Quantum Gravitational Corrections to the Nonrelativistic Scattering Potential of Two Masses

N.E.JBjerrum-Bohra

JohnF.Donoghueb,andBarryR.Holsteinb

a

DepartmentofPhysics

UniversityofCaliforniaatLosAngelesLosAngeles,CA90095-1574,U.S.A∗

and

TheNielsBohrInstitute,Blegdamsvej17,CopenhagenØ,

DK-2100,Denmarkb

DepartmentofPhysics-LGRTUniversityofMassachusetts

Amherst,MA01003

February1,2008

Abstract

Wetreatgeneralrelativityasaneffectivefieldtheory,obtainingthefullnonanalyticcomponentofthescatteringmatrixpotentialtoone-looporder.Thelowestordervertexrulesfortheresultingeffectivefieldtheoryarepresentedandtheone-loopdiagramswhichyieldtheleadingnonrelativisticpost-NewtonianandquantumcorrectionstothegravitationalscatteringamplitudetosecondorderinGarecalculatedindetail.TheFouriertransformedamplitudesyieldanonrelativisticpotentialandourresultisdiscussedinrelationtopreviouscalculations.Thedefinitionofapotentialisdiscussedaswellandweshowhowtheambiguityofthepotentialundercoordinatechangesisresolved.

1Introduction

Theideathatafieldtheoryneednotbestrictlyrenormalizableinthetra-ditionalsenseyetcanstillyieldusefulquantumpredictionswhentreatedasaneffectivefieldtheory[1]hasbeenclearlydemonstratedinchiralpertur-bationtheoryandinotherapplications[2].Quantumloopcalculationsleadtowelldefinedresultsinthelowenergylimit.Interestingly,suchmethodscanalsobeappliedtogeneralrelativity.Asaneffectivefieldtheory,thequantizationofgeneralrelativitycanbecarriedoutinaconsistentway,sincethetroublesomesingularitieswhichoccurforvarioustypesofmattersourcesintraditionalrenormalizationschemes[3,4,5,6,7]canbeabsorbedintophenomenologicalconstantswhichcharacterizetheeffectiveactionofthetheory.Thiseffectivefieldtheoryapproachoffersthenapossiblewayaroundthefamiliarrenormalizationdifficultiesofgeneralrelativityinthelowenergyregimeand,usingthisapproachwithbackgroundfieldquanti-zation[8]ofgeneralrelativity,oneofus[9,10]someyearsagoderivedtheleadingquantumandclassicalcorrectionstotheNewtonianpotentialoftwolargenon-relativisticmasses.Thiscalculationhassincebeenthefocusofanumberofpublications[11,12,13,14],andthisworkcontinues,mostrecentlyinthepaper[15].Unfortunately,duetodifficultyofthecalculationanditsmyriadoftensorindicestherehasbeenlittleagreementamongthesevariousauthors.Theclassicalcomponentofthecorrectionhaspreviouslybeendiscussedinthepapersby[16,17,18,19,20],andherethereisgen-eralagreementalthough,asweshalldiscuss,thereexistsanunavoidableambiguityindefiningthepotential.ThebasicdisagreementslieratherinthequantumcorrectionsandinthepresentpaperweshallpresentwhatwebelievetobethedefinitiveresultfortheleadingclassicalandquantumcorrectionsoforderG2,usingthefullscatteringamplitudeasthedefinition

1

ofthenon-relativisticpotential.

Wenotethat,asapreludetothiseffort,inarecentpaper[24]twoofushavedealtwiththethequantumandclassicalcorrectionstotheReissner-Nordstr¨omandKerr-Newmanmetricsofchargedscalarsandfermions.Suchquantumandclassicalcorrectionshavealsobeenconsideredfromtheview-pointofascatteringpotentialinapaper[25]byoneofus.RecentlywehavealsocalculatedthefullclassicalandquantumcorrectionstotheSchwarzshildandKerrmetricsofscalarsandfermions[26]andhaveshownindetailhowthehigherordergravitationalcontributionstothesemetricsemergefromloopcalculations.Inthepresentpaperthenweconsiderthecorrespondingcalculationofthefullscatteringamplitude.

Ofcourse,treatinggeneralrelativityasaneffectivefieldtheoryiscarriedoutatthecostofintroducinganeverendingsetofadditionalhigherderiva-tivecouplingsintothetheory.InthissenseEinstein’sgeneralrelativityisstillaperfectlyvalidtheoryforgravitationalinteractions—althoughnowitrepresentsonlythetheminimaltheory.AtsomestageadditionalderivativecouplingsmustbeappendedtotheEinsteinaction,signifyingmanifesta-tionsofthehigherenergycomponentoftheeffectivefieldtheory.However,thelowenergyscatteringpotentialisfreefromthesenewcouplingsandrepresentsamodel-independentresultforquantumgravity.

Thiscalculationispossiblebecausethepost-Newtonianandquantumcorrectionswhichweconsideraredeterminedfullybythenon-analyticpiecesoftheoneloopamplitudegeneratedbythelowestorderEinsteinaction.Ofcourse,inordertodealwiththeultravioletdivergenceswhichariseatoneloop,onemustrenormalizetheparametersofhigherderivativetermsintheaction.However,suchpieceswillonlyaffecttheanalyticpartsoftheone-loopamplitude,andwillnotcontributetoourpotential.

Wewillemploythesameconventionsasinourpreviouspapers,namely(󰀁=c=1)aswellastheMinkowskimetricconvention(+1,−1,−1,−1).Wewillbegininsection2withaveryshortintroductiontotheeffectivefieldtheoryquantizationofgeneralrelativityandfocushereespeciallyonthedistinctionbetweennon-analyticandanalyticcontributionstothescat-teringamplitude.Wealsoincludeadiscussionofthedefinitionofthenon-relativisticpotential.

Nextinsection3weevaluatethediagramswhichcontributetothescat-teringandexamineindetailtheresultsforthevariouscomponents.Theresultingnonanalyticpieceofthescatteringamplitudeisthenusedinordertoconstructtheleadingcorrectionstothenonrelativisticgravitationalpo-tential.Wealsodiscussourresultinrelationtopreviouscalculationsandattempttosortoutthevariousinconsistenciesinthepublishednumbers.

2

Finallyinaconcludingsectionwesummarizeourfindings.

2

Reviewofgeneralrelativityasaneffectivefieldtheory

Webeginwithabriefreviewofgeneralrelativity,theLagrangianofwhich(notincludingacosmologicalterm)is

L=√κ2

+Lmatter󰀈

(1)

whereκ2=32πGisthegravitationalcoupling,Rµµ

ναβΓµσαΓσµ≡∂αΓµνβ

−∂βΓνα+νβσofthemetric−ΓσβΓναisthecurvaturetensor,andgdenotesthedeterminant

fieldgµν.Here

√

−g

󰀌

2R

inthelowenergylimitoftheeffectivefieldtheory,theydominateovertheanalyticcontributionswhicharisefromthepropagationofmassivemodes.󰀍Thetypicalnon-analytictermswewillconsiderareofthetype:1/

wherep,p′istheincoming,outgoingfour-momentum.ThecorrespondingcoordinatespacerepresentationcanbefoundbytakingthenonrelativisticlimitandFourier-transforming,yieldingtheresult

󰀎

d3q1

V(x)=

2m2

E−En+iǫ

+...

󰀈

(5)

Thedefinitionoftheboundstatepotentialisdiscussedindetailin[17].Inparticular,inaHamiltoniantreatmenttherearealsotermsintheHamil-tonianinvolvingGp2/rthatcontributeatthesameorder.Therelationof

˜bs(q),inEinstein-Infeld-Hoffmanncoordinates,theboundstatepotentialV

tothelowestorderscatteringpotentialis

˜bs(r)=V(r)+7Gm1m2(m1+m2)V

q2

+β1κln(−q)+β2κ

424

m−q2

+...

󰀑

(7)

HerethecoefficientsA,B,...correspondtoanalyticpieceswhichareofno

interesttous,asthesetermswillonlydominateinthehighenergyregimeof

5

theeffectivetheory.Rather,theα,β1,β2,...coefficientscorrespondtothenonlocal,nonanalyticcontributionstotheamplitudeandaretheoneswhichweseek.Inparticular,theβ1,β2termswillyieldtheleadingpost-Newtonianandquantumcorrectionstothepotential.

3ResultsfortheFeynmandiagrams

Inthissectionwewillpresentourresults.Alldiagramshavebeenper-formedbothbyhandandbycomputer.Inordertoevaluatethediagramsbycomputer,analgorithmforMaple7(TM)1wasdeveloped.Thisprogramcontractsthevariousindicesandperformtheloopintegrations.Allresultsobtainedthiswaywereconfirmedresultsobtainedbyhand.TheresultingamplitudeswerethenFouriertransformedtoproducethescatteringpoten-tial,andonlythenonanalyticpiecesoftheamplitudewereretained.Forthispartofthecalculation,thefollowingFourierintegralsareuseful:

󰀎

d3q

1|q|2=iq·󰀎

(2π)3e

r12π2r2

(8)d3q

2πr3Afterthisbriefintroduction,weproceedtogivetheresultsforeachdiagraminturn.NotethatthebasicverticesneededforourcalculationaregiveninAppendixA.

3.1Thetreediagram

Theresultforthisdiagraminthenonrelativisticlimitisthewell-knownlowestordertree-levelresultwhichyieldstheNewtonianpotential.WedefinethediagramusingtheFeynmanrulesas:

iM1(a)(󰀭q)=τµν

1(k1,k2,m1)

󰀓iP

µναβ󰀭q

2(10)

1(a)

Figure1:ThetreediagramgivingNewtonslaw.

whoseFouriertransformproducesthescatteringpotential

V1(a)(r)=−

Gm1m2

(2π)4τµν1

(k1,k1+l,m1)τρσ1(k1+l,k2,m1)×ταβ−l,m2)τγδ

1(k3,k31(k3−l,k4,×

󰀓mil)2−m22

󰀆󰀓2)

iPµναβ

(k3−(l+q)2

󰀆

7

(12)

fortheboxand

M2b=

󰀎

d4l

P

ρσγδ

(k1+l)2−m21

󰀆·

󰀓i

l2

󰀆󰀓il·k1

(2π)2

󰀎d4l

2

l2(l+q)2((l+k1)2−m21)((l−k3)2−m22)

by

1

1

(2π)2

l·q

(2π)2

d4l

2

󰀎l2(l+q)2((l+k1)2−m21)((l−k3)2−m22)

whichsimplifiesto

2

−1

q(2π)2

(14)

(16)

Viathesesimplificationsonecanreducetheboxandcrossboxamplitudestoareducedpiececonsistingonlyofintegralswithtwoorthreepropagatorsandacomponentwiththebasicformoftheboxandcrossedboxintegrals,i.e.,withnoloopmomentumtermsinthenumerator.Then,usingtheintegralspresentedinAppendixB,performingtheabove-describedcontractionsinthetwodiagrams,andtakingthenonrelativisticlimitweendupwiththeresult

46red

M2((q󰀭)=a)+2(b)

m1m2G23

πr3

(21)

fortheirreduciblepiecesothatthetotalresultfortheboxandcrossbox

contributiontothepotentialis

totV2(a)+2(b)(r)=−

47

πr3

(22)

Theseresultsareinagreementwiththoseof[15].

3.3Thetrianglediagrams

Thenextpieceswewillconsiderarethesetoftrianglediagrams,forwhich

wefind

󰀎

d4l

M3(a)(q)=

(l+q)2

󰀆󰀓iP

µνσρ

(l+k1)2−m21

󰀆

(23)

9

3(a)3(b)

Figure3:Thesetoftriangulediagramscontributingtothescatteringpo-tential.andM3(b)(q)=

󰀎

d4l

󰀆󰀓iP

αβγδ

l2

(24)

Thecalculationofsuchdiagramsyieldsnorealcomplications—theintegralsneededarequitestraightforwardandarepresentedinAppendixB.However,asignificantsimplificationresultsfromtheuseoftheidentity

PγδσρPαβµντσρµν(k1,k2,m1)=τγδαβ(k1,k2,m1)

and,takingthenonrelativisticlimit,wefindforthesetwopieces

󰀒󰀅7

M3(a)(󰀭q)=−8G2m1m2

|󰀭q|

󰀒󰀅7

M3(b)(󰀭q)=−8G2m1m2

|󰀭q|

(25)

(l−k3)2−m22

󰀆

(26)

Ourresultsforthesediagramsagreewiththoseof[15],andtheFourier

transformedresultis

V3(a)+3(b)(r)=−4

G2m1m2(m1+m2)

πr3

(27)

3.4Thedouble-seagulldiagram

1

󰀓iP

󰀆

Wehaveforthedouble-seagulltermM4(a)(q)=

(2π)

αβγδσρµντ(k,k,m)τ(k3,k4,m2)×121224

αβµν

l2

(28)

10

4(a)

Figure4:Thedouble-seagulldiagramcontributiontothescatteringpoten-tial.

Thedouble-seagullloopdiagramisquitestraightforward,andissimplifiedbyuseoftheidentityEq.25.Note,however,thatthereexistsasymmetryfactorof1/2!.Theresultingamplitudeisfoundtobe

M4(a)(󰀭q)=44G2m1m2log󰀭q2

(29)

whoseFouriertransformyieldsthedouble-seagullcontributiontothepo-tential

(r)=−22

m1m2G2

V4(a)Thereexisttwoclassesofvertexcorrectiondiagrams:Forthemassiveloopdiagrams,showninFigures5(a)and5(b)wehave4M5(a)(q)=

dl

󰀆󰀓iP

φǫρσ

2q2

󰀆󰀓

i

lαβµνλκγδρσ(φǫ)

(2π)4

τ1(k1,l+k1,m1)τ1

(l+k1,k2,m1)τ1(k3,k4,m2)τ3(−l,q)

×

󰀓iP

αβγδ

(l+q)2󰀆󰀓iPφǫλκ(l+k1)2−m21

󰀆(32)

whileforthepuregravitonloopdiagramsshowninFigures5(c)and5(d)wehave

M5(c)(q)=

1(2π)

4τλκǫφ4,m2)ταβ

2,m1)τµνρσ(γδ)2(k3,k1(k1,k3(l,−q)×

󰀓iP

µνλκ

iP

αβγδ

(l+q)2

󰀆󰀓d4l2!

󰀎l2

󰀆󰀓iP

ρσαβ

q2

󰀆

(34)

Thevertexcorrectiondiagramsarecertainlythemostchallengingtoper-form,disregardingtheboxandcrossedboxdiagrams,andtheresultsgobacktotheoriginalcalculationof[9,10]—however,duetoanalgebraicer-ror,theoriginalresultquotedforthesuchdiagramswasinerror.Sincethattime,theresultsforthegravitationalvertexcorrectionshavebeencheckedatlengthinvariouspublications[13,15];however,until[26]thecorrectformshavenotbeengiven.Usingtheresultsof[26]andtakingthenonrelativisticlimit,wefindtheamplitudes,

MG2m1m2

󰀒π2

(m5(a)+5(b)(󰀭q)=21+m2

)3

log󰀭q2󰀅M52

5(c)+5(d)(󰀭q)=−

whoseFouriertransformyieldsthecorrectedresultsforthevertexmodifi-cationstothescatteringpotential:

V5(a)+5(b)(r)=

G2m1m2(m1+m2)

3

m1m2G23

m1m2G2

Itiscorrectlypointedoutin[15]thatthecoefficientofthetwo-gravitonvertexquotedin[9],[10]istoosmallbyafactoroftwo.However,thenumericalresultfortheloopintegralsgiventhereiniscorrectbecauseofthepresenceofthissymmetryfactor.

2

13

theamplitude

M6(a)+6(b)(q)=τρσ(k1,k2,m1)

iPρσλξ

q2

τγδ(k3,k4,m2)(38)

wherethevacuumpolarizationtensorisfoundfromtheeffectiveLagrangianobtainedby’tHooftandVeltman[3,4,5]—

L=−

1

120

󰀇

R2+

7

−

π23

log(−q2)

21

120

q4ηαβηγδ

240

q2(qαqδηβγ+qβqδηαγ+qαqγηβδ+qβqγηαδ)+

11

15

G2m1m2log󰀭q2

(42)

whichisequivalenttothatoriginallyderivedin[9,10].AfterFouriertrans-formingwefindacontributiontothescatteringpotential

V6(a)+6(b)(r)=−

43

πr3

(43)

whichisinagreementwiththatgivenby[15].

4Theresultforthegravitationalpotential

Addingupallthecorrectionstothenon-relativisticpotentialwehaveour

finalresult

󰀈

Gm1m241

+V(r)=−(44)

rr2TheclassicalterminthispotentialagreeswithEq.2.5ofIwasaki[17].

14

However,wenotethatthequantumcomponentofthepotentialisnotequivalenttoanypreviouspublishedresult[9,10,11,12,13,15].Asde-scribedabove,webelievethisresulttobetodefinitiveformofthenon-relativisticscatteringmatrixpotential.Ourresultsagreewith[15]foralldiagramsexceptthevertexcorrections3Howeverwehavemultiplechecksonthevertexcorrection,aswehaveseenthatourversionleadsexactlytotherequiredformoftheclassicalSchwarzschildmetric,forbothfermionsandbosons.

4.1APotentialAmbiguity

Thereisanimportantissueconcerningthepotentialthathasnotbeenthus-farmuchdiscussedintheliterature—thatboththeclassicalandquantumcorrectionstothepotentialareinsomesenseambiguous.Fortheclassicalcorrection,thisrealizationgoesbacktothepapersof[18,19,20],andtheseargumentsgeneralizereadilytothequantumcomponent.Inthissectionwediscussthisambiguityandarguethatatonelooporderourcalculationofthequantumcorrectiondoeshaveawelldefinedmeaning.

Asexploredin[18]theclassicalpost-Newtonianpotentialisnotinvariantunderacoordinatetransformationoftheform

󰀓G(m1+m2)r→r1+α

r

󰀇

1+c

G(m1+m2)

r

󰀇

G(m1+m2)

1+(c−α)

q2

=

1

1

Tobeprecisewearecomparingtothethirdversionof[15],availableintheelectronicarchive.Wehadseveraldisagreementswiththeoriginalversion,allofwhichexceptthevertexcorrectionhavebeencorrectedinthethirdversion.

3

15

Usingthispropagatorinordertoderivethegravitationalpotential,theresultwillingeneraldependonx,withtherelationtothecoordinatechangebeingα=−1r2dependonx,󰀑󰀐butsowillthecorrectionsoforderGc.Thisindicatesthatthepost-Newtonianpartofthestaticgravitationalpotentialisnotwell-definedinandofitself.

Thecoordinateambiguityalsogeneralizestothequantumpartofthepotential.Thereexistsacoordinateredefinition

󰀇

G󰀁

r→r1+β

󰀇

G󰀁1+d

󰀇

G󰀁

1+(d−β)

rr

m2

,

Gm

mGm2

,

Gm2

m

,3

r

,(

Gm

m

)(53)

Thereexistsanambiguitybetweenthelasttwoofthesetermsarisingfroma

coordinatechange,butthiseffectcancelsforphysicalobservables,asshown

16

explicitlyin[18].WritingtheHamiltonianinthecenterofmassframetopost-Newtonianorder,wehave

󰀒2󰀅󰀒4󰀅pp

H=−

2m28m32

󰀈

(p·ˆr)2Gm1m2

+b()−

m1m2rInthestandardEinstein-Infeld-Hoffmanncoordinates,thecoefficientsa,b,c

havethevalues

󰀈

1

a=

m1m2

1

b=

2

(55)

butunderthecoordinatetransformationofEq.45,onefindsthemodifica-tions

󰀈

1

a→

m1m2

1

b→

m1m2

1

c→−

m5

,

p4

r

󰀅

,

p2

r

󰀅2

,

Gm2

r

󰀅2

.(57)

Thelasttermhereisnoteworthybecauseitgoeslike1/r3—i.e.likethequantumcorrectioninthepotential—butitisdistinguishablebyitsmassdependenceplusthefactthatitisorderG3.Suchtermswillalsobeambigu-ousunderacoordinatetransformation,buttheeffectswillcancelamongthe

17

effectstermsofthisorder.Sothe“classical”coordinatechangeabovedoeschangethe1/r3terminthepotential,buthasaspecificformandtheeffectscancelsamongotherclassicaltermsintheHamiltonian.

Nowletusexaminethequantumeffects,keepingonlyonepowerofthequantumexpansionparameter

G󰀁

G󰀁rG󰀁m

−q2andq2ln−q2

terms.Atransformationofthesetermstocoordinatespaceallowsustointerprettheseaslongdistancecorrectionstothepotential.OurresultisdisplayedinEq.44.

18

Thequantumcorrectionsthatwefindforthescatteringpotentialholdequallyfortheboundstatepotential.Thisisbecausethetransformationbetweenthetwohasnoquantumcomponent.Thedifferencebetweenthetwoisaclassicalcorrectionthatcomesfromiteratingthelowestorderpo-tential.Dimensionalanalysisrevealsthatinthenonrelativisticlimitthisiterationhastheabilitytogenerateonlyaclassicaleffectandnotaquan-tumcorrection.

Wehavefoundaresultforthenonrelativisticpotentialwhichwebelieveisthefinalandcompleteresultforthisquantity.Thepotentialmatchestheexpectationsfromdimensionalanalysisasdiscussedpreviously[9,10]andtheknownambiguityoftheformoftheclassicalcorrectionhasbeenseentooriginatefromthepossibilityofrewritingapotentialenergytermintheHamiltonianintermsofkineticenergyandviceversa,asalsodiscussedin[18,20,19].Suchrewritingsarenotpossibletodoforthequantumtermattheordertowhichwework.Thereforethequantumcorrectiontothepotentialisadefiniteexactquantity.

Thequantumcorrectionsaretoosmalltobeobservedexperimentally.However,thefactthatthesearereliablypredictedisimportantforourunderstandingofquantumgravity.Theseeffectsareduetothelowenergypropagationofmasslessdegreesoffreedomandhenceareuniquelypredictedforanyquantumtheoryofgravitythatreducestogeneralrelativityinthelowenergylimit4.Inthissense,thesearelowenergytheoremsofquantumgravity.

Acknowledgments

N.E.J.Bjerrum-BohrwouldliketothanktheDepartmentofPhysicsandAstronomyatUCLAforitskindhospitalityandP.H.Damgaardfordis-cussions.TheworkofBRHanJFDissupportedinpartbytheNationalScienceFoundationunderawardPHY-98-01875.

Appendices

A

VerticesandPropagators

WebeginbylistingtheFeynmanruleswhichareemployedinourcalculation.Foraderivationoftheseforms,see[26].

A.1Scalarpropagator

Themassivescalarpropagatoris:

q2−m2+iǫ

A.2Gravitonpropagator

Thegravitonpropagatorinharmonicgaugecanbewrittenintheform:

where

Pαβγδ=

q2+iǫ

1

µν=τ1(p,p′,m)

where

µν

τ1(p,p′,m)=−

iκ

ηλρσ

=τ2(p,p′,m)

20

where

τηλρσ

2(p,p′)=iκ2󰀇󰀌

IηλαδIρσβ1δ󰀌

−

Iηλρσ1

2

−2

(ηαγηβδ+ηαδηβγ).

A.53-gravitonvertex

The3-gravitonvertexcanbederivedviathebackgroundfieldmethodandhastheform[9],[10]

=τ3µναβγδ(k,q)

where

τ3µνiκ

αβγδ(k,q)=−

ηµνq2

󰀈

+2qλqσ󰀇2

IµναβσλIσλγδ+IγδIµννλαβ−IµσαβIγδ−IµσγδIαβνλ+󰀇qλqµ󰀒󰀈

ηαβIγδνλ+ηγδIαβνλ󰀅+qλqν

󰀐

ηαβIµλµλγδ+ηγδIαβ

−q2󰀐

ηαβIµνγδ−ηγδIµναβ󰀑−ηµνqσqλ󰀐ηαβIγδσλ+ηγδIαβσλ󰀑󰀑󰀈+󰀇

2qλ󰀃

IµαβλσIγδσν(k−q)µ+IαβλσIµγδσ(k−q)ν−IγδλσIαβσνkµ−IγδλσIαβσkν

󰀄+q2󰀐

IµαβσIγδνσ+IµαβνσIγδσ󰀑+ηµνqσqλ󰀐IλραβIγδρσ+IλρσγδIαβρ󰀑󰀈+󰀌

(k2+(k−q)2)󰀂Iµσµσ1

αβIγδσν+IγδIαβσν−

BUsefulIntegrals

Intheevaluationofthevariousdiagramsweemploythefollowingintegrals

󰀎

d4li

=J=

l2(l+q)2

󰀓󰀆lµ

qµL+...()

(2π)432π2󰀎󰀐1󰀑d4li2

Jµν==L−qηµν−

l2(l+q)23

1

(2π)432π2m2

󰀎

d4l

i󰀎

d4l

󰀂

−L−S+...

󰀅󰀈

󰀁

(66)

Iµ=

=Iµν

=

l2(l+q)2((l+k)2−m2)

q2q224

2

S+...(67)

l2(l+q)2((l+k)2−m2)

󰀅󰀅󰀒

i11

=S+kµkν−L−S

8m2m2

󰀒󰀄󰀐1󰀃q2q21

+qµkν+qνkµL+

21

Iµνα=

=++

+

l2(l+q)2((l+k)2−m2)

󰀅󰀅󰀒

i1

S+kµkνkα−16m2

󰀅󰀒

󰀃󰀄11

L+Sqµkνkα+qνkµkα+qαkµkν

m2m2󰀒

󰀃󰀄󰀐1q2q2qµqνkα+qµqαkν+qνqαkµ−

232󰀑q2L12

󰀑󰀈󰀐󰀃󰀄1

q2S+...ηµνqα+ηµαqν+ηναqµ−

16

π2m

.−q2

󰀎

d4l

(69)

wherewehavedefinedL=ln(−q2)andS=

Onlythelowestorder

nonanalytictermsareincludedintheaboveforms.Higherordernonanalyticcontributionsaswellastheneglectedanalytictermsaredenotedbythe

2

ellipses.Thefollowingidentitiescanbeverifiedonshell,k·q=q

.

Inordertodoevaluatetheboxdiagramsthefollowinglowestorderintegralsareneeded.Thehigherordercontributionsofnon-analytictermsaswellasneglectedanalytictermsareagaindenotedbyellipses.

󰀎

d4l

K=22l2(l+q)2((l+k1)2−m21)((l−k3)−m2)

󰀑󰀈i

=L+...(70)

3m1m2

󰀎

d4l′

K=22l2(l+q)2((l+k1)2−m21)((l+k4)−m2)

󰀑󰀈i

L+...(71)=

3m1m2

2

q

Herek1·q=q,k·q=−322,wherek1−k2=

2=m2=k2togetherwithk2=m2=k2.Also,wehavek4−k3=qandk112324

definedw=(k1·k3)−m1m2andW=(k1·k4)−m1m2.

Thefollowingconstraintsforthenonanalytictermsoftheaboveintegralsholdtrueonshell:

2

2

Iµναηαβ=Iµνηµν=Jµνηµν=0

23

(72)

Iµναq=−

α

q2

22

2Jµν,

Iµνkν=

1

Iµ,Iµq=−

µ

µ

q2q2

Jµ,Jµq=−

2

J(74)

References

[1]S.Weinberg,“PhenomenologicalLagrangians,”PhysicaA96(1979)

327.[2]See,e.g.,J.GasserandH.Leutwyler,Ann.Phys.(NY)158,142(1984);

Nucl.Phys.B250,465(1985).[3]G.’tHooftandM.Veltman,69-94,Ann.Inst.HenriPoincar´e,Sec.

A20,1974.[4]M.Veltman,Gravitation,266-327,editedbyR.BalianandJ.Zinn-Justin,eds.,LesHouches,SessionXXVIII,1975,NorthHollandPub-lishingCompany,1976.[5]M.H.GoroffandA.Sagnotti,“TheUltravioletBehaviorOfEinstein

Gravity,”Nucl.Phys.B266(1986)709.[6]S.DeserandP.vanNieuwenhuizen,“OneLoopDivergencesOfQuan-tizedEinstein-MaxwellFields,”Phys.Rev.D10(1974)401.[7]S.DeserandP.vanNieuwenhuizen,“NonrenormalizabilityOfThe

QuantizedDirac-EinsteinSystem,”Phys.Rev.D10(1974)411.[8]B.S.Dewitt,“QuantumTheoryOfGravity.1.TheCanonicalTheory,”

“QuantumTheoryOfGravity.Ii.TheManifestlyCovariantTheory,”“QuantumTheoryOfGravity.Iii.ApplicationsOfTheCovariantThe-ory,”Phys.Rev.160(1967),1113;162(1967)1195;162(1967)1239.[9]J.F.Donoghue,“LeadingquantumcorrectiontotheNewtonianpoten-tial,”Phys.Rev.Lett.72(1994)2996[arXiv:gr-qc/9310024].[10]J.F.Donoghue,“GeneralRelativityAsAnEffectiveFieldTheory:The

LeadingQuantumCorrections,”Phys.Rev.D50(1994)3874[arXiv:gr-qc/9405057].

24

[11]I.J.MuzinichandS.Vokos,“Longrangeforcesinquantumgravity,”

Phys.Rev.D52,3472(1995)[arXiv:hep-th/9501083].[12]H.W.HamberandS.Liu,“OnthequantumcorrectionstotheNewto-nianpotential,”Phys.Lett.B357,51(1995)[arXiv:hep-th/9505182].[13]A.A.Akhundov,S.BellucciandA.Shiekh,“Gravitationalinteraction

tooneloopineffectivequantumgravity,”Phys.Lett.B395,16(1997)[arXiv:gr-qc/9611018].[14]N.E.J.Bjerrum-Bohr,Cand.Scient.Thesis,Univ.ofCopenhagen

(2001).[15]I.B:KhriplovichandG.G.Kirilin,“Quantumpowercorrectiontothe

Newtonlaw,”[arXiv:gr-qc/0207118][16]S.N.Gupta,“QuantizationofEinstein’sGravitationalField:General

Treatment,”Proc.Phys.Soc.LondonA65,608,1952;S.N.GuptaandS.F.Radford,“Quantumfield-theoreticalelectromagneticandgravita-tionaltwo-particlepotentials,”Phys.Rev.D,Vol21,No.8,2213,1980.[17]Y.Iwasaki,“QuantumTheoryofGravitationvs.ClassicalTheory,”

Prog.Theor.Phys.46,1587(1971).[18]K.HiidaandH.Okamura,“GaugeTransformationandGravitational

Potentials,”Prog.Theor.Phys.47,1743(1972).[19]B.M.BarkerandR.F.O’Connell,“PostnewtonianTwoBodyAndN

BodyProblemsWithElectricChargeInGeneralRelativity,”J.Math.Phys.18,1818(1977)[Erratum-ibid.19,1231(1978)].[20]B.M.BarkerandR.F.O’Connell,“GravitationalTwoBodyProblem

WithArbitraryMasses,Spins,AndQuadrupoleMoments,”Phys.Rev.D12(1975)329.[21]G.Modanese,“Potentialenergyinquantumgravity,”Nucl.Phys.B

434(1995)697[arXiv:hep-th/9408103].[22]DiegoA.R.DalvitandFranciscoD.Mazzitelli,Phys.Rev.D50:1001-1009,1994[arXiv:gr-qc/9402003].[23]K.A.Kazakov,“Onthenotionofpotentialinquantumgravity,”Phys.

Rev.D63,044004(2001)[arXiv:hep-th/0009220].

25

[24]J.F.Donoghue,B.R.Holstein,B.GarbrechtandT.Konstandin,

“QuantumcorrectionstotheReissner-NordstroemandKerr-Newmanmetrics,”Phys.Lett.B529(2002)132[arXiv:hep-th/0112237].[25]N.E.J.Bjerrum-Bohr,“Leadingquantumgravitationalcorrectionsto

scalarQED,”Phys.Rev.D66,084023(2002)[arXiv:hep-th/0206236].[26]N.E.J.Bjerrum-Bohr,J.F.Donoghue,andB.R.Holstein,”Quantum

CorrectionstotheSchwartzschildandKerrMetrics,”inpreparation.[27]J.F.DonoghueandT.Torma,“Onthepowercountingofloopdia-gramsingeneralrelativity,”Phys.Rev.D,4963(1996)[arXiv:hep-th/9602121].[28]K.BeckerandM.Becker,“Quantumgravitycorrectionsfor

Schwarzschildblackholes,”Phys.Rev.D60,026003(1999)[arXiv:hep-th/9810050].

26