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D. Benedetti, F. Caravelli, “The Local potential approximation in quantum gravity,” JHEP 1206
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J. A. Dietz and T. R. Morris, “Asymptotic safety in the f ⁡ ( R ) f(R) approximation,” JHEP 1301
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D. Becker and M. Reuter, “En route to Background Independence: Broken split-symmetry, and how to restore it with bi-metric average actions,” Annals Phys. 350
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Then

H. Gies, B. Knorr, S. Lippoldt, “Generalized Parametrization Dependence in Quantum Gravity,” Phys. Rev. D 92
Original
2015
Later among the works it cites.
A. Eichhorn, “The Renormalization Group flow of unimodular f(R) gravity,” JHEP 1504
Original
2015
Later among the works it cites.
M. Demmel, F. Saueressig and O. Zanusso, “Fixed-Functionals of three-dimensional Quantum Einstein Gravity,” JHEP 1211
Original
2015
Later among the works it cites.
P. Labus, T. R. Morris and Z. H. Slade, “Background independence in a background dependent renormalization group,” Phys. Rev. D 94
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Closest in time.
A. Nink, “Field Parametrization Dependence in Asymptotically Safe Quantum Gravity,” Phys. Rev. D 91

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2016

The background scale Ward identity in quantum gravity

Percacci, Roberto, Vacca, Gian Paolo

Understand

We show that with suitable choices of parametrization, gauge fixing and cutoff, the anomalous variation of the effective action under global rescalings of the background metric is identical to the derivative with respect to the cutoff, i.e.

to the beta functional, as defined by the exact RG equation.
The Ward identity and the RG equation can be combined, resulting in a modified flow equation that is manifestly invariant under global background rescalings.