Investigating the Ultraviolet Properties of Gravity with a W(16)

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

coe cientsasbefore,con rmingthatthebetafunctionsofthedimensionlesscouplingsarescheme–independent.

Inordertohavemoreexplicitformulae,andinnumericalwork,oneneedstocalculatealsothescheme–dependentQ-functionals.Thisrequires xingthepro leRk.Inthispaperwewillmostlyusetheso–calledoptimizedcuto (A21)inwhichtheintegralsarereadilyevaluated,seeequations(A22,A23,A24).Thiscuto hastheveryconvenientpropertythatQ n tRk

dt= tRk+ R′ E

k

i

dt=a(n)

ik4 n,

wherea(n)areconstants.Then,thebetafunctionsofthedimensionlessvariablesg (n)

ii=