[YETI] Asymptotic Safety: a pathway to Quantum Gravity

The following are the (incomplete, possibly erroneous) notes I took during Daniel Litim’s talk at the YETI 2017 conference in Durham.

Why Quantum Gravity?

The SM is perturbatively renormalizable and predictive, with quantum fluctuations modifying the strength of interactions as a function of energy or distance (running couplings). In fact, asymptotic freedom, or the realization that $\alpha_s$ decreases as $Q$ [GeV] increases, is a triumph of QFT.

On the other end of the physics spectrum lies GR, which, not having been tested above $10^{28}$ cm or below $10^{-2}$ cm, leaves ample space for new physics. Singularities (black holes, early Universe…), the near-perfect homogeneity of the CMB and a small $\Lambda$ are evidence that GR is ultimately incomplete. As classical gravity is sourced by quantum matter, we have to expect quantum effects $\propto T_{\mu\nu}$.

The physics of QG

We start with the action

and perform dimensional analysis to define the “Planck constants” :

such that we expect quantum modifications at energy scales $\sim M_{\text{Pl}}$. Broadly speaking, the idea is to consider the metric $g_{\mu\nu}$ as a propagating quantum field.

The trouble with gravity

We note two main problems : first, the structure of UV divergences. Given $[g_{\mu\nu}]=0$, $[\text{Ricci}]=2$ and $[G_N]=2-d$, we define the effective expansion parameter $g_{\text{eff}}\equiv G_NE^2\sim\frac{E^2}{M_{\text{Pl}}^2}$; and so the $N$-loop Feynman diagram is $\sim\int\mathrm{d}pp^{A-[G]N}$. We can classify the renormalizability of such theories as :

The perturbative non-renormalizability of GR is indeed the second issue, and has only been tentatively addressed by either the addition of matter interactions or Goroff-Sagnotti terms.

Some hints from pertubation theory

We can formulate an effective theory for gravity that allows us to compute quantum corrections for energies $g_{\text{eff}}\ll 1$; knowledge of UV completion is not required there.

Alternatively, one can add higher derivative terms to the action, thereby making the theory pertubatively renormalizable; however, it must be noted that unitarity issues arise at high energies (i.e. probabilities not conserved etc.). This issue was later resolved by Weinberg and Gomis, who proposed adding not only a few, but all possible higher derivative operators, restoring unitarity at high energyies, but at the same time losing the predictive power of the theory.

Asymptotic safety builds on these caveats, and on Wilson’s work in the seventies that showed that a QFT could be fundamentally defined by its UV fixed point. Asymptotic freedom is understood to be the phenomenon whereby the theory becomes free at high energies (Gaussian UV fixed point); but technically, one only need asymptotic safety with an interacting UV fixed point (so that the theory remains interacting at high energies).

Principles of Asymptotic Safety

Renormalization group

We consider gravitons in $D=2+\epsilon$ dimensions with a coupling $\alpha=G_N(\mu)\mu^{D-2}$, where $\mu$ is the renormalization group scale and $G_N(\mu)$ expresses the energy dependence of $G_N$, understood as the beta function of couplings. With the parameterization

we get

RG flow

where we clearly observe that for $A>0$ there is a zero of the $\beta$-function other than the IR fixed point : this one we define as the UV fixed point. If we look instead at the plot of $\alpha$ versus $\mu$, we see that above some typical energy scale $\Lambda_T$ we move from the GR regime ($G(\mu)\approx G_N$) to one where gravity weakens ($G(\mu)\approx\alpha_*/\mu^{D-2}$) :

Gravity weakens near the UV fixed point

As always in Wilsonian renormalization theory, the UV behaviour is characterized by relevant/marginal/irrelevant invariants, but only for finitely many relevant invariants is the theory said to be predictive. Data from observations should tell us which RG trajectory corresponds to our $(G_N,\Lambda)$. Exact asymptotic safety is now possible in $D=4$ for certain classes of gauge-Yukawa theories (perhaps for the SM?).

In the diagram below, the two relevant couplings are $g=G_k\cdot k^2$ and $\lambda=\Lambda_k/k^2$ :

Gravitation phase diagram in 4D

4D quantum gravity

With large anomalous dimensions, one expects large couplings and so non-perturbative tools are mandatory. One such tool is functional Wilsonian renormalization (diagram below), where one introduces a momentum scale $k$ in the path integral, and slowly varies $k$ to $0$ (classical GR is regained in the limit $k\to 0$), thereby shifting the problem to the computing of a derivative of the path integral – which has a much more controlled behaviour.

Functional Wilsonian renormalization

Bootstrap search strategy

Start from the hypothesis that the relevancy of invariants follows their canonical mass dimensions, and apply the following algorithm :

  1. retain invariants up to dimension $D$;
  2. compute (using RG, lattice, holography, etc.);
  3. enhance $D$;
  4. iterate.

Convergence of the iteration is then interpreted as supporting the hypothesis, non-convergence as refuting it. As an example, in $f(R)$ gravity we can write down 3 relevant invariants : $\Lambda$, $R$ and $R^2$; which have (ir)relevancy $\sim D=2(N-1)$ :

Convergence and relevant eigenvalues in f(R) gravity

More online

Daniel unfortunately didn’t have time to go through his entire lecture, which was supposed to include as well experimental tests of asymptotic safety in the physical world, in cosmology, particle physics and black holes.

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