Investigating the Ultraviolet Properties of Gravity with a W(15)

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

=1

(4π)2

+

+11 d4x√ Pk Pk

32π2 d4x√

dt=nS

2Tr(M)Pk( 2) tRk( 2) nDPk( 2)+R

Pk( 2) .(22)

Expandingeachtraceasin

(A10),collectingtermswiththesamenumberofderivativesofthemetric,andkeepingtermsuptofourderivativesweget

dΓk1 tRkg(nS 4nD+2nM)Q22

tRk tRkQ1Q1nD263Pk 1 tRk+ Q2Pk 180

+5nSR2+12(nS+nD 3nM) 2R +.... (3nS+18nD+36nM)C2 (nS+11nD+62nM)E(23)

Weseethatthetermslinearincurvature,whichcontributetothebetafunctionofNew-ton’sconstant,havechanged.However,thetermsquadraticincurvaturehavethesame