### Asymptotic Safety

Once upon a time, I wrote a blog post about the proposal by Reuter and various collaborators that quantum gravity in four dimension is controlled by a UV fixed point. Below some cutoff (at the Planck scale, if not below it), gravity is described by an effective theory. This effective description breaks down at the cutoff scale, and the theory is ill-defined unless one of two things happens

- New degrees of freedom enter (as in String Theory).
- The UV physics is controlled by a fixed point, so the apparently infinite number of coupling are actually not independent, but rather lie on an (IR-repulsive) trajectory emanating from the fixed point set.

The latter is known to occur for pure gravity in $2+\epsilon$ dimensions. The hope is that the same holds true in 4 dimensions. The technique used to study this is the so-called Exact Renormalization Group, truncated to some finite-dimensional space of couplings. Reuter’s original work truncated to a 2-dimensional subspace, consisting of the cosmological constant and Einstein-Hilbert term. The existence of a fixed point in that (brutal) truncation was not terribly convincing.

Since I wrote my post, there have been many followup papers, by various authors (see the recent review by Percacci), purporting to adduce further evidence, by including various other couplings, and checking to see whether the fixed point persists. For instance, this paper considers adding a polynomial (up to 6th order) in the curvature scalar. Reuter considered adding the square of the Weyl tensor.

The trouble with all of these papers is that they really don’t address the issue in a meaningful way.

The terms considered vanish on-shell (in flat space) and in conventional perturbation theory, any divergence in these terms can be absorbed by a field redefinition. The ${(\text{Weyl})}^2$ term is a slight exception. But it can be rewritten as the Gauss-Bonnet density plus terms which vanish on-shell. The Gauss-Bonnet density, being a topological invariant, receives no corrections.

The *first* term which *can* receive a nontrivial renormalization in pure gravity, and hence which *would* actually serve as an acute test of whether the fixed point really exists, is cubic in the Riemann tensor. Goroff and Sagnotti did the perturbative computation to show that it, in fact, received a log-divergent correction at 2-loops. This is the *first* divergence in pure gravity; the 1-loop divergences can be absorbed by field redefinitions, by the argument of the previous paragraph.

So the first *nontrivial* test of the asymptotic safety proposal will come when someone computes the ERGE for
$S = \int d^4 x \sqrt{-g} \left(M^4 c_1 + \tfrac{M^2}{c_2} R + \tfrac{c_3}{M^2} \tensor{R}{_\mu_\nu_^\alpha^\beta}\tensor{R}{_\alpha_\beta_^\rho^\sigma}\tensor{R}{_\rho_\sigma_^\mu^\nu}\right)$

Now, I’ve thought about doing this computation myself. But

- Goroff and Sagnotti’s computation was
*hard*. And using the ERGE approach can’t make it any easier. - It’s a pretty foregone conclusion what the result will be: there is no fixed point for any finite value of $c_3$.

So maybe I should throw this out there for the readers of this blog. Anyone want to attain fame and fortune by performing the first nontrivial test of the gravitational asymptotic safety hypothesis?

Posted by distler at January 30, 2008 10:43 AM
## Re: Asymptotic Safety

Yes, I want to do this. Even if it is a foregone conclusion, it needs to be checked.

It may take me some time though, as I will have to go over the papers first and review renormalization group flow.

p.s

I am a high energy theory graduate student looking for a real problem to work on.