In [181] the coupling flow (4.45
) implied by the Einstein–Hilbert truncation has been analyzed in detail,
using both analytical and numerical methods. In particular all trajectories of this system of equations have
been classified, and examples have been computed numerically. The most important classes of trajectories in
the phase portrait on the
–
plane are shown in Figure 2
. The trajectories were obtained by
numerically solving the system (4.50
) for a sharp cutoff; using a smooth one all qualitative features remain
unchanged.
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The RG flow is dominated by two fixed points : a Gaussian fixed point (GFP) at
, and a non-Gaussian fixed point (NGFP) with
and
. There are three classes
of trajectories emanating from the NGFP: Trajectories of Type Ia and IIIa run towards negative and
positive cosmological constants, respectively, and the single trajectory of Type IIa (“separatrix”) hits the
GFP for
. The short–distance properties of Quantum Einstein Gravity are governed
by the NGFP; for
, in Figure 2
all RG trajectories on the half–plane
run
into this fixed point – its unstable manifold is two-dimensional. Note that near the NGFP the
dimensionful Newton constant vanishes for
according to
. The
conjectured nonperturbative renormalizability of Quantum Einstein Gravity is due to this NGFP: If it
was present in the untruncated RG flow it could be used to construct a microscopic quantum
theory of gravity by taking the limit of infinite UV cutoff along one of the trajectories running
into the NGFP, implying that the theory does not develop uncontrolled singularities at high
energies [227
].
The trajectories of Type IIIa cannot be continued all the way down to the infrared () but rather
terminate at a finite scale
. (This feature is not resolved in Figure 2
.) At this scale the
-functions diverge. As a result, the flow equations cannot be integrated beyond this point. The value of
depends on the trajectory considered. The trajectory terminates when the dimensionless
cosmological constant reaches the value
. This is due to the fact that the functions
and
– for any admissible choice of
– have a singularity at
, and because
in all terms of the
–functions. In Equations (4.50
) the divergence at
is seen
explicitly. The phenomenon of terminating RG trajectories is familiar from simpler theories, such
as Yang–Mills theories. It usually indicates that the truncation becomes insufficient at small
.
Here we collect the evidence for asymptotic safety obtained from the Einstein–Hilbert and
truncations, Equation (4.34
) and Equation (4.35
), respectively, of the flow equations in
Section 4.2 [133
, 131
].
The details of the flow pattern depend on a number of ad-hoc choices. It is crucial that the properties of
the flow which point towards the asymptotic safety scenario are robust upon alterations of these choices.
This robustness of the qualitative features will be discussed in more detail below. Here let us only
recapitulate the three main ingredients of the (truncated) flow equations that can be varied: The shape
functions in Equation (4.31
) can be varied, the gauge parameter
in Equation (4.33
) can be
varied, and the vector and transversal parts in the traceless tensor modes can be treated differently (type
and
cutoffs).
Picking a specific value for the gauge parameter has a somewhat different status than the other two
choices. The truncations are actually one-parameter families of truncations labelled by ; in
a more refined treatment
would be a running parameter itself determined by the
FRGE.
In practice the shape function was varied within the class (4.51
) of exponential cutoffs and a
similar one-parameter class of cutoffs with compact support [133
, 131
]. Changing the cutoff
function
at fixed
may be thought of as analogous to a change of scheme in perturbation
theory.
The main qualitative properties of the coupling flow can be summarized as follows:
We proceeded to discuss various aspects of the evidence for asymptotic safety in more detail, namely the structure of unstable manifold and the robustness of the qualitative features of the flow. Finally we offer some comments on the full FRGE dynamics.
This can be studied in the vicinity of the fixed point by a standard linearized stability analysis. We
summarize the results for the non-Gaussian fixed point, first in the Einstein–Hilbert truncation and then in
the more general truncation. To set the notation recall that for a flow equation of the form
the linearized flow near the fixed point is governed by the stability matrix with
components
,
As explained in Section 2.1 it is often convenient to set (which is to be read as
in the presence of an ultraviolet cutoff
) and ask “where a coarse graining trajectory
comes from” by formally sending
to
(while the coarse graining flow is in the direction of
increasing
). The tangent space to the unstable manifold has its maximal dimension if all the essential
couplings taken into account hit the fixed point as
is sent to
: The fixed point is
ultraviolet stable in the direction opposite to the coarse graining. This is the case for both
the Einstein–Hilbert truncation and the
truncation, as we shall describe now in more
detail.
Linearizing the flow equation (4.45) according to Equation (4.54
) we obtain a pair of complex conjugate
eigenvalues
with negative real part
and imaginary parts
. In terms of
the general solution to the linearized flow equations reads
Solving the full, nonlinear flow equations numerically [181] shows that the asymptotic scaling region
where the linearization (4.56
) is valid extends from
down to about
with the Planck
mass defined as
. Here
marks the lower boundary of the asymptotic scaling
region. We set
so that the asymptotic scaling regime extends from about
to
.
The non-Gaussian fixed point of the -truncation likewise proves to be ultraviolet attractive in any
of the three directions of the
tangent space for all cutoffs used. The linearized flow in its
vicinity is always governed by a pair of complex conjugate eigenvalues
with
, and a real negative one
. The linearized solution may be expressed as
For any cutoff, the numerical results have several quite remarkable properties. They all indicate that,
close to the non-Gaussian fixed point, the flow is rather well approximated by the Einstein–Hilbert
truncation:
Due to the large value of , the new scaling field is ‘very relevant’. However, when we
start at the fixed point
and raise
it is only at the low energy(!) scale
that
reaches unity, and only then, i.e. far away from the fixed point,
the new scaling field starts growing rapidly.
Summarizing the three points above we can say that very close to the fixed point the flow seems to
be essentially two-dimensional, and that this two-dimensional flow is well approximated by the coupling
flow of the Einstein–Hilbert truncation. In Figure 3
we show a typical trajectory which has all
three normal modes excited with equal strength
,
. All its
way down from
to about
it is confined to a very thin box surrounding the
-plane.
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As explained before the details of the coupling flow produced by the various truncations of Equation (4.19)
depend on the choice of the cutoff action (
, type A vs. B) and the gauge parameter
. Remarkably
the qualitative properties of the flow, in particular those features pointing towards the asymptotic safety
scenario are unchanged upon alterations of the computational scheme. Here we discuss these robustness
properties in more detail. The degree of insensitivity of quantities expected to be “universal” can serve as a
measure for the reliability of a truncation.
We begin with the very existence of a non-Gaussian fixed point. Importantly, both for type
A and type B cutoffs the non-Gaussian fixed point is found to exists for all shape functions
. This generalizes earlier results in [205]. Indeed, it seems impossible to find an admissible
mode-cutoff which destroys the fixed point in
. This is nontrivial since in higher dimensions
the fixed point exists for some but does not exist for other mode-cutoffs [181
] (see
however [79
]).
Within the Einstein–Hilbert truncation also a RG formalism different from (and in fact much simpler
than) that of the average action was used [39]. The fixed point was found to exist already in a simple RG
improved 1-loop calculation with a proper time cutoff.
We take this as an indication that the fixed point seen in the Einstein–Hilbert [204, 133, 136, 39] and
the
truncations [131
] is the projection of a genuine fixed point and not just an artifact of an
insufficient truncation.
Support for this interpretation comes from considering the product of the fixed point coordinates.
Recall from Section 2.3.2 that the product
is a dimensionless essential coupling invariant under
constant rescalings of the metric [116]. One would expect that this combination is also more robust with
respect to scheme changes.
In Figure 4 we show the fixed point coordinates
for the family of shape functions (4.51
)
and the type B cutoff. For every shape parameter
, the values of
and
are almost the same as
those obtained with the Einstein–Hilbert truncation. Despite the rather strong scheme dependence of
and
separately, their product has almost no visible
-dependence for not too small values of
! For
, for instance, one obtains
from the Einstein–Hilbert truncation and
from the generalized truncation. One can also see that the
coupling
at the fixed point is uniformly small throughout the family of exponential shape
functions (4.51
).
A similar situation is found upon variation of the gauge parameter . Within the Einstein–Hilbert
truncation the analysis has been performed in ref. [133
] for an arbitary constant gauge parameter
,
including the ‘physical’ value
. For example one finds
Next we consider the (in)dependence of the “critical exponents”
,
in Equation (4.56
,
4.57
). Within the Einstein–Hilbert truncation the eigenvalues are found to be reasonably constant within
about a factor of 2. For
and
, for instance, they assume values in the ranges
,
and
,
, respectively. The
corresponding results for the
truncation are shown in Figure 5
. It presents the
dependence of
the critical exponents, using the family of shape functions (4.51
). For the cutoffs employed
and
assume values in the ranges
and
, respectively. While the
scheme dependence of
is weaker than in the case of the Einstein–Hilbert truncation one finds that it is
slightly larger for
. The exponent
suffers from relatively strong variations as the
cutoff is changed,
, but it is always significantly larger in modulus than
.
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In summary, the qualitative properties listed above (,
, etc.) hold for all cutoffs.
The
’s have a stronger scheme dependence than
, however. This is most probably due to having
neglected further relevant operators in the truncation so that the
matrix we are diagonalizing is still
too small.
Finally one can study the dimension dependence of these results. The beta functions produced by the
truncated FRGE are continuous functions of the spacetime dimension and it is instructive to analyze
them for
. This was done for the Einstein–Hilbert truncation in [181
, 79
], with the result that
the coupling flow is quantitatively similar to the 4-dimensional one for not too large
. The
robustness features have been explored with varios cutoffs with the result that the sensitity
on the cutoff parameters increases with increasing
. In [181] a strong cutoff dependence
was found for
larger than approximately
, for two versions of the sharp cutoff (with
) and for the exponential cutoff with
. In [79] a number of different cutoffs
were employed and no sharp increase in sensitivity to the cutoff parameters was reported for
.
Close to the results of the
-expansion are recovered. Indeed, the fixed point of
Section 1 originally found in the
-expansion is recovered in the present framework [179
],
This concludes our analysis of the robustness properties of the truncated RG flow. For further details the
reader is referred to Lauscher et al. [133, 131
, 132]. On the basis of these robustness properties we believe
that the non-Gaussian fixed point seen in the Einstein–Hilbert and
truncations is very unlikely to be
an artifact of the truncations. On the contrary there are good reasons to view it as the projection of a fixed
point of the full FRGE dynamics. It is especially gratifying to see that within the scheme dependence the
additional
-term has a quantitatively small impact on the location of fixed point and its unstable
manifold.
In summary, we interpret the above results and their mutual consistency as quite nontrivial indications supporting the conjecture that 4-dimensional Quantum Einstein Gravity possesses a RG fixed point with precisely the properties needed for its asymptotic safety.
The generalization of the previous results to more complex truncations would be highly desirable, but for time being it is out of computational reach. We therefore add some comments on what one can reasonably expect to happen.
The key issue obviously is the dimension and the structure of the unstable manifold. For simplicity let us
restrict the discussion to the ansatz (4.29, 4.33
) in which the bi-metric character of the functionals and the
evolution of the ghost sector are neglected. Morally speaking the following remarks should however apply
equally to generic functionals
. Within the restricted functional space (4.29
, 4.33
) only the
ansatz for
can be successively generalized. A generic finite-dimensional truncation ansatz for
has the form
Let us first briefly recall the scaling pattern based on the perturbative Gaussian fixed point. As
described in Section 3.3 in a perturbative construction of the effective action the divergent part of the
loop contribution is always local and thus can be added as a counter term to a local bare
action
, where the sum is over local curvature invariants
. The scaling
pattern of the monomials
with respect to the perturbative Gaussian fixed point will thus
reflect those of the
in the effective action and vice versa. As explained in Section 2.3 the
short-distance behavior of the perturbatively defined theory will be dominated by the
’s
with the largest number of derivatives acting upon
. In a local invariant containing the
Riemann tensor to the
th power and
covariant derivatives acting on it, the number of
derivatives acting on
is
. If one starts with just a few
’s and performs loop
calculations one discovers that higher
’s are needed as counter terms. As a consequence the
high energy behavior is dominated by the bottomless chain of invariants with more and more
derivatives.
As already argued in Section 2.3 in an asymptotically safe Quantum Gravidynamics the situation is different. The absence of a blow-up in the couplings is part of the defining property. The dominance of the high energy behavior by the bottomless chain of high derivative local invariants is replaced with the expectation that all invariants should be about equally important in the extreme ultraviolet.
This can be seen from the FRGE for the effective average action via the following heuristic argument.
Assume that , where the sum runs over a (dynamically determined) subset
of all local and nonlocal invariants. The existence of a nontrivial fixed point means that the
dimensionless couplings
approach constant values
for
. As a
consequence, the dimensionful couplings have the following
-dependence in the fixed point regime:
Clearly the above argument can be generalized to action functionals depending on all the fields
,
,
. Also the choice of field variables is inessential and the argument should carry over
to other types of flow equations. It suggests that there could indeed be a fixed point action
which is well-defined when expressed in terms of
dimensionless quantities and which describes the extreme ultraviolet dynamics of Quantum
Einstein Gravity. By construction the unstable manifold of this fixed point action would be
nontrivial.
Without further insight unfortunately little can be said about its dimension. Among the local invariants
in Equation (4.57) arguably only a finite number should be relevant. This is because the power counting
dimensions
of the infinite set of irrelevant local invariants may receive large positive corrections
which makes them relevant with respect to the NGFP. An example for this phenomenon is provided by the
invariant. It is power counting marginal (
) but with respect to the NGFP
the scaling dimension of the associated dimensionless coupling is shifted to a large positive value
of
. Nevertheless it seems implausible that this will happen to couplings with
arbitrarily large negative power counting dimension as correspondingly large corrections would be
required.
On the other hand this reasoning does not apply to nonlocal invariants. For example arbitrary functions
of the above
are likewise power counting marginal, and on account of a similar positive
shift they too would become relevant with respect to the NGFP. While such terms would not occur in the
perturbative evaluation of the effective action, in the present framework they are admissible
and their importance has to be estimated by computation. As a similar an ansatz of the form
Also scalar modes like the conformal factor have vanishing power counting dimension. The way how such dimensionless scalars enter the effective action is then not constrained by the above ‘implausibility’ argument. An unconstrained functional occurance however opens the door to a potentially infinite-dimensional unstable manifold.
Another core issue are of course the positivity properties of the Quantum Einstein Gravity defined
through the FRGE. As already explained in Section 1.5 the notorious problems with positivity and
causality which arise within standard perturbation theory around flat space in higher-derivative
theories of Lorentzian gravity are not an issue in the FRG approach. For example if is of the
type, the running inverse propagator
when expanded around flat space has ghosts
similar to those in perturbation theory. For the FRG flow this is irrelevant, however, since in
the derivation of the beta functions no background needs to be specified explicitly. All one
needs is that the RG trajectories are well defined down to
. This requires only that
is a positive operator for all
. In the exact theory this is believed to be the
case.
A rather encouraging first result in this direction comes from the truncation [131
]. In the FRG
formalism the problem of the higher derivative ghosts is to some extent related to the negative
factors
discussed in Section 2.1. It was found that, contrary to the Einstein–Hilbert truncation, the
truncation has only positive
factors in the fixed point regime
. Hence in this truncation the
existence of ‘safe’ couplings appears to be compatible with the absence of unphysical propagating modes, as
required by the scenario.
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