Manifest Ultraviolet Behavior for the Three-Loop Four-Point

Using the method of maximal cuts, we obtain a form of the three-loop four-point scattering amplitude of N=8 supergravity in which all ultraviolet cancellations are made manifest. The Feynman loop integrals that appear have a graphical representation with o

UCLA/08/TEP/24SLAC–PUB–13361ManifestUltravioletBehaviorfortheThree-LoopFour-Point

AmplitudeofN=8Supergravity

Z.Berna,J.J.M.Carrascoa,L.J.Dixonb,H.JohanssonaandR.Roibanc

a

arXiv:0808.4112v1 [hep-th] 29 Aug 2008DepartmentofPhysicsandAstronomy,UCLA,LosAngeles,CA90095-1547,USAbStanfordLinearAcceleratorCenter,StanfordUniversity,Stanford,CA94309,USAcDepartmentofPhysics,PennsylvaniaStateUniversity,UniversityPark,PA16802,USA(Dated:August,2008)AbstractUsingthemethodofmaximalcuts,weobtainaformofthethree-loopfour-pointscatteringamplitudeofN=8supergravityinwhichallultravioletcancellationsaremademanifest.TheFeynmanloopintegralsthatappearhaveagraphicalrepresentationwithonlycubicvertices,andnumeratorfactorsthatarequadraticintheloopmomenta,ratherthanquarticasinthepreviousform.Thisquadraticbehaviorre ectscancellationsbeyondthoserequiredfor niteness,andmatchesthequadraticbehaviorofthethree-loopfour-pointscatteringamplitudeinN=4super-Yang-Millstheory.Bydirectintegrationwecon rmthatnoadditionalcancellationsremainintheN=8supergravityamplitude,thusdemonstratingthatthecriticaldimensioninwhichthe rstultravioletpergenceoccursatthreeloopsisDc=6.Wealsogivethevaluesofthethree-

looppergencesinD=7,9,11.Inaddition,wepresenttheexplicitlycolor-dressedthree-loopfour-pointamplitudeofN=4super-Yang-Millstheory.

PACSnumbers:04.65.+e,11.15.Bt,11.30.Pb,11.55.Bq

Using the method of maximal cuts, we obtain a form of the three-loop four-point scattering amplitude of N=8 supergravity in which all ultraviolet cancellations are made manifest. The Feynman loop integrals that appear have a graphical representation with o

I.INTRODUCTION

Recently,thewidelyheldbeliefthatitisimpossibletoconstructaconsistentfour-dimensionalpoint-liketheoryofquantumgravityhasbeencalledintoquestionformaxi-mallysupersymmetric[1]N=8supergravity.Inparticular,anintegralrepresentationofthethree-loopfour-pointamplitude[2]inthistheoryhasbeenobtainedexplicitly,whichexhibitscancellationsbeyondthoseneededforultraviolet niteness.Moreover,foraclassoftermsaccessiblebyisolatingone-loopsubdiagramsviageneralizedunitarity[3],theone-loop“no-triangle”property[4,5,6,7,8]–recentlyproveninrefs.[9,10]–showsthatatleastasubsetofthesecancellationspersisttoalllooporders[11].Interestingly,Mtheoryandstringtheoryhavealsobeenusedtoargueeitherforthe nitenessofN=8supergrav-ity[12],orthatpergencesaredelayedthroughatleastnineloops[13,14],thoughissueswithdecouplingtowersofmassivestates[15]mayaltertheseconclusions.Ifaperturbativelyultraviolet- nitepoint-liketheoryofquantumgravitycouldbeconstructed,theunderlyingmechanismresponsiblefortherequiredcancellationswouldhaveaprofoundimpactonourunderstandingofgravity.

Overtheyearssupersymmetryhasbeenstudiedextensivelyasamechanismfordelayingtheonsetofpergencesingravitytheories(seee.g.refs.[16,17]).Inparticular,theexistence,orconjecturedexistence,ofvariouso -shellsuperspaceformalismsrestrictstheformanddimensionsofpotentialcounterterms,leadingtoboundsonthe rstpermissiblelooporderatwhichanultravioletpergencemightappearinN=8supergravity.Theprecisebounddependsonthedetailedsetofassumptions.Forexample,ifano -shellsuperspacewithN=6supersymmetriesmanifestweretoexist,potentialD=4pergenceswouldbedelayedtoatleast veloops[17],whiletheexistenceofasuperspacewithN=7supersymmetriesmanifestwoulddelaythe rstpotentialpergencetoatleastsixloops[17].Similarly,ifoneweretoassumetheexistenceofafullycovarianto -shellsuperspacewithN=8supersymmetriesmanifest,thenthe rstpotentialpergencewouldbepushedtothesevenlooporder[18].Fullsuperspaceinvariants,whichcouldactaspotentialcounterterms,havebeenconstructedateightloops,suggestingthatapergencemightappearatthislooporder,ifitdoesnotappearearlier[19].This rstpotentialpergencecanevenbepushedtotheninelooporder,withanadditionalspeculativeassumptionthatall eldsrespectten-dimensionalgeneralcoordinateinvariance[20].Thisboundcoincideswiththe

Using the method of maximal cuts, we obtain a form of the three-loop four-point scattering amplitude of N=8 supergravity in which all ultraviolet cancellations are made manifest. The Feynman loop integrals that appear have a graphical representation with o

oneargued[14]fromthetypeIIstringtheorynon-renormalizationtheoremofBerkovits[13].Beyondthisorder,nopurelysupersymmetricmechanismhasbeensuggestedforpreventingtheonsetofpergences.Infact,ondimensionalgroundsonecanarguethat,foranysupergravitytheorytobeultraviolet nitetoalllooporders,novelcancellationsbeyondtheknownsupersymmetriconesmustexist.

Surprisingly,cancellationsbeyondthoseimpliedbynaiveloop-momentumpower-countingappeartobegenericingravitytheories,assuggestedbytheone-loopstudyofref.[21].Thisreferencedemonstratedthatthesenovelone-loopcancellationsaredirectlyconnectedtotheremarkablygoodhigh-energybehaviorofgravitytreeampli-tudesunderthecomplexdeformationsusedtoproveon-shellrecursionrelationsingrav-ity[10,22,23,24,25,26].Recently[27,28],thesetree-levelpropertieshavebeenunderstoodintermsofaspace-likegaugesimilartolight-conegauge[29],whereanenhancedLorentzsymmetrywasshowntoexistunderlargecomplexdeformations.

Fornon-supersymmetrictheories,thesecancellationsareinsu cienttorenderthetheoryultraviolet nite.Indeed,itisaclassicalresultthatgravitycoupledtomattergenericallypergesatoneloop[30,31,32].PureEinsteingravitydoesnotpossessaviablecountertermatoneloop,delayingthepergencetotwoloops[30,33,34].Thepresenceofatwo-looppergenceinpureEinsteingravitywasestablishedbyGoro andSagnottiandbyvandeVen,throughdirectcomputation[35,36].

Supersymmetriccancellationscanoftenactontopofanygenericcancellations.Inpartic-ular,inthecaseofN=8supergravityatoneloop,suchcancellationscombinetocausethevanishingofthecoe ci …… 此处隐藏：28930字，全部文档内容请下载后查看。喜欢就下载吧 ……