Investigating the Ultraviolet Properties of Gravity with a W(2)

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

Abstract

Wereviewandextendinseveraldirectionsrecentresultsontheasymptoticsafetyapproachtoquantumgravity.ThecentralissueinthisapproachisthesearchofaFixedPointhavingsuitableproperties,andthetoolthatisusedisatypeofWilsonianrenormalizationgroupequation.Webeginbydiscussingvariouscuto schemes,i.e.waysofimplementingtheWilsoniancuto procedure.Wecomparethebetafunctionsofthegravitationalcouplingsobtainedwithdi erentschemes,studying rstthecontributionofmatter eldsandthentheso–calledEinstein–Hilberttruncation,whereonlythecosmologicalconstantandNewton’sconstantareretained.Inthiscontextwemakeconnectionwitholdresults,inparticularwereproducetheresultsoftheepsilonexpansionandtheperturbativeoneloopdivergences.WethenapplytheRenormalizationGrouptohigherderivativegravity.Inthecaseofageneralactionquadraticincurvaturewerecover,withincertainapproximations,theknownasymptoticfreedomofthefour–derivativeterms,whileNewton’sconstantandofthecosmologicalconstanthaveanontrivial xedpoint.Inthecaseofactionsthatarepolynomialsinthescalarcurvatureofdegreeuptoeightwe ndthatthetheoryhasa xedpointwiththreeUV–attractivedirections,sothattherequirementofhavingacontinuumlimitconstrainsthecouplingstolieinathree–dimensionalsubspace,whoseequationisexplicitlygiven.Weemphasizethroughoutthedi erencebetweenscheme–dependentandscheme–independentresults,andprovideseveralexamplesofthefactthatonlydimensionlesscouplingscanhave“universal”behavior.