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In the Asym Safe safe approach to quantum geometry both G and Lambda run with scale.

In particular Lambda gets large, as k→∞ and the dimensionless version Λ/k

^{2}=λ→λ

_{*}goes to a finite fixedpoint limit.

So it is of interest that Hamber says that Lambda cannot run unless you want to give up general covariance.

As I see it, if Hamber Toriumi's finding is sustained this effectively shoots Asymptotic Safety down in the quantum cosmology (QC) arena. This makes me a bit sad---I've harbored considerable hope and enthusiasm for it. AFAICS the quality of Hamber Toriumi's paper is high, maybe someone else will take a look and offer a second opinion.

http://arxiv.org/abs/1301.6259

**Inconsistencies from a Running Cosmological Constant**

Herbert W. Hamber, Reiko Toriumi

(Submitted on 26 Jan 2013)

We examine the general issue of whether a scale dependent cosmological constant can be consistent with general covariance, a problem that arises naturally in the treatment of quantum gravitation where coupling constants generally run as a consequence of renormalization group effects. The issue is approached from several points of view, which include the manifestly covariant functional integral formulation, covariant continuum perturbation theory about two dimensions, the lattice formulation of gravity, and the non-local effective action and effective field equation methods. In all cases we find that the cosmological constant cannot run with scale, unless general covariance is explicitly broken by the regularization procedure. Our results are expected to have some bearing on current quantum gravity calculations, but more generally should apply to phenomenological approaches to the cosmological vacuum energy problem.

34 pages.