11institutetext: Frank Saueressig 22institutetext: Institute for Mathematics,
Astrophysics and Particle Physics, Radboud University Nijmegen, Heyendaalseweg
135, 6525 AJ Nijmegen, The Netherlands
22email<EMAIL_ADDRESS>
# The Functional Renormalization Group
in Quantum Gravity††thanks:  Invited chapter for the ”Handbook of Quantum
Gravity” (Eds. C. Bambi, L. Modesto and I.L. Shapiro, Springer Singapore,
expected in 2023)

Frank Saueressig

###### Abstract

The gravitational asymptotic safety program envisions a high-energy completion
of the gravitational interactions by an interacting renormalization group
fixed point, the Reuter fixed point. The primary tool for investigating this
scenario are functional renormalization group equations, foremost the
Wetterich equation. This equation implements the idea of the Wilsonian
renormalization group by integrating out quantum fluctuations shell-by-shell
in momentum space and gives access to the theory’s renormalization group flow
beyond the realm of perturbation theory. This chapter gives a pedagogical
introduction to the gravitational asymptotic safety program with a specific
focus on clarifying conceptual points which led to confusion in the past. We
provide a step-by-step introduction to the Wetterich equation and its most
commonly used non-perturbative approximations. This exposition also introduces
recent developments including the minimal essential scheme and $N$-type
cutoffs. The use of the Wetterich equation in explicit computations is
illustrated within the Einstein-Hilbert truncation which constitutes the
simplest non-perturbative approximation of the gravitational renormalization
group flow. We conclude with a brief summary and comments on recent
developments originating from other quantum gravity programs.

## Keywords

Quantum gravity, asymptotic safety, renormalization group, Wetterich equation,
Reuter fixed point, Einstein-Hilbert truncation, phase diagram

##  1 Introduction

Our theoretical understanding of nature rests on two pillars. The electroweak
and strong force and their interactions with the elementary particles are
described by the standard model of particle physics. This theory is formulated
as a relativistic quantum field theory in Minkowski space. The description of
gravity is provided by general relativity, a classical field theory which
encodes the gravitational interactions in the dynamics of spacetime.
Conceptually, these theories are on very different footing and the
construction of a framework unifying gravity with the laws of quantum
mechanics is one of the key open questions in theoretical high-energy physics
to date.

An important insight along these lines is that the quantization techniques
successful in the case of the standard model of particle physics do not extend
to gravity in a straightforward way: the perturbative quantization of general
relativity leads to a perturbatively non-renormalizable quantum field theory
with new infinities appearing at every order in perturbation theory
'tHooft:1974bx ; Goroff:1985sz ; Goroff:1985th ; vandeVen:1991gw . This has
led to the advance of several physics principles which deviate from the
principles of continuum quantum field theory in more or less radical ways, see
Armas:2021yut ; Loll:2022ibq  for recent non-technical accounts.

The gravitational asymptotic safety program is one particular line of quantum
gravity research. The program is conservative in the sense that it strives for
a consistent and predictive theory of the gravitational interactions within
the framework of quantum field theory by seeking a non-perturbative high-
energy completion. Its core assumptions are that the gravitational degrees of
freedom are encoded in the spacetime metric also at trans-Planckian scales.
Moreover, the theory retains invariance under coordinate
transformations.111This assumption distinguishes the gravitational asymptotic
safety program from Hořava-Lifshitz gravity Horava:2009uw  where this symmetry
requirement is reduced to foliation-preserving diffeomorphisms, see
Rechenberger:2012dt  for a pedagogical discussion. The asymptotic safety
hypothesis then stipulates that

  1. 1.

these ingredients give rise to an interacting renormalization group fixed
point – called the Reuter fixed point.

  2. 2.

this fixed point controls the gravitational dynamics at trans-Planckian
scales.

From a phenomenological perspective one also requires that the renormalization
group flow emanating from the Reuter fixed point connects to a low-energy
regime where the dynamics matches the one of general relativity to a good
approximation.

We stress that the central element of the gravitational asymptotic safety
program – the existence of the Reuter fixed point coming with suitable
properties – is not an input. It must be established based on first-principle
computations. At the technical level, this requires tools applicable to
quantum field theory beyond the realm of perturbation theory. This is a highly
non-trivial endeavor. It took about 20 years from Weinberg’s first formulation
of the asymptotic safety hypothesis Weinberg:1976xy ; Weinberg:1980gg  to the
advent of renormalization group techniques which could be used to investigate
this hypothesis in a systematic way Reuter:1996cp .

Nowadays, there are two complementary computational approaches which naturally
lend themselves to the exploration of the asymptotic safety mechanism in the
context of gravity. Causal Dynamical Triangulations Ambjorn:2012jv ;
Loll:2019rdj  and Euclidean Dynamical Triangulations Ambjorn:2013eha ;
Coumbe:2014nea ; Rindlisbacher:2015ewa ; Bassler:2021pzt ; Asaduzzaman:2022kxz
use Monte Carlo techniques to investigate the phase space of quantum
geometries resulting from the gravitational path integral. In this setting,
the Reuter fixed point may manifest itself as a second-order phase transition
Ambjorn:2011cg  which allows to take the continuum limit in a controlled way.
Alternatively, the Reuter fixed point can manifest itself in (approximate)
solutions of the Wetterich equation Wetterich:1992yh ; Morris:1993qb .

This chapter will provide a basic introduction to the ideas underlying the
gravitational asymptotic safety program (Sec. 2) before introducing the
Wetterich equation Wetterich:1992yh ; Morris:1993qb  and its adaptation to
gravity Reuter:1996cp  as one of the main computational tools in the program
(Sec. 3). Sec. 4 illustrates how this tool is used in practical computations
by working out the example of the Einstein-Hilbert truncation in a modern,
background-independent way. Sec. 5 provides our conclusion and a brief
comments on renormalization group techniques implemented by other approaches
to quantum gravity.

We stress that the exposition in this chapter is necessarily incomplete since
it seeks to provide a concise introduction to the gravitational asymptotic
safety program and the functional renormalization group which is accessible to
a broader quantum gravity audience. For further details the reader is invited
to consult the text books Percacci:2017fkn ; Reuter:2019byg , lecture notes
Nagy:2012ef ; Reichert:2020mja , and general reviews Niedermaier:2006wt ;
Codello:2008vh ; Reuter:2012id . General introductions to the functional
renormalization group are provided in Berges:2000ew ; Gies:2006wv ;
Pawlowski:2005xe ; Dupuis:2020fhh  and there are topical reviews focusing on
asymptotic safety in the presence of matter fields Eichhorn:2018yfc , the
fluctuation approach to asymptotic safety Pawlowski:2020qer , and its
applications in the context of black holes Koch:2014cqa  and cosmology
Bonanno:2017pkg . Open issues have been discussed in the community report
Bonanno:2020bil .

##  2 The Asymptotic Safety Mechanism

The insight that gravity could be asymptotically safe dates back to the
seminal work of Weinberg Weinberg:1976xy ; Weinberg:1980gg . This initial
proposal advocated asymptotic safety as a mechanism which renders physical
scattering amplitudes finite (but non-vanishing) at energy scales exceeding
the Planck scale. Motivated by computations showing that gravity in
$d=2+\epsilon$ spacetime dimensions possesses a non-trivial renormalization
group (RG) fixed point Gastmans:1977ad ; Christensen:1978sc , it was suggested
that this family of fixed points admits an analytic continuation up to $d=4$
where the corresponding fixed point should provide the high-energy completion
of the gravitational interactions. The link between scattering amplitudes
being finite and the RG fixed point builds on the insight that at such a fixed
point all dimensionless quantities remain finite. If the fixed point controls
the high-energy behavior, this property will also carry over to scattering
amplitudes, which by themselves are dimensionless objects. This heuristic
argument implies that it is not necessary that all dimensionless couplings
remain finite. It suffices that the subset of couplings entering into physical
observables (called essential couplings) attain their fixed-point values, as
this is sufficient to ensure that the observables are well-behaved. A more
detailed analysis of this scenario within the amplitude approach to asymptotic
safety Draper:2020bop ; Draper:2020knh ; Knorr:2021iwv  revealed that there
must be intricate relations between couplings and propagators. Most likely,
these arise as a consequence of quantum scale symmetry realized at the fixed
point Wetterich:2019qzx .

The starting point for developing the idea of Asymptotic Safety is the
functional integral over all Euclidean metrics,

$Z=\int\mathcal{D}h\,e^{-S[h]}\,,$ (1)

which would allow to determine all physical quantities of interest. In this
respect, Asymptotic Safety shares the same starting point as Monte Carlo
approaches to quantum gravity, foremost the Causal Dynamical Triangulation
Ambjorn:2012jv ; Loll:2019rdj  and Euclidean Dynamical Triagulation
Ambjorn:2013eha ; Coumbe:2014nea ; Rindlisbacher:2015ewa ; Bassler:2021pzt ;
Asaduzzaman:2022kxz  programs as well as Quantum Regge Calculus Rocek:1981ama
; Hamber:2009mt .

The functional renormalization group then recasts the problem of performing
this functional integral into the problem of solving a functional differential
equation, the Wetterich equation for the effective average action $\Gamma_{k}$
Wetterich:1992yh ; Morris:1993qb ; Reuter:1993kw ; Reuter:1996cp  (derived in
Sec. 3):

$k\partial_{k}\Gamma_{k}=\frac{1}{2}{\rm
Tr}\left[\left(\Gamma^{(2)}_{k}+\mathcal{R}_{k}\right)^{-1}k\partial_{k}\mathcal{R}_{k}\right]\,.$
(2)

Here $k$ is the coarse-graining scale and the trace contains an integration
over loop momenta. The Wetterich equation implements the Wilsonian picture of
renormalization in the following way: The regulator $\mathcal{R}_{k}$
appearing on the right-hand side separates the fluctuations into low- and
high-momentum modes with respect to $k$. The change of $\Gamma_{k}$ is then
governed by integrating out quantum fluctuations with momenta $p^{2}\approx
k^{2}$. In this way, one arrives at a formulation that is much better behaved
as the initial problem of solving the functional integral (1) in one stroke.

By construction, the propagators and vertices in the effective average action
$\Gamma_{k}$ include the quantum corrections due to the high-momentum
fluctuations. In this sense, it provides an effective description of physics
at length scales $l\sim k^{-1}$. This makes $\Gamma_{k}$ a quite complicated
object. Its natural habitat is the theory space $\mathcal{T}$. By definition,
this space consists of all action functionals $A[\cdot]$ which can be
constructed from the field content of the theory and meets its symmetry
requirements. In the context of gravity, where the field content is given by
(Euclidean) spacetime metrics $g_{\mu\nu}$, prototypical examples for these
building blocks include the terms appearing in the Einstein-Hilbert action,

$\mathcal{O}_{1}=\int d^{d}x\sqrt{g}\,,\qquad\mathcal{O}_{2}=\int
d^{d}x\sqrt{g}R\,,$ (3)

where $\sqrt{g}\equiv\sqrt{\det(g)}$ and $R$ is the Ricci scalar constructed
from $g_{\mu\nu}$ (also see Table 1 for further examples). Given a basis
$\\{\mathcal{O}_{i}\\}$ for these monomials, the effective average action can
be expanded in this basis

$\Gamma_{k}=\sum_{i}\,\bar{u}^{i}(k)\,\mathcal{O}_{i}\,.$ (4)

The dependence on the coarse-graining scale is captured by the dimensionful
couplings $\bar{u}^{i}(k)$. For the purpose of studying RG flows it is useful
to trade these dimensionful couplings with their dimensionless counterparts
obtained by rescaling with $k$,

$u^{i}(k)\equiv\bar{u}^{i}(k)\,k^{-d_{i}}\,,$ (5)

where $d_{i}\equiv[\bar{u}^{i}]$ is the mass-dimension of the coupling. The
couplings $u^{i}$ then serve as coordinates on $\mathcal{T}$.

Evaluating (2) for the expansion (4) gives the component form of the
functional renormalization group equation

$k\partial_{k}\,u^{i}(k)=\beta^{i}(\\{u^{j}\\})\,.$ (6)

The beta functions $\beta^{i}(\\{u^{j}\\})$ capture the dependence of the
dimensionless couplings on the coarse-graining scale. Dimensional analysis
entails that the functions $\beta^{i}(\\{u^{j}\\})$ are independent of $k$,
since this is the only dimensionful object in the construction. Thus Eq. (6)
constitutes an infinite-dimensional system of coupled, autonomous, first order
differential equations. Its solutions are called RG trajectories. The problem
of performing the functional integral (1) is then translated into finding
globally well-defined RG trajectories

$k\rightarrow\Gamma_{k}\,,\qquad k\in[0,\infty]\,,$ (7)

which exist for all values of the coarse-graining scale $k$.

Figure 1:  Illustration of theory space and its structures: by definition, the
theory space contains all action functionals $A[\cdot]$ which can be
constructed from a given field content and obey the desired symmetries. The
theory space comes with a vector field, the beta functions $\beta$. The
integral curves of this vector field (RG trajectories) are exemplified by the
black solid curve. The example emanates from a fixed point (red dot) with one
UV-attractive (Re$\theta_{I}>0$) and one UV-repulsive (Re$\theta_{I}<0$)
eigendirection. The endpoint of the RG trajectory at $k=0$ coincides with the
effective action $\Gamma$. Conventionally, all arrows point towards a lower
coarse-graining scale, i.e., in the direction of integrating out fluctuation
modes.

By definition, RG fixed points $\\{u^{j}_{*}\\}$ are stationary points of the
system (6), satisfying

$\beta^{i}(\\{u^{j}_{*}\\})=0\,,\qquad\forall\,i\,.$ (8)

As a consequence, it takes infinite amount of “RG-time” for an RG trajectory
to actually reach the fixed point. In this way fixed points can provide a
well-defined limit $k\rightarrow\infty$ in which all dimensionless couplings
$u^{i}(k)\rightarrow u^{i}_{*}$ remain finite. Thus, fixed points are natural
candidates for providing a well-defined high-energy completion of a theory. It
is this concept that underlies the Wilsonian picture of renormalization.

At this point it is interesting to inquire about the conditions for an RG
trajectory being dragged into a fixed point as $k\rightarrow\infty$. This
question is closely related to the predictive power of the construction. In
the vicinity of a fixed point $\\{u^{i}_{*}\\}$, the properties of the RG flow
can be studied by linearizing the system (6),

$k\partial_{k}u^{i}(k)=\sum_{j}B^{i}{}_{j}\,\left(u^{j}(k)-u_{*}^{j}\right)+O(u^{2})\,.$
(9)

Here

$B^{i}{}_{j}\equiv\left.\frac{\partial}{\partial
u^{j}}\beta^{i}\right|_{u=u^{*}}$ (10)

is the stability matrix associated with the fixed point. The solutions of (9)
are readily given in terms of the right-eigenvectors $V_{I}$ and stability
coefficients $\theta_{I}$ of $B$,

$\sum_{j}B^{i}{}_{j}\,V^{j}_{I}=-\theta_{I}\,V_{I}^{i}\,,\qquad\forall\,I\,,$
(11)

and take the form

$u^{i}(k)=u^{i}_{*}+\sum_{J}C_{J}\,V_{J}^{i}\left(\frac{k_{0}}{k}\right)^{\theta_{J}}\,.$
(12)

Here $C_{J}$ are constants of integration and $k_{0}$ denotes an arbitrary
reference scale.

Inspecting (12) reveals that eigendirections with Re$(\theta_{I})>0$ are
attracted by the fixed point as $k\rightarrow\infty$ while the ones with
Re$(\theta_{I})<0$ are repulsive in this limit. The corresponding scaling
operators are called “UV-relevant” and “UV-irrelevant”, respectively. This
suggests splitting the set $\\{C_{I}\\}$ according to

$\\{C_{I}^{\rm relevant}\\}=\\{C_{I}\,|\,{\rm
Re}(\theta_{I})>0\\}\,,\qquad\\{C_{I}^{\rm irrelevant}\\}=\\{C_{I}\,|\,{\rm
Re}(\theta_{I})<0\\}\,.$ (13)

The case Re$\theta_{I}$=0 corresponds to a marginal direction. Determining
whether this direction is UV-attractive or UV-repulsive requires going beyond
the linear approximation (12) and will not be discussed in detail here.

The condition that the fixed point controls the UV-behavior of the RG-
trajectory then enforces $C_{I}^{\rm irrelevant}=0$, for all $I$. The
solutions meeting this condition span the UV-critical hypersurface of the
fixed point. The $\\{C_{I}^{\rm relevant}\\}$ are the free parameters of the
construction and label the solutions within this hypersurface. Their value is
unconstrained by demanding a well-defined UV-completion and must be determined
by other theoretical considerations or experimental input. This discussion
also shows that fixed points with a lower-dimensional UV-critical hypersurface
have a higher predictive power.

Up to this point, our discussion of a high-energy completion referred to a
generic renormalization fixed point. It is then customary to distinguish among
a Gaussian fixed point (GFP) and a non-Gaussian fixed point (NGFP). The
definition of the former is that the critical exponents of its stability
matrix agree with the canonical mass-dimension of the corresponding coupling
$\theta_{I}=d_{I}$. This signals that the underlying theory is the free
theory. At a NGFP, the stability coefficients receive quantum corrections,

$\theta_{I}=d_{I}+\text{quantum corrections}\,.$ (14)

The latter indicate that the theory linked to the fixed point is interacting.
Notably, this definition of a Gaussian and non-Gaussian fixed point is not
based on the values $\\{u_{*}^{i}\\}$. Since the spectrum of the stability
matrix is invariant under a redefinition
$u^{i}\mapsto\tilde{u}^{i}(\\{u^{j}\\})$, this characterization is independent
of a specific choice of “coordinate system” on $\mathcal{T}$. An important
subset of NGFPs are “almost-Gaussian” NGFPs. In this case the quantum
corrections in (14) are weak in the sense that the $\theta_{I}$ are dominated
by their classical part. This implies that classical power-counting is still a
valid guiding principle for determining whether a scaling operator is relevant
or irrelevant. Beyond the class of “almost Gaussian” NGFPs, there could also
be fixed points where the critical exponents are dominated by quantum effects.
The systematic investigation of this possibility is beyond the scope of most
of current searches for RG fixed points based on functional renormalization
group equations though.222Some insights on potential stability patterns
associated with such fixed points have recently be discussed based on the
composite operator equation Houthoff:2020zqy ; Kurov:2020csd , indicating that
studying such fixed points requires approximations at a significant level of
complexity as well as dedicated search strategies. Depending on whether the
high-energy completion is provided by a GFP or a NGFP, the theory is termed
“asymptotically free” or “asymptotically safe”. A prototypical example of the
former case is Quantum Chromodynamics while the latter case is realized by
gravity in $d=2+\epsilon$ spacetime dimensions Gastmans:1977ad ;
Christensen:1978sc .

We conclude this section with two clarifications. For a globally well-defined
RG trajectory, the solutions (7) interpolate between the microscopic dynamics
determined by the RG fixed point for $k\rightarrow\infty$ and the standard
effective action $\lim_{k\rightarrow 0}\Gamma_{k}=\Gamma$. _All physics should
then be extracted from $\Gamma$ using its quantum corrected propagators and
vertices._ Similarly to (4), $\Gamma$ can be expanded in a basis of the theory
space

$\Gamma=\sum_{i}\,\bar{u}^{i}_{\rm eff}\,\mathcal{O}_{i}\,,$ (15)

with the relation between the couplings being $\bar{u}^{i}_{\rm
eff}=\lim_{k\rightarrow 0}\bar{u}^{i}(k)$. This expansion is similar to the
one encountered in effective field theory where the $\mathcal{O}_{i}$ are
organized according to their canonical mass-dimension and the sum is truncated
at a given order. The key difference to the effective field theory approach is
that the RG flow determines the effective couplings in terms of the free
coefficients $\\{C_{I}^{\rm relevant}\\}$:

$\bar{u}^{i}_{\rm eff}=\bar{u}^{i}_{\rm eff}(C_{I}^{\rm relevant})\,.$ (16)

Provided that there are more couplings $\bar{u}^{i}_{\rm eff}$ than free
parameters $C_{I}^{\rm relevant}$, the high-energy completion induces a
(potentially infinite number of) relations between the effective couplings.
These provide predictions which can be confronted with theoretical consistency
requirements and experimental data. On this basis one can deduce whether a
given RG fixed point leads to low-energy physics compatible with nature. This
also allows to falsify the construction, provided that the properties of the
fixed point and its UV-critical surface are known at a sufficient level of
detail.

We also stress that the dependence of couplings on the coarse-graining scale
$k$ should not be identified with the running of a coupling with respect to a
physical energy scale, see Donoghue:2019clr ; Bonanno:2020bil  for instructive
examples. Generically, the couplings appearing in (16) are not constant but
come in the form of form factors depending on the momenta of the fields in a
non-trivial way. In the simplest case (cf. (72)) this dependence contains a
single momentum scale

$\bar{u}^{i}_{\rm eff}\rightarrow\bar{u}^{i}_{\rm eff}(p^{2})\,.$ (17)

In practice, the value of the coupling is then measured at a fixed momentum
scale $\mu^{2}$. The non-trivial $p$-dependence then induces the “running” of
the coupling with respect to its value determined at the reference scale. In
this simplest case, this is the logarithmic running of a dimensionless
coupling seen in perturbation theory, but the momentum dependence can be
significantly more involved than that.

##  3 The Functional Renormalization Group

The basic idea of a functional renormalization group equation (FRGE) is to
recast the functional integral over quantum fluctuations in terms of a
functional differential equation. The FRGE implements Wilson’s modern
viewpoint on renormalization Wilson:1973jj : in contrast to a perturbative
approach based on evaluating Feynman diagrams, quantum fluctuations are not
integrated over in one stroke. Instead they are integrated out “shell-by-
shell” in momentum space starting with the most energetic ones. This leads to
a one-parameter family of effective actions $\Gamma_{k}$ whose propagators and
vertices already contain the quantum corrections from fluctuations with
momenta $p^{2}\gtrsim k^{2}$. The textbook effective action $\Gamma$ is
recovered in the limit where all fluctuations are integrated out,
$\Gamma=\lim_{k\rightarrow 0}\Gamma_{k}$.

The FRGE most frequently used in hands-on computations is the Wetterich
equation Wetterich:1992yh ; Morris:1993qb ; Reuter:1996ub ; Reuter:1996cp .
This section reviews its construction for scalar fields (Sec. 3.1) before
extending the formalism to gravity (Sec. 3.2). The most common non-
perturbative approximation techniques to this equation are introduced in Sec.
3.3 and important extensions giving structural insights to the gravitational
renormalization group flow are summarized in Sec. 3.4.

###  3.1 The Wetterich equation for scalar field theory

The Wetterich equation is a universal tool for studying the RG flow of
theories built from essentially any field content Dupuis:2020fhh . In order to
introduce this tool with the absolute minimum of technicalities, we first
focus on a real scalar field $\varphi$ living on a $d$-dimensional Euclidean
spacetime $(\mathbb{R}^{d},\delta_{\mu\nu})$. For pedagogical reasons, we
first review the construction of the effective action in this setting before
introducing the effective average action and its FRGE.

We start from the generating functional of correlation functions (path
integral)

$Z[J]\equiv\frac{1}{N}\int\mathcal{D}\varphi\,\exp\left\\{-S[\varphi]+\int
d^{d}x\,J(x)\varphi(x)\right\\}\,.$ (18)

Here $N\equiv\int\mathcal{D}\varphi\,\exp\left\\{-S[\varphi]\right\\}$ is a
normalization factor and $J(x)$ a source coupling to the quantum field. The
dynamics of the field is governed by the bare action $S[\varphi]$ which is
kept arbitrary at this point. Generically, this generating functional diverges
and we implicitly assume that it has been suitably regularized by including an
UV-cutoff. Eq. (18) allows to construct expectation values of operators
$\mathcal{O}$

$\left\langle\mathcal{O}[\varphi]\right\rangle\equiv\frac{1}{N}\int\mathcal{D}\varphi\,\mathcal{O}[\varphi]\,\exp\left\\{-S[\varphi]\right\\}\,.$
(19)

In particular, expectation values of operators polynomial in $\varphi$ can be
obtained by taking functional derivatives with respect to the source and
subsequently setting $J$ to zero

$\langle\varphi(x_{1})\cdots\varphi(x_{n})\rangle=\left.\frac{\delta^{n}Z[J]}{\delta
J(x_{1})\cdots\delta J(x_{n})}\right|_{J=0}\,.$ (20)

Here, the normalization factors are chosen such that
$\langle\mathbb{1}\rangle=1$. Based on the path integral (18), one obtains the
functional $W[J]$ generating all connected Green’s functions by setting

$Z[J]\equiv e^{W[J]}\,.$ (21)

We then introduce the mean field $\phi(x)$ as the expectation value of
$\varphi(x)$:

$\phi(x)=\langle\varphi(x)\rangle=\frac{\delta W[J]}{\delta J(x)}\,.$ (22)

Finally, one constructs the effective action $\Gamma[\phi]$ as the Legendre
transform of $W[J]$. If the relation (22) can be solved for the source, giving
$J[\phi]$, it takes the form333In the general case, the effective action is
obtained as the Legendre-Fenchel transform $\Gamma[\phi]=\sup_{J(x)}\left(\int
d^{d}x\,J[\phi](x)\phi(x)-W[J[\phi]]\right)$. In the sequel, formulas are
understood to include the supremum if needed.

$\Gamma[\phi]=\int d^{d}x\,J[\phi](x)\phi(x)-W[J[\phi]]\,.$ (23)

The fact that $W[J]$ and $\Gamma[\phi]$ are related by a Legendre transform
implies that

$\int d^{d}y\,\frac{\delta^{2}W[J]}{\delta J(x_{1})\delta
J(y)}\,\frac{\delta^{2}\Gamma[\phi]}{\delta\phi(y)\delta\phi(x_{2})}=\delta^{d}(x_{1}-x_{2})\,.$
(24)

The effective action provides the equation of motion for the mean field in the
presence of a source,

$\frac{\delta\Gamma[\phi]}{\delta\phi(x)}=J(x)\,.$ (25)

Higher order functional derivatives generate the one-particle irreducible
($1$PI) $n$-point functions

$\Gamma^{(n)}[\phi]\equiv\frac{\delta^{n}\Gamma[\phi]}{\delta\phi(x_{1})\cdots\delta\phi(x_{n})}=\langle\varphi(x_{1})\cdots\varphi(x_{n})\rangle_{1{\rm
PI}}\,.$ (26)

Eq. (24) then entails that the second functional derivative of $\Gamma[\phi]$
encodes the quantum corrected propagator

$\left(\Gamma^{(2)}(x_{1},x_{2})\right)^{-1}=W^{(2)}(x_{1},x_{2})=G(x_{1},x_{2})\,.$
(27)

Scattering processes are described by tree-level Feynman diagrams constructed
from the propagators and vertices extracted from $\Gamma[\phi]$. In this
sense, the effective action is the quantum analog of the classical action,
since it encodes the quantum physics at tree level. Determining $\Gamma[\phi]$
is therefore often considered as equivalent to solving the quantum theory.

The construction of the effective average action $\Gamma_{k}[\phi]$ proceeds
along very similar lines. The key modification occurs at the level of the
generating functional (18) which is supplemented by an IR-regulator

$\Delta S_{k}[\varphi]=\frac{1}{2}\int
d^{d}x\,\varphi(x)R_{k}(-\partial^{2})\varphi(x)\,.$ (28)

The purpose of this extra ingredient is to provide a $k$-dependent mass-term
for quantum fluctuations with moments $p^{2}\ll k^{2}$. In the simplest case,
this is implemented by requiring that the regulator $R_{k}(p^{2})$ satisfies

$R_{k}(p^{2})\approx\left\\{\begin{array}[]{ll}k^{2}&\quad\text{for}\;p^{2}\ll
k^{2}\,,\\\ 0&\quad\text{for}\;p^{2}\gg k^{2}\,.\end{array}\right.$ (29)

Examples of regulators used in practical computations include the (smooth)
exponential cutoff,

$R_{k}(p^{2})=p^{2}\left(\exp(p^{2}/k^{2})-1\right)^{-1}\,,$ (30)

and Litim-type regulators,

$R_{k}(p^{2})=(k^{2}-p^{2})\Theta(1-p^{2}/k^{2})\,,$ (31)

where $\Theta(x)$ is the Heaviside step function. Adding (28) to the weight in
the generating functional (18) induces a dependence on the scale $k$

$Z_{k}[J]=\frac{1}{N}\int\mathcal{D}\varphi\,\exp\left\\{-S[\varphi]-\Delta
S_{k}[\varphi]+\int d^{d}x\,J(x)\varphi(x)\right\\}\,.$ (32)

The effect is that the contribution of modes with $p^{2}\ll k^{2}$ to the
generating functional becomes suppressed while the modes with $p^{2}\gg k^{2}$
are integrated out in the usual way. Thus $k$ acquires a natural
interpretation as a coarse-graining scale, marking the scale up to which
microscopic quantum fluctuations are included in the generating functional.

Following the steps leading to the effective action, we then define the (now
$k$-dependent) generating functional for connected Green’s functions
$W_{k}[J]$ by

$Z_{k}[J]=\exp[W_{k}[J]]\,.$ (33)

By definition, the effective average action is then given by a modified
Legendre transform of $W_{k}[J]$:

$\Gamma_{k}[\phi]\equiv\int d^{d}x\,J[\phi](x)\phi(x)-W_{k}[J]-\Delta
S_{k}[\phi]\,.$ (34)

For $k=0$ the IR regulator in the definition of $W_{k}[J]$ as well as $\Delta
S_{k}[\phi]$ vanish and (34) agrees with the definition of the effective
action (23):

$\lim_{k\rightarrow 0}\Gamma_{k}[\phi]=\Gamma[\phi]\,.$ (35)

The key virtue of the effective average action is that its $k$-dependence is
governed by a functional renormalization group equation, the Wetterich
equation. This equation is formally exact in the sense that no approximations
are made in its derivation. The construction of the Wetterich equation then
proceeds along the following lines. We start by introducing the RG time
$t\equiv\ln k/k_{0}$, with $k_{0}$ being an arbitrary reference scale, so that
$\partial_{t}=k\partial_{k}$. We then consider the auxiliary generating
functional

$\tilde{\Gamma}_{k}[\phi]\equiv\int d^{d}x\,J[\phi](x)\,\phi(x)-W_{k}[J]\,.$
(36)

Taking a partial derivative of this definition with respect to RG time yields

$\begin{split}\partial_{t}\tilde{\Gamma}_{k}[\phi]=&-\partial_{t}W_{k}[J]=\frac{1}{2}\int
d^{d}x\int
d^{d}y\;\langle\varphi(x)\varphi(y)\rangle\;\partial_{t}R_{k}(x,y)\,.\end{split}$
(37)

Here we have used that $\partial_{t}W_{k}[J]=\partial_{t}\ln Z_{k}[J]$ with

$\begin{split}\partial_{t}\ln
Z_{k}[J]&\,=-\frac{1}{2Z_{k}}\int\mathcal{D}\varphi\int d^{d}x\int
d^{d}y\,\varphi(x)\,\partial_{t}R_{k}(x,y)\,\varphi(y)\,\times\\\
&\qquad\times\exp\left\\{-S[\varphi]-\Delta S_{k}[\varphi]+\int
d^{d}x\,J(x)\varphi(x)\right\\}\\\ &\,=-\frac{1}{2}\int d^{d}x\int
d^{d}y\,\,\langle\varphi(x)\varphi(y)\rangle\,\partial_{t}R_{k}(x,y)\,\,,\end{split}$
(38)

in the second step. We then introduce the ($k$-dependent) mean field

$\phi(x)=\langle\varphi(x)\rangle=\frac{\delta W_{k}[J]}{\delta J(x)}\,,$ (39)

together with the two-point functions

$\langle\varphi(x)\varphi(y)\rangle_{\rm
c}\equiv\frac{\delta^{2}W_{k}[J]}{\delta J(x)\delta
J(y)}\,,\qquad\tilde{\Gamma}_{k}^{(2)}(x,y)\equiv\frac{\delta^{2}\tilde{\Gamma}_{k}[\phi]}{\delta\phi(x)\delta\phi(y)}\,.$
(40)

Since $\tilde{\Gamma}_{k}[\phi]$ and $W_{k}[J]$ are again related by a
Legendre transform, these functionals are again each others inverse, cf. Eq.
(24). This allows to express the two-point function appearing in the relation
(37) in terms of $\tilde{\Gamma}_{k}^{(2)}(x,y)$

$\begin{split}\langle\varphi(x)\varphi(y)\rangle=&\,\langle\varphi(x)\varphi(y)\rangle_{\rm
c}+\langle\varphi(x)\rangle\,\langle\varphi(y)\rangle\,\\\
=&\left(\tilde{\Gamma}_{k}^{(2)}(x,y)\right)^{-1}+\phi(x)\phi(y)\,.\end{split}$
(41)

Here we used the definition of the (now $k$-dependent) mean field when
recasting the last term. Substituting this relation into (37) then yields

$\partial_{t}\tilde{\Gamma}_{k}=\frac{1}{2}\int d^{d}x\int
d^{d}y\,\left[\left(\tilde{\Gamma}_{k}^{(2)}(x,y)\right)^{-1}\,\partial_{t}R_{k}(x,y)\right]+\partial_{t}\Delta
S_{k}[\phi]\,.$ (42)

Bringing the second term to the left-hand side and using that
$\Gamma_{k}=\tilde{\Gamma}_{k}-\Delta S_{k}$ allows to rewrite this equation
in terms of the effective average action

$\partial_{t}\Gamma_{k}[\phi]=\frac{1}{2}\int d^{d}x\int
d^{d}y\,\left[\,\left(\Gamma_{k}^{(2)}+R_{k}\right)^{-1}\,\partial_{t}R_{k}\,\right]\,.$
(43)

Noticing that the integrals on the right-hand side actually correspond to
taking the trace of the argument, we arrive at the Wetterich equation in its
iconic form

$\partial_{t}\Gamma_{k}=\frac{1}{2}{\rm
Tr}\left[\left(\Gamma_{k}^{(2)}+R_{k}\right)^{-1}\,\partial_{t}R_{k}\right]\,.$
(44)

The Wetterich equation exhibits several remarkable features arising from the
interplay of $R_{k}(p^{2})$ in the numerator and denominator of the trace
argument. In the propagator term $\left(\Gamma_{k}^{(2)}+R_{k}\right)^{-1}$,
the regulator provides a mass to the fluctuations, ensuring the absence of IR-
singularities as long as $k$ is finite. In the numerator, the condition
$R_{k}(p^{2})\rightarrow 0$ for $p^{2}\gg k^{2}$ entails that the trace
argument vanishes for high-momentum modes. As a consequence the right-hand
side is IR and UV-finite and any UV-regulator implicit in the definition of
the initial functional integral can be removed trivially.444In some practical
computations, as e.g. in the computation of spectral flows Braun:2022mgx , one
may want to resort to regulators $R_{k}(p^{2})$ where this fall-off property
in the UV does not hold. In this case, the flow equation must be supplemented
by additional counterterms absorbing the UV-divergences.

The regulator structure furthermore entails that the trace argument is peaked
at momenta $p^{2}\approx k^{2}$. Hence the flow of $\Gamma_{k}[\phi]$ is
driven by integrating out quantum fluctuations whose momenta are comparable to
the coarse-graining scale $k$. In this way the Wetterich equation implements
the Wilsonian picture of renormalization, integrating out quantum fluctuations
shell-by-shell in momentum space. Notably, Eq. (44) allows to start from any
initial condition $\Gamma_{\Lambda}$ and integrate its RG flow towards the
infrared. Thus the Wetterich equation does not require specifying a bare
action a priori. These are obtained as the fixed points of the RG flow through
the reconstruction problem Manrique:2008zw .

We also observe that the combination of propagator and regulator within the
trace induces a projective feature. Any $k$-independent rescaling of the
fluctuation field affects the regulator and propagator in the same way, so
that such rescalings drop out from the right-hand side of the equation. This
renders the flow equation invariant with respect to certain classes of field
redefinitions.

###  3.2 The Wetterich equation for gravity

In the previous section, we derived the Wetterich equation (44) for a real
scalar field. Its extension to gauge fields and fermions is conceptually
straightforward. In the context of gravity the construction faces two
conceptual obstacles though. Firstly, our understanding of classical gravity
based on general relativity indicates that gravitational interactions are
mediated through the curvature of spacetime. This implies that spacetime
itself becomes a dynamical and, in the context of the quantum theory, also
fluctuating object. Hence, the concept of a fixed, non-dynamical spacetime
providing the stage for the dynamics is lost at this point. This raises the
question about how to define the coarse-graining scale $k$. Secondly, gravity
shares some properties of a gauge theory. The Einstein-Hilbert action, for
example, is invariant under coordinate transformations which act on the metric
according to

$\delta
g_{\mu\nu}\equiv\mathcal{L}_{v}g_{\mu\nu}=v^{\rho}\partial_{\rho}g_{\mu\nu}+(\partial_{\mu}v^{\rho})g_{\rho\nu}+(\partial_{\nu}v^{\rho})g_{\rho\mu}\,.$
(45)

Here $\mathcal{L}_{v}$ denotes the Lie derivative along the generating vector
field $v^{\mu}$. In order to ensure that the generating functional $Z_{k}$
sums over physically inequivalent configurations only, one has to introduce a
suitable gauge-fixing condition. By construction, the gauge-fixing term breaks
the invariance under the transformations (45). As a consequence, the effective
(average) action may loose this symmetry, leading to a proliferation of
interaction monomials which could be generated along the RG flow.

Following the seminal work by Reuter Reuter:1996cp , both of these conceptual
difficulties can be overcome by resorting to the background field method. This
procedure splits the (Euclidean) quantum metric $g_{\mu\nu}$ into a generic
(but non-fluctuating) background metric $\bar{g}_{\mu\nu}$ and fluctuations
around this background $h_{\mu\nu}$. There is no requirement that the latter
are small. The decomposition can then be implemented either through a linear
or an exponential split (see Ohta:2016npm ; Ohta:2016jvw  for a detailed
discussion):

$g_{\mu\nu}=\bar{g}_{\mu\nu}+h_{\mu\nu}\,,\qquad
g_{\mu\nu}=\bar{g}_{\mu\alpha}\left(e^{h}\right)^{\alpha}{}_{\nu}\,.$ (46)

Here we follow the standard convention that indices are raised and lowered
with the background metric, i.e.,
$h^{\mu}{}_{\nu}=\bar{g}^{\mu\alpha}h_{\alpha\nu}$, etc. While these
decompositions agree to leading order in the fluctuation field, they actually
define different theories, since they do not cover the same space of quantum
fluctuations. Heuristically, this can be argued based on the observation that
the linear split allows for $g_{\mu\nu}$ and $\bar{g}_{\mu\nu}$ having
different signatures while in the exponential split this is not the case
Demmel:2015zfa . This is also confirmed by computing properties of the Reuter
fixed point in $d=2+\epsilon$ dimensions Nink:2015lmq . In the following, we
will adopt the linear split for simplicity.

The background metric then allows to quantize metric fluctuations along the
lines of quantum field theory in a curved spacetime. Moreover, it allows to
circumvent the conceptual difficulties discussed above as follows. Firstly, it
provides the basis for separating fluctuations into “high-” and “low-”momentum
modes relative to the coarse-graining scale in a purely geometric way. Taking
the background to be compact and introducing the Laplacian
$\Delta\equiv-\bar{g}^{\mu\nu}\bar{D}_{\mu}\bar{D}_{\nu}$ constructed from the
background metric, one can obtain the ordered set of eigenmodes

$\Delta h^{n}_{\mu\nu}=E_{n}\,h^{n}_{\mu\nu}\,,\qquad n=0,1,\cdots\,,$ (47)

with $E_{0}\leq E_{1}\leq E_{2}\leq\cdots$. Fluctuations with $E_{n}\lesssim
k^{2}$ are then considered “long-range” and are suppressed by the regulator
while “short-range” fluctuations characterized by $E_{n}\gtrsim k^{2}$ are
integrated out without suppression factor. Practically, this is achieved by
generalizing (28) to

$\Delta S_{k}[h;\bar{g}]=\frac{1}{2}\int
d^{d}x\sqrt{\bar{g}}\left[h_{\mu\nu}(x)\,\mathcal{R}_{k}^{\mu\nu\alpha\beta}(\Delta)\,h_{\alpha\beta}(x)\right]\,.$
(48)

The switch from $R_{k}$ to $\mathcal{R}_{k}$ anticipates that, in general, the
regulator is a matrix in field space carrying a non-trivial tensor structure.
Note that (48) is quadratic in the fluctuation field: specifically,
$\mathcal{R}_{k}^{\mu\nu\alpha\beta}(\Delta)$ is independent of the
fluctuation field and depend on $\bar{g}_{\mu\nu}$ only. This property is
essential in order to arrive at a FRGE of the form (44).

Secondly, the linear split allows to realize the transformation (45) in two
distinct ways. Quantum gauge transformations ($Q$) keep $\bar{g}_{\mu\nu}$
fixed and attribute the transformation of $g_{\mu\nu}$ to the fluctuation
field

$\delta^{Q}\bar{g}_{\mu\nu}=0\,,\qquad\delta^{Q}h_{\mu\nu}=\mathcal{L}_{v}(\bar{g}_{\mu\nu}+h_{\mu\nu})\,.$
(49)

It is this transformation that must be gauge-fixed. In addition, one can
define background gauge transformations ($\delta^{B}$) where each field
transforms as a tensor of the corresponding rank

$\delta^{B}\bar{g}_{\mu\nu}=\mathcal{L}_{v}\bar{g}_{\mu\nu}\,,\qquad\delta^{B}h_{\mu\nu}=\mathcal{L}_{v}h_{\mu\nu}\,.$
(50)

This transformation can be maintained as an auxiliary symmetry by resorting to
the class of background covariant gauges. Following the Faddeev-Popov
procedure, the gauge-fixing is implemented by supplementing the gravitational
action $S[g]$ by a gauge-fixing term

$S^{\rm gf}[h;\bar{g}]=\frac{1}{2\alpha}\int
d^{d}x\sqrt{\bar{g}}\,\bar{g}^{\mu\nu}F_{\mu}\,F_{\nu}\,.$ (51)

Here, $\alpha$ is a free parameter and the gauge-fixing condition
$F_{\mu}[h;\bar{g}]$ transforms as a rank-one tensor with respect to (50).

The gauge-fixing term is accompanied by the action for the Faddeev-Popov ghost
and anti-ghost fields $C^{\mu}$ and $\bar{C}_{\mu}$

$S^{\rm ghost}[h,\bar{C},C;\bar{g}]=-\sqrt{2}\int
d^{d}x\sqrt{\bar{g}}\,\bar{C}_{\mu}\,\bar{g}^{\mu\nu}\,\frac{\delta
F_{\nu}}{\delta
h_{\alpha\beta}}\,\mathcal{L}_{C}(\bar{g}_{\alpha\beta}+h_{\alpha\beta})\,.$
(52)

This action exponentiates the Faddeev-Popov determinant

$\det\mathcal{M}=\det\left[\frac{\delta F_{\mu}}{\delta
v^{\nu}}\right]=\int\mathcal{D}C^{\mu}\mathcal{D}\bar{C}_{\nu}\,e^{-\int\bar{C}\mathcal{M}C}\,.$
(53)

At this point we have all the ingredients to write down the analogue of the
generating functional (33) in the context of gravity

$\begin{split}\exp(W_{k}[J;\bar{g}])=\frac{1}{N}\int&\,\mathcal{D}h_{\alpha\beta}\mathcal{D}C^{\mu}\mathcal{D}\bar{C}_{\nu}\,\exp\Big{\\{}-S[\bar{g}+h]-S^{\rm
gf}[h;\bar{g}]\\\ &-S^{\rm ghost}[h,\bar{C},C;\bar{g}]-\Delta
S_{k}[h,\bar{C},C;\bar{g}]+S^{\rm source}\Big{\\}}\,.\end{split}$ (54)

Here $S[g]$ denotes a generic action built from the metric $g_{\mu\nu}$,
invariant under (45), $S^{\rm gf}[h;\bar{g}]$ and $S^{\rm
ghost}[h,\bar{C},C;\bar{g}]$ are the gauge-fixing and ghost actions given in
Eqs. (51) and (52), and $\Delta S_{k}[h,\bar{C},C;\bar{g}]$ is the IR
regulator (48) extended by a $k$-dependent mass term for the ghost fields.
Finally,

$S^{\rm source}=\int
d^{d}x\sqrt{\bar{g}}\left\\{t^{\mu\nu}h_{\mu\nu}+\bar{\sigma}_{\mu}C^{\mu}+\sigma^{\mu}\bar{C}_{\mu}\right\\}$
(55)

introduces sources for the quantum field, which we collectively label by
$J\equiv(t^{\mu\nu},\sigma^{\mu},\bar{\sigma}_{\mu})$.

The construction of the effective average action then proceeds analogously to
the scalar case. Taking functional derivatives of $W_{k}[J;\bar{g}]$ with
respect to the sources gives the expectation values of the fluctuation fields

$\langle h_{\mu\nu}\rangle=\frac{1}{\sqrt{\bar{g}}}\frac{\delta W_{k}}{\delta
t^{\mu\nu}}\,,\quad\langle\bar{C}_{\mu}\rangle=\frac{1}{\sqrt{\bar{g}}}\frac{\delta
W_{k}}{\delta\sigma^{\mu}}\,,\quad\langle
C^{\mu}\rangle=\frac{1}{\sqrt{\bar{g}}}\frac{\delta
W_{k}}{\bar{\sigma}_{\mu}}\,.$ (56)

In a slight abuse of notation we then use the same labels for the mean- and
quantum fields, identifying

$h_{\mu\nu}=\langle h_{\mu\nu}\rangle\,,\qquad C^{\mu}=\langle
C^{\mu}\rangle\,,\qquad\bar{C}_{\mu}=\langle\bar{C}_{\mu}\rangle\,,\qquad
g_{\mu\nu}=\langle\bar{g}_{\mu\nu}+h_{\mu\nu}\rangle\,.$ (57)

We then assume again that the field-source relations (56) can be solved for
the sources as functions of the mean field. The effective average action is
then again defined as the modified Legendre transform of $W_{k}$:

$\Gamma_{k}[\Phi;\bar{g}]=\int
d^{d}x\sqrt{\bar{g}}\left\\{t^{\mu\nu}h_{\mu\nu}+\bar{\sigma}_{\mu}C^{\mu}+\sigma^{\mu}\bar{C}_{\mu}\right\\}-W_{k}[J;\bar{g}]-\Delta
S_{k}[\Phi;\bar{g}]\,.$ (58)

Here we used $\Phi=(h,\bar{C}_{\mu},C^{\mu})$ to denote the collection of
expectation values.

The key property of the effective average action (58) is that its dependence
on the coarse-graining scale $k$ is again governed by a formally exact
functional renormalization equation taking the form (44). Its derivation
essentially follows the one for the scalar theory. Taking the derivative of
(58) with respect to the RG time $t$ and expressing the right-hand side in
terms of the Hessian of $\Gamma_{k}[\Phi;\bar{g}]$ one finds Reuter:1996cp

$\begin{split}\partial_{t}\Gamma_{k}[\Phi;\bar{g}]=&\,\frac{1}{2}{\rm
Tr}\left[\left(\Gamma_{k}^{(2)}+\mathcal{R}_{k}\right)^{-1}_{hh}\left(\partial_{t}\mathcal{R}_{k}\right)_{hh}\right]\\\
&-\frac{1}{2}{\rm
Tr}\left[\left\\{\left(\Gamma_{k}^{(2)}+\mathcal{R}_{k}\right)^{-1}_{\bar{C}C}-\left(\Gamma_{k}^{(2)}+\mathcal{R}_{k}\right)^{-1}_{C\bar{C}}\right\\}(\partial_{t}\mathcal{R}_{k})_{\bar{C}C}\right]\,.\end{split}$
(59)

Here the matrix elements constituting the Hessian of $\Gamma_{k}$ are defined
via

$\left(\Gamma_{k}^{(2)}\right)_{ij}(x,y)\equiv\frac{1}{\sqrt{\bar{g}(x)}\sqrt{\bar{g}(y)}}\,\frac{\delta^{2}\Gamma_{k}}{\delta\Phi^{i}(x)\delta\Phi^{j}(y)}\,.$
(60)

For the Grassmann-valued (anti-commuting) fields in the ghost sector, we adopt
the convention that matrix elements are defined in terms of left-derivatives,
i.e.,

$\left(\left(\Gamma_{k}^{(2)}\right)_{\bar{C}C}\right)_{\mu}{}^{\nu}(x,y)=\frac{1}{\sqrt{\bar{g}(x)}}\frac{\delta}{\delta
C^{\mu}(x)}\frac{1}{\sqrt{\bar{g}(y)}}\frac{\delta}{\delta\bar{C}_{\nu}(y)}\,\Gamma_{k}[\Phi;\bar{g}]\,.$
(61)

Introducing a supertrace STr which includes a sum over all fluctuation fields
as well as a minus sign for Grassmann-valued degrees of freedom, Eq. (59) can
again be written in compact form,

$\partial_{t}\Gamma_{k}[\Phi;\bar{g}]=\frac{1}{2}{\rm
STr}\left[\left(\Gamma_{k}^{(2)}+\mathcal{R}_{k}\right)^{-1}\,\partial_{t}\mathcal{R}_{k}\right]\,.$
(62)

This equation maintains all the properties discussed in the context of the
scalar theory. It is the central result of this section and constitutes the
starting point for investigating the Wilsonian renormalization group flow of
gravity. Notably, its use is not limited to the case where the gravitational
degrees of freedom are encoded in metric fluctuations. It is also applicable
to formulations building on different sets of degrees of freedom, including
unimodular gravity, the Hilbert-Palatini formulation, the Arnowitt-Deser-
Misner (ADM) decomposition of the metric degrees of freedom, and also Hořava-
Lifshitz gravity. This makes (62) a powerful and rather universal tool to
study the quantum properties of gravity beyond perturbation theory and its use
in practical computations will be discussed in Sec. 4.

At this point the following conceptual clarifications are in order. At first
sight the introduction of a background metric seems to contradict the idea of
background independence intrinsic to general relativity. This is not the case
though. Keeping $\bar{g}_{\mu\nu}$ generic essentially corresponds to
quantizing the theory in all backgrounds simultaneously. Subsequently, one can
then evoke a dynamical principle determining $\bar{g}_{\mu\nu}$. In this way
one retains background independence even in the presence of a background
metric. This viewpoint underlies the concept of self-consistent backgrounds
developed in Becker:2014pea ; Pagani:2019vfm .

###  3.3 Common approximation schemes

The Wetterich equation (62) constitutes a formally exact equation. Finding
exact solutions to it is equivalent to carrying out the functional integral
(1). This is extremely ambitious though and usually can not be carried out
exactly. Thus, one has to resort to approximations.

Probably, the most prominent approximation is perturbation theory. In this
case the standard result is recovered by neglecting the $k$-dependence of
$\Gamma_{k}^{(2)}$ on the right-hand side of Eq. (62) and approximating
$\Gamma_{k}^{(2)}\rightarrow S_{\Lambda}^{(2)}$ with $S_{\Lambda}$ the bare
action defined at the UV-scale $\Lambda$. This approximation turns the trace
into a total derivative

$\partial_{t}\Gamma_{k}\simeq\frac{1}{2}\partial_{t}{\rm
Tr}\left[\ln\left(S_{\Lambda}^{(2)}+\mathcal{R}_{k}\right)\right]\,.$ (63)

Here and in the following we use $\simeq$ to indicate an approximation of the
exact flow. Integrating this equation from the UV-scale down to $k=0$ and
assuming that the regulator vanishes at the boundaries then yields the
standard formula for the one-loop effective action

$\Gamma^{\rm 1-loop}=S_{\Lambda}+\frac{1}{2}{\rm Tr}\left[\ln
S_{\Lambda}^{(2)}\right]\,.$ (64)

The investigation of RG fixed points typically builds on non-perturbative
approximation schemes though. The basic idea is to start from the exact flow
and project it onto a subspace spanned by a finite (or even infinite) set of
interaction monomials $\mathcal{O}_{i}$. In the setup introduced in Sec. 2,
this amounts to truncating the sum in eq. (4) to a finite set

$\Gamma_{k}\simeq\sum_{i=1}^{N}\,\bar{u}_{i}(k)\,\mathcal{O}_{i}\,.$ (65)

These types of approximations can be set up systematically, either in the form
of a derivative expansion or a vertex expansion. These commonly used non-
perturbative approximation schemes will be discussed in Sects. 3.3 and 3.3,
respectively.

#### Derivative and curvature expansion

When developing non-perturbative approximation schemes, it is important to
appreciate that $\Gamma_{k}$ depends on two metric arguments $g_{\mu\nu}$ and
$\bar{g}_{\mu\nu}$. The dependence on $g_{\mu\nu}$ can be traded for the
fluctuations $h_{\mu\nu}$ by substituting the linear split (46). Structurally,
it is then convenient to organize the contributions in $\Gamma_{k}$ according
to their transformation properties with respect to the background and quantum
gauge transformations

$\Gamma_{k}[g,\bar{g},\bar{C},C]=\bar{\Gamma}_{k}[g]+\widehat{\Gamma}_{k}[g,\bar{g}]+\Gamma^{\rm
gf}_{k}[g,\bar{g}]+\Gamma_{k}^{\rm ghost}[g,\bar{g},\bar{C},C]\,.$ (66)

Here $\Gamma_{k}^{\rm gf}[g,\bar{g}]$ and $\Gamma_{k}^{\rm
ghost}[g,\bar{g},C,\bar{C}]$ are the standard gauge-fixing and ghost terms.
The subscript $k$ thereby indicates that these sectors can contain
$k$-dependent couplings, as, e.g., a wave-function renormalization for the
ghost fields. The contribution $\bar{\Gamma}_{k}[g]$ collects all terms
constructed from $g_{\mu\nu}$ only. By construction, $\bar{\Gamma}_{k}[g]$ is
then invariant with respect to both background and quantum gauge
transformations

$\delta^{B}\bar{\Gamma}_{k}[g]=0\,,\qquad\delta^{Q}\bar{\Gamma}_{k}[g]=0\,.$
(67)

The terms contained in $\widehat{\Gamma}_{k}[g,\bar{g}]$ genuinely depend on
both arguments. It collects the “off-diagonal” contributions and satisfies

$\widehat{\Gamma}_{k}[g,g]=0\,.$ (68)

A rather broad class of approximations based on (66) truncates the effective
average action by setting $\widehat{\Gamma}_{k}[g,\bar{g}]\simeq 0$. Commonly,
these approximations are referred to as _single-metric approximations_
Manrique:2009uh ; Manrique:2010am ; Manrique:2010mq . Most approximations
along these lines also work with a classical ghost sector, setting
$\Gamma_{k}^{\rm ghost}[g,\bar{g},\bar{C},C]\simeq S^{\rm
ghost}[g,\bar{g},\bar{C},C]$.

Building on the results by Fulling, King, Wybourne, Cummins Fulling:1992vm
(further elaborated on in Decanini:2008pr ), one can systematically construct
a basis $\mathcal{O}_{i}[g]$ in which $\bar{\Gamma}_{k}[g]$ can be expanded.
The explicit construction of the independent basis elements needs to take into
account redundancies due to the Bianchi identity
$D_{[\mu}R_{\alpha\beta]\gamma\delta}=0$. In addition, low-dimensional cases
are subject to additional simplifications, e.g., due to the vanishing of the
Weyl tensor in $d=3$.

The symmetries of $\bar{\Gamma}_{k}[g]$ dictate that the corresponding
monomials are built from the Riemann tensor $R_{\mu\nu\rho\sigma}$, its
contractions, and covariant derivatives $D_{\mu}$ acting on the curvature
tensors. Convenient building blocks for the basis elements are then provided
either by the Riemann basis

$\mathcal{O}_{i}[g]=\mathcal{O}_{i}[\sqrt{g},R,R_{\mu\nu},R_{\mu\nu\rho\sigma},D_{\mu}]$
(69)

or the Weyl basis

$\mathcal{O}_{i}[g]=\mathcal{O}_{i}[\sqrt{g},R,R_{\mu\nu},C_{\mu\nu\rho\sigma},D_{\mu}]$
(70)

The two choices are related by the identity

$C_{\mu\nu\rho\sigma}=R_{\mu\nu\rho\sigma}-\frac{2}{d-2}\left(g_{\mu[\rho}R_{\sigma]\nu}-g_{\nu[\rho}R_{\sigma]\mu}\right)+\frac{2}{(d-1)(d-2)}Rg_{\mu[\rho}g_{\sigma]\nu}\,.$
(71)

In terms of structural aspects, it is often useful to work in the Weyl basis,
since this choice disentangles the contributions of the higher-derivative
terms to the flat-space graviton propagator.

The expansion of $\bar{\Gamma}_{k}[g]$ can be organized systematically by
counting the number of spacetime derivatives $n$ contained in the monomial
$\mathcal{O}_{i}[g]\equiv\int d^{d}x\sqrt{g}\tilde{\mathcal{O}}_{l}^{n}[g]$.
The index set $\\{i\\}\mapsto\\{n,l\\}$ where $l$ enumerates the basis
elements occurring at a fixed order $n$, see Table 1 for examples. This scheme
is called the _derivative expansion_ of $\bar{\Gamma}_{k}[g]$. The basis
elements appearing at the lowest orders are given in Table 1.

$n$ $l$ |  $1$ |  $2$ |  $3$ |  $4$ |  $\cdots$
---|---|---|---|---|---
$\vdots$ | $\vdots$ | $\vdots$ | $\vdots$ | $\vdots$ | $\vdots$
$8$ | $R\Delta^{2}R$ | $C_{\mu\nu\rho\sigma}\Delta^{2}C^{\mu\nu\rho\sigma}$ | $R^{4}$ | $R^{2}\,R_{\mu\nu}\,R^{\mu\nu}$ | $\cdots$
$6$ | $R\Delta R$ | $C_{\mu\nu\rho\sigma}\Delta C^{\mu\nu\rho\sigma}$ | $R^{3}$ | $R\,R_{\mu\nu}\,R^{\mu\nu}$ |  $+6$ more
$4$ | $R^{2}$ | $C_{\mu\nu\rho\sigma}C^{\mu\nu\rho\sigma}$ | $E$ |  |
$2$ | $R$ |  |  |  |
$0$ | $\mathbb{1}$ |  |  |  |
Table 1: Illustration of the interaction monomials
$\tilde{\mathcal{O}}_{l}^{n}[g]$ appearing in the derivative expansion of
$\bar{\Gamma}_{k}[g]$ at order $n$ using the Weyl basis (70). The terms listed
in the middle contribute to the graviton propagator in a four-dimensional flat
background. Terms in the right-most block contribute terms proportional to the
background curvature in $\Gamma_{k}^{(2)}[h=0;\bar{g}]$ and may be interpreted
as “potential terms”. Furthermore,
$E=R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}-4R_{\mu\nu}R^{\mu\nu}+R^{2}$
denotes the integrand of the Gauss-Bonnet term, which is topological in $d=4$.

The number of independent basis elements increases significantly with each
order in the derivative expansion. This expansion scheme provides a good
ordering principle when studying the “low-energy” properties of the theory.
For fixed points which are Gaussian or “almost-Gaussian”, the power-counting
also provide a good guiding principle whether a given operator is relevant or
irrelevant.

A conceptual shortcoming of the derivative expansion is that truncating the
series of terms contributing to the gravitational propagator induces
potentially spurious poles Becker:2017tcx . The reason is that the
approximation intrinsic to the derivative expansion leads to inverse
propagators which are polynomial in the momentum. Hence it is difficult to
address questions about stability and the potential presence of ghosts within
this approximation Platania:2020knd ; Platania:2022gtt .

As stressed in Knorr:2019atm , this feature can be bypassed by switching to a
curvature expansion. The basic idea is to collect the covariant derivatives
appearing in interaction monomials in operator-valued functions, called _form
factors_. These capture the dependence of propagators and interaction vertices
on the (generalized) momenta of the fields and can also be defined in an
arbitrary curved background spacetime. Building on the examples given in Table
1, the form factors appearing at the lowest non-trivial order in the curvature
expansion arise from combining the terms in the columns with $l=1$ and $l=2$:

$\begin{split}&\sum_{i=0}\bar{u}^{i}(k)\,R\,\Delta^{n}\,R\mapsto
R\,W^{R}_{k}(\Delta)\,R\,,\\\
&\sum_{i=0}\bar{u}^{i}(k)\,C_{\mu\nu\rho\sigma}\,\Delta^{n}\,C^{\mu\nu\rho\sigma}\mapsto
C_{\mu\nu\rho\sigma}\,W^{C}_{k}(\Delta)\,C^{\mu\nu\rho\sigma}\,.\end{split}$
(72)

Notably, there are only two form factors appearing at second order in the
spacetime curvature. A potential third function $R_{\mu\nu}\,W^{\rm
Ric}_{k}(\Delta)\,R^{\mu\nu}$ can be mapped to (72) and higher-curvature terms
by applying the Bianchi identity. The functions $W^{C}_{k=0}(\Delta)$ and
$W^{R}_{k=0}(\Delta)$ fix the graviton propagator in a flat background. From
Table 1, it is also apparent that there is no form factor at first order in
the curvature expansion. Any derivatives acting on $R$ would lead to a surface
term. As a consequence, Newton’s constant $G_{0}$ (and also the cosmological
constant $\Lambda_{0}$) can not carry a dependence on the physical momenta of
the field.

The $k$-dependence of a form factor can again be obtained by substituting the
corresponding ansatz for $\Gamma_{k}$ into the Wetterich equation and
projecting the flow on the corresponding subspace. In general, this results in
a non-linear integro-differential equation for the unknown functions, see
Table 2. Solving these equations either numerically or by employing
pseudospectral methods then allows to obtain information on the graviton
propagator and momentum dependence of interaction vertices, see Bosma:2019aiu
for pioneering work in this direction.

approximation of $\Gamma_{k}$ |  structure of RG flow |  fixed points
---|---|---
finite number of $\mathcal{O}_{i}$ | ODEs | algebraic
| field dependent functions
---
$f_{k}(R_{1},\cdots,R_{n})$
| PDEs
---
$(n+1)$ variables
| PDEs
---
$n$ variables
| momentum-dependent form factors
---
$f_{k}(p_{1},\cdots,p_{n})$
| IDEs
---
$(n+1)$ variables
| IDEs
---
$n$ variables
Table 2: Summary of the mathematical structures capturing the flow of
$\Gamma_{k}$ in different classes of approximations. Depending on the scale-
dependent terms retained in $\Gamma_{k}$, the projected flow equations are
non-linear ordinary differential equations (ODEs), partial differential
equations (PDEs), or (partial) integro- differential equations (IDEs). Since
fixed functionals are $k$-stationary solutions, their structure is encoded in
differential equations which contain one variable less than the corresponding
flow equation.

#### Incorporating higher-order interaction vertices

The background approximation evaluates the Wetterich equation at zeroth order
in the fluctuation field. This class of approximations can then be extended
systematically by taking into account higher orders of the fluctuation field.
This is the idea behind the bimetric computations initiated in Manrique:2009uh
; Manrique:2010am ; Manrique:2010mq  and the fluctuation approach reviewed in
Pawlowski:2020qer . It can be implemented systematically by performing a
vertex expansion of $\Gamma_{k}[h;\bar{g}]$ in powers of the fluctuation
field:555The discussion of the ghost contributions follows the same lines, but
is suppressed for the sake of readability.

$\Gamma_{k}[h;\bar{g}]=\sum_{n,l}\frac{1}{n!}\int
d^{d}x\;\Gamma_{k}^{l;\mu_{1}\nu_{1}\cdots\mu_{n}\nu_{n}}[\bar{g}]\;h_{\mu_{1}\nu_{1}}\cdots
h_{\mu_{n}\nu_{n}}\,.$ (73)

Here $l$ enumerates the set of independent tensor structures contracting $n$
powers of the fluctuation fields. Note that all dependence on the background
metric is stored in
$\Gamma_{k}^{l;\mu_{1}\nu_{1}\cdots\mu_{n}\nu_{n}}[\bar{g}]$. Similarly to
(69) and (70), the vertices can be build from $\sqrt{\bar{g}}$, background
curvature tensors and their contractions, as well as the background covariant
derivative. By construction
$\Gamma_{k}^{l;\mu_{1}\nu_{1}\cdots\mu_{n}\nu_{n}}[\bar{g}]$ transforms as a
tensor of the corresponding rank with respect to background gauge
transformations. Since the expansion captures contributions from both
$\bar{\Gamma}_{k}[g]$ and $\widehat{\Gamma}_{k}[g,\bar{g}]$, quantum gauge
invariance is broken and the classification of admissible vertices is
significantly more complicated than in the single-metric case. Prototypical
examples of terms appearing in the vertex expansion can be obtained from
expanding the gauge-fixed Einstein-Hilbert action in powers of $h_{\mu\nu}$.
Explicit examples can then be found in Eqs. (95) and (96).

The $k$-dependence of the vertices appearing in (73) can again be obtained
from the Wetterich equation. Taking functional derivatives of (2) with respect
to the fluctuation fields gives a hierarchy of equations determining of the
schematic form

$\partial_{t}\Gamma^{(n)}_{k}[\bar{g}]=\text{Flow}\left[\Gamma^{(2)}_{k}[\bar{g}],\cdots,\Gamma^{(n+2)}_{k}[\bar{g}]\right]\,.$
(74)

Here the superscript indicates the $n$-th functional derivative of
$\Gamma_{k}$ with respect to the fluctuation fields, cf. (60). Background
computations evaluate this hierarchy at zeroth order in $n$. Note that the
right-hand side also depends on the higher-order vertices
$\Gamma^{(n+1)}_{k}[\bar{g}]$ and $\Gamma^{(n+2)}_{k}[\bar{g}]$. The
truncation of the system to a finite set of tensor structures then requires an
assumption on these higher-order vertices in order to close the system. A
typical strategy is to approximate the couplings appearing at the orders
$(n+1)$ and $(n+2)$ by the ones appearing at the lower orders in the
hierarchy.

In practice, computations maintaining information about the fluctuation fields
have mainly been carried out in a flat background, setting
$\bar{g}_{\mu\nu}=\delta_{\mu\nu}$. This choice gives access to powerful
momentum space techniques and the hierarchy (74) can then be evaluated by
employing standard Feynman diagram techniques. In particular, eq. (73)
simplifies to

$\Gamma_{k}[h;\delta]=\sum_{n,l}\frac{1}{n!}\left(\prod_{n}\int\frac{d^{d}p}{(2\pi)^{d}}\right)\,\Gamma_{k}^{l;\mu_{1}\nu_{1}\cdots\mu_{n}\nu_{n}}(p_{1},\cdots,p_{n})h_{\mu_{1}\nu_{1}}(p_{1})\cdots
h_{\mu_{n}\nu_{n}}(p_{n})\,,$ (75)

where the $p_{i}$ are the momenta of the fluctuation fields. This has led to
significant insights on the momentum-dependence of the graviton two-point
function Christiansen:2014raa ; Bonanno:2021squ  and resolving the momentum-
dependence of three- and four-point vertices Christiansen:2015rva ;
Denz:2016qks .

###  3.4 Further developments

The discussion of the Wetterich equation and its properties mainly followed
the initial constructions Wetterich:1992yh ; Reuter:1993kw ; Reuter:1996cp .
We complete our exposition by briefly introducing two recent developments, the
_minimal essential scheme_ Baldazzi:2021ydj  (Sec. 3.4) and the $N$-type
cutoffs Becker:2020mjl ; Becker:2021pwo  (Sec. 3.4).

#### The minimal essential scheme

Ultimately, the goal of the gravitational asymptotic safety program is the
construction of observables. From this perspective, it turns out that the
theory space introduced in Sec. 2, spanned by all possible interaction
monomials $\mathcal{O}_{i}$, contains redundancies in the sense that not all
couplings appearing in this basis will also enter into the observables. A
prototypical example is the wave-function renormalization of a field, which
drops out from the construction of scattering amplitudes. On this basis one
distinguishes between _essential couplings_ which enter into the expressions
for physical observables and _inessential couplings_ whose values can be
changed without affecting the predictions of the theory.

Typically, a change in an inessential coupling can be absorbed into a
reparameterization of the dynamical variables. Considering an infinitesimal
change in the field, $\chi\mapsto\chi+\xi[\chi]$, the underlying action
transforms as666We use the “$\cdot$” to indicate an integral over spacetime
and potentially a sum over internal indices labeling the fields.

$S[\chi]\mapsto S[\chi]+\xi[\chi]\cdot\frac{\delta}{\delta\chi}S[\chi]\,.$
(76)

This underlies the general statement that operators which are proportional to
the equations of motion can be removed by a field redefinition and are thus
linked with inessential couplings tHooft:1973pz . Generically, one can also
consider finite frame transformations to a new field parameterization,

$\phi(x)=\phi[\chi](x),$ (77)

requiring that the map is quasi-local and invertible.

Implementing the procedure of removing inessential couplings at the level of
the functional renormalization group is slightly more complicated. Since the
corresponding couplings depend on the coarse-graining scale $k$, the field-
redefinitions required in this process inherit this scale-dependence. Thus the
frame transformation (77) is promoted to be $k$-dependent

$\phi_{k}(x)=\phi_{k}[\chi](x),$ (78)

This effect can be accommodated by formulating the Wetterich equation in a
frame-covariant way Pawlowski:2005xe

$\left(\partial_{t}+\Psi_{k}[\phi]\,\frac{\delta}{\delta\phi}\right)\Gamma_{k}[\phi]=\frac{1}{2}{\rm
Tr}\left[\left(\Gamma^{(2)}_{k}+\mathcal{R}_{k}\right)^{-1}\left(\partial_{t}+2\,\Psi_{k}[\phi]\frac{\delta}{\delta\phi}\right)\mathcal{R}_{k}\right]\,.$
(79)

The renormalization group kernel

$\Psi_{k}[\phi]\equiv\partial_{t}\phi_{k}[\chi]$ (80)

thereby accounts for the $k$-dependence of the frame transformation.

In order to illustrate the working of the minimal essential scheme, we return
to the example of a scalar field theory. Explicitly, we set

$\Gamma_{k}[\chi]=\int
d^{d}x\left\\{\frac{Z_{k}}{2}\chi\left[-\partial^{2}+m_{k}^{2}\right]\chi+\frac{Z_{k}^{2}\lambda_{k}}{12}\chi^{4}+\cdots\right\\}\,.$
(81)

Here $m_{k}$ and $\lambda_{k}$ are scale-dependent couplings, $Z_{k}$ is the
wave-function of the field, and the dots symbolizes additional interaction
terms. The wave-function renormalization constitutes an inessential coupling
and we seek to remove it by a $k$-dependent frame transformation. Inspecting
(81) indicates that this can be achieved by a $k$-dependent frame-
transformation which is linear in the field

$\phi_{k}=Z_{k}^{1/2}\,\chi\,.$ (82)

The kernel (80) then evaluates to
$\Psi_{k}[\phi]=-\frac{1}{2}\eta_{k}\phi_{k}$ where
$\eta_{k}\equiv-\partial_{t}\ln Z_{k}$ is the anomalous dimension of the
field. Evaluating (79) then yields

$\left(\partial_{t}-\frac{1}{2}\eta_{k}\,\phi\,\frac{\delta}{\delta\phi}\right)\Gamma_{k}[\phi]=\frac{1}{2}{\rm
Tr}\left[\left(\Gamma^{(2)}_{k}[\phi]+\mathcal{R}_{k}\right)^{-1}\left(\partial_{t}\mathcal{R}_{k}-\eta_{k}\,\mathcal{R}_{k}\right)\right]\,.$
(83)

The new functional $\Gamma_{k}[\phi]$ is then independent of $Z_{k}$. More
precisely, the inessential coupling has been fixed to $Z_{k}=1$ at all scales.
The result (83) furthermore shows that $\eta_{k}$ depends on the essential
couplings of the theory only.

As pointed out in Baldazzi:2021ydj  and illustrated by our explicit example
above, the use of the frame-covariant flow equation in combination with the
minimal-essential scheme may lead to significant technical simplifications
when constructing solutions to the flow equation. In practice, these
simplifications can be exploited systematically by parameterizing the kernel
$\Psi_{k}[\phi]$ in terms of $k$-dependent $\gamma$-functions Baldazzi:2021orb
; Knorr:2022ilz . The freedom gained in this way can then be used to fix the
inessential coupling constants to specific values. The scale-dependence of the
theory is then captured by the $\beta$-functions (governing the $k$-dependence
of the essential couplings) and the $\gamma$-functions (governing the
$k$-dependence of the inessential ones). Both sets of equations depend on the
essential couplings only. The last property then simplifies the search for RG
fixed points in a significant way.

#### Flows in terms of $N$-type cutoffs

Recently, a novel regularization scheme via dimensionless $N$-type cutoffs has
been introduced Becker:2020mjl ; Becker:2021pwo ; Banerjee:2023ztr , which may
constitute a more physical alternative to the usually employed dimensionful UV
cutoffs. The motivation for the introduction of a scale-free regularization
scheme is the construction of regularized quantum systems, which have the
potential of being physically realizable themselves. In this way, physical
properties of the theory, which conventionally are to be studied in the
quantum field theory limit, could already be probed at the level of the
regularized system. Moreover, this scale-free regularization scheme is
designed in a way, such that self-consistent background geometries can easily
be accessed.

Schematically, the $N$-type cutoff regularizes the path integral (1) as
follows. One expands the field in the eigenbasis of a suitable self-adjoint
operator, e.g., the background Laplacian, such that the corresponding
eigenvalues increase with $n\in\mathbb{N}$ (or $n\in\mathbb{R}^{+}$), cf. Eq.
(47). Then the path integral is regularized by restricting the domain of
integration to the field modes $h^{n}$ with $n\leq N$. As a result, one
obtains $N$-sequences of regularized quantum systems, which in principle are
physically realizable.

As a first application, the self-consistent spherical background geometries
stemming from summing up vacuum energy of a scalar field Becker:2020mjl  as
well as metric fluctuations Becker:2021pwo  have been studied. The striking
result, which is due to background independence, is that the self-consistent
scalar curvatures $R(N)$ vanished for $N\rightarrow\infty$ in both cases. This
is precisely the opposite behavior of the commonly perceived cosmological
constant problem, according to which the background curvature, and therewith
the total cosmological constant, should diverge when removing the UV
regulator. Another striking result of this regularization scheme
Becker:2020mjl  is that $N$-type cutoffs give an explanation of the
microscopical degrees of freedom which the Bekenstein-Hawking entropy of de
Sitter space counts.

##  4 The Einstein-Hilbert truncation

We proceed by giving an explicit example, illustrating how the Wetterich
equation (62) is used to extract non-perturbative information about the
gravitational RG flow. The discussion is based on the arguably simplest
approximation for the effective average action $\Gamma_{k}$, the Einstein-
Hilbert truncation. Starting from the seminal paper Reuter:1996cp , this
projection has been studied in detail in a series of works Souma:1999at ;
Reuter:2001ag ; Lauscher:2001ya ; Litim:2003vp ; Gies:2015tca . It still forms
an integral part of studying the RG flow in many gravity-matter systems. The
present exposition differs from the historical computations where the
background metric has been set to the one of the maximally symmetric
$d$-sphere $S^{d}$. Instead, we combine the idea of the universal RG machine
Benedetti:2010nr ; Groh:2011vn  with off-diagonal heat-kernel techniques
Gorbar:2002pw ; Gorbar:2003yt ; Decanini:2005gt ; Benedetti:2010nr ;
Codello:2012kq  and carry out the derivation of the beta functions _without
specifying the background metric_ $\bar{g}_{\mu\nu}$. This stresses the
background-independent nature of the computation and emphasizes the modern
viewpoint on evaluating the FRGE in the context of gravity. In order to keep
technical complications at the minimum, we adopt the harmonic gauge. The beta
functions resulting from this setting are computed in Sec. 4.1 and the
resulting fixed point structure and phase diagram is presented in Sec. 4.2.
Results obtained by generalizing this computation by resorting to additional
field decompositions and generalizing the gauge-fixing and regularization
prescription have been obtained in Gies:2015tca  and corroborate the findings
reviewed in this section.

###  4.1 Deriving the beta functions

The Einstein-Hilbert (EH) truncation works in the background approximation.
Thus the flow is obtained at zeroth order in the fluctuation fields. As a
consequence only terms of zeroth and second order in the fluctuations are
needed in the evaluation of the Wetterich equation. The projection of the flow
equation tracks the scale-dependence of the (background) Newton’s coupling
$G_{k}$ and the cosmological constant $\Lambda_{k}$. The gravitational part of
the effective average action is approximated by the Einstein-Hilbert action

$\Gamma_{k}^{\rm EH}[g]=\frac{1}{16\pi G_{k}}\int
d^{d}x\sqrt{g}\left(-R+2\Lambda_{k}\right)\,,$ (84)

with the couplings depending on the coarse-graining scale $k$. In view of the
upcoming computation, it is convenient to introduce the dimensionless
counterparts of Newton’s coupling and the cosmological constant as well as the
anomalous dimension of Newton’s coupling

$g_{k}\equiv k^{d-2}G_{k}\,,\qquad\lambda_{k}\equiv
k^{-2}\Lambda_{k}\,,\qquad\eta_{N}(k)\equiv(G_{k})^{-1}\partial_{t}G_{k}\,.$
(85)

Furthermore, geometrical quantities constructed from $\bar{g}_{\mu\nu}$ are
distinguished by a bar. E.g., $\bar{D}_{\mu}$ is the covariant derivative
constructed from the background metric.

In order to obtain well-defined propagators, $\Gamma^{\rm EH}_{k}$ must be
supplemented by a gauge-fixing term and the corresponding ghost action.
Concretely, we implement a background gauge-fixing

$\Gamma_{k}^{\rm gf}[h;\bar{g}]=\frac{1}{32\pi G_{k}\alpha}\int
d^{d}x\sqrt{\bar{g}}\,\bar{g}^{\mu\nu}F_{\mu}F_{\nu}\,,$ (86)

where the gauge-fixing condition is taken to be linear in the fluctuation
field

$F_{\mu}[h;\bar{g}]=\left[\delta_{\mu}^{\alpha}\bar{D}^{\beta}-\beta\bar{g}^{\alpha\beta}\bar{D}_{\mu}\right]\,h_{\alpha\beta}\,.$
(87)

Here $\alpha$ and $\beta$ are two gauge-parameters which can largely be chosen
arbitrary Gies:2015tca . The ghost action accompanying (86) is found in the
standard way and reads

$S^{\rm ghost}[h,\bar{C},C;\bar{g}]=-\sqrt{2}\int
d^{d}x\sqrt{\bar{g}}\,\bar{C}_{\mu}\,\mathcal{M}[g,\bar{g}]^{\mu}{}_{\nu}\,C^{\nu}\,,$
(88)

with the Faddeev-Popov operator being

$\mathcal{M}[g,\bar{g}]^{\mu}{}_{\nu}=\bar{g}^{\mu\rho}\bar{D}^{\sigma}\left(g_{\rho\nu}D_{\sigma}+g_{\sigma\nu}D_{\rho}\right)-2\beta\bar{g}^{\rho\sigma}\bar{D}^{\mu}g_{\sigma\nu}D_{\rho}\,.$
(89)

Landau-type gauge fixings correspond to the limit $\alpha\rightarrow 0$ (with
$\beta=1/d$ being a preferred choice implementing the geometric gauge). The
harmonic gauge adopted in the present computation sets $\alpha=1$ (Feynman-
type gauge) and $\beta=1/2$. This has the technical advantage that all
derivatives appear in the form of the background Laplace operator
$\Delta=-\bar{g}^{\mu\nu}\bar{D}_{\mu}\bar{D}_{\nu}$.

For the background computation ahead, it suffices to know the ghost-action to
second order in the fluctuation fields. Adopting harmonic gauge and evaluating
$\mathcal{M}[g,\bar{g}]^{\mu}{}_{\nu}|_{g=\bar{g}}$ shows that the relevant
contributions are captured by

$S^{\rm ghost}[h=0,\bar{C},C;\bar{g}]=\sqrt{2}\int
d^{d}x\sqrt{\bar{g}}\,\bar{C}_{\mu}\,\left[\,\delta^{\mu}_{\nu}\Delta-\bar{R}^{\mu}{}_{\nu}\,\right]\,C^{\nu}\,.$
(90)

Here, we used the commutator of two background-covariant derivatives evaluated
on vectors in order to combine the last two terms in (89) into the background
Ricci scalar $\bar{R}_{\mu\nu}$. The approximation for the effective average
action then combines the $\bar{\Gamma}_{k}[g]$ given in (84) with the gauge-
fixing term (86) and the ghost action (88)

$\Gamma_{k}[h,\bar{C},C;\bar{g}]\simeq\Gamma_{k}^{\rm EH}[g]+\Gamma_{k}^{\rm
gf}[h;\bar{g}]+S^{\rm ghost}[h,C,\bar{C};\bar{g}]\,.$ (91)

At this stage a comment on the projection prescription is in order.
Substituting (91) into its left-hand side and setting $g=\bar{g}$ afterwards
indicates that the scale-dependence of $G_{k}$ and $\Lambda_{k}$ can be read
off from the coefficients multiplying

$\mathcal{O}_{0}=\int d^{d}x\sqrt{\bar{g}}\,,\qquad\mathcal{O}_{1}=\int
d^{d}x\sqrt{\bar{g}}\bar{R}\,.$ (92)

All other interaction monomials spanning the gravitational theory space do not
contribute to the computation. This entails the following, profound
consequence. Eq. (92) corresponds to a derivative expansion truncated at first
order in the spacetime curvature. Hence all terms containing two or more
curvature tensors are outside the subspace spanned by our approximation.
Moreover, (92) does not contain derivatives of a curvature tensor. Hence,
there is no need to track such terms in the present computation. These
considerations allow to formulate projection rules, stating that

$\bar{D}_{\mu}\bar{R}_{\alpha\beta\gamma\delta}\simeq 0\,,\qquad
O(\bar{R}^{2})\simeq 0\,.$ (93)

We stress that these rules should not be read as restrictions on
$\bar{g}_{\mu\nu}$. They merely identify structures which do not contribute to
the computation. As a corollary of these relations, we conclude that we can
freely commute covariant derivatives and curvature tensors, since the
commutators just produce terms outside of the projection spanned by (92).

The first step in evaluating the trace appearing within the FRGE (62) consists
in expanding (91) to second order in the fluctuation fields. In the ghost-
sector the result is already given in (90). For the gravitational
fluctuations, we expand

$\Gamma_{k}[\bar{g}+h,\bar{g}]=\Gamma_{k}[\bar{g},\bar{g}]+O(h)+\Gamma^{\rm
quad}_{k}[h;\bar{g}]+O(h^{3})\,.$ (94)

The relevant coefficient $\Gamma^{\rm quad}_{k}[h;\bar{g}]$ is readily found
using computer algebra packages like xAct Brizuela:2008ra  and has the form

$\Gamma^{\rm quad}_{k}=\frac{1}{32\pi G_{k}}\int
d^{d}x\sqrt{\bar{g}}\;\frac{1}{2}\,h_{\mu\nu}\left[K^{\mu\nu}{}_{\alpha\beta}\left(\Delta-2\Lambda_{k}\right)+V^{\mu\nu}{}_{\alpha\beta}\right]\,h^{\alpha\beta}\,.$
(95)

Here the “kinetic” and “potential” parts have the explicit form

$\begin{split}K^{\mu\nu}{}_{\alpha\beta}=&\,\frac{1}{2}\left(\delta^{\mu}_{\alpha}\delta^{\nu}_{\beta}+\delta^{\mu}_{\beta}\delta^{\nu}_{\alpha}-\bar{g}^{\mu\nu}\bar{g}_{\alpha\beta}\right)\,,\\\
V^{\mu\nu}{}_{\alpha\beta}=&\,\bar{R}\,K^{\mu\nu}{}_{\alpha\beta}+\left(\bar{g}^{\mu\nu}\bar{R}_{\alpha\beta}+\bar{R}^{\mu\nu}\bar{g}_{\alpha\beta}\right)-2\delta^{(\mu}_{(\alpha}\bar{R}^{\nu)}_{\beta)}-2\bar{R}^{(\mu}{}_{(\alpha}{}^{\nu)}{}_{\beta)}\,.\\\
\end{split}$ (96)

The potential $V$ collects all terms containing the spacetime curvature and is
of first order in a curvature expansion.

In the next step, we would like to diagonalize the kinetic terms in the
quadratic form (95). This can be achieved by decomposing $h_{\mu\nu}$ into
component fields, resorting to the transverse-traceless decomposition
York:1973ia ; Lauscher:2001ya . In the present case, it suffices to split the
fluctuations into their trace- and traceless part

$h_{\mu\nu}=\hat{h}_{\mu\nu}+\frac{1}{d}\bar{g}_{\mu\nu}h\,,\qquad\bar{g}^{\mu\nu}\hat{h}_{\mu\nu}=0\,.$
(97)

Substituting this decomposition into (95) then yields

$\begin{split}\Gamma^{\rm quad}_{k}[h;\bar{g}]=\frac{1}{32\pi G_{k}}&\int
d^{d}x\sqrt{\bar{g}}\bigg{[}\frac{1}{2}\hat{h}_{\mu\nu}\left[\Delta-2\Lambda_{k}+\bar{R}\right]\hat{h}^{\mu\nu}\\\
&-\left(\frac{d-2}{4d}\right)\,h\,\left[\Delta-2\Lambda_{k}+\frac{d-4}{d}\bar{R}\right]\,h\\\
&-\bar{R}_{\mu\nu}\hat{h}^{\mu\alpha}\hat{h}_{\alpha}{}^{\nu}-\bar{R}_{\mu\nu\alpha\beta}\hat{h}^{\mu\alpha}\hat{h}^{\nu\beta}+\frac{d-4}{d}\bar{R}_{\mu\nu}\,h\,\hat{h}^{\mu\nu}\bigg{]}\,.\end{split}$
(98)

At this point, we are ready to specify the explicit form of the regulator
$\mathcal{R}_{k}$. We dress up the Laplacians according to

$\Delta\mapsto\Delta+R_{k}\,,$ (99)

where $R_{k}(\Delta)=k^{2}R^{(0)}(\Delta/k^{2})$ is the dimensionful cutoff
function and $R^{(0)}(z)$ the corresponding profile. In the nomenclature of
the review Codello:2008vh  this corresponds to a cutoff of type I. This choice
implements the initial idea of supplying the fluctuation field with a
$k$-dependent mass term. The resulting $\mathcal{R}_{k}$ is then diagonal in
field space with its matrix elements given by

$\mathcal{R}_{k}^{\hat{h}\hat{h}}=\frac{1}{32\pi
G_{k}}R_{k}\,\mathbb{1}_{2T}\,,\quad\mathcal{R}_{k}^{hh}=-\frac{1}{32\pi
G_{k}}\left(\frac{d-2}{2d}\right)R_{k}\,,\quad\mathcal{R}_{k}^{\bar{C}C}=\sqrt{2}R_{k}\,\mathbb{1}_{1}\,.$
(100)

Here

$\mathbb{1}_{2T}^{\mu\nu}{}_{\alpha\beta}=\frac{1}{2}\left(\delta^{\mu}_{\alpha}\delta^{\nu}_{\beta}+\delta^{\mu}_{\beta}\delta^{\nu}_{\alpha}\right)-\frac{1}{d}\bar{g}^{\mu\nu}\bar{g}_{\alpha\beta}\,,\qquad\mathbb{1}_{1}^{\mu}{}_{\nu}=\delta^{\mu}_{\nu}\,,$
(101)

are the units on the space of symmetric traceless two-tensors (2T) and vectors
(1), respectively.

We now proceed by constructing the inverse of the regularized Hessian. For the
gravitational degrees of freedom, we encounter the two-by-two matrix

$\left[\Gamma^{(2)}_{k}+\mathcal{R}_{k}\right]^{ij}=\left[\begin{array}[]{cc}K_{2T}(\Delta)\,\mathbb{1}_{2T}+V_{2T}&V_{\times}\\\
V_{\times}^{\dagger}&K_{0}(\Delta)\,\mathbb{1}_{0}+V_{0}\end{array}\right]\,.$
(102)

Here $i,j=\\{\hat{h},h\\}$ labels the fields and we suppress all spacetime
indices for the sake of readability. The explicit form of the kinetic
functions $K$ and the potentials $V$ can be read off from Eqs. (98) and (90)
and read

$\begin{split}K_{2T}(\Delta)=&\,\frac{1}{32\pi
G_{k}}\left(\Delta+R_{k}-2\Lambda_{k}\right)\,,\\\
K_{0}(\Delta)=&\,-\frac{1}{32\pi
G_{k}}\left(\frac{d-2}{2d}\right)\,\left(\Delta+R_{k}-2\Lambda_{k}\right)\,,\\\
K_{1}(\Delta)=&\,\sqrt{2}\,\Delta\,,\end{split}$ (103)

and

$\begin{split}&V_{2T}^{\mu\nu}{}_{\alpha\beta}=\frac{1}{32\pi
G_{k}}\left(\bar{R}\,\mathbb{1}_{2T}^{\mu\nu}{}_{\alpha\beta}-2\bar{R}^{(\mu}_{(\alpha}\delta^{\nu)}_{\beta)}-2\bar{R}_{(\alpha}{}^{(\mu}{}_{\beta)}{}^{\nu)}\right)\,,\\\
&V_{0}=-\frac{1}{32\pi
G_{k}}\,\left(\frac{d-2}{2d}\right)\left(\frac{d-4}{d}\right)\,\bar{R}\,,\\\
&V_{\times}{}_{\mu\nu}=\frac{1}{32\pi
G_{k}}\left(\frac{d-4}{d}\right)\,\left(\bar{R}_{\mu\nu}-\frac{1}{d}\bar{g}_{\mu\nu}\bar{R}\right)\,,\\\
&V_{1}{}^{\mu}{}_{\nu}=-\sqrt{2}\,\bar{R}^{\mu}{}_{\nu}\,.\end{split}$ (104)

Constructing the inverse of (102) builds on the exact inversion formula for
block matrices

$\left[\begin{array}[]{cc}A&B\\\
C&D\end{array}\right]^{-1}=\left[\begin{array}[]{cc}\left(A-BD^{-1}C\right)^{-1}&-A^{-1}B\left(D-CA^{-1}B\right)^{-1}\\\
-D^{-1}C\left(A-BD^{-1}C\right)^{-1}&\left(D-CA^{-1}B\right)^{-1}\end{array}\right]\,.$
(105)

Since the potentials (104) contain at least one power of the spacetime
curvature, each entry can be constructed as a power series in $V$. The
projection prescription (93) then indicates that it is sufficient to retain
the terms up to one power of $V$. This implies, in particular, that the off-
diagonal terms $V_{\times}$ do not enter into the present computation since
they start to contribute at second order in $V$ only. Taking into account that
the regulator $\partial_{t}\mathcal{R}_{k}$ is diagonal in field space, it is
sufficient to consider the diagonal entries in (105). Explicitly, the
corresponding inverses are given by

$\begin{split}(32\pi
G_{k})^{-1}\left[\Gamma_{k}^{(2)}+\mathcal{R}_{k}\right]^{-1}_{\hat{h}\hat{h}}\simeq&\,\frac{1}{K_{2T}}-\frac{1}{K_{2T}}V_{2T}\frac{1}{K_{2T}}+O(V^{2})\,,\\\
(32\pi
G_{k})^{-1}\left[\Gamma_{k}^{(2)}+\mathcal{R}_{k}\right]^{-1}_{hh}\simeq&\,\frac{1}{K_{0}}-\frac{1}{K_{0}}V_{0}\frac{1}{K_{0}}+O(V^{2})\,.\end{split}$
(106)

Based on these preliminary considerations, we can now write down the projected
flow equation

$\begin{split}\partial_{t}\Gamma_{k}=&\,\frac{1}{2}{\rm
Tr}_{2T}\left[\frac{1}{K_{2T}}\partial_{t}\mathcal{R}_{k}^{\hat{h}\hat{h}}\right]-\frac{1}{2}{\rm
Tr}_{2T}\left[\frac{1}{K_{2T}}V_{2T}\frac{1}{K_{2T}}\partial_{t}\mathcal{R}_{k}^{\hat{h}\hat{h}}\right]\\\
&\,+\frac{1}{2}{\rm
Tr}_{0}\left[\frac{1}{K_{0}}\partial_{t}\mathcal{R}_{k}^{hh}\right]-\frac{1}{2}{\rm
Tr}_{0}\left[\frac{1}{K_{0}}V_{0}\frac{1}{K_{0}}\partial_{t}\mathcal{R}_{k}^{hh}\right]\\\
&\,-{\rm
Tr}_{1}\left[\frac{1}{K_{1}}\partial_{t}\mathcal{R}_{k}^{\bar{C}C}\right]+{\rm
Tr}_{1}\left[\frac{1}{K_{1}}V_{1}\frac{1}{K_{1}}\partial_{t}\mathcal{R}_{k}^{\bar{C}C}\right]\,.\end{split}$
(107)

Here the subscripts $s=\\{2T,0,1\\}$ indicate that the traces are over
traceless, symmetric matrices, scalars, and vectors, respectively.

Structurally, the traces (107) can be separated in traces without and with
operator insertion $V$. In order to evaluate the resulting expressions of the
first type, we use the early-time expansion of the heat-kernel
Vassilevich:2003xt

${\rm Tr}_{s}\left[e^{-s\Delta}\right]=\frac{1}{(4\pi s)^{d/2}}\,{\rm
tr}(\mathbb{1}_{s})\,\int
d^{d}x\sqrt{\bar{g}}\left(1+\frac{1}{6}s\bar{R}\right)+O(\bar{R}^{2})\,.$
(108)

The trace tr$(\mathbb{1}_{s})$ counts the number of independent field
components in each sector, i.e.,

$\begin{split}{\rm tr}(\mathbb{1}_{0})=1\,,\quad{\rm
tr}(\mathbb{1}_{1})=d\,,\quad{\rm
tr}(\mathbb{1}_{2T})=\frac{1}{2}(d-1)(d+2)\,.\end{split}$ (109)

The heat-kernel (108) can readily be extended to traces including functions of
the Laplacian $W(\Delta)$. Formally introducing the (inverse) Laplace
transform $\widetilde{W}(s)$ through
$W(z)=\int_{0}^{\infty}ds\,\widetilde{W}(s)\,e^{-sz}$, we write

${\rm
Tr}_{s}\left[W(\Delta)\right]=\int_{0}^{\infty}ds\,\widetilde{W}(s)\,{\rm
Tr}_{s}\left[e^{-s\Delta}\right]\,.$ (110)

Substituting the early-time expansion (108), then yields

${\rm Tr}_{s}\left[W(\Delta)\right]=\frac{1}{(4\pi)^{d/2}}\,{\rm
tr}(\mathbb{1}_{s})\,\int
d^{d}x\sqrt{\bar{g}}\,\left(Q_{d/2}[W]\,+\frac{1}{6}\,Q_{d/2-1}[W]\,\bar{R}\right)+O(\bar{R}^{2})\,,$
(111)

where the $Q$-functionals are defined by

$\begin{split}Q_{n}[W]\equiv\int_{0}^{\infty}ds\,s^{-n}\,\widetilde{W}(s)\,.\end{split}$
(112)

These functionals can be re-written in terms of the original function $W(z)$:

$\begin{split}Q_{n}[W]=&\,\frac{1}{\Gamma(n)}\int_{0}^{\infty}dz\,z^{n-1}W(z)\,,\qquad
n>0\,,\\\ Q_{0}[W]=&W(0)\,.\end{split}$ (113)

For $n<0$ one can always choose an integer $k$ such that $n+k>0$. Integrating
by parts, one then establishes that

$Q_{n}[W]=\frac{(-1)^{k}}{\Gamma(n+k)}\int_{0}^{\infty}dz\,z^{n+k-1}\,W^{(k)}(z)\,,\qquad
n<0,\quad n+k>0\,.$ (114)

At this point we note that the functions $W(z)$ appearing in (107) have the
generic form

$W(z)=\frac{G_{k}}{(z+R_{k}+w)^{p}}\,\partial_{t}\left(\frac{1}{G_{k}}R_{k}\right)\,.$
(115)

In this case, it is then convenient to trade the dimensionful $Q$-functionals
with the dimensionless threshold functions

$\begin{split}\Phi^{p}_{n}(w)\equiv&\,\frac{1}{\Gamma(n)}\int_{0}^{\infty}dz\,z^{n-1}\frac{R^{(0)}-zR^{(0)^{\prime}}(z)}{(z+R^{(0)}+w)^{p}}\,,\\\
\widetilde{\Phi}^{p}_{n}(w)\equiv&\,\frac{1}{\Gamma(n)}\int_{0}^{\infty}dz\,z^{n-1}\frac{R^{(0)}(z)}{(z+R^{(0)}(z)+w)^{p}}\,,\end{split}$
(116)

where $R^{(0)}(p^{2}/k^{2})$ is the dimensionless profile function associated
with the regulator $R_{k}(p^{2})=k^{2}R^{(0)}(p^{2}/k^{2})$. It is then
readily verified that

$Q_{n}\left[\frac{G_{k}}{(z+R_{k}+w)^{p}}\,\partial_{t}\left(\frac{1}{G_{k}}R_{k}\right)\right]=k^{2(n-p+1)}\left(2\Phi^{p}_{n}(w/k^{2})-\eta_{N}\widetilde{\Phi}^{p}_{n}(w/k^{2})\right)\,,$
(117)

where the anomalous dimension $\eta_{N}$ has been introduced in (85). For the
traces in the ghost sector, the $G_{k}$-dependence in (115) is absent, so that
the terms proportional to $\eta_{N}$ do not appear in this sector.

Starting from (107) the traces without potential insertions are readily
evaluated by combining Eq. (111) with the result for the Q-functionals (117).
The traces including the insertion of a potential can be evaluated along the
same lines. Formally, such traces can be evaluated using the off-diagonal
heat-kernel formulas provided in Benedetti:2010nr . Since the potentials (104)
do not contain any covariant derivatives and, owed to the projection
prescription (93), can be treated as covariantly constant leads to significant
simplifications though. In this case the relevant contributions are given by
the leading term in the early-time expansion (108) with ${\rm
tr}\left[\mathbb{1}_{s}\right]\rightarrow{\rm tr}\left[V_{s}\right]$. A brief
computation establishes that

$\begin{split}{\rm tr}_{1}\left[V_{1}\right]=&-\sqrt{2}\bar{R}\,,\\\ {\rm
tr}_{2T}\left[V_{2T}\right]=&\frac{1}{32\pi
G_{k}}\frac{(d+2)(d^{2}-3d+4)}{2d}\,\bar{R}\,.\end{split}$ (118)

The remaining terms in (107) are then found by pulling the contributions (118)
out of the operator trace and evaluating the latter by again combining Eqs.
(111) and (117). In this way one obtains the explicit form of the right-hand
side of (107). Reading off the coefficients multiplying the interaction
monomials (92) gives the equations governing the scale-dependence of the
dimensionful couplings $G_{k}$ and $\Lambda_{k}$.

In order to study the renormalization group fixed points of the system, it is
then natural to convert the dimensionful couplings to their dimensionless
counterparts (85). The scale-dependence of $g_{k}$ and $\lambda_{k}$ is
encoded in the beta functions

$\partial_{t}g_{k}=\beta_{g}(g_{k},\lambda_{k})\,,\qquad\partial_{t}\lambda_{k}=\beta_{\lambda}(g_{k},\lambda_{k})\,.$
(119)

The explicit computation yields Reuter:1996cp

$\begin{split}\beta_{g}(g,\lambda)=&\left(d-2+\eta_{N}\right)\,g\,,\\\
\beta_{\lambda}(g,\lambda)=&-(2-\eta_{N})\lambda+\frac{g}{2(4\pi)^{d/2-1}}\Big{(}2d(d+1)\Phi^{1}_{d/2}(-2\lambda)\\\
&\qquad\qquad-8d\Phi^{1}_{d/2}(0)-d(d+1)\eta_{N}\widetilde{\Phi}^{1}_{d/2}(-2\lambda)\Big{)}\,.\end{split}$
(120)

The anomalous dimension $\eta_{N}$ takes the form

$\eta_{N}(g,\lambda)=\frac{gB_{1}(\lambda)}{1-gB_{2}(\lambda)}\,,$ (121)

with

$\begin{split}B_{1}(\lambda)=&\frac{1}{3}(4\pi)^{1-d/2}\Big{(}d(d+1)\Phi^{1}_{d/2-1}(-2\lambda)-6d(d-1)\Phi^{2}_{d/2}(-2\lambda)\\\
&\qquad\qquad\qquad-4d\Phi^{1}_{d/2-1}(0)-24\Phi^{2}_{d/2}(0)\Big{)}\,,\\\
B_{2}(\lambda)=&-\frac{1}{6}(4\pi)^{1-d/2}\Big{(}d(d+1)\widetilde{\Phi}^{1}_{d/2-1}(-2\lambda)-6d(d-1)\widetilde{\Phi}^{2}_{d/2}(-2\lambda)\Big{)}\,.\end{split}$
(122)

At this point we have completed the explicit derivation of the beta functions
(120) governing the scale-dependence of $\\{g_{k},\lambda_{k}\\}$. The result
agrees with the initial derivation Reuter:1996cp , employing a maximally
symmetric background. The present derivation shows, however, that _this result
is actually background independent_. Apart from general properties related to
the existence of the heat-kernel (108), we never specified an explicit
background $\bar{g}_{\mu\nu}$. Assuming its mere existence is sufficient to
arrive at the final result.

###  4.2 Fixed points, RG trajectories, and phase diagram

The beta functions (119) encode the dependence of Newton’s coupling and the
cosmological constant on the coarse-graining scale $k$. These have been
derived for a generic regulator $R_{k}$. In order to investigate the resulting
fixed point structure and phase diagram, we specify the regulator to be of
Litim-type (31). In this case, the integrals appearing in the threshold
functions (116) can be evaluated analytically, yielding

$\Phi^{p}_{n}(w)^{\rm
Litim}=\frac{1}{\Gamma(n+1)}\frac{1}{(1+w)^{p}}\,,\quad\widetilde{\Phi}^{p}_{n}(w)^{\rm
Litim}=\frac{1}{\Gamma(n+2)}\frac{1}{(1+w)^{p}}\,.$ (123)

Upon substituting these expressions, the flow of $\\{g_{k},\lambda_{k}\\}$ is
governed by the coupled, non-linear, autonomous, first-order differential
equations (119) with

$\begin{split}\beta_{g}=&\left(2+\eta_{N}\right)\,g\,,\\\
\beta_{\lambda}=&-(2-\eta_{N})\lambda+\frac{g}{8\pi}\Big{(}\frac{20}{1-2\lambda}-16-\frac{5}{3}\,\eta_{N}\,\frac{1}{1-2\lambda}\Big{)}\,.\end{split}$
(124)

and

$\eta_{N}=\frac{g\left(\frac{5}{1-2\lambda}-\frac{9}{(1-2\lambda)^{2}}-7\right)}{3\pi\left(1+\frac{g}{12\pi}\left(\frac{5}{1-2\lambda}-\frac{6}{(1-2\lambda)^{2}}\right)\right)}\,.$
(125)

Here we have specified $d=4$ for explicitness.

We then determine the fixed points of this system. Since the truncation
retains a finite number of interaction monomials $\mathcal{O}_{i}$ only, this
search turns into the algebraic problem of finding the roots of the system
$\\{\beta_{\lambda}=0,\beta_{g}=0\\}$, cf. Table 2. Inspecting (124), one
finds two fixed points

$\begin{array}[]{ll}\text{GFP:}&\quad\\{g_{*}=0,\;\lambda_{*}=0\\}\,,\\\\[5.59721pt]
\text{NGFP:}&\quad\\{g_{*}=0.707,\;\lambda_{*}=0.193\\}\,.\end{array}$ (126)

These correspond to a free and interacting theory, respectively. The NGFP is
_the projection_ of the Reuter fixed point onto the subspace spanned by the
ansatz (84).

The stability properties of the RG flow in the vicinity of these fixed point
are readily obtained by evaluating the stability matrix (10) for the beta
functions (124). This yields

$\begin{array}[]{ll}\text{GFP:}&\quad\\{\theta_{1}=2,\;\theta_{2}=-2\\}\,,\\\\[5.59721pt]
\text{NGFP:}&\quad\\{\theta_{1,2}=1.48\pm 3.04i\\}\,.\end{array}$ (127)

Thus the GFP constitutes a saddle point with one UV-attractive and one UV-
repulsive eigendirection. Analyzing the corresponding eigenvectors shows that
RG trajectories with a non-vanishing Newton’s coupling are repelled by the GFP
as $k\rightarrow\infty$. Hence this fixed point cannot act as the UV-
completion of gravity. In contrast, the NGFP is UV-attractive for both $g_{k}$
and $\lambda_{k}$. The complex stability coefficients indicate that the RG
flow spirals into the fixed point as $k\rightarrow\infty$. Thus this fixed
point acts as UV-completion for the RG trajectories entering its vicinity.

|  $d$ |  $g_{*}$ |  $\lambda_{*}$ |  $\theta_{1}$ |  $\theta_{2}$
---|---|---|---|---|---
GFP | $d$ | $0$ | $0$ | $2-d$ | $2$
NGFP | $2+\epsilon$ | $\frac{3}{38}\epsilon$ | $-\frac{3}{38}\epsilon$ | $\epsilon$ | $2+\frac{1}{19}\epsilon$
NGFP | $3$ | $0.20$ | $0.06$ | $1.15\pm 0.83i$
NGFP | $4$ | $0.71$ | $0.19$ | $1.48\pm 3.04i$
NGFP | $5$ | $2.85$ | $0.24$ | $2.69\pm 5.15i$

Table 3:  Characteristics of the family of NGFPs in various dimensions $d$
Souma:1999at ; Reuter:2001ag . The table on the left gives the position and
stability coefficients of the fixed points for selected values of $d$. The
diagram to the right displays that the family emerges from the Gaussian fixed
point in $d=2+\epsilon$ and continuously connects to the Reuter fixed point in
$d=4$. Whether there is an upper critical dimension where the family of fixed
points ceases to exist is currently an open question.

Treating the dimension $d$ as a continuous parameter, one can trace the
properties of the NGFP when performing an analytic continuation of the
spacetime dimension. The results are summarized in Table 3. The table in the
left panel gives the position and stability coefficients of the NGFP for
selected values $d$ while the diagram in the right panel shows the position
$g_{*}(d)$. The latter illustrates that the family of NGFPs emerges from the
GFP in $d=2+\epsilon$ dimensions. It can then be analytically continued up to
$d=4$. Thus the Reuter fixed point is the analytic continuation of the NGFP
seen in the $\epsilon$-expansion around the free theory at the lower critical
dimension $d=2$ Souma:1999at ; Reuter:2001ag . For $d>4$, the system (124)
suggests that the NGFP continues to exist for all dimensions $d>2$
Litim:2003vp . At $d\gtrsim 5$, the existence of the fixed point turns into a
regulator-dependent statement though Reuter:2001ag . Hence, it is currently
unclear if there is an upper critical dimensions where the family of NGFPs
ceases to exist.

The system (124) is readily integrated numerically. The resulting phase
diagram is governed by the interplay of the fixed points (126) and shown in
Fig. 2.

Figure 2:  Phase diagram constructed from integrating the beta functions (119)
in $d=4$. All arrows point towards lower coarse-graining scale $k$. The GFP
and NGFP are marked by the black dots while the magenta line displays the
locus where the anomalous dimension $\eta_{N}$ diverges. The NGFP acts as an
UV-attractor capturing all trajectories in its vicinity. Lowering the coarse-
graining scale, the flow undergoes a crossover towards the GFP. The separatrix
connecting the two fixed points is highlighted in blue. This solution leads to
a vanishing cosmological constant $\Lambda_{0}=0$. Trajectories to the left of
this line, exemplified by the orange trajectory, have been classified as Type
Ia and are characterized by $\Lambda_{0}<0$. RG trajectories to its right
constitute the Type IIIa solutions (represented by the green trajectory). They
terminate at a finite value of $k$ and lead to positive values
$\Lambda_{k}>0$. (Initially constructed in Reuter:2001ag ).

Here the magenta line indicates the position of a singular locus where
$\eta_{N}$ diverges. The physically relevant part of the phase diagram
consists of the RG trajectories with emanate from the NGFP in the UV and cross
over to the GFP as $k$ decreases. A special role is thereby played by the
separatrix (blue line) connecting the two fixed points. This trajectory leads
to a vanishing cosmological constant $\lim_{k\rightarrow 0}\Lambda_{k}=0$. The
trajectories to the right of this line are classified as Type Ia (orange
line). Their characteristic feature is a negative cosmological constant,
$\lim_{k\rightarrow 0}\Lambda_{k}<0$. The trajectories to the right of the
separatrix (green line) have been labeled Type IIIa. They terminate at the
singular locus at a finite value of $k$. In the vicinity of the GFP they
exhibit a regime where $\Lambda_{k}$ is constant and positive. It is expected
that nature is described by an RG trajectory within this class Reuter:2004nx .
This trajectory is special in the sense that it almost hits the GFP. Only at
the very last moment, it takes a turn flowing away from the fixed point. In
this way the trajectory accommodates the tiny value of the cosmological
constant observed in cosmology. Thus, the Einstein-Hilbert truncation does not
predict the value of the cosmological constant. It is a function of the free
parameters labeling the RG trajectories leaving the NGFP. The cosmological
constant then has the role of an experimental input which is used to identify
the RG trajectory realized in nature.

Figure 3:  Dependence of the dimensionful Newton’s coupling (left panel) and
cosmological constant (right panel) on the coarse-graining scale along a
typical RG trajectory of Type IIIa. The flow interpolates between the
classical regime ($k\ll 1$) where $G_{k}$ and $\Lambda_{k}$ are constant and
the fixed point regime ($k\gg 1$) where $G_{k}\propto k^{-2}$ and
$\Lambda_{k}\propto k^{2}$. By definition, the cross-over between these
scaling regimes occurs at the Planck scale $M_{\rm Pl}\equiv G_{0}^{-1/2}$
which is generated dynamically when flowing away from the NGFP. Notably,
$\Lambda_{k}$ exhibits an intermediate scaling regime where
$\Lambda_{k}\propto k^{4}$. All quantities are measured in units of $M_{\rm
Pl}$. (Adaptation from Gubitosi:2018gsl ).

At this stage, it is instructive to pick a generic RG trajectory of Type IIIa
and illustrate the $k$-dependence of the dimensionful couplings. The resulting
flow of $G_{k}$ and $\Lambda_{k}$ is exemplified in Fig. 3 where all
dimensionful quantities are given in units of the Planck scale $M_{\rm
Pl}=G_{0}^{-1/2}$. For $k>1$ the scale-dependence is governed by the NGFP
while for $k<1$ the flow is controlled by the GFP. As a result, the flow of
the couplings interpolates between the scaling regimes

$\begin{array}[]{llll}\text{NGFP:}&\quad G_{k}\simeq
g_{*}k^{-2}\,,&\quad\Lambda_{k}\simeq\lambda_{*}k^{2}&\quad
k>1\,,\\\\[5.16663pt] \text{GFP:}&\quad G_{k}\simeq
G_{0}\,,&\quad\Lambda_{k}\simeq\Lambda_{0}&\quad k<1\,.\end{array}$ (128)

The crossover between the two regimes occurs at the Planck scale. This scale
is generated dynamically when moving away from the NGFP. Classical general
relativity (in the sense of a low-energy effective field theory) is then
recovered in the vicinity of the GFP.

We conclude by stressing that the exact fixed point action $\Gamma_{*}$
associated with the Reuter fixed point does (most likely) not coincide with
the Einstein-Hilbert action. While this may be suggested by the analysis of
this section, one has to account for the fact that we have been working within
a projection of the full theory space to this two-dimensional subspace.
Additional contributions to $\Gamma_{*}$, as, e.g., higher-derivative terms,
are not visible in this analysis.

###  4.3 Further reading

The Einstein-Hilbert truncation discussed in this section constitutes the
starting point for understanding the theory space of gravity, its RG fixed
points, and their mutual relations. By now, this exploration has made
significant progress in moving beyond this basic example. At the level of the
background approximation $f(R)$-type projections have been studied in
polynomial approximations to very high order Codello:2007bd ; Machado:2007ea ;
Falls:2013bv ; Falls:2017lst ; Falls:2018ylp  and it has been shown that the
NGFP also persists once the two-loop counterterm identified by Goroff and
Sagnotti Goroff:1985sz ; Goroff:1985th  is included in the projection
Gies:2016con . Within the fluctuation approach there has been significant
progress on understanding the momentum-structure of the graviton propagator
Christiansen:2014raa ; Bonanno:2021squ  and resolving the momentum-dependence
of three- and four-point vertices Christiansen:2015rva ; Denz:2016qks . In
parallel, a program geared towards developing asymptotically safe amplitudes
has been initiated in Draper:2020bop . Covering these developments in detail
is beyond the scope of this introductory chapter and the interested reader may
consult the more advanced chapters of this volume for further information.

##  5 Concluding comments

The Wetterich equation (2) constitutes an essential tool in developing the
gravitational asymptotic safety program. Starting from its adaption to gravity
Reuter:1996cp , it has provided substantial evidence for the existence of a
viable interacting renormalization group fixed point – the Reuter fixed point
– which could provide a consistent and predictive high-energy completion of
the gravitational interactions.

The present chapter focused on the case where the gravitational degrees of
freedom are carried by the metric field. The applicability of the functional
renormalization group and in particular the Wetterich equation is not limited
to this setting though. It has readily been extended to other sets of fields
which, at the classical level, encode the same gravitational dynamics as
general relativity. Notably, this includes the case where the gravitational
degrees of freedom are encoded in the vielbein (“tetraed only”-formulation)
Harst:2012ni ; Dona:2012am , the Palatini formalism Harst:2014vca ;
Harst:2015eha ; Pagani:2015ema ; Gies:2022ikv , the Arnowitt-Deser-Misner
decomposition Manrique:2011jc ; Rechenberger:2012dt ; Biemans:2016rvp ;
Biemans:2017zca ; Houthoff:2017oam , and unimodular gravity Eichhorn:2013xr ;
Eichhorn:2015bna ; Percacci:2017fsy ; deBrito:2020xhy . While the exploration
of the corresponding theory spaces is far less developed than the one for the
metric theory, there are indications that these settings also possess
interacting renormalization group fixed points suitable for rendering the
construction asymptotically safe. In the case of unimodular gravity, there are
arguments that the theory is in the same universality class as the metric
formulation deBrito:2021pmw . Whether the other fixed points are in the
universality class of the Reuter fixed point is an open question though.

Notably, there also has been progress aiming at the implementation of the
renormalization group on discrete geometries. In the context of the Causal
Dynamical Triangulation program Ambjorn:2012jv ; Loll:2019rdj ,
renormalization group flows have been constructed in Ambjorn:2014gsa . The
underlying idea is to pick an observable whose value is held constant when
varying the parameters of the Monte Carlo simulation. This led to the
surprising conclusion that the phase-transition line expected to provide the
high-energy completion of the theory actually appears to be approached in the
infrared.

So far, our discussion has focused on gravitational degrees of freedom only.
It is rather straightforward to extend this construction by including
additional matter fields as well as all the building blocks of the standard
model of particle physics Eichhorn:2018yfc ; Eichhorn:2022gku . Many of the
gravity-matter systems investigated to date exhibit interacting
renormalization group fixed points whose properties are very similar to the
ones found for the Reuter fixed point. Since the beta functions encoding the
fixed point structure of these systems depend on the number of matter fields
in a continuous way, it is likely that the Reuter fixed point is part of a
continuous web of interacting fixed points. Since it is unlikely that these
encode the same universality class, it is suggestive to refer to these as
deformed Reuter fixed points, highlighting that the systems actually realize
different (albeit related) universal behaviors. A detailed summary of the
state-of-the-art in investigating asymptotically safe gravity-matter systems
is beyond the scope of this elementary introduction and we refer to the recent
reviews Eichhorn:2018yfc ; Eichhorn:2022gku  as well as other chapters of this
book. In short, it is conceivable though that the asymptotic safety mechanism
may lead to a unified theory incorporating the standard model of particle
physics and gravity within the framework of a relativistic quantum field
theory. This exciting perspective certainly warrants further investigation.

## Acknowledgements

My understanding of the functional renormalization group and its applications
in the context of gravity has benefited enormously from countless discussions
with many colleagues. It is therefore my pleasure to thank J. Ambjørn, D.
Becker, W. Beenakker, A. Bonanno, L. Bosma, T. Budd, L. Buoninfante, J.
Donoghue, T. Draper, A. Eichhorn, R. Ferrero, G. Gubitosi, R. Kleiss, A.
Koshelev, R. Loll, M. Niedermaier, R. Ooijer, C. Pagani, J. M. Pawlowski, R.
Percacci, A. D. Pereira, S. Pirlo, A. Platania, M. Reichert, M. Schiffer, O.
Zanusso, and C. Wetterich for sharing their insights and views. In addition, I
want to thank M. Becker and A. Ferreiro for their insightful comments on the
manuscript and B. Knorr and C. Ripken for their close collaboration on many
aspects presented in this review. Finally, I would like to thank M. Reuter for
introducing me to this subject and his continual advice and support.

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