asymptotic safety



Quantum field theory




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

  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. 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).


Higgs particle self-coupling

The near criticality of the Higgs field vacuum (see there at Higgs mass and vacuum (in-)stability) implies that the coefficient λ\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 1910^{19} GeV (e.g. BDGGSSS 13, p. 17-18):

In fact also the beta function β λ\beta_\lambda of the quartic coupling λ\lambda (i.e. its logarithmic derivative with respect to scale) is close to zero around the Planck scale of about 10 1910^{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 λ\lambda and its RGE-derivative β λ\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 β λ\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 1710^{17}-10 1810^{18} GeV. In order to quantify the degree of cancellation in the β-function, we plot in fig. 2 (lower right) β λ\beta_\lambda in units of its pure top quark contribution. The vanishing of β λ\beta_\lambda looks more like an accidental cancellation between various large contributions, rather than an asymptotic approach to zero.


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)

Review of the asymptotic safety program includes these:

The following review provides concise background on actual technical details:

  • Assaf Shomer, A pedagogical explanation for the non-renormalizability of gravity (arXiv:0709.3555)

See also

  • Roberto Percacci, Asymptotic Safety FAQs (web)

  • Jorn Biemans, Alessia Platania, Frank Saueressig, Renormalization group fixed points of foliated gravity-matter systems (arXiv:1702.06539)

The suggestion that asymptotic safety explains the observed mass of the Higgs particle is attributed to

but see p. 17 of

and for more see at Higgs field – Asymptotic safety?

Last revised on December 1, 2018 at 02:46:39. See the history of this page for a list of all contributions to it.