Background independent exact renormalization group for conformally reduced gravity

Abstract

Within the conformally reduced gravity model, where the metric is parametrised by a function $f(\varphi )$ of the conformal factor $\varphi$, we keep dependence on both the background and fluctuation fields, to local potential approximation and $\mathcal{O}({\partial }^2)$ respectively, making no other approximation. Explicit appearances of the background metric are then dictated by realising a remnant diffeomorphism invariance. The standard non-perturbative Renormalization Group (RG) scale $k$ is inherently background dependent, which we show in general forbids the existence of RG fixed points with respect to $k$. By utilising transformations that follow from combining the flow equations with the modified split Ward identity, we uncover a unique background independent notion of RG scale, $\hat{k}$. The corresponding RG flow equations are then not only explicitly background independent along the entire RG flow but also explicitly independent of the form of $f$. In general $f(\varphi )$ is forced to be scale dependent and needs to be renormalised, but if this is avoided then $k$-fixed points are allowed and furthermore they coincide with $\hat{k}$-fixed points.

53 pages, broken reference corrected

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