Investigating the Ultraviolet Properties of Gravity with a W(10)

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

TheERGEcanbeseenformallyasaRG–improvedoneloopequation.Toseethis,recallthatgivenabareactionS(forabosonic eld),theoneloope ectiveactionΓ(1)is

1Γ(1)=S+.δφδφ

LetusaddtoSthecuto term(7);thefunctional

Γk=S+(1)(12)1

δφδφ+Rk, (13)

maybecalledthe“oneloopaveragee ectiveaction”.Itsatis estheequation

k(1)dΓk

2Tr δ2S

dk,(14)

whichisformallyidenticalto(9)exceptthatinther.h.s.therenormalizedrunningcou-plingsgi(k)arereplacedeverywherebythe“bare”couplingsgi,appearinginS.Thusthe“RGimprovement”intheERGEconsistsinreplacingthebarecouplingsbytherunningrenormalizedcouplings.Inthisconnection,notethatingeneralthecuto functionRkmaycontainthecouplingsgiandthereforethetermkd

dkRkinther.h.s.goestozeroformomentagreaterthankand