Abstract
We employ the exponential parametrization of the metric and a "physical" gauge fixing procedure to write a functional flow equation for the gravitational effective average action in an $f(R)$ truncation. The background metric is a four-sphere and the coarse-graining procedure contains three free parameters. We look for scaling solutions, i.e. non-Gaussian fixed points for the function $f$. For a discrete set of values of the parameters, we find simple global solutions of quadratic polynomial form. For other values, global solutions can be found numerically. Such solutions can be extended in certain regions of parameter space and have two relevant directions. We discuss the merits and the shortcomings of this procedure.
Highlights
In order to make the critical theory at the fixed point (FP) physically meaningful, the corresponding conformal field theory should have a finite number of relevant deformations so that a finite number of measurements at low energy would be sufficient to completely fix the theory, i.e. all the couplings at any scale
The search of a gravitational FP has been conducted mostly using some approximation of the functional renormalization group equation (FRGE)
For the sake of comparison of our flow equation with others in the literature, we report the ninth order polynomial solutions of the fixed point equations with the popular type I cutoff, where the reference operator is −∇ ̄ 2 for all modes, and a type II cutoff where the reference operator contains precisely the R-terms that are present in the Hessian
Summary
In order to make the critical theory at the FP physically meaningful, the corresponding conformal field theory should have a finite number of relevant deformations so that a finite number of measurements at low energy would be sufficient to completely fix the theory, i.e. all the (infinitely many) couplings at any scale. To decide if the latter fundamental property may be achieved, any kind of investigation in this direction should start from a sufficiently large (possibly infinite-dimensional) theory space, show the existence of one or more critical theories and eventually should be able to identify which one has the required properties under deformations that may lead to a definition of a UV complete and predictive theory. While the number of couplings considered simultaneously has become quite high, these analyses still fall short of exploiting the full power of the FRGE machinery, which, as the name suggests, is designed to deal with the renormalization of whole functions, or equivalently infinitely many couplings
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