Investigating the Ultraviolet Properties of Gravity with a W(5)

We review and extend in several directions recent results on the asymptotic safety approach to quantum gravity. The central issue in this approach is the search of a Fixed Point having suitable properties, and the tool that is used is a type of Wilsonian r

TheattractivitypropertiesofaFParedeterminedbythesignsofthecriticalexponents i,de nedtobeminustheeigenvaluesofM.Thecouplingscorrespondingtonegativeeigenvalues(positivecriticalexponent)arecalledrelevantandparametrizetheUVcriticalsurface;theyareattractedtowardstheFPfork→∞andcanhavearbitraryvalues.Theonesthatcorrespondtopositiveeigenvalues(negativecriticalexponents)arecalledirrelevant;theyarerepelledbytheFPandmustbesettozero.

Afreetheory(zerocouplings)hasvanishingbetafunctions,sotheorigininQisaFP,calledtheGaußianFP.IntheneighborhoodoftheGaußianFPonecanapplyperturbationtheory,andonecanshowthatthecriticalexponentsarethenequaltothecanonicaldimen-sions( i=di),sotherelevantcouplingsaretheonesthatarepower–countingrenormalizable3.Inalocaltheorytheyareusually niteinnumber.Thus,aQFTisperturbativelyrenor-malizableandasymptoticallyfreeifandonlyifthecriticalsurfaceoftheGaußianFPis nitedimensional.PointsoutsideC owtoin nity,ortootherFP’s.Atheorywiththesepropertiesmakessensetoarbitrarilyhighenergies,becausethecouplingsdonotdivergeintheUV,andispredictive,becauseallbuta nitenumberofparametersare xedbytheconditionoflyingonC.Asymptoticsafetyisaformofnonperturbativerenormalizability.Itgeneralizesthispicture,replacingtheGaußianFPbyanarbitraryFP.Anasymptoticallysafetheorywouldhavethesamegoodpropertiesofarenormalizableandasymptoticallyfreeone:thecouplingswouldhavea niteUVlimitandtheconditionoflyingonCwouldleaveonlya nitenumberofparameterstobedeterminedbyexperiment.Ingeneral,studyingthepropertiesofsuchtheoriesrequirestheuseofnonperturbativetools.IfthenontrivialFPissu cientlyclosetotheGaußianone,itspropertiescanalsobestudiedinperturbationtheory,butunlikeinasymptoticallyfreetheories,theresultsofperturbationtheorydonotbecomebetterandbetterathigherenergies.

Inordertoestablishwhethergravityisasymptoticallysafe,severalauthors[5,6,7]ap-pliedthe expansionaroundtwodimensions,whichisthecriticaldimensionwhereNewton’sconstantisdimensionless.ThebetafunctionofNewton’sconstantthenhastheform

2βG = G+B1G,(5)

=Gk andB1<0[5,6],sothereisaFPatG = /B1>0.UnfortunatelythiswhereG

resultisonlyreliableforsmall anditisnotclearwhetheritwillextendtofourdimensions.