Wave Function Renormalization and Flow of Couplings in Asymptotically Safe Quantum Gravity

The importance of the proper treatment of the wave function renormalization in the renormalization group analysis of quantum gravity is pointed out. The renormalization factor, sometimes called an inessential coupling, can be used to fix any one of the coupling constants, with the exception of the coupling constants that remain unchanged by the rescaling of the field. Choosing to fix the cosmological constant, we propose to use a new regulator to obtain the renormalization group equations for invariant couplings which tell us the flow of the Newton and $R^2$ couplings. We find that the Newton coupling reaches a nontrivial ultraviolet fixed point (FP) and becomes small in the low energy, but find only asymptotically free FP of the $R^2$ couplings for the higher-derivative gauge fixing and regulator. For the asymptotically free FP, we find that both of the two independent terms are relevant operators in the high energy. It is noted that the existence of nontrivial FPs may depend on the choice of the gauge and regulator.