Asymptotic safety in quantum gravity

Trajectories of the renormalization group flow in theory space, parametrized by infinitely many coupling constants. By convention, the arrows of the vector field (and the one on the green trajectory) point from UV to IR scales. The set of actions which lie inside the theory space and are pulled into the fixed point under the inverse RG flow (i.e., going in the direction opposite to the arrows) is referred to as UV critical surface. The asymptotic safety hypothesis is that a trajectory can only be realized in Nature if it is contained in the UV critical surface since only then it has a well-behaved high energy limit (orange, blue, and magenta trajectories, by way of example). Trajectories outside this surface escape theory space for ${\displaystyle k\rightarrow \infty }$ since they develop unacceptable divergences in the UV, while going to lower scales they approach the UV critical surface. This situation is represented by the green trajectory which lies above the surface and runs away from it for increasing RG scale (opposite to the green arrow).

The asymptotic safety program adopts a modern Wilsonian viewpoint on quantum field theory. Here the basic input data to be fixed at the beginning are, firstly, the kinds of quantum fields carrying the theory's degrees of freedom and, secondly, the underlying symmetries. For any theory considered, these data determine the stage the renormalization group dynamics takes place on, the so-called theory space. It consists of all possible action functionals depending on the fields selected and respecting the prescribed symmetry principles. Each point in this theory space thus represents one possible action. Often one may think of the space as spanned by all suitable field monomials. In this sense any action in theory space is a linear combination of field monomials, where the corresponding coefficients are the coupling constants, ${\displaystyle \{g_{\alpha }\}}$. (Here all couplings are assumed to be dimensionless. Couplings can always be made dimensionless by multiplication with a suitable power of the RG scale.)

The renormalization group (RG) describes the change of a physical system due to smoothing or averaging out microscopic details when going to a lower resolution. This brings into play a notion of scale dependence for the action functionals of interest. Infinitesimal RG transformations map actions to nearby ones, thus giving rise to a vector field on theory space. The scale dependence of an action is encoded in a "running" of the coupling constants parametrizing this action, ${\displaystyle \{g_{\alpha }\}\equiv \{g_{\alpha }(k)\}}$, with the RG scale ${\displaystyle k}$. This gives rise to a trajectory in theory space (RG trajectory), describing the evolution of an action functional with respect to the scale. Which of all possible trajectories is realized in Nature has to be determined by measurements.

The construction of a quantum field theory amounts to finding an RG trajectory which is infinitely extended in the sense that the action functional described by ${\displaystyle \{g_{\alpha }(k)\}}$ is well-behaved for all values of the momentum scale parameter ${\displaystyle k}$, including the infrared limit ${\displaystyle k\rightarrow 0}$ and the ultraviolet (UV) limit ${\displaystyle k\rightarrow \infty }$. Asymptotic safety is a way of dealing with the latter limit. Its fundamental requirement is the existence of a fixed point of the RG flow. By definition this is a point ${\displaystyle \{g_{\alpha }^{*}\}}$ in the theory space where the running of all couplings stops, or, in other words, a zero of all beta-functions: ${\displaystyle \beta _{\gamma }(\{g_{\alpha }^{*}\})=0}$ for all ${\displaystyle \gamma }$. In addition that fixed point must have at least one UV-attractive direction. This ensures that there are one or more RG trajectories which run into the fixed point for increasing scale. The set of all points in the theory space that are "pulled" into the UV fixed point by going to larger scales is referred to as UV critical surface. Thus the UV critical surface consists of all those trajectories which are safe from UV divergences in the sense that all couplings approach finite fixed point values as ${\displaystyle k\rightarrow \infty }$. The key hypothesis underlying asymptotic safety is that only trajectories running entirely within the UV critical surface of an appropriate fixed point can be infinitely extended and thus define a fundamental quantum field theory. It is obvious that such trajectories are well-behaved in the UV limit as the existence of a fixed point allows them to "stay at a point" for an infinitely long RG "time".

With regard to the fixed point, UV-attractive directions are called relevant, UV-repulsive ones irrelevant, since the corresponding scaling fields increase and decrease, respectively, when the scale is lowered. Therefore, the dimensionality of the UV critical surface equals the number of relevant couplings. An asymptotically safe theory is thus the more predictive the smaller is the dimensionality of the corresponding UV critical surface.

For instance, if the UV critical surface has the finite dimension ${\displaystyle n}$ it is sufficient to perform only ${\displaystyle n}$ measurements in order to uniquely identify Nature's RG trajectory. Once the ${\displaystyle n}$ relevant couplings are measured, the requirement of asymptotic safety fixes all other couplings since the latter have to be adjusted in such a way that the RG trajectory lies within the UV critical surface. In this spirit the theory is highly predictive as infinitely many parameters are fixed by a finite number of measurements.

In contrast to other approaches, a bare action which should be promoted to a quantum theory is not needed as an input here. It is the theory space and the RG flow equations that determine possible UV fixed points. Since such a fixed point, in turn, corresponds to a bare action, one can consider the bare action a prediction in the asymptotic safety program. This may be thought of as a systematic search strategy among theories that are already "quantum" which identifies the "islands" of physically acceptable theories in the "sea" of unacceptable ones plagued by short distance singularities.

A fixed point is called Gaussian if it corresponds to a free theory. Its critical exponents agree with the canonical mass dimensions of the corresponding operators which usually amounts to the trivial fixed point values ${\displaystyle g_{\alpha }^{*}=0}$ for all essential couplings ${\displaystyle g_{\alpha }}$. Thus standard perturbation theory is applicable only in the vicinity of a Gaussian fixed point. In this regard asymptotic safety at the Gaussian fixed point is equivalent to perturbative renormalizability plus asymptotic freedom. Due to the arguments presented in the introductory sections, however, this possibility is ruled out for gravity.

In contrast, a nontrivial fixed point, that is, a fixed point whose critical exponents differ from the canonical ones, is referred to as non-Gaussian. Usually this requires ${\displaystyle g_{\alpha }^{*}\neq 0}$ for at least one essential ${\displaystyle g_{\alpha }}$. It is such a non-Gaussian fixed point that provides a possible scenario for quantum gravity. As yet, studies on this subject thus mainly focused on establishing its existence.

Quantum Einstein Gravity (QEG) is the generic name for any quantum field theory of gravity that (regardless of its bare action) takes the spacetime metric as the dynamical field variable and whose symmetry is given by diffeomorphism invariance. This fixes the theory space and an RG flow of the effective average action defined over it, but it does not single out a priori any specific action functional. However, the flow equation determines a vector field on that theory space which can be investigated. If it displays a non-Gaussian fixed point by means of which the UV limit can be taken in the "asymptotically safe" way, this point acquires the status of the bare action.

The primary tool for investigating the gravitational RG flow with respect to the energy scale ${\displaystyle k}$ at the nonperturbative level is the effective average action ${\displaystyle \Gamma _{k}}$ for gravity.[14] It is the scale dependent version of the effective action where in the underlying functional integral field modes with covariant momenta below ${\displaystyle k}$ are suppressed while only the remaining are integrated out. For a given theory space, let ${\displaystyle \Phi }$ and ${\displaystyle {\bar {\Phi }}}$ denote the set of dynamical and background fields, respectively. Then ${\displaystyle \Gamma _{k}}$ satisfies the following Wetterich-Morris-type functional RG equation (FRGE):[10][11]

${\displaystyle k\partial _{k}\Gamma _{k}{\big [}\Phi ,{\bar {\Phi }}{\big ]}={\frac {1}{2}}\,{\mbox{STr}}{\Big [}{\big (}\Gamma _{k}^{(2)}{\big [}\Phi ,{\bar {\Phi }}{\big ]}+{\mathcal {R}}_{k}[{\bar {\Phi }}]{\big )}^{-1}k\partial _{k}{\mathcal {R}}_{k}[{\bar {\Phi }}]{\Big ]}.}$

Here ${\displaystyle \Gamma _{k}^{(2)}}$ is the second functional derivative of ${\displaystyle \Gamma _{k}}$ with respect to the quantum fields ${\displaystyle \Phi }$ at fixed ${\displaystyle {\bar {\Phi }}}$. The mode suppression operator ${\displaystyle {\mathcal {R}}_{k}[{\bar {\Phi }}]}$ provides a ${\displaystyle k}$-dependent mass-term for fluctuations with covariant momenta ${\displaystyle p^{2}\ll k^{2}}$ and vanishes for ${\displaystyle p^{2}\gg k^{2}}$. Its appearance in the numerator and denominator renders the supertrace ${\displaystyle ({\mbox{STr}})}$ both infrared and UV finite, peaking at momenta ${\displaystyle p^{2}\approx k^{2}}$. The FRGE is an exact equation without any perturbative approximations. Given an initial condition it determines ${\displaystyle \Gamma _{k}}$ for all scales uniquely.

The solutions ${\displaystyle \Gamma _{k}}$ of the FRGE interpolate between the bare (microscopic) action at ${\displaystyle k\rightarrow \infty }$ and the effective action ${\displaystyle \Gamma [\Phi ]=\Gamma _{k=0}{\big [}\Phi ,{\bar {\Phi }}=\Phi {\big ]}}$ at ${\displaystyle k\rightarrow 0}$. They can be visualized as trajectories in the underlying theory space. Note that the FRGE itself is independent of the bare action. In the case of an asymptotically safe theory, the bare action is determined by the fixed point functional ${\displaystyle \Gamma _{*}=\Gamma _{k\rightarrow \infty }}$.

Let us assume there is a set of basis functionals ${\displaystyle \{P_{\alpha }[\,\cdot \,]\}}$ spanning the theory space under consideration so that any action functional, i.e. any point of this theory space, can be written as a linear combination of the ${\displaystyle P_{\alpha }}$'s. Then solutions ${\displaystyle \Gamma _{k}}$ of the FRGE have expansions of the form

${\displaystyle \Gamma _{k}[\Phi ,{\bar {\Phi }}]=\sum \limits _{\alpha =1}^{\infty }g_{\alpha }(k)P_{\alpha }[\Phi ,{\bar {\Phi }}].}$

Inserting this expansion into the FRGE and expanding the trace on its right-hand side in order to extract the beta-functions, one obtains the exact RG equation in component form: ${\displaystyle k\partial _{k}g_{\alpha }(k)=\beta _{\alpha }(g_{1},g_{2},\cdots )}$. Together with the corresponding initial conditions these equations fix the evolution of the running couplings ${\displaystyle g_{\alpha }(k)}$, and thus determine ${\displaystyle \Gamma _{k}}$ completely. As one can see, the FRGE gives rise to a system of infinitely many coupled differential equations since there are infinitely many couplings, and the ${\displaystyle \beta }$-functions can depend on all of them. This makes it very hard to solve the system in general.

A possible way out is to restrict the analysis on a finite-dimensional subspace as an approximation of the full theory space. In other words, such a truncation of the theory space sets all but a finite number of couplings to zero, considering only the reduced basis ${\displaystyle \{P_{\alpha }[\,\cdot \,]\}}$ with ${\displaystyle \alpha =1,\cdots ,N}$. This amounts to the ansatz

${\displaystyle \Gamma _{k}[\Phi ,{\bar {\Phi }}]=\sum \limits _{\alpha =1}^{N}g_{\alpha }(k)P_{\alpha }[\Phi ,{\bar {\Phi }}],}$

leading to a system of finitely many coupled differential equations, ${\displaystyle k\partial _{k}g_{\alpha }(k)=\beta _{\alpha }(g_{1},\cdots ,g_{N})}$, which can now be solved employing analytical or numerical techniques.

Clearly a truncation should be chosen such that it incorporates as many features of the exact flow as possible. Although it is an approximation, the truncated flow still exhibits the nonperturbative character of the FRGE, and the ${\displaystyle \beta }$-functions can contain contributions from all powers of the couplings.

As described in the previous section, the FRGE lends itself to a systematic construction of nonperturbative approximations to the gravitational beta-functions by projecting the exact RG flow onto subspaces spanned by a suitable ansatz for ${\displaystyle \Gamma _{k}}$. In its simplest form, such an ansatz is given by the Einstein–Hilbert action where Newton's constant ${\displaystyle G_{k}}$ and the cosmological constant ${\displaystyle \Lambda _{k}}$ depend on the RG scale ${\displaystyle k}$. Let ${\displaystyle g_{\mu \nu }}$ and ${\displaystyle {\bar {g}}_{\mu \nu }}$ denote the dynamical and the background metric, respectively. Then ${\displaystyle \Gamma _{k}}$ reads, for arbitrary spacetime dimension ${\displaystyle d}$,

${\displaystyle \Gamma _{k}[g,{\bar {g}},\xi ,{\bar {\xi }}]={\frac {1}{16\pi G_{k}}}\int {\text{d}}^{d}x\,{\sqrt {g}}\,{\big (}-R(g)+2\Lambda _{k}{\big )}+\Gamma _{k}^{\text{gf}}[g,{\bar {g}}]+\Gamma _{k}^{\text{gh}}[g,{\bar {g}},\xi ,{\bar {\xi }}].}$

Phase portrait for the Einstein–Hilbert truncation. Shown are the RG trajectories corresponding to the flow diagram on the left-hand side. (First obtained in Ref.[17])

Here ${\displaystyle R(g)}$ is the scalar curvature constructed from the metric ${\displaystyle g_{\mu \nu }}$. Furthermore, ${\displaystyle \Gamma _{k}^{\text{gf}}}$ denotes the gauge fixing action, and ${\displaystyle \Gamma _{k}^{\text{gh}}}$ the ghost action with the ghost fields ${\displaystyle \xi }$ and ${\displaystyle {\bar {\xi }}}$.

The corresponding ${\displaystyle \beta }$-functions, describing the evolution of the dimensionless Newton constant ${\displaystyle g_{k}=k^{d-2}G_{k}}$ and the dimensionless cosmological constant ${\displaystyle \lambda _{k}=k^{-2}\Lambda _{k}}$, have been derived for the first time in reference[14] for any value of the spacetime dimensionality, including the cases of ${\displaystyle d}$ below and above ${\displaystyle 4}$ dimensions. In particular, in ${\displaystyle d=4}$ dimensions they give rise to the RG flow diagram shown on the left-hand side. The most important result is the existence of a non-Gaussian fixed point suitable for asymptotic safety. It is UV-attractive both in ${\displaystyle g}$- and in ${\displaystyle \lambda }$-direction.

This fixed point is related to the one found in ${\displaystyle d=2+\epsilon }$ dimensions by perturbative methods in the sense that it is recovered in the nonperturbative approach presented here by inserting ${\displaystyle d=2+\epsilon }$ into the ${\displaystyle \beta }$-functions and expanding in powers of ${\displaystyle \epsilon }$.[14] Since the ${\displaystyle \beta }$-functions were shown to exist and explicitly computed for any real, i.e., not necessarily integer value of ${\displaystyle d}$, no analytic continuation is involved here. The fixed point in ${\displaystyle d=4}$ dimensions, too, is a direct result of the nonperturbative flow equations, and, in contrast to the earlier attempts, no extrapolation in ${\displaystyle \epsilon }$ is required.

Subsequently, the existence of the fixed point found within the Einstein–Hilbert truncation has been confirmed in subspaces of successively increasing complexity. The next step in this development was the inclusion of an ${\displaystyle R^{2}}$-term in the truncation ansatz.[18] This has been extended further by taking into account polynomials of the scalar curvature ${\displaystyle R}$ (so-called ${\displaystyle f(R)}$-truncations),[19] and the square of the Weyl curvature tensor.[20][21] Also, f(R) theories have been investigated in the Local Potential Approximation finding nonperturbative fixed points in support of the Asymptotic Safety scenario.[22] Moreover, the impact of various kinds of matter fields has been investigated.[15] Also computations based on a field reparametrization invariant effective average action seem to recover the crucial fixed point.[23] In combination these results constitute strong evidence that gravity in four dimensions is a nonperturbatively renormalizable quantum field theory, indeed with a UV critical surface of reduced dimensionality, coordinatized by only a few relevant couplings.[16]