The New Improved Asymptotic Safety FAQs


Last updated November 2012

Thanks to Astrid Eichhorn for revisiting these FAQs and suggesting a number of improvements and updates.

 

Q: What is asymptotic safety?

A: It's a way in which a quantum field theory could be well defined at all energies without being perturbatively renormalizable. A quantum field theory is said to be asymptotically safe if it corresponds to a trajectory of the renormalization group that ends at a fixed point in the UV.

 

Q: Why is a fixed point good for the theory?

A: In quantum field theory, observable quantities such as decay rates and cross sections can be expressed as functions of the couplings. Generically, if the couplings are finite, also the observable quantities will be finite. So a way of ensuring that our description of the world has a good ultraviolet limit is to require that it lies on a renormalization group trajectory for which all couplings remain finite when the energy goes to infinity. The simplest way of achieving this is to demand that the trajectory flows towards a fixed point. More complicated situations such as limit cycles are also possible.

 

Q: So, if gravity was asymptotically safe, Newton’s constant would have a finite limit?

A: No. The couplings of familiar theories such as the standard model are dimensionless and for them the definition of a fixed point is completely straightforward. For dimensionful couplings the definition of a fixed point is a bit more involved. To motivate the correct definition, consider the following argument.

 

In Einstein's theory, the strength of the gravitational coupling is the number Ǧ=G k2, where G is Newton's constant and k is some momentum scale of the process being considered. The reason why the energy scale k appears is that gravity couples to mass, and energy is mass; the higher the energy of a particle the stronger its gravitational coupling. (Technically this manifests itself in the fact that all the couplings of gravitons contain derivatives.) The number Ǧ  is very small at all accessible energies but this formula implies that when k goes to infinity also the strength of the coupling goes to infinity at the same rate.

 

The renormalization group will change this. Newton’s constant becomes a running coupling G(k) and it is conceivable that for large k it behaves like k-2 ; then Ǧ  would tend to a constant. This is what is meant by a fixed point for Newton’s constant. It implies that at some point the dimensionless strength of the gravitational coupling would cease growing with the energy and would tend to a finite limit.

 

More generally, in any field theory there will be in principle infinitely many couplings [g]. If a coupling g has canonical mass dimension d we define the dimensionless number ǧ = g k-d . The set of all these variables parametrizes a space that we may call "theory space", because it parametrizes all the possible actions. Parametrizing theory space with the dimensionless couplings [ǧ] just means that we are using the cutoff k as a unit of mass. Naively all the ǧ whose corresponding g has negative mass dimension would seem to diverge when k goes to infinity. Instead, on a trajectory approaching a fixed point, they would reach finite limits.

 

Q: Do we really have to consider all possible couplings?

A: Actually, if a coupling can be eliminated by a field redefinition it cannot enter into any physical observable and therefore it need not have a finite UV limit. Such couplings are called redundant or inessential. By fixing a field parametrization one can divide the set of all couplings into essential and inessential, and in general there will be infinitely many of each type. At a fixed point all the essential couplings must have the behavior described above.

(Note also that it might happen that not all essential couplings in our theory enter observables independently. If, e.g. certain couplings enter observables in a certain combination, only this combination has to show a fixed point. Thus potentially the theory could yield finite predictions even if some couplings would diverge at a certain scale, but the combination in which they enter observables stayed finite.)

 

Q: If it is desirable that the renormalization group flow tends to a fixed point, then the best theory should be one for which all points in theory space are attracted towards it?

A: If all points in theory space were attracted towards the fixed point, then we would have a good UV limit irrespective of the initial conditions. This would leave us with infinitely many arbitrary couplings, each one of which would have to be determined by experiment, and the theory would lose predictivity.

 

Instead, we want to use the condition of having a good UV limit as a way of selecting physically acceptable trajectories. From this point of view the ideal case would be that in which a single trajectory reached the fixed point. This would pin down the theory uniquely. More generally, if the FP has a finite number n of attractive directions, then the set of trajectories that are physically acceptable has dimension n-1. The set of all points belonging to these trajectories is a n – dimensional surface called the “UV-critical surface” or the “unstable manifold”. In order to have a good UV limit the initial point of the RG flow must lie in this surface.

 

In an asymptotically safe theory we would have to perform n experiments at some given energy scale k to pin down completely our position in theory space. Everything else could then be computed, at least in principle, and would constitute genuine predictions that could be verified experimentally.

 

Q: How can we study the critical surface?

A: We can determine the tangent space to the critical surface at the fixed point, by linearizing the flow at the fixed point. The directions that are attracted towards the fixed point in the ultraviolet are called “relevant”. This is described more technically here (in .pdf format) (alternatively here in .ps format) . This is enough to make predictions that hold at high energies. As we move towards lower energies the surface deviates from linearity and it gets very hard to say much. (Note that this is true also in the case of perturbatively renormalizable theories.)

 

Q: What does asymptotic safety correspond to in perturbation theory?

A: Standard perturbation theory corresponds to the case when the fixed point is the Gaussian (free theory) fixed point. In this case the linearization of the flow is described in technical terms here (in .pdf format) (alternatively here in .ps format) . The tangent space to the critical surface is spanned by the couplings that have positive or vanishing mass dimension, i.e. those that are power counting renormalizable (or marginal).  Thus asymptotic safety at the Gaussian fixed point is equivalent to perturbative renormalizability plus asymptotic freedom. It is widely agreed that a theory with these properties makes sense up to arbitrarily high energies and therefore can be regarded as a fundamental theory. Asymptotic safety is a generalization of this behaviour to the case when the fixed point is not a free theory.

 

Q: If we cannot use perturbation theory we cannot calculate anything so we may as well give up.

A: There are important examples in nature where physics simply becomes non-perturbative. This is not a question of our choice or taste. As an example, QCD at low energies exhibits non-perturbative phenomena such as confinement and chiral symmetry, which are accompanied by an explicit breakdown of perturbation theory. There are non-perturbative methods such as lattice gauge theory, Dyson-Schwinger equations and Functional Renormalisation that can be used to gain understanding of the non-perturbative regime.


It should also be said that if a nontrivial fixed point i
s not too distant from the free one, in some suitable sense, it may be possible to study its properties using perturbation theory.

 

Q: Do we know any QFT that is asymptotically safe?

A: QCD is asymptotically safe: it has a fixed point (the free theory) and only finitely many couplings are attracted towards it in the UV, namely the Yang-Mills coupling and the quark masses. All other couplings are repelled from it, so in order to have a good UV limit one sets them to zero. (To be more precise, one sets them to zero in perturbation theory. i.e. in a neighborhood of the fixed point; further away low orders in perturbation theory may be insufficient, the critical surface may be curved and the relevant couplings could contain some admixture of operators with negative mass dimension).

 

Q. Do we know any QFT that is asymptotically safe without being asymptotically free?

A: The Gross-Neveu model in two dimensions with a p-2+ε propagator, which is perturbatively nonrenormalizable, has been rigorously shown to be renormalizable at a nontrivial fixed point. See Gawedzki and Kupiainen (1985a,b,c). A similar result in three dimensions has been proven by de Calan et al (1991). For a more recent discussion see Braun, Gies and Scherer (2010).

 

Q. Are there other examples of systems that are governed by a nontrivial fixed point?

A: A scalar field theory in three dimensions has a nontrivial fixed point which gives a rather accurate description of many critical phenomena. Of course in this case one does not need to go to arbitrarily high energies because condensed matter systems have a natural cutoff at the atomic scale. However the existence of the fixed point guarantees that there can exist systems where the ratio between the overall size of the system and the UV scale is arbitrarily large.

 

Q: What is the status of gravity in perturbation theory?

A: If we restrict the action to contain only the Hilbert term there is a “Gaussian” fixed point that corresponds to vanishing Newton's constant. Perturbation theory describes a neighborhood of this point. Calculations in perturbation theory (‘t Hooft and Veltman 1974, Goroff and Sagnotti 1986, van de Ven 1992) indicate that quantum corrections generate terms cubic in curvature which cannot be absorbed by field redefinitions, so this theory is not perturbatively renormalizable.

 

If we enlarge the set of actions to include terms quadratic in curvature then perturbation theory shows that no new terms are generated by the quantum corrections, so the theory is perturbatively renormalizable (Stelle 1977, Voronov and Tyutin 1984). The reason why this is not generally hailed as the quantum theory of gravity is that actions quadratic in curvature generically describe, in addition to the graviton, also other massive spin two particles with the wrong sign in the propagator (ghosts). It is not at all clear that the presence of ghosts in the action near the fixed point is fatal; some early comments in this sense can be found in the papers of Julve and Tonin 1982, Salam and Strathdee. But neither do we have a proof of unitarity, so the point remains for the time being obscure.

 

Another potential problem is that the proof of renormalizability relies on flat space perturbation theory and hence requires that the cosmological constant vanishes identically. This seems to be inconsistent with the Wilsonian flow.

 

Q: Why can one not neglect the cosmological constant?

A: Essentially because the cosmological constant runs too. We can fix our renormalization conditions in such a way that it vanishes at some energy scale but then it will be nonzero at other scales. A more technical explanation can be found here (in .pdf format) (alternatively here in .ps format). It shows that the flow in the neighbourhood of the Gaussian fixed point must have the structure illustrated in the following figure:

 

 

Fig.1: the flow in the perturbative region.

 

The point here is the slant in the eigenvectors at the origin. If both eigenvectors were directed along the axes, it would be consistent to have nonzero Newton’s constant and zero cosmological constant. But due to the slant, if we choose our initial conditions to correspond to a point on the positive Ǧ - axis (i.e. a small and positive Newton’s constant and zero cosmological constant) we see that the flow will generate a positive cosmological constant when going towards higher energies.


Possibly, a unimodular quantum theory could also become asymptotically safe, in which case the cosmological constant would not be a running coupling. This option is so far unexplored.

 

Q: Where does the flow lead to?

A: According to all the calculations done up to now, it leads to a nontrivial fixed point located somewhere in the upper right quadrant, corresponding to positive cosmological constant and Newton’s constant. Of course there are also other couplings, so the location of the fixed point, if it really exists, has to be described in a higher dimensional space.

Note that neither the microscopic cosmological constant nor the microscopic Newton coupling need to be positive in order to be consistent with observations: It suffices that they flow towards the observed positive values at smaller momentum scales. Since we know that higher-curvature operators also have non-zero couplings at the fixed point, the positivity of Newton's coupling is not required for the stability of the theory. So far, we do however not know an example where the flow crosses the line G=0, which suggests that the fixed-point value of Newton's coupling should be positive in order for the flow to be compatible with observations in the infrared.

 

Q: To study the flow further from the Gaussian fixed point we cannot use perturbation theory. What can be used?

A: The most powerful instrument is the Exact Renormalization Group Equation or ERGE. There are different version of it; a popular one is the Polchinski equation (Polchinski 1983). It is a functional equation obeyed by the interaction part of the Wilsonian action. For some applications it has proven more convenient to use another equation for the Legendre transform or effective average action Γk[Φ] (Wetterich 1993, Morris 1994a,b).

 

The basic idea behind these equations goes back to Wilson: the effective action (effective in the sense of “effective field theory”) describing physics at scale k is the result of functionally integrating out all degrees of freedom of the field with momenta greater than k. In practice in the ERGE this is achieved by integrating over all degrees of freedom but inserting a suppression factor for the fluctuations with momenta lower than k. Thus k acts effectively like an infrared cutoff.

 

An important aspect of all this is the “cutoff” function which is used to suppress the contribution of the low momentum modes. This function is required to go quickly to zero for momenta larger than k and to some finite value for momenta going to zero. Aside from this, the cutoff function is arbitrary. In general, results obtained from the ERGE will depend on the choice of this function. However, physical results must be independent of the shape of the cutoff.

 

Q: I am not familiar with the ERGE. Does it agree with the familiar results when perturbation theory is applicable?

A: The ERGE gives the renormalization group flow of the functional Γk[Φ] and this functional contains in principle infinitely many couplings. So the ERGE contains the beta functions of all these couplings. It can be regarded as the “beta functional” of the theory.

 

If the average effective action Γk[Φ] is written as a sum of generic operators On[Φ] with coefficients gn , then the coefficient of On[Φ]  in the ERGE is the beta function of gn . With some work one can extract the beta functions of individual couplings from the ERGE. When applied to scalar, fermion and gauge theories, this procedure can be used to reproduce the results that are well known from perturbation theory.

 

As mentioned above, in general the beta functions depend on the shape of the cutoff. This is similar to the dependence on the renormalization scheme in perturbation theory. However the leading terms in the beta functions of dimensionless couplings (quartic scalar coupling, Yukawa coupling, gauge coupling) turn out actually to be independent of the shape of the cutoff (they are “universal”) and agree with the results of perturbation theory.

 

Q: The ERGE uses a cutoff. Does this not conflict with gauge invariance?

A: There is a well established way of quantizing gauge theories that preserves a form of gauge invariance: the background field method. One can define the ERGE for a gauge theory using the background field gauge and using the background field in the definition of the cutoff function. The average effective action is then a functional of two fields Γk[Ā;A], where Ā is the background field and A is the “classical” field introduced with the Legendre transform. This functional is gauge invariant when both arguments are gauge transformed. The background field is arbitrary and at the end of the day one can set Ā=A. In this way one obtains a gauge invariant functional of a single gauge field, whose flow can be calculated using the ERGE. As mentioned above, this reproduces the beta function of the Yang-Mills coupling in the case of a gauge theory. In the case of gravity this has been discussed in Reuter 1998.


Q: How do you define the scale k, when you want to integrate over all possible metric configurations? The definition of k requires the notion of a metric.
A: Here the background field method is crucial: Picking a background metric allows to define the notion of a scale k, according to which we define metric fluctuations to be "high energetic" or "low energetic" modes. A priori, any metric is as good as any other. However we should note that one distinguished metric emerges dynamically: it is the solution of the quantum equations of motion in the limit k-->0, where all quantum fluctuations have been integrated out.

Q: What ultraviolet regularization do you use?

A: There is no need to use any ultraviolet regularization. The ERGE gives automatically finite results for the beta functions because it receives contributions only from field modes with momenta comparable to k .

 

Technically this is due to the fact that the k-derivative of the IR cutoff in the r.h.s. of the ERGE is a function that tends to zero very rapidly when the momentum is greater than k. This effectively provides an UV cutoff for the trace in the r.h.s. of the ERGE.

 

Yes, it is somewhat confusing that an IR cutoff should work as an UV cutoff, but this is one of the nice properties of the ERGE.

 

Q: If k is an infrared cutoff, how can it be used to study UV physics?

A: It is an infrared cutoff in the sense that in the definition of the generating functional it sets a lower limit to the functional integration. It does not mean that k has to be small in any sense. Once the beta functions of all couplings have been calculated, they can be used to study the limit when k goes to infinity.

 

Q: Still, the definition of the functional Γk[Φ] requires some form of UV regularization at scales Λ >> k . So, you cannot study the limit when k goes to infinity without having previously defined the limit when Λ goes to infinity.

A: Indeed, if we wanted to define the functional Γk[Φ] from the functional integral, then we would have to use some ultraviolet cutoff  Λ >> k. But the derivative of Γk[Φ] with respect to k does not need any UV regulator, as I said before. So, we apply the following logic:

1.     We know that if the functional Γk[Φ] exists it obeys the ERGE

2.     We assume that a functional Γk[Φ] exists

3.     We study the flow of Γk[Φ] using the ERGE, which is well defined

4.     If the flow has a FP with the desired properties, then the UV limit in k is well defined

 

Q: What does the ERGE say about the running of Newton’s constant?

A: First of all one must clarify that since the background field effective action is a functional of two metrics (or equivalently a metric and a fluctuation field), there are actually two separate notions of Newton's constant. Most of the work has been done on the coupling that multiplies the background Einstein-Hilbert action, but the available evidence suggests that the "other" Newton's constant has a qualitatively similar behavior.
 
Starting from very low energies, where perturbation theory is applicable, one finds a RG running of Newton’s constant of the form G(k) = G0(1+ω G0k2+…),
with a value of ω that is negative and of order one. This implies that G decreases when one goes to higher energies, i.e. gravity is antiscreening. If there is a fixed point, this antiscreening behavior persists to arbitrarily high energy and in the UV limit one has G(k) Ǧ k-2 . In the one loop approximation one has a fixed point at Ǧ=-1/ ω>0 .

 

Of course the perturbative calculation cannot be trusted when Ǧ  becomes of order one. However, the ERGE is not similarly limited. In fact with the ERGE one can compute also all the higher powers of G0  in the beta function. This leads to a finite shift in the position of the fixed point, but qualitatively the beta function is similar to the one loop result.

 

Q: Is there a heuristic understanding of the gravitational antiscreening?

A: Yes, and it closely parallels the explanation of antiscreening in Yang-Mills theory. Recall that the graviton kinetic operator has the structure of a covariant Laplacian plus terms linear in curvature. Antiscreening can be understood as the dominance of the "paramagnetic" effects, encoded in the curvature terms, over the "diamagnetic" effect of the Laplacian. This is discussed by Nink and Reuter (2012).


Q: If G(k) Ǧ k-2, could one not say that Newton’s constant is asymptotically free?

A: To some extent this is a matter of terminology, but I find that this statement would be misleading. Newton’s constant, being dimensionful, does not have a value. Only the ratio of Newton’s constant to some unit of area u has a mathematical value and a physical meaning. Ultimately any unit will be reducible to some other dimensionful coupling appearing in the theory, which will itself be subject to renormalization group flow (see Kawai and Ninomiya 1990. Percacci 2007a). The ratio G/u may tend to zero, to a finite limit or to infinity depending on the choice of the unit. As discussed above, the common practice in renormalization theory is to use the independent variable k as a unit of mass. Having made this choice, the proper meaning of the statement that “Newton’s constant is asymptotically free” would be that Ǧ goes to zero in the ultraviolet, and this is not what calculations lead us to believe.

 

Q: What does the ERGE predict for the ultraviolet limit?

A: So far we have mentioned only Newton’s constant. However, a theory of gravity could contain in principle infinitely many other couplings. Let us call “QGD” for “Quantum GraviDynamics” a general metric theory of gravity, meaning that the dynamical variable of the theory is the metric. (A slight generalization of QGD may contain also an independent connection, but we will not discuss this here). In the spirit of effective field theories, the action of QGD will contain the most general diffeomorphism invariant functional of the metric. In a derivative expansion, the action can be written as a sum of arbitrary powers of curvatures and their covariant derivatives, with arbitrary coefficients. Thus the theory space of QGD is parametrized by all these couplings (made dimensionless with k, as described above). A fixed point for QGD requires that the beta functions of all these couplings vanish simultaneously.

 

So far a nontrivial fixed point has been found using each of the following approximations to QGD:

1) The ε expansion of Einstein’s theory around two dimensions

2) The leading order of the 1/N expansion

3) The Einstein-Hilbert truncation using a variety of gauges and cutoff functions, also in the presence of matter fields, both minimally and non-minimally coupled.

4) A truncation including all terms with up to four derivatives of the metric

5) A truncation involving polynomials up to the eighth power in the Ricci scalar

6) The reduction of general relativity to metrics admitting two Killing vectors
7) An infinite truncation with action of the form f (R).
8) Truncations involving a non-trivial gauge fixing sector (bimetric truncations) and Faddeev-Popov ghost sector.
9) A calculation using the Vilkovisky-DeWitt effective action

 

Q: What does the ε expansion teach us?

A: This was historically the first hint that there may be a nontrivial fixed point for Newton’s constant (Weinberg 1979). In 2+ε dimensions Newton’s constant has mass dimension -ε , so Ǧ=G k ε . The beta function of Ǧ  can be calculated and is εǦ-(38/3) Ǧ2.  This has an UV attractive fixed point at Ǧ = (3/38) ε. The beta function is plotted below in the case ε= 2 (d = 4).

 

 

Fig.2: the beta function of Newton’s constant in the epsilon expansion.

 

For further information on this approximation see the series of papers by the Japanese groups Kawai and Ninomiya 1990, Kawai et al 1993a,b, 1996, Aida et al 1994, 1997, Nishimura et al 1994.

 

Unfortunately it is not at all clear from this procedure that the result can be trusted when ε is so large. One reason is that dimensional regularization was used and one may expect poles at rational values of the dimension between 2 and 4. However, this result has been subsequently vindicated by the ERGE, in the following sense. In the ERGE there is no need for any ultraviolet regulator so one can calculate the beta function in any dimension d and follow the dependence of the results on d without encountering any singularity. It turns out that a fixed point exists for d>2 and the derivative of the fixed point with respect to d, at d=2 is exactly 3/38. Furthermore, the number 3/38 is independent of the shape of the cutoff function.

 

Q: What do you mean by a truncation?

A: Since we cannot in practice solve the ERGE exactly, the most common approximation consists of retaining just a finite number of terms in the functional Γk[Φ] and extracting their beta functions from the ERGE without any further approximation. Such a procedure is called a truncation. For example the Einstein-Hilbert truncation consists in retaining only the cosmological constant and Newton’s constant. Unless further approximations are made, it is clear that this takes into account nonperturbative effects.

 

When we truncate the action we neglect the effect that other couplings can have on the running of the couplings that are retained. There is no easy and direct estimate of the errors that are being introduced in this way. One indirect guess can be based on the scheme-dependence of the results. We know that in the exact equation “universal” quantities such as critical exponents must be independent of the cutoff function. So the amount by which they vary when the theory is truncated can be taken as a measure of the quality of the truncation. This kind of argument had been used by Reuter and collaborators to argue that the results of the Einstein-Hilbert truncation must be stable against the addition of new couplings.

 

Of course the best test is to calculate the effect of other couplings on the results of the earlier truncation. For the Einstein-Hilbert action, Codello Percacci and Rahmede 2007 have checked that the addition of operators up to eight powers in the Ricci scalar changes the fixed point and the critical exponents only by few percent, confirming the validity of the earlier conjectures.

 

Q: What do we learn from the Einstein-Hilbert truncation?

A: The results of the Einstein-Hilbert truncation are summarized in this flow diagram, resulting from a numerical integration of the beta functions:

 

 

Fig.3: the flow of the dimensionless cosmological constant and Newton constant in the Einstein-Hilbert truncation.

 

It shows the Gaussian fixed point in the origin, with the behaviour anticipated in fig.1, and a nontrivial fixed point at positive cosmological constant and Newton’s constant. The critical exponents at the nontrivial fixed point are a complex conjugate pair, giving the spiralling approach to the fixed point.

 

The position of the fixed point depends on the cutoff function. However it can be verified that the dimensionless product of the cosmological constant and Newton’s constant, which had been argued by Kawai and Ninomiya 1990 to be a universal quantity, is indeed quite stable at the numerical level. Also the critical exponents are rather stable under changes of cutoff.


Q: Does it make sense to have complex critical exponents?
A: It is only the real part of complex critical exponent that decides about the relevance of a coupling. The imaginary part signals that the flow approaches the fixed point in a spiral-type shape. Although exotic, they are not unknown in condensed-matter systems.
Whether some of the critical exponents remain complex beyond the Einstein-Hilbert truncation is actually unclear.


Q: What is known about curvature squared terms?

A: The beta functions of higher derivative gravity (including terms with four derivatives of the metric) had been calculated earlier by several authors using dimensional regularization (see Peixoto and Shapiro 2006 for the state of the art of those calculations). It had been found that the dimensionless couplings (the inverses of the coefficients of the four-derivative terms) are asymptotically free. However, the status of the cosmological constant and Newton’s constant was not very clear.

 

These calculations have been repeated  using the ERGE by Codello and Percacci (2006). Technical complications have further required the use of the one loop approximation and an expansion in powers of the cosmological constant. The beta functions of previous authors have been exactly reproduced, but some additional terms appeared in the beta functions of the cosmological constant and Newton’s constant. In a conventional calculation of the effective action, the new terms would correspond to quadratic and quartic divergences. Obviously such terms had been discarded in dimensional regularization but they prove essential in generating a nontrivial fixed point for the cosmological constant and Newton’s constant. The flow in this plane, assuming that the other couplings have been set to their fixed point values, is given in the following figure.

 

 

Fig.4: the flow of the dimensionless cosmological constant and Newton constant in a higher derivative truncation at one loop.

 

 

It is therefore found that the ERGE reproduces the universal beta functions of dimensionless couplings as computed in perturbative approaches. On the other hand the ERGE is uniquely suited to discuss the behaviour of the dimensionful couplings, which appear to reach a nontrivial fixed point.

 

More recently Benedetti et al (2009) have calculated these beta functions without some of the approximations made before. In a four parameter truncation (the flow of the topological term is not calculated) the critical surface appears to be three dimensional. As expected, all couplings tend to finite nonzero limits. This behavior persists also in the presence of a scalar field.

 

Q: What can be said about higher powers of curvature ?

A: So far only higher powers of the curvature scalar have been studied. Codello et al (2007) and Machado and Saueressig (2007) have studied Lagrangians of the type f(R), with f a polynomial of order up to six (later extended to eight), and shown that a fixed point exists and has a three dimensional critical surface.


The flow equation can be written for the whole function f but initially it proved too hard to solve. A simplified form of the flow has been later proposed by Benedetti and Caravelli (2012) and fixed point solutions have been found by Dietz and Morris (2012). This is a very powerful extension of previous truncations because it shows that the fixed point exists even when one considers an infinite number of couplings. This has the same level of sophistication of studies of scalar theories in the "local potential approximation", where one allows the potential to be completely general.

 

Q: Quite impressive, but is it good enough? We need a fixed point for all possible couplings. How can you think of ever proving its existence ?

A: This looks hard indeed. But there exists at least some approximation where one can prove the existence of the fixed point for all couplings: it is the leading order of the 1/N approximation.

 

In a gravitational context N is the number of matter fields coupled to the metric. Each one of these matter fields gives a contribution to the gravitational effective action and hence to the gravitational beta functions. If the number of such fields goes to infinity the contribution of graviton loops can be neglected relative to that of matter loops. Notice that this may actually be a reasonably good approximation in the real world, where there is only one graviton and dozens of matter fields (in GUTs there may even be hundreds of matter fields).

 

Tomboulis 1978, 1980 has shown how to turn this into a systematic expansion. Here we are only interested in the leading order of this expansion, which consists in neglecting the graviton contribution altogether. This simplifies calculations dramatically: the beta functions are just constants. This allows one to establish the existence of the fixed point at all orders in the derivative expansion of the gravitational action.

 

For a generic cutoff function one finds that all the couplings have nonzero values at the fixed point, with the exception of the dimensionless ones (the coefficients of the curvature squared terms) which tend logarithmically to infinity (equivalently, their inverses are asymptotically free); the critical exponents are equal to the canonical dimensions, so the dimension of the critical surface is five. There is a special class of cutoff functions such that the coefficients of Rn with n>2 all vanish. (This is related to a remark made by Chamseddine and Connes (1996) in the context of the spectral action of noncommutative geometry.)

 

This gives us a mechanism that can be easily understood and generates a fixed point for the whole infinite set of couplings of the derivative expansion.

 

Q: What does the two-Killing vector reduction teach us?

A: In this reduction one keeps the most general form for the gravitational action but cuts the number of degrees of freedom by imposing the existence of two Killing vectors. In this case gravity becomes a specific sigma model which has been shown by Niedermaier 2002, 2003 to be asymptotically safe.

 

Q: What do we know about the critical surface beyond linearization?

A: In the Einstein-Hilbert truncation there is a critical trajectory that flows from the nontrivial fixed point to the Gaussian fixed point, as energies are lowered. See Fig. 3. This critical trajectory obviously lies in the critical surface. It can be studied numerically and is there for all choices of gauge and cutoff. It is is also there at one loop in the presence of curvature squared terms, see fig.4 , but it is not known whether it exists also with higher truncations (some results on this issue in the R^2 truncation have been given by Rechenberger and Saueressig (2012)).

 

Q: What is the behavior of the graviton at high energy in asymptotically safe gravity?

A: In linearized Einstein theory, the wave function renormalization constant of the graviton is Z = 1/(16 π G) . The anomalous dimension of the graviton is therefore η = -β/G, where β is the beta function of G. Since at a gravitational fixed point β=(2-d) G , where d is the spacetime dimension, we conclude that at a fixed point η = d-2 . It means that the graviton propagator in the UV limit behaves like p-2-η=p-d . Such a large, integer anomalous dimension appears also in other gauge theories away from the critical dimension, see e.g. Gies (2003) or Kazakov (2003).

 

Q: If gravity is asymptotically safe, what does spacetime look like at short distances?

A: One general conclusion that can be drawn is that spacetime geometry cannot be understood in terms of a single metric: rather, one should use a different effective metric at each momentum scale. This had been suggested by Floreanini and Percacci 1995a,b, who calculated the scale dependence of the metric using an effective potential for the conformal factor. Such a potential will be present in the effective action Γk[Φ] before the background metric is identified with the classical metric (as mentioned in section 1.2). In the context of the ERGE, Lauscher and Reuter 2005 have discussed a running metric emerging as a solution of a running average effective action in the Einstein-Hilbert truncation.

 

A set of metrics depending on the scale can be seen as a description of the fractal geometry of spacetime. Quite generally, from dimensional analysis it follows that the effective metric must run as k-2 (here we assume the coordinates to be dimensionless). This affects the propagation of other fields. A phenomenon characterized by an energy scale k will “see” the effective metric at scale k . For a (generally off-shell) free particle with four--momentum p (as measured by some fiducial metric) it is natural to use k = p . Its inverse propagator will then behave at high energy as p-4 , independent of spacetime dimension. In four dimensions this agrees with the result derived above for the propagator of graviton. That these two arguments should give the same result only in four dimensions is quite intriguing.


Note that the anomalous dimension for matter fields typically is not -2, so this would suggest that matter propagators do actually not show a scaling that leads to a logarithmic real space propagator as in 2 d.

It is also important to realize that there are many different notions of dimensionality, and what we usually think of as dimension is indeed the topological one. This does not show any change in asymptotic safety from IR to UV and is always 4. What does show a scale-dependence is the spectral dimension. This definition of the dimension relies on the properties of a fictitious particle probing the Euclidean space-time in a random walk without any back reaction. This dimension shows a scale dependence and runs towards 2 in the ultraviolet. This result is shared by several other approaches to quantum gravity, but it is important to realize that it does not mean, that space-time becomes two-dimensional in a naive way, i.e. quantum gravity in the UV is not simply described by some theory with topological dimension 2 (i.e. \int d^2 x).

 

Q: Can we test this fractal structure in some way?

A: We cannot currently test it in the lab but we can do so in numerical simulations. Ambjřrn, Jurkiewicz and Loll 2005a,b have studied the propagation of a scalar particle in a dynamically triangulated spacetime, with a discretized version of the Einstein dynamics. They have found that for short time scales the spectral dimension of the triangulated spacetime is two, while for long time scales it tends to four. This could be related to the result mentioned above by observing that a propagator that decays like p-4  in Fourier space decays logarithmically in coordinate space, and this is the typical behaviour of propagators in two dimensions.

 

Q: I believe that spacetime must be discrete at short distances.

A: Before answering the question whether spacetime is discrete or continuous, one has to specify how lengths are going to be measured. In general, units of length can be traced to some combination of the couplings appearing in the action. For example, in Planck units one takes the square root of Newton's constant as a unit of length. Because the couplings run, when the cutoff is sent to infinity the distance between two given points could go to zero, to a finite limit or to infinity depending on the asymptotic behaviour of the unit. In principle it seems possible that spacetime looks discrete in certain units and continuous in others. Then, even if asymptotic safety is based on continuum quantum field theory methods, it need not be in conflict with models where spacetime is discrete. (For example, the statement that when k goes to infinity Newton’s constant tends to fixed point G(k) k2 = Ǧ*  can also be read as the statement that, when k goes to infinity its value in Planck units goes to a constant. This suggests that the world may look discrete when described in Planck units.)


Recent work of Bianca Dittrich supports this conclusion from another perspective. It is possible to define discrete theories where the dynamics correspond exactly to the dynamics of a continuum theory. The question whether space-time is discrete or continuous may therefore very well be only a matter of description, and not a physical distinction.

 

Q: If gravity is asymptotically safe, almost certainly it will contain higher derivative terms. Is this not going to spoil unitary?

A: A theory that contains R^2-type operators (we disregard the tensor structure here), is non-unitary around a flat background in perturbation theory. There are several ways in which this conclusion could be avoided:
- The couplings in the propagator are running couplings: Depending on how they actually run, the ghost pole of the propagator could actually never be reached (see Floreanini and Percacci (1995), Benedetti, Machado, Saueressig (2009)).
- The fixed-point action will presumably contain infinitely many non-zero couplings, thus it will not be a simple polynomial in R. The question of ghost poles then boils down to the question where the full propagator has poles. It may well be that a finite truncation of the effective action shows fictitious poles in the propagator, that are not present in the full propagator.
- The flat background is probably not a solution of the effective equations of motion. The question of unitarity requires us to first find the correct vacuum, and then look at the second order in fluctuations around this background.


So the answer is that we do not know. Available information on the FP action is certainly not enough to decide.

 

Q: OK, let’s assume that all this is true. How do we relate it to the low energy world that we are familiar with?

A: The fixed point action describes the world at Planckian scales and beyond. In order to relate it to known phenomenology we should evolve the renormalization group flow down to the energy scales of particle physics. Reuter and Weyer (2004) have discussed this in the context of the Einstein-Hilbert truncation. They show that the existence of an extended semiclassical regime, where the dimensionful cosmological constant and Newton’s constant are essentially constant, implies that the world must lie on a trajectory that is very close to the critical trajectory of fig.1 and then turns toward the right (when going towards the infrared).

 

The leftmost point on this trajectory, which is called the turning point, must correspond to an energy scale of the order of the milli-eV. The trajectory then apparently hits an infrared singularity when the dimensionless cosmological constant becomes equal to ˝. This happens at energies of order of the Hubble scale. The infrared singularity could be an indication that the Einstein-Hilbert truncation becomes insufficient at these scales. More recently there have been hints that there is an IR fixed point there.

 

Q: Can there be observable consequences for high energy physics?

A: As usual with quantum gravity, strong observable effects will only manifest themselves at the Planck scale. If the world is four dimensional there is at present little hope of observing directly the consequences of the asymptotically safe behaviour in the laboratory. However, in theories with higher dimensions the Planck scale could be at the TeV scale. Then, signals of asymptotic safety could be observed at colliders. Hewett and Rizzo 2007, Litim 2007, Koch 2007 have discussed several processes including graviton-mediated Drell-Yan processes, graviton production and black hole production at LHC, both for large extra dimensions and warped extra dimensions scenarios. They find improvements in the high energy behaviour of the cross sections due to asymptotic safety, which provide distinctive signatures for this scenario.


At lower scales and in four dimensions, the effects are suppressed by powers of the energy scale over the Planck scale. Thus another way to access quantum gravity is to reach a very high experimental precision and/or to concentrate on processes that would be otherwise forbidden, or processes that could accumulate over large distances, as with the propagation of light from distant sources.


In connection with these scaling relations, it is important to realise that canonical power counting does not hold at a non-Gaussian fixed point. Operators that by canonical power counting would be suppressed by some power of the Planck scale could indeed show a large anomalous dimension that would make them less suppressed than we think naively.

Another way to indirectly test asymptotic safety would arise if the existence of the fixed point required certain properties of the matter sector, such as limitations of the number and types of allowed fields.

 

Q: Can there be observable consequences for astrophysics?

A:  An application of the running Newton constant near the core of a black hole gives an interesting prediction for the final stages of the Hawking evaporation (Bonanno and Reuter 1999, Ward 2004). There have also been speculations about a possible weakening of Newton’s constant at galactic scales being responsible for the flattening of the rotation curves. (However this would be an infrared effect, so not directly related to asymptotic safety.)

 

Q: Can there be observable consequences for cosmology?

A: Several authors have tried to use the renormalization group running of gravitational couplings in a cosmological context. However not all these attempts are necessarily related to asymptotic safety. In some cases beta functions have been postulated that have no relation to those calculated from the ERGE.

 

The starting point is always to link the renormalization scale k to some cosmological parameter. The most popular choice seems to be the Hubble scale, which is related to the curvature of the universe at large scales. Then, the running Newton constant and running cosmological constant are inserted in the Friedmann equations. This produces new nonlinearities, because these parameters now depend on the scale factor in some way which is prescribed by the renormalization group.

 

Since asymptotic safety is concerned with the ultraviolet behaviour of the theory, its impact in cosmology will probably be most important at the very earliest stages of cosmic history, e.g. the inflationary era. In fact, it has been shown that asymptotic safety could generate inflation with approximately 60 e-foldings and an automatic exit without the introduction of an extra scalar field and a highly fine-tuned potential (Bonanno, 2012). Furthermore, the cosmological constant runs down to arbitrarily low energy scales and this has been linked by Bonanno and Reuter (2007) to the generation of the cosmic entropy.


Q: What is the relation of asymptotic safety to other approaches to quantum gravity?
A: This is a question that cannot be answered definitely at the moment. The most promising similarities are with Causal Dynamical Triangulations, which shares the same general ideology and exhibits similar behavior of the spectral dimension.
Also there exist approaches such as group field theory that postulate that at the fundamental level, gravitational degrees of freedom are fields on an abstract group manifold. Spacetime emerges effectively when the theory goes through a second-order phase transition. A point of contact to asymptotic safety would arise, if this second-order phase transition could be related to the fixed point seen in the RG flow, and the critical exponents of the second-order phase transition were those that have been found for the fixed point in the RG flow.

Some similarities between the functional RG flow for gravity and the holographic RG flow have been pointed out in Litim, Percacci and Rachwal (2010). The relation between asymptotic safety and the notion of "classicalization" has been discussed to some extent by Percacci and Rachwal (2011).

Q: Has the functional RG, on which most results about asymptotic safety in gravity rely, been tested for other theories?
A: The functional RG has been tested extensively in other physical models, ranging from QCD in the non-perturbative regime over scalar O(N) theories to phase transitions in ultracold quantum gases.
In all of these examples, other methods and experimental results exist to which the results from the FRG can be compared, and these comparisons show that the FRG is a reliable method. As with all methods in the non-perturbative regime, it is highly non-trivial to control the systematic error of the truncation, but in principle, the results from the FRG can be extremely accurate, as the example of critical exponents in scalar O(N) theories demonstrates.

Q: Is it possible to have more exotic UV completions?
A: In principle, the physical requirement is that observables stay finite for all finite values of the RG scale k. This could also be accomplished by a limit cycle (see Litim, Satz (2012) and Bonanno, Guarnieri (2012)), or some more complicated attractor in theory space.
Furthermore, even a trajectory on which the couplings stay finite at all finite momentum scales and only diverge at k -> \infty provides for a UV complete theory (an example of a toy model that admits such a trajectory in a simple truncation is axion electrodynamics (Eichhorn, Gies, Roscher (2012))).

Q: Can asymptotic safety solve the triviality problem of the Standard Model?
A: The triviality problem occurs in QED as well as scalar Φ4 theory in four dimensions (thus presumably affecting the Higgs sector of the standard model). The fact that the running coupling hits a Landau pole at a finite RG scale implies that if the cutoff of the theory is to be taken to infinity (thus making it a fundamental as opposed to an effective theory), the IR value of the coupling will be zero. Thus the theory requires a more complicated UV completion with further degrees of freedom.
In principle, the Landau pole could be removed due to the effect of quantum gravity fluctuations and the RG flow of, e.g. gravity+QED could approach a common fixed point in the UV, thus solving the triviality problem.

Q: Can asymptotic safety help to reduce the number of free parameters of the standard model?
A: At a non-Gaussian fixed point, canonical power-counting arguments break down. Thus the relevance of a coupling is not determined by its canonical dimensionality alone, but depends on the effect of quantum fluctuations. Thus operators that we classify as relevant in perturbation theory (such as the gauge couplings and masses in the standard model), could in principle become irrelevant at a non-Gaussian fixed point for the standard-model+gravity system. Then some of the free parameters of the standard model could be predicted. An intriguing example that this might be the case has been found in Reuter, Harst (2011), where a U(1) gauge theory coupled to gravity admits a non-Gaussian fixed point where the gauge coupling becomes irrelevant. In principle, this could lead to a prediction of the fine structure constant.

Q: Is there an infrared fixed point? What does this have to do with asymptotic safety?

A: A number of papers have recently found hints of a possible infrared fixed point (Nagy, Kriszan, Sailer (2012), Rechenberger, Saueressig (2012), Litim, Nicolai, Pawlowski, Rodigast (2012)). These results were based on the same functional RG methods that are used to study asymptotic safety, but but if this infrared fixed point exists, then it is independent of the UV completion for gravity.

Q: How many degrees of freedom are there in asymptotically safe quantum gravity?

A: Within the asymptotic-safety scenario the degrees of freedom are not fixed a priori. They are only constrained in that we insist that they should be carried by certain fields (the metric, but possibly a tetrad and possibly even an independent connection). In fact, the degrees of freedom then depend on the operators in the fixed-point action. In principle, further, lower spin degrees of freedom could become dynamical in addition to the massless spin-2 mode of Einstein gravity.

Q: Could we also use a first-order formulation in terms of vielbeins? Could torsion be dynamical in asymptotic safety?

A: Asymptotic safety is a proposal for a UV completion of any quantum field theory, and a priori comes with no requirements about the fields that carry the degrees of freedom. Thus the standard metric theory space as well as a theory space using vielbeins as the carriers for the gravitational degrees of freedom could admit a non-Gaussian fixed point.


First explorations of vielbein theory spaces with dynamical torsion have been carried out by Daum and Reuter (2011), where the RG flow of the Newton coupling, cosmological constant and the Immirzi parameter was studied and a non-trivial fixed point was found. The analogue of the Einstein-Hilbert truncation in a vielbein theory space was investigated in Harst and Reuter (2012) and Dona', Percacci (2012). They found a non-trivial fixed point for the Newton coupling and the cosmological constant, however with a more severe scheme-dependence than in the metric theory space.