# nLab asymptotic safety

Contents

### Context

#### Gravity

gravity, supergravity

# Contents

## Idea

In the context of quantum field theory (QFT) the term asymptotic safety (Weinberg 79) refers to the situation where a QFT may not be renormalizable (is not defined on all scales by the choice of a finite number of coupling constants), but has the property that its renormalization group flow has a non-trivial non-perturbative fixed point such that the subspace of the space of coupling constants whose RG-flow converges to this fixed point is finite dimensional.

One defines the “UV-critical hypersurface” as the set of all those points in the infinite-dimensional theory space which are “pulled” into the fixed point by the inverse RG flow: trajectories lying in this surface approach the fixed point for increasing momentum scales. General arguments and known examples suggest that the UV-critical hypersurface has a finite dimensionality. This dimensionality equals the number of (infrared-) relevant couplings, i.e. couplings which get attracted to the fixed point in the UV. The important point is that once the value of these (few) couplings are known at some scale all other (irrelevant) couplings are fixed by requiring an asymptotically safe theory, that is a trajectory which lies entirely in the UV-critical hypersurface. By this means we achieve that, first, the couplings are determined by a finite number of measurements rendering the theory predictive, and, second, the UV behavior is unproblematic without any unphysical divergences. (Nink-Reuter 12, p. 2)

A key example of a non-renormalizable QFT is Einstein gravity, and there is speculation that it might be asymptotically safe, and that this might be the solution to the construction of quantum gravity (Reuter 96).

## Issues

Issues that the program of asymptotic safety of gravity is facing include the following:

### Theoretical issues

1. All existing computations that see hints for a UV-fixed point do so by first applying a drastic truncation to the space of couplings, and then checking only whether there is a UV-fixed point for the RG-flow in the remaining small subspace. It seems unclear to which extent these approximate considerations may be extrapolated.

2. Most existing computations consider only pure Einstein gravity without matter coupling. It seems unclear to which extent these results may be extrapolated to the situation where matter is taken account of (but see Biemans-Platania-Saueressig 17).

3. Donoghue 19 claims technical problems with the handling of renormalization group flow in Newton’s constant $G_N$ as currently practiced in the asymptotic safety program:

1. The “running$G_N(E)$ comes from power-divergent corrections that vanish in dimensional regularization.

2. Computing different Minkowski spacetime processes gives different power-divergent corrections: there is no generic common $G_N(E)$.

3. There is no generic energy $E$: some processes depend on $s \sim + E^2$, others on $t \sim -E^2$ and (unlike in the usual log running) this makes a difference.

4. If one assumes (as is widely, but no generally believed) that Bekenstein-Hawking entropy seen in classical gravity is to correspond to a microscopic entropy of its quantum degrees of freedom, then the scaling of this entropy with area as opposed to volume contradicts the assumption that quantum gravity is a local field theory at small scales and higher energies (Shomer 07, section IV) and hence then it contradicts the asymptotic safety of gravity.

Similarly, if one trusts the AdS/CFT correspondence then gravity is fundamentally not a local field theory, only its boundary CFT is, in contradiction with asymptotic safety of gravity (Shomer 07, section IV).

### Experimental issues

The near criticality of the Higgs field vacuum (see there at Higgs mass and vacuum (in-)stability) implies that the coefficient $\lambda$ of the quartic part of the Higgs potential is close to zero after renormalization group flow (“RGE”) to around the Planck scale of about $10^{19}$ GeV (e.g. BDGGSSS 13, p. 17-18):

In fact also the beta function $\beta_\lambda$ of the quartic coupling $\lambda$ (i.e. its logarithmic derivative with respect to scale) is close to zero around the Planck scale of about $10^{19}$ GeV (BDGGSSS 13, p. 18):

Earlier it has been suggested that this reflects the principle of asymptotic safety (Shaposhnikov-Wetterich 09). But this would mean that not only $\lambda$ and its RGE-derivative $\beta_\lambda$ vanish around the Planck scale, but that in fact all higher derivatives do, too (see e.g Niedermaier 06, equation (1.5)) hence that $\beta_\lambda$ asymptotes to zero. But this does not seem to be the case; in (BDGGSSS 13, p. 17-18) it says:

As shown in fig. 2 (upper right), the corresponding Higgs quartic beta-function vanishes at a scale of about $10^{17}$-$10^{18}$ GeV. In order to quantify the degree of cancellation in the β-function, we plot in fig. 2 (lower right) $\beta_\lambda$ in units of its pure top quark contribution. The vanishing of $\beta_\lambda$ looks more like an accidental cancellation between various large contributions, rather than an asymptotic approach to zero.

## References

The idea of asymptotic safety as such and as a cure for quantum gravity is due to

• Steven Weinberg, Ultraviolet divergences in quantum theories of gravitation, in “General Relativity: An Einstein centenary survey”, ed. S. W. Hawking and W. Israel. Cambridge University Press. pp. 790–831 (1979) (spire)

It gained new popularity with this result:

An attempt to conceptually explain why gravity might have a UV-fixed point is in this article:

• Andreas Nink, Martin Reuter, On quantum gravity, Asymptotic Safety, and paramagnetic dominance, Int. J. Mod. Phys. D22 (2013) 1330008 (arXiv:1212.4325)

Observation of a special role of spacetime dimension 2:

and speculation of this being related to the Connes-Lott model (see here):

Review of the asymptotic safety program includes these:

Critical review: