Towards a $C$-function in 4D quantum gravity

Understand

We develop a generally applicable method for constructing functions, $C$, which have properties similar to Zamolodchikov's $C$-function, and are geometrically natural objects related to the theory space explored by non-perturbative functional renormalization group (RG) equations.

• Employing the Euclidean framework of the Effective Average Action (EAA), we propose a $C$-function which can be defined for arbitrary systems of gravitational, Yang-Mills, ghost, and bosonic matter fields, and in any number of spacetime dimensions.
• It becomes stationary both at critical points and in classical regimes, and decreases monotonically along RG trajectories provided the breaking of the split-symmetry which relates background and quantum fields is sufficiently weak.
• Within the Asymptotic Safety approach we test the proposal for Quantum Einstein Gravity in $d>2$ dimensions, performing detailed numerical investigations in $d=4$.