsforthegravitonstructurefunctionsondeSitterbackground.Allpreviousresults[ 86 99 ]havebeenspecializedtonon-covincidentpoints,andsocannotbeusedintheeffectiveeldequations. 90

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Oursecondform( 7 )ismanifestlytransverse,asrequiredbygaugeinvariance.ItisalsodeSitterinvariant,despitethefactthatthemassless,minimallycoupledscalarpropagatorbreaksdeSitterinvariance[ 62 ],becausethedeSitterbreakingtermdropsoutofmixedsecondderivatives( 2 ).Ourresultagreeswiththeatspacelimit[ 61 ].Andthedivergentpartsofthecountertermsweusedtosubtractoffthedivergencesagreewiththosefoundlongagoby`tHooftandVeltman[ 9 ].WeactuallyincludedniterenormalizationsofNewton'sconstantandofthecosmologicalconstant.SuchrenormalizationsarepresumablynecessarywhenconsideringtheeffectiveeldequationsofquantumgravityiftheparametersandGaretohavetheircorrectphysicalmeanings.Thepointofthisexerciseistodiscoverwhetherornottheinationaryproductionofscalarshasasignicanteffectongravitationalradiationandtheforceofgravity.Inordertocheckthiswehaveemployedthequantum-correctedlinearizedEinsteinequation, p )]TJ ET q .478 w 79.12 -327.39 m 85.89 -327.39 l S Q BT /F3 11.955 Tf 79.12 -334.71 Td[(gDh(x))]TJ /F12 11.955 Tf 11.96 16.27 Td[(Zd4x0hreni(x;x0)h(x0)=1 2p )]TJ ET q .478 w 357.63 -327.39 m 364.39 -327.39 l S Q BT /F3 11.955 Tf 357.63 -334.71 Td[(gTlin(x),(7)whereDistheLichnerowiczoperator( 3 )specializedtodeSitterbackground.Becauseweonlyknowtheself-energyatorder2,allwecandoistosolve( 6 )perturbativelybyexpandingthegravitoneldandtheself-energyinpowersof2, h(x)=h(0)(x)+2h(1)(x)+O(4), (7) hreni(x;x0)=2h1i(x;x0)+O(4). (7) Ofcourseh(0)(x)obeystheclassical,linearizedEinsteinequation.Giventhissolution,thecorrespondingoneloopcorrectionisdenedbytheequation, p )]TJ ET q .478 w 117.13 -569.47 m 123.89 -569.47 l S Q BT /F3 11.955 Tf 117.13 -576.78 Td[(g(x)Dh(1)(x)=Zd4x0h1i(x;x0)h(0)(x0).(7) 91

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Foradynamicalgravitonofwavevector~k,theclassical0thordersolutiontakestheform[ 70 ], h(0)(x)=(~k)a2u(,k)ei~k~x,(7)wherethetreeordermodefunctionis, u(,k)=H p 2k3h1)]TJ /F3 11.955 Tf 15.64 8.09 Td[(ik Haiexphik Hai,(7)andthepolarizationtensorobeysallthesamerelationsasinatspace, 0=0=kiij=jjandijij=1.(7)OurresultisthattheinationaryproductionofMMCscalarshasnoeffectondynamicalgravitonsatonelooporder.Thereisnothingverysurprisingaboutthisresult.Itisexactlywhathappensinatspace[ 9 ]andwehaverevieweditinSection5.2.3.Althoughthescalarcontributiontothegravitonself-energyisenormouslymorecomplexindeSitterthaninatspace,weshowedinsection6.2thatallofthiscomplexitycanbeabsorbedintosurfaceintegrationsattheinitialtime.Itisplausiblethatthesesurfaceintegrationscanberegardedasperturbativeredenitionsoftheinitialstatewhichinvolvetwoscalarsandonegraviton.Thenulleffectofatspacecertainlyhasthisinterpretation,whichimpliesthesameforthehighestderivativepartofthedeSitterresult.Whathasyettobeprovedandsomustbelabeledaconjectureisthatthelowerderivative,intrinsicallydeSitterpartshavethesameinterpretation.Checkingthisrequiresacomputationlikethatrecentlycompletedfortheself-interactingscalar[ 7 ].AnotherwayofunderstandingthisresultistoconsiderthenumberoftheMMCscalars.Theoneloopcorrectionsweseektocomputerepresenttheresponse(ofeitherdynamicalgravitonsortheforceofgravity)tothevastensembleofinfraredscalarswhichareproducedbyination.Itissimpletoshowthattheoccupationnumberforeach 92

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modewithwavenumber~kgrowslike[ 49 ](asreviewedintheintroductionsection1.2.3), N(k,)=Ha() 2k2(7)Thisgrowthisbalancedbyexpansionofthe3-volumesothatthenumberdensityofinfraredparticleswith0
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Thesamereductionprocedureswelaidoutinsection6.2canbeappliedinthiscaseexceptthat: ThespinzeroprojectorP(x0)doesnotannihilate( 7 );and Thelinearizedstresstensordoesnotvanish.ThecomputationhasbeenreducedtoasingleintegrationwhichIwillcompletesoon.Itshouldbenotedthatthevirtualscalarsofatspacedoinduceacorrectiontotheclassicalpotential[ 43 44 ]whichisreviewedinsection5.2.5,andweexpectoneaswellondeSitterbackground.Ondimensionalgroundstheatspaceresultmust(anddoes)taketheform, at=)]TJ /F3 11.955 Tf 10.49 8.09 Td[(GM r(1+constantG r2+O(G2)).(7)OndeSitterbackgroundthereisadimensionallyconsistentalternativeprovidedbytheHubbleconstantHandbytheseculargrowthdrivenbycontinuousparticleproduction, dS=)]TJ /F3 11.955 Tf 10.49 8.09 Td[(GM r(1+constantGH2ln(a)+O(G2)).(7)IfsuchacorrectionweretooccuritsnaturalinterpretationwouldbeasatimedependentrenormalizationoftheNewtonconstant.Thephysicaloriginoftheeffect(ifitispresent)wouldbethatvirtualinfraredquantawhichemergenearthesourcetendtocollapsetoit,leadingtoaprogressiveincreaseinthesource.Bothmathandphysicssuggestthatinationarygravitonsmightdosomethinginterestingtoothergravitons.Thegravitoncontributiontothegravitonself-energyhasbeenderivedatonelooporder[ 60 ]sothecomputationcanbemade.Ofcourseonecanreducetheeffecttoatemporalsurfaceterm,aswedidinsection6.2,butitseemslikelythatthissurfacetermwilldependupontheobservationtimesothatitcannotbeabsorbedintoaperturbativecorrectiontotheinitialstate.ThereasonforthisisthatthegravitoncontributioncontainsdeSitter-breaking,infraredlogarithms[ 60 ],unlikethescalarcontribution.Thephysicalprincipleinvolvedwouldbethatgravitonspossessspinandevenveryinfraredgravitonscontinuetointeractviathespin-spincouplingwhich 94

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doesn'texistforscalars.Thisispresumablywhyinationarygravitonsinduceasecularenhancementoftheeldstrengthofmasslessfermions[ 97 ].Finalcommentconcernsthepossiblecomparisonwithcosmologicaldata.OneconsequenceofourcomputationwouldbeonthetensorcomponentoftheanisotropiesintheCMB(CosmicMicrowaveBackgroundradiation).Thishasnotbeenresolvedyetbuthighprecisionmeasurementsoftheso-calledB-modesoftheCMBpolarization,forexamplefromthePlancksatellite,mayallowustoprobeit[ 106 107 ].TherearealsothreeNASA-fundedballoonprobessettogoupoverthecourseofthenextyear[ 108 ].NoneofthesemeasurementswouldbesensitivetotheoneloopcorrectionsIamcomputing,buttheyarerststeps.Ontheverylongseveraldecadesterm,thereisenoughdatatoresolveoneloopcorrectionsfromveryearlyproto-structures[ 109 ].Theobservableisthetensorpowerspectrum,whichisk3=22timesthespatialFouriertransformof.Myresultisthatasinglescalarloopdoesnotgiveanycorrectiontothegravitonmodefunctions.Howeveraswehavediscussedinthepreviousparagraph,itseemslikelythattheoneloopeffectsfromgravitonsmightbesignicantbecausegravitonscaninteractthroughtheirspins,whichdonotredshift. 95

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[108] http://www.astro.caltech.edu/lgg/spider/spiderfront.htm [109] https://portfolio.du.edu/portfolio/getportfoliole?uid=167700 [110] A.F.Radkowski,Ann.Phys.56(1970)319;D.M.Capper,M.J.DuffandL.Halperin,Phys.Rev.D10(1974)461;D.M.CapperandM.J.Duff,Nucl.Phys.B82(1974)147;D.M.Capper,NuovoCimentoA25(1975)29;M.J.Duff,Phys.Rev.D9(1974)1837;H.HamberandS.Liu,Phys.Lett.B357(1995)51,hep-th/9505182;M.J.DuffandJ.T.Liu,Phys.Rev.Lett.85(2000)2052,hep-th/0003237;J.F.Donoghue,Phys.Rev.Lett.72(1994)2996,gr-qc/9310024;Phys.Rev.D50(1994)3874,gr-qc/9405057;I.J.MuzinichandS.Kokos,Phys.Rev.D52(1995)3472,hep-th/9501083;A.Akhundov,S.BelucciandA.Shiekh,Phys.Lett.B395(1997)16,gr-qc/9611018;I.B.KhriplovichandG.G.Kirilin,J.Exp.Theor.Phys.95(2002)981,gr-qc/0207118;J.Exp.Theor.Phys.98(2004)1063,gr-qc/0402018.N.E.J.Bjerrum-Bohr,J.F.DonoghueandB.R.Holstein,Phys.Rev.D67(2003)084033,Erratum-ibid.D71(2005)069903,hep-th/0211072;N.E.J.Bjerrum-Bohr,Phys.Rev.D66(2002)084023,hep-th/0206236;B.R.HolsteinandA.Ross,SpinEffectsinLongRangeGravitationalScattering,arXiv:0802.0716;A.CamposandE.Verdaguer,Phys.Rev.D49(1994)1861,gr-qc/9307027;F.C.LombardoandF.D.Mazzitelli,Phys.Rev.D55(1997)3889,gr-qc/9609073;R.MartinandE.Verdaguer,Phys.Rev.D61(2000)124024,gr-qc/0001098;A.Satz,F.D.MazzitelliandE.Alvarez,Phys.Rev.D71(2005)064001,gr-qc/0411046;D.A.R.DalvitandF.D.Mazzitelli,Phys.Rev.D50(1994)1001,gr-qc/9402003;E.D.Carlson,P.R.Anderson,A.Fabbri,S.Fagnocchi,W.H.HirschandS.Klyap,Phys.Rev.D82(2010)124070,arXiv:1008.1433;A.MarunovicandT.Prokopec,Phys.Rev.D83(2011)104039,arXiv:1101.5059. 102

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BIOGRAPHICALSKETCH SohyunParkcamefromSouthKorea.SheearnedherundergraduatedegreeinmathematicsatPusanNationalUniversity(PNU).Inthelastyearofherundergraduatestudy,shehadachancetoparticipateincomputationalbiologyresearchwhichledhertodohermaster'sinbiologywiththesistitle,ImplementationofCellularAutomatafortheSpatio-TemporalAnalysisofPopulationDynamicsandDiffusionModelofPineNeedleGallMidge.Itwasagoodexperiencefromwhichshelearnedmathematicalmodelingandcomputersimulationtechniques.However,shefoundherselfhavingmoreinterestinfundamentaltheoryandshedecidedtostudyhighenergyphysics.ThusshenallyswitchedtothedepartmentofphysicsatPNUandtookanothermaster'sdegreeinphysicswithathesisentitled,AStudyontheSnyder-YangDiscreteSpace-timeunderthedirectionofProf.ChangGilHan.Aftercompletinghermaster's,shehadteachingjobsineducationalinstitutesandatechnicalcollegewhileplanningtostudyabroad.ShecametotheUniversityofFloridaintheFallof2007.Inherrstyear,shetookthegraduatecorecourses,ParticlePhysicsandQuantumFieldTheory.Inhersecondandthirdyear,shetookGeneralRelativity,StandardModelandaSpecialTopicscourseonDarkMatter.IntheFallof2009,shestartedherdoctoralresearchonquantumeldtheoryincurvedspaceunderthedirectionofProf.RichardWoodardwhichledtothisdissertation.InFall2011,shewonaFermiNationalAcceleratorLaboratory(Fermilab)FellowshiptospendherlastyearofgraduatestudydoingresearchunderthedirectionofDr.ScottDodelsonatFermilab.Herwinningproposalwastoworkoutstructureformationinanewtypeofmodiedgravitytheorieswhichhavebeensuggestedtoexplainthecurrentphaseofcosmologicalacceleration.SohyunreceivedherPh.D.intheSummerof2012. 103