# The functional $f(R)$ approximation

Tim R. Morris and Dalius Stulga
<EMAIL_ADDRESS><EMAIL_ADDRESS>
STAG Research Centre, Department of Physics and Astronomy,
University of Southampton, Highfield, Southampton, SO17 1BJ, U.K.

###### Abstract

This article is a review of functional $f(R)$ approximations in the asymptotic
safety approach to quantum gravity. It mostly focusses on a formulation that
uses a non-adaptive cutoff, resulting in a second order differential equation.
This formulation is used as an example to give a detailed explanation for how
asymptotic analysis and Sturm-Liouville analysis can be used to uncover some
of its most important properties. In particular, if defined appropriately for
all values $-\infty<R<\infty$, one can use these methods to establish that
there are at most a discrete number of fixed points, that these support a
finite number of relevant operators, and that the scaling dimension of high
dimension operators is universal up to parametric dependence inherited from
the single-metric approximation. Formulations using adaptive cutoffs, are also
reviewed, and the main differences are highlighted.

Keywords— Quantum gravity, Renormalization group, Asymptotic safety, $f(R)$
approximation, Sturm-Liouville, Asymptotic analysis

###### Contents

  1. 1 Introduction
  2. 2 Flow equations
  3. 3 Cutoff functions
  4. 4 Flow equations with adaptive cutoff
  5. 5 Evaluating traces
    1. 5.1 Sphere
    2. 5.2 Hyperboloid
    3. 5.3 Flat space
  6. 6 Fixed point solutions
    1. 6.1 Asymptotic analysis
    2. 6.2 Numerical solution
  7. 7 Eigenoperators
    1. 7.1 Asymptotic analysis
    2. 7.2 Sturm-Liouville theory

##  1 Introduction

One attempted route to a quantum theory of gravity is through the asymptotic
safety programme [1, 2, 3, 4]. Although quantum gravity based on the Einstein-
Hilbert action is plagued by ultraviolet infinities that are perturbatively
non-renormalizable (implying the need for an infinite number of coupling
constants), a sensible theory of quantum gravity might be recovered if there
exists a suitable ultraviolet fixed point [1].

The task is not just that of searching for an ultraviolet fixed point. They
must also have the correct properties. Perturbatively renormalizable ones
exist for example “Conformal gravity”, based on the square of the Weyl tensor,
which thus corresponds to a Gaussian ultraviolet fixed point [5]. It is
apparently not suitable however, because the theory is not unitary. Suitable
unitary fixed points, if they exist, have to be non-perturbative.  They must
also satisfy phenomenological constraints,  for example they have to allow a
renormalized trajectory with classical-like behaviour in the infrared, since
General Relativity is confirmed by observation across many phenomena and to
impressive precision. Of particular relevance for this chapter is that there
should be a fixed point with a finite number of relevant directions (otherwise
it would be no more predictive than the perturbatively defined theory).
Preferably the theory should have only one fixed point, or at least only a
finite number (otherwise again we lose predictivity).

Functional RG (renormalization group) equation [6, 7, 8, 9, 10, 11] studies,
first introduced by Wilson and Wegner many years ago [6, 7] (and called by
them the “exact RG”), have flourished into a powerful approach for
investigating this possibility.  These equations describe the flow of the
Wilsonian effective action for some quantum field theory, under changes in an
effective cutoff scale $k$. The asymptotic safety literature uses almost
exclusively the flow equation for $\Gamma_{k}$ which is, modulo minor details,
the Legendre effective action (the generator of one-particle irreducible
diagrams) cut off in the infrared by $k$. It was also formulated long ago [9]
(in the sharp cutoff limit) and then rediscovered for smooth cutoffs much
later in refs. [10, 11]. Following ref. [10], $\Gamma_{k}$ is sometimes called
the “effective average action”, however in this chapter it will simply be
called an effective action.

It is not practical to solve the full functional RG equations exactly. In a
situation such as this, where there are no useful small parameters, one can
only proceed by considering model approximations. These always proceed from
the following observation: Wilsonian effective actions can be written as a sum
over operators, where the coefficients are the couplings for these operators
and they evolve with the scale $k$.

In fact this sum should be restricted to local operators. This is the
requirement of quasi-locality, which comes from the short range nature of the
Kadanoff blocking step in Wilsonian RG [6], when implemented in the continuum
[12, 13]. A related point is that the Wilsonian RG is performed in euclidean
signature, so that “short range” has a sensible meaning.

The problem is that for any general solution, this sum is infinite, over all
possible local operators allowed by the symmetries (the space of all such
couplings being known as “theory space”). However, this motivates the simplest
model approximation which is to truncate drastically the infinite dimensional
theory space to a handful of operators. An example is the original truncation
studied by Reuter [2]:

$\Gamma_{k}[g_{\mu\nu}]=\int
d^{4}x\sqrt{g}\big{(}u_{0}(k)+u_{1}(k)R\big{)}\,,$ (1)

which retains only the cosmological constant term and the scalar curvature $R$
term. For obvious reasons this is called the “Einstein-Hilbert truncation”.
Classically $u_{0}=-\lambda_{cc}/(8\pi G)$ and $u_{1}=-1/(16\pi G)$, where
$\lambda_{cc}$ is the cosmological constant and $G$ is Newton’s constant, but
after quantum corrections these couplings run with $k$ in the functional RG.
The minus sign in $u_{1}$ comes from working in euclidean signature.

Apart from RG symmetry, these truncations destroy pretty well all the
properties that ought to hold. For example scheme independence (i.e.
independence on choice of cutoff, or more generally universality), and
modified BRST invariance [14, 2] (which encodes diffeomorphism invariance for
the quantum field under influence of the cutoff) cannot then be recovered.
Furthermore, only by keeping an infinite number of local operators can the
non-local long-range nature of the (one-particle irreducible) Green’s
functions be recovered (see e.g. ref. [15]). One has to trust that by
considering ever less restrictive truncations the description gets closer to
the truth. There are some examples that go well beyond the Einstein-Hilbert
truncation by keeping a large number of operators [16, 17, 18, 19]. These are
based around polynomial truncations, i.e. where everything is discarded except
powers of some suitable local operators, typically the scalar curvature $R$
again, up to some maximum degree. They appear to show convergence, in
particular the number of relevant operators is found to be three.

Another approximation in the asymptotic safety literature that is necessary in
order to formulate diffeomorphism invariant truncations, such as eqn. (1),
conflates the true (quantum) metric with the background metric. It is called
the “single metric” or “background field” approximation,  and will be
described in the next section. It is harder to relax this approximation in any
substantive way, although see refs. [20, 21, 22, 23, 24, 25, 26, 27, 28] for
some approaches.

Whilst very encouraging results are found from multiple studies of such finite
order truncations (see e.g. the review [29]), successful implementations of
more powerful approximations would build confidence in the scenario. The next
step is to keep an infinite number of operators. Arguably the simplest such
truncation is to keep a full function $f(R)$, making the ansatz [30, 31, 32,
33, 34, 35, 36, 37, 38, 39, 40, 23, 22, 41, 24, 42, 43]

$\Gamma_{k}[g]=\int\\!d^{4}x\sqrt{g}\,f_{k}(R)\,.$ (2)

This is the functional $f(R)$ approximation which is the subject of this
chapter.  It is achieved by specialising to a maximally symmetric background
manifold, either a four-sphere or four-hyperboloid.

Closely related approximations have been studied in scalar-tensor [44, 45] and
unimodular [46] gravity, and in three space-time dimensions [37]. In fact, the
high order finite dimensional truncations [16, 17, 18, 19] were developed by
taking examples of these $f(R)$ equations and then further approximating to
polynomial truncations.

Note that the functional $f(R)$ approximation actually goes beyond keeping a
countably infinite number of couplings, the Taylor expansion coefficients
$g_{n}=f^{(n)}(0)$, because a priori the large field parts of $f(R)$ contain
degrees of freedom that are unrelated to all these $g_{n}$. For example
suppose that at large $R$ one finds that $f(R)\approx\exp(-a/R^{2})$, where
$a>0$ is some parameter. Such an $f(R)$ is in the form of a standard counter-
example in mathematical analysis. It has the property that $g_{n}=0$ for all
$n$.

As ref. [33] emphasised, the truncation (2) is as close as one can get to the
Local Potential Approximation (LPA)  [47, 48], a successful approximation for
scalar field theory in which only a general potential $V(\varphi)$ is kept for
a scalar field $\varphi$ (see e.g. [47, 48, 49, 50, 51, 52]). The LPA can be
viewed as the start of a systematic derivative expansion [49], in which case
this lowest order corresponds to regarding the field $\varphi$ as constant. In
rough analogy, an approximation of form (2) may be derived by working on a
euclidean signature space of maximal symmetry, where the scalar curvature $R$
is constant. (Typically a four-sphere is chosen.) In particular, techniques
that have proved successful in scalar field theory [48, 49, 50, 51, 53, 52]
have been adapted to this very different context, and used to gain substantial
insight [54, 55, 56, 57, 43].

The functional truncation (2) still has the problems that were highlighted
earlier for its finite dimensional counterparts. However, again one can hope
that it is closer to the truth. One hint that this is in fact the case is
covered at the end of this chapter. Assuming that the most recent version [43]
does have a fixed point solution, then it turns out that operators with high
scaling dimension do begin to display universality – unfortunately up to an
annoying parameter that remains which is clearly caused by the single-metric
approximation.

In this chapter, it will be explained how to construct functional $f(R)$
approximations and how to interpret them. Important properties of formulations
that use an adaptive cutoff [30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 23,
22, 41, 24] will be reviewed. These result in third order differential
equations, with fixed singularities and problematic asymptotic behaviour.
Mostly the chapter will focus on a non-adaptive cutoff formulation [42, 43]
that results in a second order differential equation, using it as an example
to give a detailed exposition of the techniques, especially asymptotic
analysis and Sturm-Liouville analysis, that can be used to prove properties of
functional $f(R)$ approximations. In particular, if the second order
formulation is taken to apply to only one of the two spaces (sphere or
hyperboloid), the fixed point solutions form a continuous set and the
eigenoperator spectrum is not quantised. However, if these spaces are joined
together smoothly (through flat space at their boundary), these methods
establish that there are at most a discrete number of fixed points, that the
fixed points support a finite number of relevant operators, and yield the
result above for operators of high scaling dimension. They do not establish
that such fixed points actually exist however. Such a demonstration requires
more powerful numerical analysis and/or simpler fixed point formulations [43].

##  2 Flow equations

The starting point is the functional RG flow equation [10, 11]:

$\partial_{t}\Gamma_{k}=\frac{1}{2}\text{STr}\Big{[}(\Gamma^{(2)}_{k}+\mathcal{R}_{k})^{-1}\partial_{t}\mathcal{R}_{k}\Big{]},$
(3)

where Str is a functional trace over spacetime coordinates and indices that
takes into account statistics of the fields. In momentum space this is an
integral over one loop momentum. The right hand side is a one-loop integral if
$\Gamma_{k}$ is taken to be classical. It includes all higher loops because
$\Gamma_{k}$ is actually given by the full (all orders) effective action.
$\Gamma_{k}^{(2)}$ (Hessian) is the second variation of the effective action
with respect to the fields and $\mathcal{R}_{k}$ is an IR (infrared) cutoff
for these fields. The cutoff scale $k$ is related to RG “time” $t$ via
$t=\ln(k/\mu)$, where $\mu$ is the standard arbitrary physical energy scale
that appears in RG treatments (including in perturbative quantum field
theory).

An essential step in Wilsonian RG is to introduce dimensionless variables by
multiplication of appropriate powers of the cut-off scale $k$. In the $f(R)$
approximation the appropriate powers are just the canonical (a.k.a.
engineering) ones:

$\tilde{f}_{k}(\tilde{R})\equiv\tilde{f}(\tilde{R},t)=k^{-4}f_{k}(k^{2}\tilde{R}),\qquad\tilde{R}={R}/{k^{2}}\,.$
(4)

In Wilsonian RG, one integrates out modes, starting with the high momentum
modes first, by a coarse-graining procedure. Traditionally, after integrating
out the modes, one has to rescale the action back to the original UV cut-off
of the theory to see how the couplings change. By working with dimensionless
quantities this is taken care of automatically.

(From this point onwards it is convenient to drop the tilde denoting
dimensionless quantities, unless otherwise specified, but the reader should
assume that all the quantities are dimensionless.)

In this way solutions to the flow equation will reveal all the  fixed points
of the theory, i.e. $t$ independent solutions ${f}({R},t)={f}({R})$. Fixed
points are characterized by the number of eigenoperators ${v}({R})$ (operators
of definite scaling dimension) that flow into the fixed point when we increase
the cutoff scale $k$.  These are called relevant eigenoperators. Conversely
irrelevant operators are the ones that flow away from the fixed point.  The
terms relevant (irrelevant) are common in the Wilsonian RG literature.
Following Weinberg [1], in asymptotic safety literature often they are
referred to equivalently as essential (inessential). Eigenoperators whose
couplings do not flow (in some approximation or exactly) are called marginal.
We do not need to discuss them in this chapter. Exceptionally eigenoperators
can appear that are “redundant”, corresponding to a change of variables in the
theory [58, 55, 59].

Eigenoperators are found by linearising the flow equations around the fixed
point and separating variables:

$f_{k}(R)=f(R)+\epsilon\,v(R)\,\mathrm{e}^{-\theta t}$ (5)

where $\epsilon$ is a small parameter.  This turns the flow equation into an
eigenvalue problem where the RG eigenvalue $\theta$ is often called a
“critical exponent” in the asymptotic safety literature.  From its associated
$v(R)$ it can be similarly classified as relevant, irrelevant, marginal or
redundant. Thus if $\real\theta>0$ then it is relevant, whilst if
$\real\theta<0$ it is irrelevant. In statistical physics, non-redundant
$\theta$ can be straightforwardly related to experimentally defined and
measurable critical exponents, see e.g. [60]. If computed correctly an
important property of a non-redundant $\theta$ is that it is universal, which
means in particular that its value is independent of the regularisation scheme
and the choice of flow equation [58].

As already intimated, one is generally interested in those fixed points that
have finitely many relevant operators, because their couplings become the free
parameters in the theory, and will have to be fixed by experiments. Thus,
theories based around these points are predictive and are safe from UV
divergences when $k\rightarrow\infty$. The goal of the asymptotic safety
program is to verify if such points exist for gravity, analyse their
properties and deduce their consequences, both qualitatively and
quantitatively.

Actually, the flow equation (3) requires a significant amount of adaptation to
deal with the fact that quantum gravity is a gauge theory. In standard
fashion, it therefore requires gauge fixing. This is commonly done by
employing the  background field method where the full (a.k.a. total) metric
$\hat{g}_{\mu\nu}$ is split into a background $g_{\mu\nu}$ plus fluctuations
(the quantum field):

$\hat{g}_{\mu\nu}={g}_{\mu\nu}+h_{\mu\nu}\,.$ (6)

In common with most of the literature, this chapter will only use a linear
split, although other, non-perturbatively better motivated, splits are
possible [39, 29]. Then the gauge fixing is imposed on the quantum field
$h_{\mu\nu}$ in such a way that diffeomorphism invariance of the background
metric ${g}_{\mu\nu}$ is retained:

$F_{\mu}={\nabla}^{\nu}h_{\mu\nu}-\frac{1}{4}{\nabla}_{\mu}h^{\nu}_{\nu}\,,$
(7)

where the covariant derivative and raised indices, are defined using the
background metric. The process of fixing a gauge, adds the gauge fixing term

$\Gamma_{gf}=\frac{1}{2\alpha}\int d^{4}x\sqrt{{g}}{g}^{\mu\nu}F_{\mu}F_{\nu}$
(8)

to the effective action, and leads also to a ghost action. In practice the
Landau gauge is chosen: $\alpha\to 0$. Finally, it proves useful to make a
change of variables, this is explained in (12), and this leads to further,
auxiliary, fields.

The true solution involves arbitrarily complicated interactions to arbitrarily
high order between all these fields, molified only by the symmetries (in
particular background field diffeomorphism invariance and modified BRST
invariance [14, 61]). The next steps in the approximation drastically
truncates all of this [2]. It can be summarised as follows. Only the one-loop
contributions from the bilinear ghost and auxiliary field and fluctuation
field actions are retained, i.e. on the right hand side of the flow equation
(3) only the Hessian from the classical action for these fields is used. The
flow of the bit of the effective action that only depends on the background
metric, is therefore reproduced correctly at one loop. For the part beyond one
loop, the correct Hessian in (3) for the metric,

$\frac{\delta^{2}\Gamma_{k}}{\delta h_{\mu\nu}(x)\,\delta
h_{\alpha\beta}(y)}\,,$ (9)

is replaced by one in which the functional derivatives are with respect to the
background field instead:

$\frac{\delta^{2}\Gamma_{k}}{\delta{g}_{\mu\nu}(x)\,\delta{g}_{\alpha\beta}(y)}\,.$
(10)

This is the single metric, or background field, approximation.  It is almost
always applied in asymptotic safety investigations. The review [28] covers
exceptions. It should be emphasised that already at one loop the single metric
approximation is not correct, because the dependence of the effective action
on $h_{\mu\nu}$ has no direct relation to its dependence on $g_{\mu\nu}$. The
replacement above would be correct only if the effective action were a
functional of the full metric (6) alone, but that relation is broken at the
classical level by the gauge fixing term (8) (and corresponding ghost action).
Nevertheless the replacement is attractive as a model, because it leaves us
with a flow equation for $\Gamma_{k}[g]$ that depends only on the background
metric and in a diffeomorphism invariant way.

Now by choosing the background manifold to be one of maximal symmetry, all
diffeomorphism invariants can be related either to the volume or the scalar
curvature $R$, which is a constant: $\partial_{\mu}R=0$. In this way the
effective action has been reduced to (2): the functional $f(R)$ approximation.

Plugging this with appropriately scaled fields (4) (and coordinates
$\tilde{x}^{\mu}=kx^{\mu}$), into the flow equation (3), one readily derives
the form of the left hand side:

$\partial_{t}\Gamma_{k}=\int\\!\\!d^{4}{x}\sqrt{g}\big{[}\,\partial_{t}{f}_{k}({R})+4{f}_{k}({R})-2{R}{f}^{\prime}_{k}({R})\,\big{]}.$
(11)

The right hand side of the flow equation depends on the detailed way the
quantum corrections are handled, which differs between authors [30, 31, 32,
33, 34, 35, 42, 36, 37, 38, 39, 40, 23, 22, 41, 24, 43]. For this we need to
compute the second variation of $\Gamma_{k}$ with respect to the fields.
First, the gauge fixing term (8) is chosen and the ghost action is derived.
Then the transverse  traceless (a.k.a. York) decomposition of the metric [62]
is used:

$h_{\mu\nu}=h^{T}_{\mu\nu}+{\nabla}_{\mu}\xi_{\nu}+{\nabla}_{\nu}\xi_{\mu}+{\nabla}_{\mu}{\nabla}_{\nu}\sigma+\frac{1}{d}{g}_{\mu\nu}\bar{h}\,,$
(12)

which separates physical degrees of freedom, viz. $h^{T}_{\mu\nu}$ and
$\bar{h}$, from the unphysical ones associated with gauge degrees of freedom,
namely $\xi_{\mu}$ and $\sigma$. These fields satisfy

${h^{T}}_{\mu}^{\mu}=0,\quad{\nabla}^{\mu}h^{T}_{\mu\nu}=0,\quad{\nabla}^{\mu}\xi_{\mu}=0,\quad\bar{h}=h-{\nabla}^{2}\sigma.$
(13)

Expressing $\sqrt{\hat{g}}$ and $\hat{R}$, where the latter is the curvature
of the full metric (6), to quadratic order in these fields, the elements of
the Hessian can be determined for these components. For example for the
physical components one finds

${\Gamma}^{(2)}_{h^{T}_{\mu\nu}h^{T}_{\alpha\beta}}=-\frac{1}{2}\bigg{[}f^{\prime}_{k}(R)\bigg{(}-{\nabla}^{2}+\frac{1}{6}{R}\bigg{)}+\bigg{(}f_{k}-\frac{1}{2}{R}f^{\prime}_{k}\bigg{)}\bigg{]}\delta^{\mu\nu,\alpha\beta}\,,$
(14)
${\Gamma}^{(2)}_{\bar{h}\bar{h}}=\frac{1}{16}\bigg{[}9f^{\prime\prime}_{k}\bigg{(}-{\nabla}^{2}-\frac{{R}}{3}\bigg{)}^{2}+3f^{\prime}_{k}\bigg{(}-{\nabla}^{2}-\frac{{R}}{3}\bigg{)}-\bigg{(}{R}f^{\prime}_{k}-2f_{k}\bigg{)}\bigg{]}\,,$
(15)

where the right hand side is evaluated at $\hat{g}_{\mu\nu}=g_{\mu\nu}$, in
preparation for the single metric approximation.  We can write these more
compactly if we introduce

$E_{k}(R)=2f_{k}(R)-Rf^{\prime}_{k}(R)\,,$ (16)

which is the equation of motion that follows from the action (2), and express
them instead using the natural Laplacian $\Delta_{s}$ for a spin $s$ component
field (on a maximally symmetric background) [33]:

$\Delta_{0}=-\nabla^{2}-\frac{R}{3},\quad\Delta_{1}=-\nabla^{2}-\frac{R}{4},\quad\Delta_{2}=-\nabla^{2}+\frac{R}{6}\,.$
(17)

A similar decomposition is applied to the ghost action. In the formulation of
ref. [33] the contribution to the Hessian coming from the gauge degrees of
freedom from the metric and the ghosts cancel each other exactly. Finally,
including the contributions of the auxiliary fields that encode the Jacobians
due to the transverse traceless decomposition of the metric and the ghost
fields, gives the full flow equation (3) in the single metric and functional
$f(R)$ approximation:

$V\big{(}\partial_{t}f_{k}(R)+2E_{k}(R)\big{)}=\mathcal{T}_{2}+\mathcal{T}_{0}^{\bar{h}}+\mathcal{T}_{1}^{Jac}+\mathcal{T}_{0}^{Jac}\,,$
(18)

where $V=\int d^{4}x\sqrt{g}$ is the volume of the manifold, and the
$\mathcal{T}$ objects are the following spacetime traces:

$\mathcal{T}_{2}=\text{Tr}\Big{[}\frac{d_{t}\mathcal{R}^{T}_{k}}{-f^{\prime}_{k}(R)\Delta_{2}-E_{k}(R)/2+2\mathcal{R}^{T}_{k}}\Big{]}\,,$
(19)
$\mathcal{T}_{0}^{\bar{h}}=\text{Tr}\Big{[}\frac{8\,d_{t}\mathcal{R}^{\bar{h}}_{k}}{9f^{\prime\prime}_{k}(R)\Delta_{0}^{2}+3f^{\prime}_{k}(R)\Delta_{0}+E_{k}(R)+16\mathcal{R}^{\bar{h}}_{k}}\Big{]}\,,$
(20)
$\mathcal{T}_{1}^{Jac}=-\frac{1}{2}\text{Tr}\Big{[}\frac{d_{t}\mathcal{R}_{k}^{V}}{\Delta_{1}+\mathcal{R}^{V}_{k}}\Big{]}\,,$
(21)
$\mathcal{T}_{0}^{Jac}=\frac{1}{2}\text{Tr}\Big{[}\frac{d_{t}\mathcal{R}_{S_{1}}^{V}}{\Delta_{0}+R/3+\mathcal{R}^{S_{1}}_{k}}\Big{]}-\text{Tr}\Big{[}\frac{2\,d_{t}\mathcal{R}_{S_{2}}^{V}}{(3\Delta_{0}+R)\Delta_{0}+4\mathcal{R}^{S_{2}}_{k}}\Big{]}\,.$
(22)

The right hand side of (18) has been subdivided into contributions coming from
fields of different spins. The first two come from the physical spin-2
traceless part of the metric and the spin-0 trace of the metric, as the reader
can see by using (14) and (15) in (3). The last two are spin-1 and spin-0
parts coming from field redefinitions.

##  3 Cutoff functions

One place where crucial differences occur between the different
implementations is in the choice of cutoff $\mathcal{R}_{k}$. There is quite a
lot of freedom as these functions only need to satisfy a few key properties
which ensure that they behave like momentum dependent mass terms suppressing
low momentum modes:

$\lim_{p^{2}\rightarrow
0}\mathcal{R}_{k}(p^{2})>0,\quad\lim_{p^{2}\rightarrow\infty}\mathcal{R}_{k}(p^{2})=0,\quad\lim_{k\rightarrow
0}\mathcal{R}_{k}(p^{2})=0\,.$ (23)

The first two conditions ensure that we integrate out the UV modes first and
ignore the IR modes. The last condition ensures that we are left with the
standard definition of the effective action once the cutoff scale is sent to
zero.

An apparently attractive strategy is to choose cutoffs that simplify the flow
equations as much as possible.  “Adaptive cutoffs” are introduced partly with
that aim [30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 23, 22, 41, 24]. They
implement the following rule for all appearances of the Laplacian operator
$-\nabla^{2}$:

$-\nabla^{2}\mapsto-\nabla^{2}+k^{2}r(-\nabla^{2}/k^{2})\,,$ (24)

where $r(z)$ is a cutoff profile function.

Such a choice also seemingly solves an awkward feature of euclidean quantum
gravity, which is that the euclidean signature Einstein-Hilbert action (1) has
a wrong-sign  kinetic term and propagator for $\bar{h}$, the so-called
conformal instability[63].  This can be seen in the negative coefficient for
$\Delta_{0}$ in (20) in this case. By implementing (24), the cutoff
automatically adapts to this wrong sign, so that it continues to modify the
propagator in the intended way: by adding a momentum dependent mass term.
Indeed if this were not done, the cutoff and kinetic term would have opposite
signs, resulting in a singular propagator. However, this trick does not
entirely cure the problem since it results in poor asymptotic (large $R$)
behaviour.  This issue will be briefly touched on below and in sec. 4. For
further discussion, see refs.[2, 54, 64, 65, 66, 43].

Technically the above replacement rule is implemented by setting

$\mathcal{R}^{\phi}_{k}=\Gamma^{(2)}_{k}\big{[}-\nabla^{2}+k^{2}r(-\nabla^{2}/k^{2})\big{]}-\Gamma^{(2)}_{k}[-\nabla^{2}]\,,$
(25)

for each mode $\phi$, so that the desired effect is created for
$\Gamma^{(2)}_{k}[-\nabla^{2}]+\mathcal{R}_{k}$ in the flow equation (3).
Notice that the cutoff function is then of the same form as the Hessian
elements themselves and thus now also depends on $f(R)$. This has a particular
consequence for the scalar $\bar{h}$ mode, since
$\Gamma^{(2)}_{\bar{h}\bar{h}}$ contains $f^{\prime\prime}_{k}(R)$, cf. eqn.
(15). It means that plugging this type of cutoff into the flow equation will
result in the appearance of $Rf^{\prime\prime\prime}_{k}(R)$, due to the
presence of $d_{t}\mathcal{R}^{\bar{h}}_{k}$ in the numerator in (20) and the
definition (4) of $f_{k}(R)$. This makes the flow equation a third order
differential equation, which unfortunately lacks the powerful properties found
in a second order formulation (as covered in sec. 4). Furthermore, the factor
of $R$ leads to a so-called “fixed singularity” at $R=0$.  Third order
formulations suffer from further fixed singularities and, as already
mentioned, poor asymptotic behaviour, this latter leading to continuous
eigenoperator spectra [54].  These problems will be further covered in sec. 4.

When using an adaptive cutoff, the cutoff profile function
$r(-\nabla^{2}/k^{2})$ is almost always chosen to be the “optimised” profile
[67]

$r(z)=(1-z)\,\theta(1-z).$ (26)

The advantage of using this setup is that
$d_{t}\mathcal{R}_{k}\propto\theta(1+\nabla^{2}/k^{2})$, and thus the
eigenvalues of $-\nabla^{2}$ are restricted to be less than $k^{2}$. This
means that in denominators one can simply ignore the $\theta$ and thus
$k^{2}r(-\nabla^{2}/k^{2})\equiv k^{2}+\nabla^{2}$. Therefore the net effect
in denominators is just to replace $\Gamma^{(2)}_{k}[-\nabla^{2}]$ with
$\Gamma^{(2)}_{k}[k^{2}]$, massively simplifying the computation of spacetime
traces.

The second order formulation [42, 43] chooses a non-adaptive cutoff function
of the form

$\mathcal{R}^{\phi}_{k}=k^{m_{\phi}}c_{\phi}r({\Delta}_{s}+\alpha_{s}{R})$
(27)

where $s$ is the spin of the mode $\phi$, $m_{\phi}$ is set such that the
cutoff has the same dimension as $\Gamma^{(2)}$ for this mode, and $c_{\phi}$
is a number. In this chapter the $c_{\phi}$ will be taken to be positive for
all fields. This is a problem for developing solutions $f_{k}(R)$ that
approximate the perturbative quantisation of the Einstein-Hilbert action (1)
because the $\bar{h}$ Hessian has the wrong sign  there (as noted above). But
again the alternative choice $c_{\bar{h}}<0$ leads to poor asymptotic
behaviour at large $R$, resulting in a continuous spectrum of eigenoperators
[43].

Notice that the cutoffs (27) have been chosen to depend on $\Delta_{s}$,
rather than simply the $-\nabla^{2}$ part [33], and furthermore include an
“endomorphism”,  a curvature correction with endomorphism coefficient
$\alpha_{s}$ [42]. In refs. [42, 43] the traces are computed directly as a sum
over modes. The $\alpha_{s}$ are there to ensure that

$\Delta_{s}+\alpha_{s}R>0\,,$ (28)

for all modes, which in turn ensures that they are all integrated out as
$k\rightarrow 0$, and that the flow equation does not suffer from fixed
singularities.  For these non-adaptive cutoffs, the optimised cutoff profile
(26) brings no particular advantage. In fact on a sphere the trace is a
discrete sum and sharp cutoff profiles  would lead to a staircase behaviour
[33], with an ill-defined limit as $R\rightarrow 0$. Hence, a smooth
(infinitely differentiable) cutoff profile  is used, such as [10]

$r(z)=\frac{z}{\exp(az^{b})-1},\qquad a>0,b\geq 1\,.$ (29)

##  4 Flow equations with adaptive cutoff

In those formulations that use an adaptive cutoff, spacetime traces are
evaluated using a heat-kernel asymptotic expansion,  apart from ref. [33]
which uses a direct spectral sum together with a smoothing procedure (to get
over the aforementioned staircase problem). As an illustration, the result of
the earliest four such formulations [31, 30, 32] for the flow of $f\equiv
f(R,t)$ on a four-sphere, can be summarised as:

$\displaystyle 384\pi^{2}\left(\partial_{t}f+4f-2Rf^{\prime}\right)=$ (30)
$\displaystyle\quad\Big{[}5R^{2}\theta\left(1-\tfrac{R}{3}\right)-\left(12+4\,R-\tfrac{61}{90}\,R^{2}\right)\Big{]}\Big{[}1-\tfrac{R}{3}\Big{]}^{-1}+\Sigma$
$\displaystyle+\Big{[}10\,R^{2}\,\theta(1-\tfrac{R}{4})-R^{2}\,\theta(1+\tfrac{R}{4})-\left(36+6\,R-\tfrac{67}{60}\,R^{2}\right)\Big{]}\Big{[}1-\tfrac{R}{4}\Big{]}^{-1}$
$\displaystyle+\Big{[}(\partial_{t}f^{\prime}+2f^{\prime}-2Rf^{\prime\prime})\,\left(10-5R-\tfrac{271}{36}R^{2}+\tfrac{7249}{4536}R^{3}\right)+f^{\prime}\left(60-20R-\tfrac{271}{18}R^{2}\right)\Big{]}\left[f+f^{\prime}(1-\tfrac{R}{3})\right]^{-1}$
$\displaystyle+\tfrac{5R^{2}}{2}\,\Big{[}(\partial_{t}f^{\prime}+2f^{\prime}-2Rf^{\prime\prime})\left\\{r(-\tfrac{R}{3})+2r(-\tfrac{R}{6})\right\\}+2f^{\prime}\theta(1+\tfrac{R}{3})+4f^{\prime}\theta(1+\tfrac{R}{6})\Big{]}\left[f+f^{\prime}(1-\tfrac{R}{3})\right]^{-1}$
$\displaystyle+\Big{[}(\partial_{t}f^{\prime}+2f^{\prime}-2Rf^{\prime\prime})f^{\prime}\left(6+3R+\tfrac{29}{60}R^{2}+\tfrac{37}{1512}\,R^{3}\right)$
$\displaystyle\qquad+\left(\partial_{t}f^{\prime\prime}-2Rf^{\prime\prime\prime}\right)\left(27-\tfrac{91}{20}R^{2}-\tfrac{29}{30}\,R^{3}-\tfrac{181}{3360}\,R^{4}\right)$
$\displaystyle\qquad+f^{\prime\prime}\left(216-\tfrac{91}{5}R^{2}-\tfrac{29}{15}R^{3}\right)+f^{\prime}\left(36+12R+\tfrac{29}{30}R^{2}\right)\Big{]}\Big{[}2f+3f^{\prime}(1-\tfrac{2}{3}R)+9f^{\prime\prime}(1-\tfrac{R}{3})^{2}\Big{]}^{-1}\,.$

Here the function $r$ is the optimised cutoff profile (26),  which also leads
to the appearance of the step functions (a.k.a. Heaviside $\theta$ functions).
In ref. [32] the equation is adapted to polynomial truncations only, which
means that the step functions are all set to one. The first two lines of the
right hand side are independent of $f(R,t)$ and encapsulate the contributions
from the ghosts, auxiliaries, $\xi_{\mu}$ and $\sigma$. Here we have
introduced the term $\Sigma$. The third and fourth line arises from
$h^{T}_{\mu\nu}$, whilst the final ratio is the contribution from ${h}$.
Unphysical modes are isolated differently in these implementations, but the
changes can be summarised in the different expressions

$\Sigma=0\,,\quad
10\,R^{2}\,\theta\left(1-\tfrac{R}{3}\right)\,,\quad-\frac{10R^{2}(R^{2}-20R+54)}{(R-3)(R-4)}\,,\quad\frac{10(11R-36)}{(R-3)(R-4)}\,.$
(31)

The first, third and fourth options are derived in refs. [30, 32], whilst the
second option comes from ref. [31]. We have suppressed some other details, for
more discussion see ref. [54].

Setting $\partial_{t}f=0$ in the above turns this flow equation into the
differential equation that must be satisfied by a fixed point $f(R)$. It is a
highly non-linear third-order ODE (ordinary differential equation). In the
formulation [31], the appearance of the $\theta$ functions, explicitly and in
$r$, will result in jumps in $f^{\prime\prime\prime}(R)$ across the point
where they switch on or off, but this can be accommodated.

A more important and generic feature is the existence of fixed and moveable
singularities. These concepts come from the mathematics of analysis of ODEs.
To discuss them it is helpful to cast the fixed point ODE in “normal” form:

$f^{\prime\prime\prime}(R)=rhs\,,$ (32)

where $rhs$ (right hand side) contains no $f^{\prime\prime\prime}$ terms. A
Taylor expansion about some generic point $R_{p}$ takes the form:

$f(R)=f(R_{p})+(R-R_{p})f^{\prime}(R_{p})+\frac{1}{2}(R-R_{p})^{2}f^{\prime\prime}(R_{p})+\frac{1}{6}(R-R_{p})^{3}f^{\prime\prime\prime}(R_{p})+\cdots\,.$
(33)

Since (32) determines the fourth coefficient in terms of the first three, we
see that typically (33) provides a series solution depending on three
continuous real parameters, here

$f(R_{p})\,,\quad f^{\prime}(R_{p})\quad{\rm and}\quad
f^{\prime\prime}(R_{p})\,,$ (34)

with some finite radius of convergence $\rho$ whose value also depends on
these parameters. Therefore the standard mathematical result is recovered that
around a generic point $R_{p}$ there is some domain ${\cal
D}=(R_{p}-\rho,R_{p}+\rho)$ in which there is a three-parameter set of well-
defined solutions. From here one can try to extend the solution to a larger
domain, e.g. by matching to a Taylor expansion about another point within
${\cal D}$. A typical problem, seen also in the LPA and the derivative
expansion [48, 49, 53, 50, 52] and in the second order formulation [42, 43],
is that eventually, at some point $R=R_{c}$, dependent on the parameters, the
denominator of $rhs$ develops a zero, so that as $R\to R_{c}$, (32) implies

$f^{\prime\prime\prime}(R)=2c/(R-R_{c})+\cdots\,,$ (35)

where $c$ is some constant and the ellipses contains the non-singular part.
Integrating this we see that the solution typically ends in a moveable
singularity, of form

$f(R)\sim c\,(R-R_{c})^{2}\ln|R-R_{c}|\,,$ (36)

where “$\sim$” means that less singular parts are neglected.

As already mentioned, fixed point equations derived with adaptive cutoff
present another challenge in that they also have fixed singular points
$R_{c}$. These correspond in $rhs$ to explicit algebraic poles in $R$, where
the domain of interest is $R\geq 0$ since the equations apply to the four-
sphere. Whatever the formulation there is always one fixed singularity
$R_{c}=0$, which is unavoidable when using an adaptive cutoff as we have seen
[33, 54]. Different formulations have different numbers and positions for the
other fixed singularities (see e.g. the discussion in refs. [38, 57]) but
there is always at least one more. Inspecting the example (30), we see that
$f^{\prime\prime\prime}$ appears once in the penultimate line in eqn. (30),
where it is multiplied by the polynomial

$R\left(27-\frac{91}{20}R^{2}-\frac{29}{30}\,R^{3}-\frac{181}{3360}\,R^{4}\right)\,.$
(37)

Thus, rearranging the fixed point equation into normal form (32), results in
poles from the zeroes of this polynomial. Two of these are in the required
domain, namely at $R_{c}=0$ and $R_{c}=2.0065$. There are also two further
single poles, at $R_{c}=3$ and $R_{c}=4$, from the first two lines of the
right hand side of (30).

As $R$ approaches one of these $R_{c}$, $f$ will end at a singularity of form
(36) unless the $f$–dependent parts in $rhs$ are tuned so as to conspire to
cancel the pole. Substituting the Taylor expansion (33), with $R_{p}=R_{c}$,
one sees that this requirement forces some generally non-linear combination of
$f(R_{c})$, $f^{\prime}(R_{c})$ and $f^{\prime\prime}(R_{c})$ to vanish. Thus,
a fixed singularity imposes a constraint on the solution, reducing the number
of free parameters by one.

The inevitable fixed singularity at $R_{c}=0$ can thus be seen as restoring
consistency since it reduces the three parameter set of solutions to a two
parameter set, in agreement in this respect with what is obtained from the
non-adaptive-cutoff second order formulation.

Unfortunately, since there are a further three fixed singularities, these
equations are overconstrained, and thus there are no fixed point solutions
$f(R)$ that are valid over the whole range $R\geq 0$.

However, these fixed singularities are artefacts of the regularisation
procedure: it is possible to move them and eliminate most of them. Benedetti
and Caravelli were the first to realise this, and we will refer to their
version [33] as the “BC” formulation. Before regularisation, the Jacobian
trace (21) has a denominator that vanishes if $\Delta_{1}$ vanishes. Likewise
the Jacobian trace (22) has a denominator that vanishes when $\Delta_{0}$
vanishes. Recalling the form (17) of the $\Delta_{s}$, and that the net effect
of the adaptive optimised cutoff is to replace $-\nabla^{2}$ with $k^{2}$ in
the denominator, we see that these contributions give poles $1/(1-R/4)$ and
$1/(1-R/3)$ (after using (4) to scale to dimensionless quantities). These are
the poles that are visible in the first two lines of the right hand side of
(30).

BC eliminate them by using an endomorphism,  namely by using $r(\Delta_{s})$
instead of $r(-\nabla^{2})$ [33] (a so-called cutoff of type II [32]). Then
one is left with the $R_{c}=0$ singularity, and a fixed singularity at some
positive $R_{c}$ which is due to the fact that the $\bar{h}$ trace vanishes
there [33, 54]. These fixed singularities thus reduce $f(R)$ solutions to a
one-parameter set.

Now there is still the danger of encountering a moveable singularity (36), and
this imposes further restrictions on the remaining parameter. Such a
singularity can appear at any value of $R$, and in particular at large $R$
where the equations can then be solved analytically by developing the solution
as an asymptotic expansion. In scalar field theory [48, 49, 53, 51, 50, 52]
and in the second order formulation [43], what is found is that this
asymptotic expansion has less than the full number of parameters expected. One
can also show that the missing parameters are associated with fast growing
perturbations that are incompatible with an asymptotic solution. In this way
it is possible to deduce analytically the number of constraints that moveable
singularities are responsible for imposing.

The result for scalar field theory is that the parameters are fixed, typically
to a handful of values [48, 49, 53], corresponding to a finite set of fixed
points, or in special cases a discrete infinity of fixed points [50]. However,
there is at this stage also the possibility that there are no fixed point
solutions. The actual number of solutions then needs to be determined
numerically.111Although some may be found analytically, e.g. the Gaussian
fixed point, or special cases [40]. We will see this at work in the second
order formulation in sec. 6 where we describe in detail how to find asymptotic
solutions $f_{asy}(R)$.

Unfortunately for third order formulations, asymptotic analysis typically does
not find sufficient constraints [57]. For example for the BC formulation, the
asymptotic solution turns out to have the maximum three parameters [54]:

$f_{asy}(R)=A\,R^{2}+R\left\\{\frac{3}{2}A+B\cos\ln R^{2}+C\sin\ln
R^{2}\right\\}+\cdots\,,$ (38)

where the ellipses stand for asymptotic corrections with lower powers of $R$,
and the three parameters are restricted only by the inequality:

$\frac{121}{20}A^{2}>B^{2}+C^{2}\,.$ (39)

Thus, one still expects to find one-parameter sets (i.e. lines) of global
solutions $f(R)$ in this case, and that is exactly what is found by careful
numerical analysis [54].  Asymptotic analysis also shows that the BC
formulation has continuous eigenoperator spectra. Initially it was suggested
that these effects can be attributed to the fact that all eigenoperators are
redundant  if the equation of motion (16) for the fixed point $f(R)$, has no
solution for $R$ in the required range $R\geq 0$ [55]. But it is now clear
that the poor behaviour is again associated to the scalar mode $\bar{h}$ [54,
38, 64], and is one more malign effect of the conformal instability [63, 54,
64].  In fact precisely these problems reappear in the second order
formulation if one chooses $c_{\bar{h}}<0$, as already mentioned in sec. 3.

As emphasised in ref. [57], asymptotic analysis plays three powerful rôles.
Firstly, as just sketched and discussed in detail in sec. 6.1, it allows one
to deduce the dimension of the solution space. Secondly the asymptotic
solution provides a way to validate numerical solutions since if one can
integrate out far enough, the numerical solution should match the asymptotic
solution, allowing a reliable determination of the asymptotic parameters.

Finally, the asymptotic solution actually contains only the physical part of
the fixed point effective action.  To see this, we need to return temporarily
to labelling scaled quantities with a tilde, and recall that the effective
infrared cutoff $k$ is added by hand such that the physical Legendre effective
action is recovered only in the limit that this cutoff $k\to 0$. This must be
done while holding the physical quantities such as $R$ fixed, rather than
scaled quantities $\tilde{R}$. In normal field theory, e.g. scalar field
theory, the analogous object is the universal scaling equation of state, which
for a constant field precisely at the fixed point takes the simple form

$V(\varphi)=A\,\varphi^{d/d_{\varphi}}\,,$ (40)

where $d$ is the space-time dimension and $d_{\varphi}$ is the full scaling
dimension of the field (i.e. incorporating also the anomalous dimension). In
the current case we keep fixed the constant background scalar curvature $R$.
Thus by (2) and (4), the only physical part of the fixed point action in this
approximation is:

$f({R})|_{\text{phys}}=\lim_{k\to 0}k^{4}\,\tilde{f}(R/k^{2})=\lim_{k\to
0}k^{4}\,\tilde{f}_{asy}(R/k^{2})\,.$ (41)

For example from (38), for the BC formulation one finds:

$f({R})|_{\text{phys}}=AR^{2}\,.$ (42)

This is invariant under changes of scale as it must be, and is a sensible
answer for the scaling equation of state precisely at the fixed point. We will
find the same answer from the second order formulation.

We still have the problem that since there are one-parameter sets of fixed-
point solutions, $A$ is not fixed. In third order formulations one can use the
ability to add endomorphisms to try to patch this up [38] but asymptotic
analysis then shows there is actually a whole zoo of possibilities for the
scaling equation of state and dimension of the solution space, depending on
parameter choices in the endomorphisms [57]. One can also try to extend the
solution to negative $R$. This does reduce the solution space of the BC
formulation to a discrete set but that set appears to be empty since no
numerical solutions were then found [54]. A more careful version of this
strategy is also used in the second order formulation.

Actually one can question whether the large $\tilde{R}=R/k^{2}$ regime makes
physical sense [37, 38, 40]. The problem arises when the cutoff depends on
modified Laplacians, e.g. as in (28), where the endomorphism is added to
ensure that the minimum eigenvalue is positive. It is most easily seen if we
take a sharp (step function) cutoff profile, and write the minimum eigenvalue
as $R\,\lambda_{min}$. Then once $k^{2}<R\,\lambda_{min}$, i.e.
$\tilde{R}>1/\lambda_{min}$, there are no more modes to be integrated out.
This means that the functional behaviour in this large $\tilde{R}$ regime is
meaningless since it is not describing any actual changes. However the
physical Legendre effective action is only reached by taking $k\to 0$, and
this argument would appear to imply that such a limit is inherently ill-
defined.

In fact this conundrum is another artefact of the single-metric approximation
[22]. In reality one should be integrating out over an ensemble of manifolds
described by the fluctuating full metric $\hat{g}_{\mu\nu}$. The Wilsonian RG
only makes sense when applied to such an ensemble.  Then no matter how small
$k$ is, there are always manifolds with sufficiently small curvature that
their eigenvalues remain to be integrated out. It is possible to repair the
single-metric approximation sufficiently in this case by retaining the scale
degree of freedom $h_{\mu\nu}\propto g_{\mu\nu}$ in the fluctuation field
dependence, and thus regaining an ensemble of manifolds. However the net
result of such a repair is the same type of functional RG equations again, but
now with a clear explanation for why the large $\tilde{R}$ regime should be
trusted [22, 23, 24].

We now abandon third order formulations and concentrate on a second order
formulation [42, 43], which in almost all respects has more promising
behaviour.

##  5 Evaluating traces

In the formulation [42, 43] the traces  are evaluated by a direct spectral
sum. In common with the rest of the literature one chooses a (globally)
maximally symmetric background manifold. There are three to choose from: the
four sphere $\mathbb{S}^{4}$, which has a finite volume and positive
curvature, so the spectrum of the allowed modes form a discrete set that have
to be summed over; the hyperboloid $\mathbb{H}^{4}$ which has negative
curvature and infinite volume so the spectrum is continuous; and finally flat
space $\mathbb{R}^{4}$, which is a limiting case for both of the two previous
manifolds when $R\rightarrow 0$. As we will see they all need to be
considered. Actually they become smoothly joined together in an ensemble which
thus allows the same flow equation to be defined over the entire domain
$-\infty<R<\infty$.

###  5.1 Sphere

On the sphere the traces are evaluated using

$\text{Tr}\,W(\Delta_{s})=\sum_{n}D_{n,s}W(\lambda_{n,s})$ (43)

where $\lambda_{n,s}$ are eigenvalues of the $\Delta_{s}$ defined in (17), and
$D_{n,s}$ are their multiplicities. Explicit values are shown in table 1 [33].

Spin s |  Eigenvalue $\lambda_{s,n}$ |  Multiplicity $D_{n,s}$
---|---|---
0 | $\frac{n(n+3)-4}{12}R$ | $\frac{(n+2)(n+1)(2n+3)}{6}$
1 | $\frac{n(n+3)-4}{12}R$ | $\frac{n(n+3)(2n+3)}{2}$
2 | $\frac{n(n+3)}{12}R$ | $\frac{5(n+4)(n-1)(2n+3)}{6}$
Table 1: Values of the multiplicities and eigenvalues for evaluating the
traces.

There are a few caveats. Not all the modes contribute in the sum, for example
vectors satisfying $\nabla_{\mu}\xi_{\nu}+\nabla_{\nu}\xi_{\mu}=0$ and the
scalar modes $\sigma=\text{constant}$. Because of this, the tensor mode and
the vector mode sums start at $n=2$, the scalar mode of the Jacobian starts at
$n=1$ and the $\bar{h}$ mode starts at $n=0$. Now the requirement (28) means
that $\lambda_{n,s}+\alpha_{s}R>0$ must be satisfied. For the tensor and
vector modes it is sufficient to set $\alpha_{2}=\alpha_{1}=0$, however from
table 1 we see that we must have $\alpha_{0}>1/3$.

###  5.2 Hyperboloid

As already mentioned, the hyperboloid has a negative curvature, an infinite
volume, and a continuous spectrum of eigenvalues. The traces on this manifold
are evaluated using [68]

$\displaystyle\text{Tr}\,W(\Delta_{s})=\frac{2s+1}{8\pi^{2}}\int\\!d^{4}x\sqrt{g}\
\bigg{(}-\frac{R^{2}}{12}\bigg{)}^{2}\int_{0}^{\infty}d\lambda\bigg{(}\lambda^{2}+\big{(}s+\frac{1}{2}\big{)}^{2}\bigg{)}\lambda\tanh(\pi\lambda)W(\Delta_{\lambda,s}).$
(44)

Even though there is now an infinite volume factor in the flow equation (18),
this precise factor also appears above, so the equations still make sense once
we cancel this factor from both sides. The eigenvalues of the spectrum are

$\Delta_{\lambda,s}=-\frac{R}{12}\lambda^{2}-\beta_{s}R\,,\qquad\text{where}\qquad\beta_{0}=\frac{25}{48},\quad\beta_{1}=\frac{25}{48},\quad\beta_{2}=\frac{9}{48}\,.$
(45)

Using the same flow equation, and thus the same endomorphism parameters
$\alpha_{s}$, the requirement (28) must again be satisfied. We can still take
$\alpha_{2}=\alpha_{1}=0$, but now $\alpha_{0}$ also has an upper bound
$\alpha_{0}<25/48$.

###  5.3 Flat space

Finally, evaluating traces on flat space can be achieved by taking the limit
as $R\rightarrow 0$ from positive or negative side.  If we start from the
positive side we first make a substitution $p=n\sqrt{R/12}$ then take
$R\rightarrow 0$ while keeping $p$ fixed. All Laplacians then become
$\Delta_{n,s}\rightarrow p^{2}$ and $p^{2}$ can be identified as the flat
space momentum. Plugging in our choice of the cutoff (27), and performing
these substitutions, yields

$\displaystyle\partial_{t}f_{k}(0)+4f_{k}(0)$
$\displaystyle=\frac{1}{8\pi^{2}}\int_{0}^{\infty}dpp^{3}\Bigg{[}16c_{\bar{h}}\frac{2r(p^{2})-p^{2}r^{\prime}(p^{2})}{9f_{k}^{\prime\prime}(0)p^{4}+3f_{k}^{\prime}(0)p^{2}+2f_{k}(0)+16c_{\bar{h}}r(p^{2})}$
$\displaystyle+10c_{T}\frac{r(p^{2})-p^{2}r^{\prime}(p^{2})}{-f_{k}^{\prime}(0)p^{2}-f_{k}(0)+2c_{T}r(p^{2})}-3c_{V}\frac{r(p^{2})-p^{2}r^{\prime}(p^{2})}{p^{2}+c_{V}r(p^{2})}$
$\displaystyle-4c_{S_{2}}\frac{2r(p^{2})-p^{2}r^{\prime}(p^{2})}{3p^{4}+4c_{S_{2}}r(p^{2})}+c_{S_{1}}\frac{r(p^{2})-p^{2}r^{\prime}(p^{2})}{p^{2}+c_{S_{1}}r(p^{2})}\Bigg{]}$
(46)

This same equation is arrived at if we take $R\rightarrow 0$ from the negative
side by first setting $p=\lambda\sqrt{-R/12}$ on the hyperboloid and holding
$p$ fixed. The form of these equations already give some information about the
possible solutions, and can help guide numerical searches [43]. In particular,
by inspection, it is clear that there are no fixed singularities, and choices
for $f_{k}(0)$, $f^{\prime}_{k}(0)$ and $f^{\prime\prime}_{k}(0)$ can be made
that give well defined non-singular integrals.

##  6 Fixed point solutions

The fixed point solution to the flow equation $f_{k}(R)=f(R)$ occurs when
$\partial_{t}f_{k}(R)=0$. An advantage of the non-adaptive cutoff is that
$\partial_{t}f_{k}(R)$ only appears once on the left hand side of (18), so the
fixed point equation  is

$2\,VE(R)=\mathcal{T}_{2}+\mathcal{T}_{0}^{\bar{h}}+\mathcal{T}_{1}^{Jac}+\mathcal{T}_{0}^{Jac}\,.$
(47)

Another crucial advantage is, like (46), inspection of the trace equations
(19) – (22) makes clear that there are no fixed singularities any more. The
flow equation is non-linear and very hard to work with, so solving the
equations exactly is unfeasible. The strategy is to solve analytically for
$f(R)$ around $R=0$ as a Taylor expansion and around $R=\pm\infty$ by an
asymptotic expansion. Then numerical methods can be used to try to patch in a
solution that goes smoothly from the Taylor expansion at $R=0$ to the
asymptotic solutions at $R=\pm\infty$.

###  6.1 Asymptotic analysis

We now explain in detail how to develop asymptotic solutions, using these
equations as an example. In these large $R$ limits, the equations simplify due
to rapidly decaying cutoff profiles $r(z)$. At first sight, it looks like all
the traces on the right hand side of the flow equation vanish and one is only
left with (16), the equation of motion $E(R)=0$. This is actually true on the
hyperboloid and the fixed point solution is therefore the solution of $E(R)=0$
namely

$f(R)=AR^{2},$ (48)

where $A$ is an arbitrary constant. At any finite $R$ this is then accompanied
by rapidly decaying corrections as discussed later, cf. eqn. (63).

The story is different on the sphere since upon closer inspection not all of
the terms in the sums vanish.  There are three such terms, the $n=0$ and $n=1$
components from $\mathcal{T}_{0}^{\bar{h}}$ and the $n=1$ of
$\mathcal{T}_{0}^{Jac}$. To see this for the $n=0$ case, note that from table
1, $\Delta_{0}=-R/3$. Thus, using (27), the denominator of this term in the
sum (20) is given by

$9f^{\prime\prime}(R)\Delta_{0}^{2}+3f^{\prime}(R)\Delta_{0}+E(R)+16\mathcal{R}^{\bar{h}}_{k}=R^{2}f^{\prime\prime}(R)-Rf^{\prime}(R)+E(R)+16k^{4}c_{\phi}r([\alpha_{0}-\tfrac{1}{3}]R)$
(49)

Now, assuming that the leading asymptotic behaviour is $f(R)=AR^{2}$, we see
that the first two terms cancel each other, and likewise $E(R)$ vanishes, so
we are left only with the cutoff term in the denominator. Therefore this term
takes the form of

$\frac{1}{k^{4}\,r(z)}\frac{d}{dt}\left[k^{4}r(z)\right]=4-2z\frac{d\ln
r(z)}{dz}$ (50)

with $z$ set equal to $z=[\alpha_{0}-\tfrac{1}{3}]R$.

Turning to the $n=1$ components, note that from table 1, both $\Delta_{0}$ and
$\Delta_{1}$ vanish for $n=1$. In (20), apart from the cutoff term the whole
denominator therefore vanishes (because $E(R)$ vanishes). In (22) it is the
second component that has a vanishing denominator apart from the cutoff term.
The $S_{1}$ (first) component does not suffer from the same problem because
there is also the $+R/3$ part in the denominator. However, the cutoff
dependence is the same for the $n=1$ contributions namely $r(\alpha_{0}R)$ and
the numerical factors are such that these two $n=1$ contributions exactly
cancel each other.

Altogether then, effectively the only term on the RHS (right hand side) of the
flow equation that does not vanish asymptotically is the $n=0$ component of
the $\mathcal{T}^{\bar{h}}_{0}$ trace. This is a problem however, since the
$n=0$ component of $\mathcal{T}^{\bar{h}}_{0}$ contributes a term that grows
at least as fast as $R^{2}$. This is inconsistent with the fact that the LHS
(left hand side) of flow equation has been set to vanish asymptotically.
Actually this analysis shows that $f(R)$ grows faster than $R^{2}$. For
example in the best-case scenario the RHS $\sim R^{2}$ but that implies
$f(R)\sim R^{2}\ln R$ so that the LHS is left with an $E(R)\sim R^{2}$ to
balance the contribution from the $n=0$ component of
$\mathcal{T}^{\bar{h}}_{0}$.

Therefore we now assume that $f(R)$ actually grows faster than $R^{2}$ at
large $R$. But this means we need to check again which terms in the traces
have denominators that would vanish without a cutoff. By inspection none of
the traces that depend on $f(R)$ can now have this issue. In particular the
$n=1$ component of the $\mathcal{T}^{\bar{h}}_{0}$ trace no longer has a
denominator that could vanish, because $E(R)$ no longer vanishes at large $R$,
while for the $n=0$ component the $f^{\prime\prime}(R)$ part in the
denominator now dominates at large $R$. So the only contribution that survives
on the RHS at large $R$, is now the $n=1$ $S_{2}$ component of
$\mathcal{T}^{Jac}_{0}$.

Keeping just this term it turns out one can solve the fixed point equation in
closed form, thus obtaining the correct asymptotic behaviour for general
cutoff function $r(z)$. Using the values from table 1 we have that the
multiplicity of the $n=1$ component is $D_{1,0}=5$, note that $m_{S_{2}}=4$
and that $1/V=R^{2}/384\pi^{2}$ for the four-sphere. Thus, keeping only this
leading term on the RHS of the flow equation, we have

$2f(R)-Rf^{\prime}(R)=\frac{R^{2}}{768\pi^{2}}\left[-10+5\alpha_{0}R\frac{r^{\prime}(\alpha_{0}R)}{r(\alpha_{0}R)}\right]\,.$
(51)

This is exactly soluble. Indeed dividing through by $R^{3}$ it can be
rewritten as

$-\frac{d}{dR}\left(\frac{f(R)}{R^{2}}\right)=\frac{1}{768\pi^{2}}\left[-\frac{10}{R}+5\frac{d}{dR}\ln
r(\alpha_{0}R)\right]\,,$ (52)

which can be immediately integrated to give

$f(R)=\frac{5R^{2}}{768\pi^{2}}\ln\frac{R^{2}}{r(\alpha_{0}R)}+AR^{2}+o(R^{2})\quad\text{as}\quad
R\to+\infty\,,$ (53)

where we included the integration constant $A$ and finally we noted that terms
that grow slower than $R^{2}$ will be generated by iterating this asymptotic
solution to higher orders, hence the $o(R^{2})$ part. The $\ln r$ term
actually dominates, i.e. the large $R$ behaviour is dominated by cutoff-
dependent effects. For example using the cutoff (29), gives the first three
terms in this series:

$f(R)=\frac{5a\alpha_{0}^{b}}{768\pi^{2}}R^{2+b}+\frac{5}{768\pi^{2}}R^{2}\ln
R+AR^{2}+\frac{16c_{\bar{h}}}{5ab(1+b)\alpha_{0}^{b}}\left(\alpha_{0}-\frac{1}{3}\right)\,\mathrm{e}^{-a\left(\alpha_{0}-\frac{1}{3}\right)^{b}R^{b}}+\cdots\,.$
(54)

To get the next term in the series, the solution is substituted back into the
fixed point equation and the next leading correction is isolated. This leads
to the last displayed correction above. It is exponentially decaying and comes
from the $n=0$ term in the $\mathcal{T}^{\bar{h}}_{0}$ trace. One finds that
other corrections decay faster provided that
$\alpha_{0}<\frac{5}{6}+\alpha_{1}$. This is satisfied thanks to the
restrictions on the $\alpha_{i}$ parameters discussed in secs. 5.1 and 5.2.
Substituting (54) back into the fixed point equation and proceeding similarly
one can in principle develop the whole asymptotic series. It is an infinite
series of ever faster decaying terms and is indicated by the ellipses. In
particular these terms will include a power series in $A$.

At this point we have succeeded in finding consistent asymptotics. $f(R)$ does
grow faster than $R^{2}$ on the sphere, as assumed, and using such a form in
the RHS of the fixed point equation one can see that the $n=1$ $S_{2}$
component of $\mathcal{T}^{Jac}_{0}$ dominates at large $R$, which leads back
to the above equation.

Recall that the fixed point equation is actually second order. But the
asymptotic solutions only have one free parameter $A$, even though there
should be two. To find out where the second parameter has gone we linearise
about the fixed point $f(R)+\delta f(R)$ and plug it into the flow equation
(47) to get

$-a_{2}(R)\,\delta f^{\prime\prime}(R)+a_{1}(R)\,\delta
f^{\prime}(R)+a_{0}(R)\,\delta f(R)=4\,\delta f(R)\,,$ (55)

with

$a_{2}=\frac{144c_{h}}{V}\text{Tr}\Bigg{[}\frac{\Delta_{0}^{2}(2r(\Delta_{0}+\alpha_{0}R)-(\Delta_{0}+\alpha_{0}R)r^{\prime}(\Delta_{0}+\alpha_{0}R))}{(9f^{\prime\prime}(R)\Delta_{0}^{2}+3f^{\prime}(R)\Delta_{0}+E(R)+16c_{h}r(\Delta_{0}+\alpha_{0}R))^{2}}\Bigg{]}\,,$
(56) $\displaystyle a_{1}$
$\displaystyle=2R-\frac{16c_{h}}{V}\text{Tr}\Bigg{[}\frac{(3\Delta_{0}-R)(2r(\Delta_{0}+\alpha_{0}R)-(\Delta_{0}+\alpha_{0}R)r^{\prime}(\Delta_{0}+\alpha_{0}R))}{(9f^{\prime\prime}(R)\Delta_{0}^{2}+3f^{\prime}(R)\Delta_{0}+E(R)+16c_{h}r(\Delta_{0}+\alpha_{0}R))^{2}}\Bigg{]}$
$\displaystyle+\frac{2c_{T}}{V}\text{Tr}\Bigg{[}\frac{(R/2-\Delta_{2})(2r(\Delta_{2}+\alpha_{2}R)-(\Delta_{2}+\alpha_{2}R)r^{\prime}(\Delta_{2}+\alpha_{2}R))}{(-f^{\prime}(R)\Delta_{2}-E(R)/2+2c_{T}r(\Delta_{2}+\alpha_{2}R))^{2}}\Bigg{]}\,,$
(57) $\displaystyle a_{0}$
$\displaystyle=\frac{32c_{h}}{V}\text{Tr}\Bigg{[}\frac{(2r(\Delta_{0}+\alpha_{0}R)-(\Delta_{0}+\alpha_{0}R)r^{\prime}(\Delta_{0}+\alpha_{0}R))}{(9f^{\prime\prime}(R)\Delta_{0}^{2}+3f^{\prime}(R)\Delta_{0}+E(R)+16c_{h}r(\Delta_{0}+\alpha_{0}R))^{2}}\Bigg{]}$
$\displaystyle+\frac{2c_{T}}{V}\text{Tr}\Bigg{[}\frac{(2r(\Delta_{2}+\alpha_{2}R)-(\Delta_{2}+\alpha_{2}R)r^{\prime}(\Delta_{2}+\alpha_{2}R))}{(-f^{\prime}(R)\Delta_{2}-E(R)/2+2c_{T}r(\Delta_{2}+\alpha_{2}R))^{2}}\Bigg{]}\,.$
(58)

In the large $R$ limit $a_{1}(R)\sim 2R$ and $a_{0}$ and $a_{2}$ vanish
asymptotically. Then it is tempting to simply set $a_{0}$ and $a_{2}$ to zero
to find the asymptotic solution to (55). But if this is done there is only one
solution $\delta f(R)=\delta A\,R^{2}$. In fact this is just the leading term
in an asymptotic series which is nothing but what one would derive from (54)
by differentiating with respect to $A$. (Recall that the ellipses actually
contain a power series in $A$.) This asymptotic solution is an exact series
solution to (55) where $a_{0}$ and $a_{2}$ are only involved in constructing
the subleading corrections. To find more than the one parameter $\delta A$ in
the solution to (55), $\delta f^{\prime\prime}(R)$ cannot be neglected,
implying that higher derivatives must dominate over lower ones in the large
$R$ limit. Hence, the other solution is one where $\delta f(R)$ can at first
be neglected. Then writing (55) as

$\frac{d}{dR}\ln\delta
f^{\prime}(R)=\frac{a_{1}(R)}{a_{2}(R)}\quad\implies\delta
f(R)=B\int^{R}dR^{\prime}\exp\int^{R^{\prime}}dR^{\prime\prime}\frac{a_{1}(R^{\prime\prime})}{a_{2}(R^{\prime\prime})},$
(59)

where $B$ is the second parameter. For the explicit form, $a_{2}$ is needed.
It gets its leading contribution from the same source as the last displayed
term in (54). Using the same cutoff choice, (29), asymptotically

$a_{2}(R)=\frac{24576\pi^{2}c_{\bar{h}}}{25ab(1+b)^{2}\alpha_{0}^{2b}}\left(\alpha_{0}-\frac{1}{3}\right)^{1+b}\\!\\!\\!R^{1-b}\,\mathrm{e}^{-a\left(\alpha_{0}-\frac{1}{3}\right)^{b}R^{b}}+\cdots\,.$
(60)

Recalling that $a_{1}=2R$ to leading order, the integrals can be evaluated by
successive integration by parts, as an asymptotic series and where each term
is given in closed form.

Since this strategy is used many times in this kind of asymptotic analysis let
us sketch it on the indefinite integral:

$\int\\!\\!dR\,G(R)\,\mathrm{e}^{F(R)}=\frac{G(R)}{F^{\prime}(R)}\,\mathrm{e}^{F(R)}-\int\\!\\!dR\,\left(\frac{G(R)}{F^{\prime}(R)}\right)^{\prime}\,\mathrm{e}^{F(R)}\,.$
(61)

The above equality follows by integration by parts, however if $F(R)$ grows at
least as fast as $R$ for large $R$, where $F$ is either sign, and $G(R)$ grows
or decays slower than an exponential of $R$, then the integral on the right is
subleading compared to the integral on the left. Iterating this identity then
evaluates the integral in the large $R$ limit as $\mathrm{e}^{F(R)}$ times an
asymptotic series, the first term on the RHS being the leading term.

In this way, using the cutoff (29), the solution (59) on the sphere turns out
to be

$\delta f(R)\sim
B\exp\\{\frac{12(1+b)^{2}\alpha_{0}^{2b}}{12288\pi^{2}c_{\bar{h}}}\bigg{(}\alpha_{0}-\frac{1}{3}\bigg{)}^{-1-2b}R\,\mathrm{e}^{a(\alpha_{0}-1/3)^{b}R^{b}}\bigg{\\}}\,.$
(62)

The analysis proceeds similarly on the hyperboloid [43]. As $R\to-\infty$, one
finds:

$f(R)=AR^{2}+\frac{c_{S1}}{96\sqrt{3\pi
a^{3}b^{3}}}\left(\frac{25}{48}-\alpha_{0}\right)^{\frac{5-3b}{2}}\\!\\!\\!\left(-R\right)^{2-\frac{3b}{2}}\left\\{1+O\left(|R|^{-\frac{1}{2}}\right)\right\\}\,\mathrm{e}^{-a\left[\left(\alpha_{0}-\frac{25}{48}\right)R\right]^{b}}+\cdots\,.$
(63)

The correction is again a decaying exponential because $\alpha_{0}$ is
restricted to $\alpha_{0}<25/48$. All scalar traces (thus also $A$) contribute
to the $O\left(|R|^{-\frac{1}{2}}\right)$ term, and the ellipses stand for
terms with faster decaying exponentials. The asymptotic behaviour of $a_{2}$
turns out now to be:

$a_{2}(R)=\frac{4c_{\bar{h}}}{81A^{2}\sqrt{3\pi
ab}}\left(\frac{25}{48}-\alpha_{0}\right)^{\frac{5-b}{2}}\\!\\!\\!\left(-R\right)^{1-\frac{b}{2}}\,\mathrm{e}^{-a\left[\left(\alpha_{0}-\frac{25}{48}\right)R\right]^{b}}+\cdots$
(64)

(the ellipses being faster decaying terms). And thus one finds on the
hyperboloid

$\delta f(R)\sim
B\exp\\{\frac{81A^{2}}{2c_{\bar{h}}}\sqrt{\frac{3\pi}{ab}}\bigg{(}\frac{25}{48}-\alpha_{0}\bigg{)}^{-\frac{b+5}{2}}(-R)^{1-b/2}\,\mathrm{e}^{a[(\alpha_{0}-25/48)R]^{b}}\bigg{\\}}.$
(65)

However, there is a problem here. Both these solutions (62) and (65) for
$\delta f(R)$, are rapidly growing exponentials of an exponential. In the
asymptotic regime, these perturbations are no longer small, thus invalidating
the initial linearization assumption used to derive them. Therefore, these
solutions must be discarded and thus we conclude that the fixed points have
only one free parameter on both the sphere and hyperboloid.

These results allow us to draw important conclusions. Each of the asymptotic
fixed point solutions, (54) and (63), contribute one constraint on the flow
equation.222For example at some initial very large $R$ we can set
$2f(R)=Rf^{\prime}(R)$ since this boundary condition imposes the leading
behaviour (48). Using the subleading corrections we can furnish a more
accurate Robin boundary condition at more reasonable values of $R$. There are
no boundary conditions coming from $R=0$, so we can expect one-parameter sets
of fixed point solutions on both $\mathbb{S}^{4}$ and $\mathbb{H}^{4}$. At
first sight this is a disappointing result for the asymptotic safety program.
However, if we now use $\mathbb{R}^{4}$, eqn. (46), to match smoothly between
these solutions then we have two boundary conditions on a second order
differential equation, one coming from the sphere and one from the
hyperboloid. Thus, there can now only be at most a discrete set of solutions.
In the next section more evidence will be presented for why these topologies
should be considered smoothly joined together in this way.

###  6.2 Numerical solution

Note that this does not answer the question of whether there is more than one
fixed point, or no fixed point at all, or the phenomenologically preferred
answer: a single fixed point. As already mentioned, the only way to find out
which of these latter possibilities is realised, is to perform an extensive
numerical search for such global $f(R)$ solutions. As we have just seen, the
asymptotic fixed point solutions (54) and (63), on the sphere and hyperboloid
respectively, depend on one single parameter, which we called $A$ on both
sides. Even if there is a global solution that connects them, the value of $A$
will almost surely be different in the two different topologies. On one of the
topologies, one can scan through $A$ at some large value of $R$ and solve the
equations backwards towards $R=0$. With the exponential cutoff profile (29),
solutions have been found on the sphere this way, in a narrow region around
$A=-0.01$, matching to the asymptotic series at $R\sim 10$. On the hyperboloid
no solution was found however, although a more comprehensive numerical
analysis might find one [43].

##  7 Eigenoperators

So far we have analyzed the flow equation in the $f(R)$ approximation at the
fixed point (where $\partial_{t}f(R)=0$).  Assuming that there is a global
solution, we now turn to the question of whether the theory is predictive.
This is answered by solving the eigenvalue equation and figuring out how many
relevant operators  the fixed point solution has. Relevant operators are the
ones that fall into the fixed point when increasing the cutoff scale $k$. The
number of these operators corresponds to the number of parameters that will
have to be fixed experimentally. We now prove that in this second order
formulation, if we take the equations to apply simultaneously across the three
spaces $\mathbb{S}^{4}$, $\mathbb{R}^{4}$ and $\mathbb{H}^{4}$, there are a
finite number of relevant operators [42]. Plugging (5) into the flow equation
(18) we get a second order ordinary differential eigenvalue equation:

$-a_{2}(R)\,v^{\prime\prime}(R)+a_{1}(R)\,v^{\prime}(R)+a_{0}(R)\,v(R)=\lambda
v(R)$ (66)

where the eigenvalues $\lambda=4-\theta$, $v(R)$ is the eigenoperator, and the
$a_{i}$’s are given by eqns. (56 – 58).

###  7.1 Asymptotic analysis

The first step is to apply asymptotic analysis to the eigenoperator equation.
The procedure closely follows that for the fixed point in sec. 6.1. As already
noted there, $a_{0}$ and $a_{2}$ decay exponentially fast and in the large $R$
limit $a_{1}\sim 2R$. Then the asymptotic form of the eigenvalue equation is:

$\lambda\,v(R)-2R\,v^{\prime}(R)=-a_{2}(R)\,v^{\prime\prime}(R)\,.$ (67)

Starting with the left hand side the solution is

$v(R)\propto|R|^{\frac{\lambda}{2}}+\cdots\,,$ (68)

where the ellipses stand for subleading corrections from the $a_{i}$’s, in
particular from the RHS. The solutions have one parameter, the constant of
proportionality. The missing parameter must come from a solution for which
$v^{\prime\prime}(R)$ cannot be neglected. But this implies diverging
derivatives and thus $v(R)$ can be neglected. The equation is then analogous
to what we had before where the second solution is now $v(R)\sim\delta\\!f(R)$
in (62) on the sphere and (65) on the hyperboloid.

Now we ask whether these solutions are actually valid. The linearised solution
(5) is meant to describe the RG flow ‘close’ to the fixed point. For any fixed
$\epsilon$, if $|v(R)/f(R)|\to\infty$ as $R\to\pm\infty$ that is not
necessarily true since linearisation is no longer valid. In this case one can
set

$f_{k}(R)=f(R)+\epsilon\,v_{k}(R)\,,$ (69)

and, without linearising, ask for the correct evolution for $v_{k}(R)$ at
large $R$. For large negative $R$ the RHS of the flow equation (18) can be
neglected. For large positive $R$, the RHS of the flow equation can be
neglected except for the $n=1$ $S_{2}$ component of $\mathcal{T}^{Jac}_{0}$,
which however just cancels the contributions from the LHS that grow faster
than $R^{2}$ resulting from $f(R)$, cf. (54). Since in fact the $O(R^{2})$
part of $f(R)$ also vanishes from the LHS (on both sphere and hyperboloid), in
the large $R$ regime one has

$\partial_{t}v_{k}(R)-2R\,v^{\prime}_{k}(R)+4\,v_{k}(R)=o(R^{2})\,.$ (70)

Any part of $v_{k}(R)$ growing at least as fast as $R^{2}$ is then easily
solved for, and gives mean-field evolution involving some arbitrary function
$v$:

$v_{k}(R)={\rm e}^{-4t}\,v(R\,{\rm e}^{\,2t})+o(R^{2})\,.$ (71)

It will be the same function $v$ that was introduced in the linearised
solution (5) if one requires as boundary condition, $v_{k}(R)=v(R)$ at
$k=\mu$. The question that remains is whether the RG evolution (71) is
consistent with what we found by linearising.

For the power-law solution (68), linearisation is valid at large $|R|$ if and
only if $\lambda\leq 4$. This follows from the hyperboloid fixed point
asymptotics (63), the sphere side (54) requiring only the weaker constraint,
$\lambda\leq 4+2b$. On the other hand if $\lambda>4$, one can use the general
perturbation (69), finding the solution (71). Substituting the explicit form
(68) of the boundary condition, gives:

$v_{k}(R)=v(R)\,\text{e}^{-\theta t}+o(R^{2})\,,$ (72)

where $\theta=4-\lambda$, i.e. the linearised solution (5) is reproduced. We
conclude that asymptotically, power-law eigenoperators (68) are valid
solutions for any $\lambda$. Their $t$ evolution is multiplicative and given
by the flow of a conjugate coupling $g(t)=\epsilon\,\text{e}^{-\theta t}$, cf.
(5).

On the other hand, the solutions that behave asymptotically as
$v(R)\sim\delta\\!f(R)$, are growing exponentials of exponentials.
Linearisation is not valid at large $|R|$, where the $t$ dependence is given
instead by (71). Now we cannot separate out the $t$ dependence. Therefore,
such perturbations cannot be regarded as eigenoperators evolving
multiplicatively.

Excluding them leads to quantisation of the spectrum.  This is because the
large $R$ dependence (68) provides a boundary condition on both the sphere and
the hyperboloid side, and linearity provides a further boundary condition
since one can choose a normalisation e.g. $v(0)=1$. These three conditions
over-constrain the eigenoperator equation (66) leading to quantisation of
$\lambda$, i.e. to a discrete eigenoperator spectrum.

###  7.2 Sturm-Liouville theory

Sturm-Liouville type equations take the form

$Lv(R)=\lambda w(R)v(R),$ (73)

where $L$ is the self adjoint operator

$L=-\frac{d}{dR}\Big{(}p(R)\frac{d}{dR}\cdot\Big{)}+q(R),$ (74)

with $p(R)$ and $q(R)$ being real functions and $w(R)$ also being positive.
For the second order formulation, the eigenvalue equation can be put in this
form. The properties of these equations will then allow us to draw conclusions
about the spectrum of the eigenvalues.

The weight function is defined as

$w(R)=\frac{1}{a_{2}(R)}\exp-\int^{R}\\!\\!dR^{\prime}\frac{a_{1}(R^{\prime})}{a_{2}(R^{\prime})}\,,$
(75)

since, multiplying with the eigenvalue equation (66) and rearranging, casts it
in Sturm-Liouville form:

$-\big{(}a_{2}(R)w(R)v^{\prime}(R)\big{)}^{\prime}+w(R)a_{0}(R)v(R)=\lambda
w(R)v(R)\,.$ (76)

Notice that the trace in $a_{2}$ is positive. This is because the cutoff is
monotonically decreasing, hence $r^{\prime}(z)<0$ and $r(z)>0$, so the sign of
$a_{2}$ depends on $c_{\bar{h}}$, which is positive. This implies that the
weight function $w(R)>0$ as required.

Next we check if the operator is self-adjoint. Taking $v=v_{j}(R)$,
multiplying by $v_{i}(R)$, and integrating over $R$, gives:

$-\int v_{i}Lv_{j}=-\int v_{i}\big{(}a_{2}wv^{\prime}_{j}\big{)}^{\prime}+\int
v_{i}a_{0}wv_{j}\,.$ (77)

If $L$ is self-adjoint then this should be the same for $j\leftrightarrow i$.
The first term on the RHS can be written as

$-\int\big{[}v_{i}\big{(}a_{2}wv_{j}^{\prime}\big{)}\big{]}^{\prime}+\int\big{[}v_{j}\big{(}a_{2}wv_{i}^{\prime}\big{)}\big{]}^{\prime}-\int\big{(}a_{2}wv_{i}^{\prime}\big{)}^{\prime}v_{j}\,.$
(78)

Thus what is required is that the first two terms above cancel each other.
This is automatically satisfied if $R$ is taken to have the full range since
$w(R)\rightarrow 0$ exponentially fast as $R\rightarrow\pm\infty$. If the
differential equation is restricted to either the four-sphere or four-
hyperboloid, there would be a boundary at $R=0$. The weight function does not
vanish there and thus the operator $L$ would then not be self-adjoint. This is
another powerful hint that the correct treatment is to smoothly join the three
topologies together. Note also that none of these equations would make sense
if the exponentially growing set of solutions $v(R)\sim\delta\\!f(R)$ are
included, where $\delta\\!f(R)$ is given by (62) or (65). From (59) and (61)
one can see that actually these $\delta\\!f(R)\sim 1/w(R)$ and thus such
$v(R)$ are not square integrable under the weight function $w(R)$ since
$w(R)v^{2}(R)\sim 1/w(R)$, which diverges at large $R$. Hence, this condition
only picks out the correct solutions from the eigenvalue equation and
justifies the use of Sturm-Liouville techniques.

Thus, when restricted to perturbations that grow only as a power at large
$|R|$, the eigenvalue equation (66) is of Sturm-Liouville type. The
consequences for the spectrum of the eigenvalues can be seen by a standard
transformation to Liouville normal form. Define a coordinate $x$ as

$x=\int_{0}^{R}\frac{1}{\sqrt{a_{2}(R^{\prime})}}dR^{\prime}.$ (79)

Then $x\rightarrow\pm\infty$ as $R\rightarrow\pm\infty$ because $a_{2}(R)$
vanishes at large $|R|$. Defining the ‘wave-function’

$\psi(x)=a_{2}^{\frac{1}{4}}(R)w^{\frac{1}{2}}(R)v(R),$ (80)

(66) can be transformed into

$-\frac{d^{2}\psi(x)}{dx^{2}}+U(x)\psi(x)=\lambda\psi(x),$ (81)

which is just the one-dimensional Schrödinger equation with energy $\lambda$.
The potential turns out to be [42]

$U(x)=a_{0}+\frac{a_{1}^{2}}{4a_{2}}-\frac{a^{\prime}_{1}}{2}+a^{\prime}_{2}\Big{(}\frac{a_{1}}{2a_{2}}+\frac{3a^{\prime}_{2}}{16a_{2}}\Big{)}-\frac{a^{\prime\prime}_{2}}{4}\,.$
(82)

This potential has no singularities at finite $x$. Asymptotically the term
proportional to $a_{1}^{2}$ will dominate for $x\rightarrow\pm\infty$ and thus
the potential $U(x)\rightarrow+\infty$. This then implies the following
important properties:

  * 1

The eigenvalues $\lambda_{n}$ are discrete, real and non-degenerate.

  * 2

There exists a lowest eigenvalue $\lambda_{0}$ (i.e. bounded from below).

  * 3

The only accumulation point is at infinity.

Asymptotic analysis already showed that the eigenvalues are discrete, but this
Sturm-Liouville analysis allows to conclude much more. Now it is
straightforward to see that there is a finite number of relevant operators
such that $\theta_{n}=4-\lambda_{n}\geq 0$. Indeed this is so because
$\lambda_{n}\rightarrow\infty$ as $n\rightarrow\infty$ and because there
exists a lowest eigenvalue $\lambda_{0}$.

But these results should be accepted with caution. Recall that to obtain them
some severe approximations were used, such as the single metric approximation
and the truncation to the function $f(R)$. One way to judge the validity of
the results is to check the extent to which they are scheme independent
(universal), in particular independent of the choice of cutoff. It turns out
that the critical exponents $\theta_{n}$ can be solved for analytically, again
by using asymptotic analysis, and this gives a precise way to answer the
question of scheme dependence in this regime.

From (60) and (64), the reader can see that the leading contribution to
$a_{2}$ takes the following form on both sphere and hyperboloid:

$a_{2}(R)=\frac{1}{G^{2}(R)}\,\mathrm{e}^{-2F(R)}$ (83)

where $F(R)$ is positive and proportional to $|R|^{b}$ and $G(R)$ goes like a
power of $R$. They therefore satisfy the conditions required to use the trick
(61) on the equation (79) defining $x$. Then asymptotically

$x=\frac{G(R)}{F^{\prime}(R)}\,\mathrm{e}^{F(R)}+...$ (84)

where the ellipsis stand for multiplicative subleading terms. Alternatively
this can be seen by differentiating (79) and (84) with respect to $R$. The
potential can then be approximated to leading order as

$U(x)=\frac{a_{1}^{2}}{4a_{2}}=\frac{R^{2}}{a_{2}(R)}=\big{[}RF^{\prime}(R)\big{]}^{2}x^{2}.$
(85)

Evidently $RF^{\prime}(R)=bF(R)$ and thus, taking logs of (84),

$U(x)=(bx\ln|x|)^{2}\bigg{\\{}1+O\bigg{(}\frac{\ln\ln|x|}{\ln|x|}\bigg{)}\bigg{\\}}\quad\quad
x\rightarrow\pm\infty\,,$ (86)

where in the equation above the order of the subleading correction is also
indicated. (The latter requires taking into account iterations of (61) and the
subleading corrections to $a_{2}$.) Using the WKB approximation one can then
find the critical exponents  for large $n$ [43]:

$\theta_{n}=-b\,(n\ln n)\bigg{\\{}1+O\bigg{(}\frac{\ln\ln n}{\ln
n}\bigg{)}\bigg{\\}}\quad\text{as}\quad n\to\infty\,.$ (87)

The result shows almost a linear dependence on $n$. This much is similar to
extensive numerical work done on large polynomial truncations of a third order
formulation up to $n\leq 70$ [17]. These authors find near-Gaussian scaling
dimension. They use an adaptive cutoff so there is no direct comparison, and
they use the optimised profile (26) with no free parameters in the cutoff, so
universality is not tested in this way. Indeed the scaling dimension should be
universal. The leading behaviour of this expression is independent of all
parameters in the chosen general family of cutoffs, except one, namely the
parameter $b$ in (29). Explicitly, it is independent of $a$ in (29), and of
all the $c_{\phi}$ and $\alpha_{i}$. Unfortunately the dependence on $b$ still
amounts to strong dependence.

Actually this remaining dependence is an artefact of the single-metric
approximation [2].333More generally, single-field approximations are a known
source of artefacts [56]. We have seen that it comes from the $R^{b}$
dependence of $F(R)$ in (83), equivalently (60) and (64). This in turn arises
from the cutoff dependence in eqn. (56) and in particular the cutoff profile’s
dependence on $R$ (through in fact the lowest eigenvalue). To see that the
dependence in (87) is an artefact of the single-metric approximation, imagine
for the moment that the single-metric approximation was not made and yet
somehow the initial ansatz (2) still made sense. (In reality such a simple
ansatz would no longer be possible because diffeomorphism invariance is
replaced by BRST invariance for the quantum fields and furthermore it is badly
broken, but let us overlook that for the moment.) Now the curvature in it is
the full quantum curvature $\hat{R}$, due to the full quantum metric
$\hat{g}_{\mu\nu}$ in (6). The trace and the cutoff in (56) come from summing
over modes on the background manifold in (3) so they depend on the background
curvature $R$. The Hessian in (3) will result in differentiating $f(\hat{R})$
with respect to the fluctuation field $h_{\mu\nu}$ or equivalently
differentiating with respect to $\hat{g}_{\mu\nu}$. Thus ultimately the
eigenoperator perturbation equation (55) would take the form:

$-a_{2}(R,\hat{R})\,\delta f^{\prime\prime}(\hat{R})+a_{1}(R,\hat{R})\,\delta
f^{\prime}(\hat{R})+a_{0}(R,\hat{R})\,\delta f(\hat{R})=4\,\delta f(\hat{R})$
(88)

with in particular:

$a_{2}=\frac{144c_{h}}{V}\text{Tr}\Bigg{[}\frac{\Delta_{0}^{2}(2r(\Delta_{0}+\alpha_{0}R)-(\Delta_{0}+\alpha_{0}R)r^{\prime}(\Delta_{0}+\alpha_{0}R))}{(9f^{\prime\prime}(\hat{R})\Delta_{0}^{2}+3f^{\prime}(\hat{R})\Delta_{0}+E(\hat{R})+16c_{h}r(\Delta_{0}+\alpha_{0}R))^{2}}\Bigg{]}\,.$
(89)

In deriving (87) one is interested in the large $\hat{R}$ dependence of (88).
This depends on the large $\hat{R}$ dependence of the fixed point functional
$f(\hat{R})$, and this feeds in to the coefficients $a_{i}(R,\hat{R})$. But
there is no $\exp(-a\hat{R}^{b})$ dependence because the cutoff profile $r$
depends only on the background curvature $R$, either directly or through the
Laplacians whose eigenvalues only depend on the background manifold.

## References

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