aainstitutetext: Institute for Mathematics, Astrophysics and Particle Physics
(IMAPP), Radboud University, Heyendaalseweg 135, 6525 AJ Nijmegen, The
Netherlands

# Foliated asymptotically safe gravity
in the fluctuation approach

Frank Saueressig<EMAIL_ADDRESS>a Jian Wang
<EMAIL_ADDRESS>
###### Abstract

The gravitational asymptotic safety program envisions a high-energy completion
of gravity based on a non-Gaussian renormalization group fixed point. A key
step in this program is the transition from Euclidean to Lorentzian signature
spacetimes. One way to address this challenge is to formulate the quantum
theory based on the Arnowitt-Deser-Misner decomposition of the metric field.
This equips the Euclidean spacetime with a preferred direction which may serve
as the time-direction in the Lorentzian setting. In this work we use the
Wetterich equation in order to compute the renormalization group flow of the
graviton two-point function. The resulting beta functions possess a non-
Gaussian renormalization group fixed point suitable for rendering the theory
asymptotically safe. The phase diagram underlying the flow of the two-point
function is governed by the interplay between this non-Gaussian fixed point,
the Gaussian fixed point, and an infrared fixed point. The latter ensures that
the renormalized squared graviton mass cannot take negative values. These
results are in qualitative agreement with fluctuation computations carried out
in the covariant setting. We take this as non-trivial evidence that the
asymptotic safety mechanism remains intact when considering quantum gravity on
spacetimes carrying a foliation structure. Technically, our work constitutes
the first fluctuation computation carried out within the ADM-framework.
Therefore, we also provide a detailed discussion of the conceptual framework,
highlighting the elements which differ from fluctuation computations in the
covariant setting.

##  1 Introduction

Time is an intrinsic building block in our description of nature. General
relativity implements this structure by the spacetime metric coming with
indefinite signature. Adopting the mostly plus convention, the metric of
Minkowski space is $\eta_{\mu\nu}={\rm diag}(-1,+1,+1,+1)$. This entails that
the spacetime geometry $\mathcal{M}$ possesses a “preferred direction”
inducing a foliation structure, $\mathcal{M}=\mathbb{R}\times\Sigma_{\tau}$.
Here $\tau\in\mathbb{R}$ gives the “time-coordinate” of an event and
$\Sigma_{\tau}$ are the spatial slices defined by the points in $\mathcal{M}$
with the same value of $\tau$.

The spacetime metric determines the causal order of events. The principle of
causality then states that no effect should precede its cause. In general
relativity this is realized by the propagation of signals being confined to
the lightcone. At the level of a quantum field theory in a fixed Minkowski
spacetime causality entails that correlation functions vanish outside the
light cone. The concept of causality becomes highly non-trivial once one
departs from a classical spacetime and moves into the realm of quantum
gravity. In this case, quantum fluctuations in the lightcone structure could,
e.g., lead to a violation of causality at microscopic scales Donoghue:2019ecz
. Our understanding of these effects is far from complete though. Owed to
technical reasons, many investigations of quantum gravity effects work in an
Euclidean setting, where questions related to the propagation of fields are
difficult to assess. Moreover, a generic Euclidean spacetime may not support
the additional geometrical structures required to transit from Euclidean to
Lorentzian signature settings, so that the analytic continuation only works
locally Baldazzi:2018mtl . Having a well-defined time-direction even in the
Euclidean setting requires a foliation of spacetime into spatial slices which
are then welded together to form spacetime.

Technically, the structures required for the existence of such a foliation can
be provided by the Arnowitt-Deser-Misner (ADM)-decomposition of the spacetime
metric Arnowitt:1962hi  (also see Gourgoulhon:2007ue  for a review),

$g_{\mu\nu}\mapsto(N,N_{i},\sigma_{ij})\,.$ (1)

In this case, the metric degrees of freedom are encoded in the lapse function
$N$, the shift vector $N_{i}$, and the metric on the spatial slices
$\sigma_{ij}$. The resulting foliation allows to implement the analogue of
time in the Euclidean setting. This opens the possibility of an analytic
continuation to Lorentzian signature. Moreover, transverse-traceless
fluctuations of $\sigma_{ij}$ carry two degrees of freedom (polarizations)
which are naturally associated with the two physical degrees of freedom
related to the graviton Arnowitt:1962hi ; Bardeen:1980kt ; Craps:2014wga .

As an intermediate step towards a full-fledged theory of quantum gravity
formulated at Lorentzian signature, one may study an Euclidean setting
implementing the foliation structure of spacetime. In the language of
Isham:1992ms , this corresponds to a setting in which time is identified
before quantizing. Within the gravitational asymptotic safety program,
reviewed e.g. in Percacci:2017fkn ; Reuter:2019byg ; Saueressig:2023irs ;
Reichert:2020mja ; Niedermaier:2006wt ; Codello:2008vh ; Reuter:2012id ;
Eichhorn:2018yfc ; Pawlowski:2020qer ; Bonanno:2020bil , the construction of
renormalization group (RG) flows based on ADM-variables has been developed in
Manrique2011 ; Rechenberger:2012dt ; Contillo:2013fua ; Biemans:2016rvp ;
Biemans:2017zca ; Houthoff:2017oam , also see Knorr:2018fdu  for an
implementation of a foliation structure based on a gauge-fixing construction
and Eichhorn:2019ybe ; Knorr:2022mvn  for further discussions.111For recent
steps towards formulating RG flows directly in the Lorentzian setting also see
DAngelo:2022vsh ; Banerjee:2022xvi ; DAngelo:2023tis . Conceptually, this line
differs from the covariant setting followed in e.g. Reuter:1996cp ;
Lauscher:2001ya ; Codello:2007bd ; Machado:2007ea ; Benedetti:2009rx ;
Dietz:2012ic ; Eichhorn:2012va ; Dona:2013qba ; Christiansen:2015rva ;
Gies:2016con ; Falls:2017lst ; Eichhorn:2017eht ; Eichhorn:2018akn ;
Fehre:2021eob ; Laporte:2021kyp ; Knorr:2021slg ; Knorr:2022dsx ; Pastor-
Gutierrez:2022nki ; deBrito:2023myf  in the sense that in the latter case time
plays no role at all. This also has profound consequences for the concept of
background independence. The construction of the Wetterich equation for
gravity Reuter:1996cp  makes manifest use of the background field formalism.
Background independence is then achieved by quantizing the fluctuation field
in all backgrounds simultaneously Becker:2014qya ; Dietz:2015owa ;
Pagani:2019vfm ; Becker:2020mjl , also see Labus:2016lkh ; Mandric:2022dte
for recent developments. The adaptation to the ADM-formalism also builds on
the background formalism and follows the same logic. The conceptual difference
is that the backgrounds in the ADM-formalism admit a foliation structure by
construction. In this sense, the formalism quantizes the fluctuation field in
all backgrounds carrying a foliation structure. This guarantees that the
background spacetimes possess a Lorentzian analogue. The foliation structure
is also an essential element of the Causal Dynamical Triangulations program
Ambjorn:2012jv ; Loll:2019rdj , initiated in Ambjorn:1998xu ; Ambjorn:2000dv .

In this work we follow the path initiated in Manrique:2011jc  and use the
Wetterich equation to investigate the RG flow of gravity with the
gravitational degrees of freedom carried by the ADM-variables (1). The goal of
our study is to identify the non-Gaussian fixed points (NGFPs) which could
provide a phenomenologically viable high-energy completion of the theory via
the asymptotic safety mechanism. Our work goes beyond the background
approximation by studying the flow of the graviton two-point function sourced
by the three- and four-point vertices of the theory. It constitutes the first
investigation of the asymptotic safety mechanism for foliated spacetimes in a
fluctuation computation.

Given the novel type of computation, we provide a detailed exposition on
setting up fluctuation computations in the ADM-framework. In particular, we
highlight the novel features appearing in the projection onto the vertices of
the theory which are absent in the covariant approach. Within this general
setting, we perform an explicit study of the RG flow of the graviton two-point
function obtained from the Einstein-Hilbert action and its extensions encoding
the effect of different speeds of light. As our main result, we identify a
NGFP suitable for rendering the construction asymptotically safe. While
fluctuation computations projecting the RG flow on different sets of couplings
than the background computation, we find that the stability properties of the
fixed point in these conceptually different approximations are very similar.
We interpret this result as evidence for the robustness of the background
computations as well as a first hint on the manifestation of effective
universality Eichhorn:2018akn  in the foliation setting. Moreover, it provides
a strong indication that the asymptotic safety mechanism for gravity is robust
when transiting from the Euclidean setting to backgrounds carrying a foliation
structure.

The remainder of this work is organized as follows. Sects. 2 and 3 review the
ADM-decomposition of the metric field and the Wetterich equation formulated on
foliated spacetimes, respectively. In particular, Sect. 3.3 gives a general
introduction to fluctuation field computations in this setting. The beta
functions arising from the foliated Einstein-Hilbert truncation and its non-
relativistic two-derivative extensions are derived in Sect. 4 and the
resulting fixed point structure and phase diagrams are given in Sect. 5. We
conclude with a discussion and outlook in Sect. 6. Technical details
underlying our computation have been relegated to two appendices.

##  2 Foliated spacetimes in the ADM-formalism

We start by reviewing the Arnowitt-Deser-Misner (ADM) decomposition of the
spacetime metric Arnowitt:1962hi . In this case, the gravitational degrees of
freedom are encoded in the ADM-fields $(N,N_{i},\sigma_{ij})$. Let
$\mathcal{M}$ be a $d+1$-dimensional manifold equipped with a Euclidean
signature metric $g_{\mu\nu}$ and coordinates $x^{\mu}$. Here Greek letters
$\mu,\nu,\cdots$ denotes spacetime indices taking values from $1$ to $d+1$. We
assume that $\mathcal{M}$ can be foliated by a family of spatial hypersurfaces
$\Sigma_{\tau}$, labeled by the “time”-parameter $\tau$. Points in the same
spatial slice then share the same time-coordinate $\tau$ and are labeled by
spatial coordinates $y^{i}$, $i=1,2,\cdots,d$ on $\Sigma_{\tau}$. We then
perform a change of coordinates

$x^{\mu}\mapsto(\tau,y^{i})$ (2)

and introduce the basis vectors

$t^{\alpha}\equiv\frac{\partial
x^{\alpha}}{\partial\tau}\bigg{|}_{y^{i}}\,,\qquad
e^{\alpha}_{i}\equiv\frac{\partial x^{\alpha}}{\partial
y^{i}}\bigg{|}_{\tau}.$ (3)

We further define the unit vector $n_{\alpha}$, normal to the surface
$\Sigma_{\tau}$,

$n_{\alpha}\equiv N\partial_{\alpha}\tau\,,\qquad
n_{\alpha}e_{i}^{\alpha}=0\,.$ (4)

Here the lapse function $N(\tau,y^{i})$ is introduced as a normalization
factor. The vector $t^{\alpha}$ can then be decomposed into components normal
and tangent to the surface $\Sigma_{\tau}$

$t^{\alpha}=Nn^{\alpha}+N^{i}e^{\alpha}_{i}\,,$ (5)

where $N^{i}(\tau,y^{i})$ is called the shift vector.

Next, we apply the change of coordinates (2) to the line element

$ds^{2}=g_{\mu\nu}dx^{\mu}dx^{\nu}=g_{\mu\nu}(t^{\mu}d\tau+e^{\mu}_{i}dy^{i})(t^{\nu}d\tau+e^{\nu}_{j}dy^{j})\,.$
(6)

Substituting the decomposition (5) and introducing the induced metric on
$\Sigma_{\tau}$, $\sigma_{ij}\equiv g_{\mu\nu}e^{\mu}_{i}e^{\nu}_{j}$, the
line element can be expressed in terms of the lapse, shift, and $\sigma_{ij}$:

$ds^{2}=(N^{2}+\sigma_{ij}N^{i}N^{j})d\tau^{2}+2\sigma_{ij}N^{i}d\tau
dy^{j}+\sigma_{ij}dy^{i}dy^{j}.$ (7)

Comparing eqs. (6) and (7) then allows to express the spacetime metric in
terms of the ADM-fields,

$g_{\mu\nu}=\begin{pmatrix}N^{2}+N^{i}N_{i}~{}~{}~{}&N_{j}\\\
N_{i}~{}~{}~{}&\sigma_{ij}\\\ \end{pmatrix}\,,\qquad
g^{\mu\nu}=\begin{pmatrix}\frac{1}{N^{2}}~{}~{}~{}&-\frac{N^{j}}{N^{2}}\\\
-\frac{N^{i}}{N^{2}}~{}~{}~{}&\sigma^{ij}+\frac{N^{i}N^{j}}{N^{2}}\\\
\end{pmatrix},$ (8)

where the spatial indices $i,j$ are raised and lowered with the induced metric
$\sigma_{ij}$.

Next we consider the transformation properties of the spacetime metric
$g_{\mu\nu}$ under the full diffeomorphism group. For a general
$d+1$-dimensional infinitesimal coordinate transformation
$v^{\mu}(\tau,y^{i})$, the metric transforms as

$\delta g_{\mu\nu}=\mathcal{L}_{v}g_{\mu\nu},$ (9)

where $\mathcal{L}_{v}$ is the Lie derivative of the metric with respect to
the vector $v^{\mu}$. Decomposing the infinitesimal coordinate transformation
into spatial and time-part,

$v^{\mu}(\tau,y^{i})=(f(\tau,y^{i}),\xi^{i}(\tau,y^{i}))\,,$ (10)

eq. (9) induces the following transformation of the ADM-fields

$\begin{split}\delta
N=&\partial_{\tau}(fN)+\xi^{k}\partial_{k}N-NN^{i}\partial_{i}f,\\\ \delta
N_{i}=&\partial_{\tau}(N_{i}f)+\xi^{k}\partial_{k}N_{i}+N_{k}\partial_{i}\xi^{k}+\sigma_{ki}\partial_{\tau}\xi^{k}+N_{k}N^{k}\partial_{i}f+N^{2}\partial_{i}f,\\\
\delta\sigma_{ij}=&f\partial_{\tau}\sigma_{ij}+\xi^{k}\partial_{k}\sigma_{ij}+\sigma_{jk}\partial_{i}\xi^{k}+\sigma_{ik}\partial_{j}\xi^{k}+N_{j}\partial_{i}f+N_{i}\partial_{j}f.\end{split}$
(11)

We also give the transformation for $N^{i}$ for completeness

$\delta
N^{i}=\partial_{\tau}(N^{i}f)+\xi^{k}\partial_{k}N^{i}-N_{k}\partial_{k}\xi^{i}+\partial_{\tau}\xi^{i}-N^{i}N^{j}\partial_{j}f+N^{2}\sigma^{ij}\partial_{j}f.$
(12)

The full diffeomorphism group contains an important subgroup, the foliation
preserving diffeomorphisms. Their action is obtained by restricting the
function $f(\tau,y^{i})$ to functions of $\tau$ only.

Finally, we are interested in constructing an action describing the dynamics
of the spatial metric $\sigma_{ij}$. At this point, we restrict ourselves to
interactions containing at most two derivatives with respect to the spacetime
coordinates. While this property is not invariant under the RG flow which
inevitably generates higher-order derivative interactions, the resulting
action serves as a starting point for generating the tensor structures which
will be tracked in the RG computation later on.

Since the foliation equips the manifold with a preferred time-direction, it is
natural to construct the action from the view of non-relativistic theories
where we have the kinetic term first and subsequently add a potential. To
describe how the spatial metric $\sigma_{ij}$ changes between different
spatial surfaces $\Sigma_{\tau}$, we introduce the extrinsic curvature

$K_{ij}\equiv\frac{1}{2}\mathcal{L}_{n}\sigma_{ij}=\frac{1}{2}N^{-1}(\partial_{\tau}\sigma_{ij}-D_{i}N_{j}-D_{j}N_{i})\,.$
(13)

Here $\mathcal{L}_{n}$ is the Lie derivative of spatial metric with respect to
the normal vector $n^{\alpha}$ and $D_{i}$ is the covariant derivative on the
spatial slice, carrying the Levi-Civita connection constructed from
$\sigma_{ij}$. Since the extrinsic curvature $K_{ij}$ also measures the rate
of change of the spatial metric along the time direction, one can construct
the kinetic term in terms of $K_{ij}$,

$S^{K}=\frac{1}{16\pi G}\int d\tau
d^{d}yN\sqrt{\sigma}(\alpha_{1}\,K^{ij}K_{ij}-\alpha_{2}K^{2})\,,$ (14)

with $K=K_{ij}\sigma^{ij}$ being the trace of the extrinsic curvature. Here
the coupling $G$ will turn out to be Newton’s constant and $\alpha_{1}$ and
$\alpha_{2}$ are parameters giving the relative weight between the two kinetic
terms.

The potential contains all terms which are independent of time derivatives and
compatible with the symmetries we want to impose. Insisting on diffeomorphism
invariance on the spatial slice and at most two spatial derivatives, this
limits the construction to the volume element and the integrated spatial
curvature $\leftidx{{}^{(d)}}R$. Thus,

$S^{V}=\frac{1}{16\pi G}\int d\tau
d^{d}yN\sqrt{\sigma}(-\leftidx{{}^{(d)}}R+2\Lambda)\,,$ (15)

with $\Lambda$ denoting the cosmological constant. Combining eqs. (14) we
arrive at

$S=\frac{1}{16\pi G}\int d\tau
d^{d}yN\sqrt{\sigma}{(\alpha_{1}K^{ij}K_{ij}-\alpha_{2}K^{2}-\leftidx{{}^{(d)}}R+2\Lambda)}.$
(16)

Since we started from the non-relativistic theory, all terms in the kinetic
and potential terms have their own independent couplings, and we denote the
couplings for the kinetic terms as $\alpha_{1}$ and $\alpha_{2}$. The
couplings for the spatial curvature and volume element are given by the Newton
coupling and cosmological constant respectively, in order to keep consistency
with other works. The parameters $\alpha_{1}$ and $\alpha_{2}$ introduce a
relative scaling between time and spatial directions.222This is reminiscent of
CDT Ambjorn2005 ; Ambjorn:2010hu  where the relative scaling of fundamental
(squared) length of space and time also appears. For
$\alpha_{1}=\alpha_{2}=1$, the action (16) is the Einstein-Hilbert action, and
one recovers invariance under the full diffeomorphism group. In general, the
action (16) is invariant under foliation preserving diffeomorphisms only.

We also remark that from a physics perspective, the parameterization (16) is
redundant in the sense that one of the couplings (canonically $\alpha_{1}$)
can be fixed to $\alpha_{1}=1$ by a rescaling of the lapse $N$ followed by a
redefinition of the other couplings. The second parameter in the kinetic part
has a physical meaning though. It captures a difference in the propagation
speed for the trace- and transverse-traceless modes of the spatial metric
fluctuation Loll:2014xja . At this point, we keep couplings for all operators
contained in (16) though.

##  3 The Wetterich equation on foliated spacetimes

The Wetterich equation Wetterich:1992yh ; Morris1994 ; Reuter:1993kw ;
Reuter:1996cp  encodes the change of the effective average action $\Gamma_{k}$
when integrating out quantum fluctuations with momenta close to the coarse-
graining scale $k$ Pawlowski:2005xe ; Gies2012 ; Percacci:2017fkn ;
Reuter:2019byg ; Bonanno:2020bil ; Dupuis:2020fhh ; Saueressig:2023irs . In
this way, it realizes the Wilsonian picture of renormalization. Its adaptation
to the ADM-decomposition has been made in Manrique2011  with further details
given in Rechenberger:2012dt . In this section, we review the key ingredients
of the construction and explain the setup for performing computations tracking
the fluctuation fields.

###  3.1 General background on the functional renormalization group

The most-frequently used tool for calculating RG flows in theories containing
gravitational degrees of freedom is the the Wetterich equation
Wetterich:1992yh ; Morris1994 ; Reuter:1993kw ; Reuter:1996cp . This equation
captures the dependence of the effective average action $\Gamma_{k}$ on the
coarse-graining scale $k$. It realizes the Wilsonian idea of renormalization
in the sense that it captures the change of the effective description of the
system when integrating out quantum fluctuations with momenta $p^{2}\simeq
k^{2}$. The equation takes a one-loop form and is given by Wetterich:1992yh ;
Morris1994

$\partial_{t}\Gamma_{k}=\frac{1}{2}\text{STr}\left[\left(\Gamma^{(2)}_{k}+\mathcal{R}_{k}\right)^{-1}\,\partial_{t}\mathcal{R}_{k}\right]\,.$
(17)

Here $t\equiv\text{ ln }(k/k_{0})$ denotes the RG time with $k_{0}$ being an
arbitrary reference scale and $\Gamma^{(2)}_{k}$ is the second functional
derivative of $\Gamma_{k}$ with respect to the fluctuation fields. The
regulator ${\mathcal{R}}_{k}(p^{2})$ equips the fluctuations with momenta
$p^{2}\ll k^{2}$ with a $k$-dependent mass term and vanishes for momenta
$p^{2}\gg k^{2}$. The latter property ensures that the trace on the right-hand
side is free from UV-divergences since its argument vanishes sufficiently fast
for high momenta. Both $\Gamma^{(2)}_{k}$ and ${\mathcal{R}}_{k}$ are matrix-
valued in field space. The supertrace STr then includes an integration over
loop-momenta as well as sums over all fluctuation fields and internal indices.
Moreover it provides a minus-sign to the contribution of the ghost fields.

In practice one obtains non-perturbative, approximate solutions of (17) by
projecting the exact equation to a subspace of all admissible action
functionals $\mathcal{O}_{i}$ constructable from a given field content,

$\Gamma_{k}\simeq\sum_{i}\bar{u}^{i}(k)\,\mathcal{O}_{i}\,.$ (18)

The $k$-dependence of $\Gamma_{k}$ is then captured by the dimensionful
couplings $\bar{u}^{i}(k)$. When analyzing the RG flow it is convenient to
consider the dimensionless versions of these couplings constructed with
respect to the coarse-graining scale $k$,
$u^{i}(k)=\bar{u}^{i}(k)k^{-[d_{i}]}$, where $[d_{i}]$ is the mass-dimension
of $\bar{u}^{i}$. Substituting (18) into (17) and matching the coefficients
multiplying the functionals $\mathcal{O}_{i}$ on its left and right-hand side
gives the beta functions encoding the flow of the couplings

$\partial_{t}u^{i}(k)=\beta_{u^{i}}(u^{j}(k))\,.$ (19)

The solutions of this system are called RG trajectories.

The most important properties of the beta functions are their fixed points
$(u^{i}_{*})$ where

$\beta_{u^{i}}(u^{j}_{*})=0\,,\qquad\forall\,i\,.$ (20)

Depending on whether the fixed point action corresponds to a free theory or
admits interactions, one distinguishes between a Gaussian fixed point (GFP)
and a non-Gaussian fixed point (NGFP).

In the vicinity of a fixed point, the properties of the RG flow can be
obtained by linearizing the system (19)

$\partial_{t}u^{i}(k)=\sum_{j}B^{i}{}_{j}(u^{j}-u^{j}_{*})+O((u^{j}-u^{j}_{*})^{2})\,,\quad
B^{i}{}_{j}=\left.\frac{\partial\beta_{u^{i}}}{\partial
u^{j}}\right|_{u^{i}=u^{i}_{*}}\,.$ (21)

The stability properties of the flow are captured by the stability
coefficients $\theta_{I}$, defined as minus the eigenvalues of the stability
matrix $B^{i}{}_{j}$. For eigendirections of $B^{i}{}_{j}$ associated with
coefficients with Re$\theta_{I}>0$ the flow is dragged into the fixed point as
$k\rightarrow\infty$ while eigendirections where Re$\theta_{I}<0$ are
repulsive in this limit.

The asymptotic safety hypothesis then stipulates that the high-energy
completion of gravity is provided by a NGFP Weinberg:1980gg . If the
underlying NGFP has eigendirections where Re$\theta_{I}<0$, the presupposition
that the NGFP provides the high-energy completion leads to testable
predictions in the following sense: any theory meeting this criterion must be
situated within the subspace of RG trajectories (the UV-critical hypersurface
of the fixed point) emanated from the fixed point as $k$ decreases. This
induces conditions among the coupling constants which are testable, at least
in principle. Recently, it has been argued in Pastor-Gutierrez:2022nki , that
applying this condition to the standard model of particle physics allows to
identify the interacting gravity-matter fixed point providing the UV-
completion of the theory.

###  3.2 Introducing the foliation structure

The construction of the effective average action for theories treating the
gravitational degrees of freedom at the quantum level makes manifest use of
the background field method Reuter:1996cp . The presence of the background
structure is essential for dividing the spectrum of fluctuations into long-
range and short-range with respect to the coarse-graining scale $k$. In
essence, the use of background fields allows to treat quantum fluctuations in
the metric along the lines of matter degrees of freedom quantized within the
framework of quantum field theory in a curved spacetime. On top of this
conceptual need, the background field formalism also induces auxiliary
symmetries in $\Gamma_{k}$ which constrain the type of interaction monomials
which are generated along the RG flow.

In practice, we implement the background field formalism by starting from the
ADM-fields $\chi=(N,N_{i},\sigma_{ij})$ and decomposing them into a background
part $\bar{\chi}=(\bar{N},\bar{N}_{i},\bar{\sigma}_{ij})$ and fluctuations
$\hat{\chi}=(\hat{N},\hat{N}_{i},\hat{\sigma}_{ij})$ via a linear
split333Since the ADM-decomposition of the metric (8) is non-linear in the
ADM-fields, the vertices containing a fixed power of the fluctuation fields
capture different contributions than the ones obtained in the covariant
computation building on the split $g_{\mu\nu}=\bar{g}_{\mu\nu}+h_{\mu\nu}$.

$N=\bar{N}+\hat{N}\,,\qquad
N_{i}=\bar{N}_{i}+\hat{N}_{i}\,,\qquad\sigma_{ij}=\bar{\sigma}_{ij}+\hat{\sigma}_{ij}.$
(22)

Depending on the values of $\alpha_{1}$ and $\alpha_{2}$, the action (16) is
invariant either with respect to the full diffeomorphism group or foliation
preserving diffeomorphisms. In order to obtain well-defined propagators, the
gravitational part of the effective average action must thus be supplemented
by a gauge-fixing condition and the corresponding ghost contribution. Thus the
ADM-adaptation of $\Gamma_{k}$ has the general structure

$\begin{split}\Gamma_{k}=&\,\bar{\Gamma}_{k}[N,N_{i},\sigma_{ij}]+\widehat{\Gamma}_{k}[\hat{N},\hat{N}_{i},\hat{\sigma}_{ij};\bar{N},\bar{N}_{i},\bar{\sigma}_{ij}]+\Gamma_{k}^{\rm
gf}[\hat{N},\hat{N}_{i},\hat{\sigma}_{ij};\bar{N},\bar{N}_{i},\bar{\sigma}_{ij}]\\\
&\,+\Gamma_{k}^{\rm
ghost}[\hat{N},\hat{N}_{i}\hat{\sigma}_{ij},\bar{c},\bar{b}^{i},c,b_{i};\bar{N},\bar{N}_{i},\bar{\sigma}_{ij}]\,.\end{split}$
(23)

Here $\bar{\Gamma}_{k}[N,N_{i},\sigma_{ij}]$ is the “diagonal” part of the
action depending on the background and fluctuation fields in the combination
(22) only and $\widehat{\Gamma}_{k}$ encodes the “off-diagonal” contributions
and genuinely depends on both arguments. $\Gamma_{k}^{\rm gf}$ provides the
gauge-fixing of the action and is accompanied by the action for the ghost
$(c,b^{i})$ and anti-ghost fields $(\bar{c},\bar{b}_{i})$ capturing the
contribution of the Faddeev-Popov determinant. Eq. (23) anticipates that all
sectors may contain $k$-dependent couplings.

Concretely, we follow Biemans:2017zca ; Barvinsky2016  and work within the
class of background gauge-fixings. The main idea underling our choice of gauge
fixing is to introduce a contribution bilinear in the fluctuation fields which
equips all fields with a relativistic dispersion relation. Restricting the
constructions to terms with at most two derivatives, we can parameterize the
gauge-fixing functional as

$\Gamma^{\text{gf}}_{k}=\frac{1}{32\pi G_{k}}\int d\tau
d^{d}x\bar{N}\sqrt{\bar{\sigma}}\left[F_{i}\bar{\sigma}^{ij}F_{j}+F^{2}\right]\,.$
(24)

The functionals $F$ and $F_{i}$ are linear in the fluctuation fields and
implement the gauge-fixing condition. The most general form of $F$ and $F_{i}$
is a linear combination of the fluctuation fields including one temporal or
spatial derivative. In a flat background, their generic form is given by

$\begin{split}F=&c_{1}\,\partial_{\tau}\hat{N}+c_{2}\,\partial^{i}\hat{N}_{i}+c_{3}\,\partial_{\tau}\hat{\sigma},\\\
F_{i}=&c_{4}\,\partial_{\tau}\hat{N}_{i}+c_{5}\,\partial_{i}\hat{N}+c_{6}\,\partial_{i}\hat{\sigma}+c_{7}\,\partial^{j}\hat{\sigma}_{ji}.\end{split}$
(25)

The free coefficients $c_{1},\cdots,c_{7}$ can be chosen for later
convenience. In the sequel, we will fix

$\begin{split}c_{1}=-\sqrt{2},~{}c_{2}=-\sqrt{2},~{}c_{3}=\frac{1}{\sqrt{2}},c_{4}=-\sqrt{2},~{}c_{5}=\sqrt{2},~{}c_{6}=\frac{1}{\sqrt{2}},~{}c_{7}=-\sqrt{2},\end{split}$
(26)

which implements the harmonic gauge condition at the level of the ADM-
decomposition. The ghost action exponentiating the Faddeev-Popov determinant
is obtained in the standard way

$\begin{split}\Gamma^{\text{ghost}}_{k}&=\int d\tau
d^{d}x\bar{N}\sqrt{\bar{\sigma}}\left[\bar{c}\,\frac{\delta
F}{\delta\hat{\chi}^{a}}\left(\delta_{c,b_{j}}\chi^{a}\right)+\bar{b}^{i}\,\frac{\delta
F_{i}}{\delta\hat{\chi}^{a}}\left(\delta_{c,b_{j}}\chi^{a}\right)\right]\,.\end{split}$
(27)

Here $(\delta_{c,b_{j}}\chi^{a})$ is the transformation of ADM-field
$\chi^{a}$ introduced in (11), with $f$ and $\xi^{i}$ replaced by $c$ and
$b^{i}$. For the parameters (26), the evaluation of (27) leads to a rather
lengthy expression. Its explicit form is given in App. A.3.

The dependence of $\Gamma_{k}$ on the coarse-graining scale $k$ is then
governed by the Wetterich equation (17). At this point a technical remark
about the construction of the regulator function ${\mathcal{R}}_{k}(\Box)$ is
in order (also see Codello:2008vh ; Braun:2022mgx  for a more detailed
discussion). In practice, the regulator is a non-local function of a
differential operator $\Box$ which is used to discriminate between
fluctuations coming with “high-” and “low-”momentum with respect to the coarse
graining scale $k$. Its basic property is that it provides a mass term for
low-momentum fluctuation and decays sufficiently fast for the high-momentum
modes. In the covariant setting it is natural to use the background Laplacian,
$\Box\equiv-\bar{g}^{\mu\nu}\bar{D}_{\mu}\bar{D}_{\nu}$, potentially
supplemented by an endomorphism constructed from the background curvature, in
order to “measure” the momentum of a fluctuation Codello:2008vh . In the
foliated setting, we have a natural discrimination between spatial and
“time”-derivatives. This opens more options. In particular, one can resort to
a regularization procedure where $\Box$ does not contain derivatives with
respect to the “time”-direction. The discrimination between low- and high-
momentum fluctuations can then be based on the Laplacian constructed on the
spatial slices $\Sigma_{\tau}$,
$\Box\equiv-\bar{\sigma}^{ij}\bar{D}_{i}\bar{D}_{j}$. This choice still
realizes a $k$-dependent mass term for the fluctuation fields and suffices to
render the trace on the right-hand side of eq. (17) finite. This choice comes
with the advantages that it does not induce higher-order time-derivatives in
the regularization procedure. Moreover, it orders fluctuations in a positive
semi-definite way which can be carried over to Lorentzian signature
computations. This is the route taken in the background computations
Manrique:2011jc ; Rechenberger:2012dt ; Contillo:2013fua ; Biemans:2016rvp ;
Biemans:2017zca . As a drawback, the regulator introduces a non-covariant
element in the construction which sources diffeomorphism-violating effects in
the RG flow. Since our present work is limited to the Euclidean signature
setting, we follow the construction Knorr:2018fdu  and adopt a covariant
choice for $\Box$. It turns out that this choice is also technically preferred
when carrying out fluctuation computations in the foliated setting.

###  3.3 Solving the Wetterich equation in the fluctuation approach

Before delving into the actual computation, let us first introduce the
conceptual elements underlying fluctuation computations based on the Wetterich
equation, following Pawlowski:2020qer . The general setup will be introduced
in Sect. 3.3.1 and we give an instructive example based on the Einstein-
Hilbert action in Sect. 3.3.2. Details specific to RG flows on a foliated
spacetime are discussed in Sect. 3.3.3.

####  3.3.1 Fluctuation field computations - the general setup

We start with briefly reviewing the fluctuation approach for the functional
renormalization group. Generically, we consider a theory whose field content
comprises $N$ fields collectively denoted by
$\chi\equiv(\chi_{1},\cdots,\chi_{N})$. In the ADM-formalism, $\chi$ contains
the lapse, shift, and spatial metric, as well as the ghost and anti-ghost
fields. The background field method decomposes these fields into their
background parts $\bar{\chi}\equiv(\bar{\chi}_{1},\cdots,\bar{\chi}_{N})$ and
fluctuations $\hat{\chi}\equiv(\hat{\chi}_{1},\cdots,\hat{\chi}_{N})$. This
decomposition could be implemented through the linear split
$\chi_{a}=\bar{\chi}_{a}+\hat{\chi}_{a}$, $a=1,\cdots,N$. Typically, the
background fields associated with the ghost and anti-ghost fields are taken to
be zero.

The idea of the fluctuation approach is to expand the effective average action
in powers of the fluctuation fields

$\Gamma_{k}[\hat{\chi};\bar{\chi}]=\sum_{n=0}^{\infty}P(a_{i})\int_{x}\Gamma^{(\hat{\chi}_{a_{1}}\cdots\hat{\chi}_{a_{n}})}_{k}[\bar{\chi}]~{}\hat{\chi}_{a_{1}}\cdots\hat{\chi}_{a_{n}}\,.$
(28)

The dependence of $\Gamma_{k}$ on the coarse-graining scale is carried by the
expansion coefficients
$\Gamma^{(\hat{\chi}_{a_{1}}\cdots\hat{\chi}_{a_{n}})}_{k}[\bar{\chi}]$. These
depend on background quantities only and may contain covariant derivatives
acting on the fluctuation fields. The $P(a_{i})$ denotes a combinatorial
factor ensuring that

$\Gamma^{(\hat{\chi}_{a_{1}}\cdots\hat{\chi}_{a_{n}})}_{k}[\bar{\chi}]=\left.\frac{\delta^{m}}{\delta\hat{\chi}_{a_{1}}\cdots\delta\hat{\chi}_{a_{m}}}\Gamma_{k}[\hat{\chi};\bar{\chi}]\right|_{\hat{\chi}=0}\,,$
(29)

and a sum over the indices $a_{i}$ is implied. The expansion disentangles the
contributions from the background and fluctuation fields. In explicit
computations, it is useful to extract the wave-function renormalization
factors of the fluctuation fields from the vertices,

$\Gamma^{(\hat{\chi}_{a_{1}}\cdots\hat{\chi}_{a_{n}})}_{k}[\bar{\chi}]=\left(\prod_{i=1}^{n}Z^{\frac{1}{2}}_{{\hat{\chi}}_{a_{i}}}\right)~{}\bar{\Gamma}_{k}^{({\hat{\chi}}_{a_{1}}\cdot\cdot\cdot{\hat{\chi}}_{a_{n}})}[\bar{\chi}]\,.$
(30)

In order to keep our notation light, we will keep these factors within
$\Gamma^{(\hat{\chi}_{a_{1}}\cdots\hat{\chi}_{a_{n}})}_{k}[\bar{\chi}]$.

At this stage, one can introduce a basis on the space of tensor structures
$(\mathcal{T}^{({\hat{\chi}}_{a_{1}}\cdots{\hat{\chi}}_{a_{n}})})$ contracting
$n$ fluctuation fields. In general, these basis elements carry the dependence
of the vertices on the internal indices of the fields. The expansion
coefficients
$\Gamma^{(\hat{\chi}_{a_{1}}\cdots\hat{\chi}_{a_{n}})}_{k}[\bar{\chi}]$ can
then be expanded in this basis

$\Gamma_{k}^{({\phi}_{a_{1}}\cdot\cdot\cdot{\phi}_{a_{n}})}[\bar{\chi}]=\sum_{j}\bar{u}_{n,j}(k)\,\mathcal{T}_{j}^{({\phi}_{a_{1}}\cdot\cdot\cdot{\phi}_{a_{n}})}[\bar{\chi}].$
(31)

The expansion coefficients $\bar{u}_{n,j}(k)$ are the dimensionful coupling
constants of the theory and depend on the coarse-graining scale $k$.

The dependence of the couplings $\bar{u}_{n,j}(k)$ on $k$ can then be obtained
by substituting the expansion (28) into the Wetterich equation (17) and
comparing the coefficients multiplying a given tensor structure on its left-
and right-hand side. The projection onto the tensor structures is conveniently
performed by taking functional derivatives of the initial equation with
respect to the fluctuation fields and their contracted (generalized) momenta.
Making use of (29), this leads to equations of the form

$\partial_{t}\Gamma^{(\hat{\chi}_{a_{1}}\cdots\hat{\chi}_{a_{n}})}_{k}[\bar{\chi}]=\frac{1}{2}\left.\frac{\delta^{n}}{\delta\hat{\chi}_{a_{1}}\cdots\delta\hat{\chi}_{a_{n}}}{\rm
Str}\left[(\Gamma^{(2)}_{k}[\hat{\chi};\bar{\chi}]+\mathcal{R}_{k}[\bar{\chi}])^{-1}\,\partial_{t}\mathcal{R}_{k}[\bar{\chi}]\right]\right|_{\hat{\chi}=0}\,.$
(32)

For the one-point correlation functions this general expression implies

$\displaystyle\partial_{t}\Gamma^{(\hat{\chi}_{i})}_{k}$ $\displaystyle=$
$\displaystyle-\frac{1}{2}\text{STr}\left.\left[\left(\Gamma^{(\hat{\chi}_{a}\hat{\chi}_{b})}_{k}+{\mathcal{R}}_{k}\right)^{-1}\Gamma^{(\hat{\chi}_{i}\hat{\chi}_{b}\hat{\chi}_{c})}_{k}\left(\Gamma^{(\hat{\chi}_{c}\hat{\chi}_{d})}_{k}+{\mathcal{R}}_{k}\right)^{-1}\partial_{t}{\mathcal{R}}_{k}^{(\hat{\chi}_{d}\hat{\chi}_{a})}\right]\right|_{\hat{\chi}=0}\,,$
(33)

while the flow of a two-point function has the general form

$\begin{split}\partial_{t}\Gamma^{(2)}_{k}=&\,\text{STr}\left[\left(\Gamma^{(2)}_{k}+{\mathcal{R}}_{k}\right)^{-1}\Gamma^{(3)}_{k}\left(\Gamma^{(2)}_{k}+{\mathcal{R}}_{k}\right)^{-1}\Gamma^{(3)}_{k}\left(\Gamma^{(2)}_{k}+{\mathcal{R}}_{k}\right)^{-1}\partial_{t}{\mathcal{R}}_{k}\right]\\\
&\;\;-\frac{1}{2}\text{STr}\left[\left(\Gamma^{(2)}_{k}+{\mathcal{R}}_{k}\right)^{-1}\Gamma^{(4)}_{k}\left(\Gamma^{(2)}_{k}+{\mathcal{R}}_{k}\right)^{-1}\partial_{t}{\mathcal{R}}_{k}\right]\,.\end{split}$
(34)

Thus, determining the $k$-dependence of the expansion coefficients appearing
at the $n$th order in the fluctuation fields requires knowledge about the
coefficients at order $n+1$ and $n+2$. Any approximation of the exact solution
has to supply this information in order to close the hierarchy at any finite
order $n$. Typically, this closure is provided by identifying couplings
$\bar{u}_{n,j}(k)$ appearing in the interaction vertices of different order.

####  3.3.2 Vertex structures in the covariant setting

Before discussing the intricacies related to fluctuation field computations on
a foliated spacetime, it is useful to illustrate the working of the approach
in the covariant setting. Explicit computations based on this setup have been
reported in Christiansen:2012rx ; Christiansen:2014raa ; Meibohm:2015twa ;
Christiansen:2015rva ; Denz:2016qks ; Christiansen:2017bsy ;
Christiansen:2017cxa ; Knorr:2017fus ; Knorr:2017mhu ; Eichhorn:2018ydy ;
Eichhorn:2018akn ; Eichhorn:2018nda ; Burger:2019upn ; Bonanno:2021squ ;
Knorr:2021niv ; Fehre:2021eob  and reviewed in Pawlowski2021 . We focus on the
gravitational field $g_{\mu\nu}$ which is decomposed into background and
fluctuation fields according to

$g_{\mu\nu}=\bar{g}_{\mu\nu}+\hat{g}_{\mu\nu}\,.$ (35)

Most of the computations utilize a flat, Euclidean background where
$\bar{g}_{\mu\nu}=\delta_{\mu\nu}$. This has the advantage that standard
momentum-space techniques are available. Derivatives within the expansion
coefficients can then be replaced by the momenta of the fluctuation fields. We
will adopt this choice in the sequel.

Subsequently, we consider the Einstein-Hilbert action supplemented by the
harmonic gauge condition

$\Gamma_{k}[\hat{g};\bar{g}]=-\frac{1}{16\pi G_{k}}\int
d^{d}x\sqrt{g}R+\frac{1}{32\pi G_{k}}\int
d^{d}x\sqrt{\bar{g}}\,\bar{g}^{\mu\nu}F_{\mu}F_{\nu}\,,$ (36)

where

$F_{\mu}=\left[\bar{D}^{\alpha}\delta^{\beta}_{\mu}-\frac{1}{2}\,\bar{g}^{\alpha\beta}\bar{D}_{\mu}\right]\hat{g}_{\alpha\beta}\,.$
(37)

Expanding (36) in a flat background, the first non-trivial term appears at
second order in the fluctuation fields

$\Gamma^{\rm quad}_{k}[\hat{g};\bar{g}]=\frac{1}{128\pi
G_{k}}\int\frac{d^{d}p}{(2\pi)^{d}}\,p^{2}\,\left[\delta^{\mu\alpha}\delta^{\nu\beta}+\delta^{\mu\beta}\delta^{\nu\alpha}-\delta^{\mu\nu}\delta^{\alpha\beta}\right]\hat{g}_{\mu\nu}(p)\hat{g}_{\alpha\beta}(-p)\,.$
(38)

Comparing this result with (28) gives the expansion coefficient

$\Gamma_{k}^{(\hat{g}\hat{g})}=\frac{1}{64\pi
G_{k}}\,p^{2}\,\left[\delta^{\mu\alpha}\delta^{\nu\beta}+\delta^{\mu\beta}\delta^{\nu\alpha}-\delta^{\mu\nu}\delta^{\alpha\beta}\right]\,.$
(39)

The tensor structure appearing in brackets can be understood as a linear
combination of the two orthogonal tensor structures

$\mathcal{T}^{(\hat{g}\hat{g})}_{1}=\frac{1}{2}\left(\delta^{\mu}_{\alpha}\delta^{\nu}_{\beta}+\delta^{\nu}_{\alpha}\delta^{\mu}_{\beta}\right)-\frac{1}{d}\delta^{\mu\nu}\delta_{\alpha\beta}\,,\qquad\mathcal{T}^{(\hat{g}\hat{g})}_{2}=\frac{1}{d}\delta^{\mu\nu}\delta_{\alpha\beta}\,,$
(40)

which can be built from a flat metric without resorting to the momentum of the
field. The two $\mathcal{T}$’s project a symmetric, rank-2 tensor onto its
traceless and trace part, respectively. Eq. (39) can then be recast in the
form (31),

$\Gamma_{k}^{(\hat{g}\hat{g})}=\bar{u}_{2}(p^{2};k)\left[\mathcal{T}^{(\hat{g}\hat{g})}_{1}+\frac{d-2}{2}\,\mathcal{T}^{(\hat{g}\hat{g})}_{2}\right]$
(41)

where

$\bar{u}_{2}(p^{2};k)\equiv\,\frac{1}{32\pi G_{k}}\,p^{2}\,.$ (42)

At this stage, the following remarks are in order. Eq. (42) indicates that
_the avatar of Newton’s coupling_ associated with the graviton two-point
function appears at order $p^{2}$ in the momentum of the fluctuation field.
While this coupling is derived from the gauge-fixed Einstein-Hilbert action
(36), the expansion in terms of tensor structures introduced in eqs. (28) and
(31) indicates that this coupling _is different from_ Newton’s coupling which
arises at zeroth order in the fluctuation field by evaluating (36) at
$\hat{g}=0$ Manrique:2009uh ; Manrique:2010am .

Secondly, our definition of $\bar{u}_{2}(p^{2};k)$ promotes Newton’s coupling
to a form factor depending on the squared momentum $p^{2}$ of the fluctuation
fields. The form factor then captures the non-trivial momentum dependence of
the graviton two-point function.444Structurally, this is very similar to the
form factors appearing in the curvature expansion of the effective average
action Bosma:2019aiu ; Knorr:2019atm ; Draper:2020bop ; Draper:2020knh
reviewed in Knorr:2021iwv ; Knorr:2022dsx . The Wetterich equation allows to
compute the dependence of $\bar{u}_{2}$ on $p^{2}$. In the covariant approach
this “reconstruction of the graviton propagator” has been carried out in
Bonanno:2021squ . In Fehre:2021eob  it was reported that the two-point
function interpolates between $\bar{u}_{2}\propto\text{const}$ and
$\bar{u}_{2}\propto k^{\eta_{h}}$, $\eta_{h}=0.96$, for small and large
momenta, respectively.

####  3.3.3 Vertex structures on a foliated spacetime

In comparison to the covariant setting, the foliation present in the ADM-
formalism provides an additional structure which allows to distinguish the
spatial and time-components of a covariant object. In particular, the squared
momentum can be written as

$p^{2}=p_{0}^{2}+\vec{p}^{\,2}\,.$ (43)

It is now instructive to insert this relation into (42)

$\bar{u}_{2}(p_{0}^{2},\vec{p}^{\,2};k)=\,\frac{1}{32\pi
G_{k}}\,\left(p_{0}^{2}+\vec{p}^{\,2}\right)\,.$ (44)

Generically, the coefficients in front of the two terms on the right-hand side
can be different. This implies that the presence of the foliation allows to
avatars of Newton’s coupling associated with the spatial and “time”-like
contributions to the squared momentum

$\bar{u}_{2}=\frac{1}{32\pi
G_{k}}p^{2}\qquad\begin{array}[]{ll}\nearrow&\bar{u}_{2a}=\frac{1}{32\pi
G_{k}}\,p_{0}^{2}\,,\\\\[5.16663pt] \searrow&\bar{u}_{2b}=\frac{1}{32\pi
G_{k}}\,\vec{p}^{\,2}\,.\end{array}$ (45)

At the level of the action (16), these avatars are implemented through the
couplings $\alpha_{1}$ and $\alpha_{2}$. The symmetries of the covariant
setting fix $\alpha_{1}=\alpha_{2}=1$, indicating that the avatars generated
in the split (45) should be identified.

The Wetterich equation adapted to the ADM-formalism incorporates contributions
which break Lorentz covariance explicitly. Hence the flow of the avatars
generated in the split (45) is expected to be different. Conceptually, reading
off the flow of a coupling from the terms containing $p_{0}^{2}$ or
$\vec{p}^{\,2}$, should be considered as computing the beta functions of _two
different couplings_. The conceptual consequences of this setting are
illustrated in Fig. 1.

Figure 1:  Projection of a three-dimensional approximation of theory space to
two-dimensional subspaces. The NGFP and its projections are marked by black
dots. The arrows indicate eigendirections of the stability matrix, describing
the RG flow in the vicinity of the fixed point. The value of the critical
exponent is encoded in the length of the arrow. Notably, projections to
different subspaces will see different projections of these arrows,
illustrating that the critical exponents do not lend themselves to a
meaningful comparison when considering subspaces spanned by different
couplings a priori.

The figure depicts a generic RG fixed point and its projection to lower-
dimensional subspaces spanned by a set of dimensionless couplings $u_{i}$
(black dots). The fixed point should be visible in all projections (straight
lines). The stability coefficients accessed in the projections capture
different properties of the RG flow in the vicinity of the fixed point though.
Thus there is no a priori reason that their values found in different
projections actually agree. This applies specifically for the two projections
described in eq. (45). Convergence of stability coefficients requires studying
extended projections which include the same subsystem. Thus, it is meaningful
to compare the stability coefficients obtained in the three-dimensional and
each of the two-dimensional subsystems, while a comparison among the different
two-dimensional systems may not show identical stability properties.

##  4 RG flows at second order in the derivative expansion

Upon introducing the general framework underlying fluctuation field
computations in the ADM-formalism, we work out a specific example and compute
the scale-dependence of the graviton two-point function resulting from the
two-derivative action (16). We start by giving the gauge-fixed propagators for
the fluctuation fields in Sect. 4.1. The projection of the flow on these
tensor structures and the resulting beta functions encoding the $k$-dependence
of the couplings are given in Sect. 4.2. Many technical details are relegated
to App. A. Throughout the computation we work in a four-dimensional flat
Euclidean background and we manifestly make use of momentum-space methods in
order to facilitate the computation.

###  4.1 Gauge-fixed two-point functions

We start from the action (16) which contains all terms constructed from at
most two derivatives. We then adopt this action as the gravitational part of
the effective average action,

$\Gamma_{k}^{\text{grav}}[N,N_{i},\sigma_{ij}]=\frac{1}{16\pi G_{k}}\int d\tau
d^{3}y\,N\sqrt{\sigma}\,(\alpha_{1}K^{ij}K_{ij}-\alpha_{2}K^{2}-\leftidx{{}^{(d)}}R+2\Lambda_{k})\,.$
(46)

Here the couplings $\alpha_{1},\alpha_{2},G_{k}$, and $\Lambda_{k}$ have been
promoted to depend on the coarse-graining scale $k$. Background computations
along the lines of Rechenberger:2012dt ; Contillo:2013fua ; Biemans:2017zca
start from a similar ansatz and subsequently evaluate the Wetterich equation
(17) at zeroth order in the fluctuation field.

In our fluctuation computation, eq. (46) serves as the generating functional
for the tensor structures onto which we project the flow equation. In order to
obtain these structures, we first supplement (46) by the gauge-fixing and
ghost terms given in eqs. (24) and (27) (also see App. A.3 for explicit
expressions). We then substitute (22) and expand the resulting functional in
powers of the fluctuation fields in the flat background. In order to bring the
Hessian $\Gamma_{k}^{(2)}$ into (almost) diagonal form, we implement a
transverse-traceless decomposition of the fluctuation fields York1973

$\begin{split}\hat{\sigma}_{ij}=&h_{ij}+\partial_{i}v_{j}+\partial_{j}v_{i}+\partial_{i}\partial_{j}E-\frac{1}{3}\delta_{ij}\partial^{2}E+\frac{1}{3}\delta_{ij}\Psi,\\\
\hat{N}_{i}=&u_{i}+\partial_{i}B\,,\end{split}$ (47)

where the component fields are subject to the constraints

$\partial^{i}h_{ij}=0\,,\quad\delta^{ij}h_{ij}=0\,,\quad\partial^{i}v_{i}=0\,,\quad\partial^{i}u_{i}=0\,,\quad\Psi=\delta^{ij}\hat{\sigma}_{ij}\,.$
(48)

These constraints ensure that the two-point functions depend on contracted
spatial derivatives $\partial^{2}\equiv\partial_{i}\partial^{i}$ only
Lauscher:2001ya ; Benedetti:2010nr . The field redefinition (47) has a non-
trivial Jacobian. This extra term is conveniently accounted for by rescaling
the fluctuations according to

$\begin{split}v_{i}\rightarrow&\frac{1}{\sqrt{-\partial^{2}}}v_{i}\,,\quad
E\rightarrow\frac{1}{(-\partial^{2})}E\,,\quad
B\rightarrow\frac{1}{\sqrt{-\partial^{2}}}B\,.\end{split}$ (49)

The projection of the fluctuation fields $\hat{\sigma}_{ij},\hat{N}_{i}$ onto
the subspaces spanned by the component fields is readily achieved through
projection operators. Introducing the unit on the space of symmetric $3\times
3$-matrices

$\mathbb{1}^{ij}{}_{kl}\equiv\frac{1}{2}\left(\delta^{i}_{k}\delta^{j}_{l}+\delta^{j}_{k}\delta^{i}_{l}\right)\,,$
(50)

the subspaces for the component fields in (47) are spanned by

$\begin{split}{\Pi_{\Psi}}^{ij}{}_{kl}&=\frac{1}{3}\sigma^{ij}\sigma_{kl}\,,\\\
{\Pi_{E}}^{ij}{}_{kl}&=(\partial^{i}\partial^{j}-\frac{1}{3}\delta^{ij}\partial^{2})(\frac{2}{3}\partial^{4})^{-1}(\partial_{k}\partial_{l}-\frac{1}{3}\delta_{kl}\partial^{2})\,,\\\
{\Pi_{v}}^{ij}_{~{}~{}kl}&=2\Big{(}\delta^{(j}_{(l}\,\partial^{i)}\,\partial^{-2}\,\partial_{k)}\Big{)}-2\partial^{i}\partial^{j}\partial^{-4}\partial_{k}\partial_{l},\\\
{\Pi_{h}}^{ij}_{~{}~{}kl}&=\mathbb{1}^{ij}_{~{}~{}kl}-{\Pi_{\Psi}}^{ij}_{~{}~{}~{}kl}-{\Pi_{E}}^{ij}_{~{}~{}kl}-{\Pi_{v}}^{ij}_{~{}~{}kl}\,.\end{split}$
(51)

Similarly, we also give the projection tensors for the space of vector fields

$\begin{split}{\Pi_{B}}{}^{i}{}_{j}\equiv\partial^{i}\,\partial^{-2}\,\partial_{j}\,,\qquad{\Pi_{u}}{}^{i}{}_{j}\equiv\delta^{i}_{j}-\partial^{i}\,\partial^{-2}\,\partial_{j}\,.\end{split}$
(52)

fields $(i,j)$ | $\Gamma^{{\rm grav}\,(ij)}_{k}+\frac{1}{2}\Gamma^{{\text{gf}\,(ij)}}_{k}$
---|---
$h_{ij}h^{kl}$ | $\frac{1}{32\pi G_{k}}\left((\alpha_{1}p_{0}^{2}+\vec{p}^{\,2})-2\Lambda_{k}\right)\,{\Pi_{h}}^{ij}_{~{}~{}kl}$
$v_{i}v^{j}$ | $\frac{1}{16\pi G_{k}}\left((\alpha_{1}p_{0}^{2}+\vec{p}^{\,2})-2\Lambda_{k}\right){\Pi_{u}}{}^{i}{}_{j}$
$EE$ | $\frac{1}{48\pi G_{k}}\left((\alpha_{1}p_{0}^{2}+\vec{p}^{\,2})-2\Lambda_{k}\right)$
$\Psi\Psi$ | $-\frac{1}{192\pi G_{k}}\left((6\alpha_{2}-2\alpha_{1}-3)p_{0}^{2}+\vec{p}^{\,2}-2\Lambda_{k}\right)$
$\hat{N}\hat{N}$ | $\frac{1}{16\pi G_{k}}(p_{0}^{2}+\vec{p}^{\,2})$
$\Psi\hat{N}$ | $-\frac{1}{16\pi G_{k}}\left(p_{0}^{2}+\vec{p}^{\,2}-2\Lambda_{k}\right)$
$u^{i}u_{j}$ | $\frac{1}{16\pi G_{k}}(p_{0}^{2}+\alpha_{1}\vec{p}^{\,2}){\Pi_{u}}{}^{i}{}_{j}$
$BB$ | $\frac{1}{16\pi G_{k}}\left(p_{0}^{2}+(1+2\alpha_{1}-2\alpha_{2})\,\vec{p}^{\,2}\right)$
$\bar{c}c$ | $\sqrt{2}\,(p^{2}_{0}+\vec{p}^{\,2})$
$\bar{b}^{i}b_{i}$ | $\sqrt{2}\,(p^{2}_{0}+\vec{p}^{\,2})\,\Pi_{u}{}_{j}{}^{i}$
Table 1:  Matrix elements of the Hessian $\Gamma^{(2)}_{k}$. The first line is
singled out since this is the tensor structure onto which we are going to
project the flow equation. The second block gives the propagators associated
with the metric degrees of freedom while the third block captures the
information about the propagators in the ghost sector.

At this point we have all the ingredients to write down the matrix elements of
the Hessian $\Gamma_{k}^{(2)}$. Including the contribution of the gauge-fixing
terms, these are tabulated in Table 1. At this point a closer inspection of
these expressions in order. All two-point functions are proportional to the
projection operators restricting the fluctuation fields to the corresponding
transverse (and traceless) subspaces. For scalars, this projection is trivial
and the corresponding projectors are omitted. For $\alpha_{1}=\alpha_{2}=1$,
the gauge-fixed (inverse) propagators all come with a _relativistic dispersion
relation_. In this case, the spatial and time-components of the momentum
combine into a relativistic four-momentum $p_{0}^{2}+\vec{p}^{\,2}=p^{2}$.
This singles out the gauge-fixing adopted in eq. (26) Biemans:2016rvp . The
cosmological constant then plays the role of a mass term in the two-point
function.

At this point we have all the ingredients to specify the components of the
regulator ${\mathcal{R}}_{k}(\Box)$. Following the nomenclature of
Codello:2008vh , we implement a Type I regularization, which fixes
${\mathcal{R}}_{k}$ through the substitution rule

$\Box\rightarrow P_{k}(\Box)\equiv\Box+R_{k}(\Box)\,,$ (53)

where $R_{k}(\Box)$ is a scalar cutoff function. Throughout this work, we
chose the cutoff function to be of Litim-type Litim:2000ci ; Litim:2001up ,

$R_{k}(\Box)=(k^{2}-\Box)\Theta(k^{2}-\Box)\,,$ (54)

where $\Theta(x)$ is the Heaviside step function. Inspecting Table 1, one
identifies various candidates for the coarse-graining operator $\Box$.
Adopting a momentum-space representation, these are within the class

$\Box_{\alpha}=\alpha\,p_{0}^{2}+\vec{p}^{\,2}\,,$ (55)

where $\alpha$ depends on $k$. Based on the kinetic terms for the various
component fields, we would then encounter as many different forms of
$\Box_{\alpha}$ as there are distinct dispersion relations.

The $\alpha$-dependence of $\Box$ introduces a significant number of technical
complications in the computation. This can be understood from noticing that
each operator $\Box_{\alpha}$ leads to a different stepfunction
$\Theta(k^{2}-\Box_{\alpha})$. Loop integrals then involve different step
functions which results in quite complicated domains when integrating over
loop momenta Becker:2017tcx . We avoid this complication by employing a
modified (but perfectly admissible) replacement rule for the regularization

$\Box_{\alpha}\rightarrow\Box_{\alpha}+R_{k}(\Box_{\alpha=1})\,.$ (56)

This results in the matrix elements for ${\mathcal{R}}_{k}$

$\begin{split}{\mathcal{R}}_{k}^{hh}=&\frac{1}{32\pi
G_{k}}R_{k}(p^{2})\,\Pi_{h}{}^{ij}_{kl}\,,\qquad{\mathcal{R}}_{k}^{uu}=\frac{\alpha_{1}}{32\pi
G_{k}}R_{k}(p^{2})\,\Pi_{u}{}^{i}_{j}\,,\end{split}$ (57)

and similar for the other matrix entries. We will adopt the choice (56) in the
sequel.555In principle, one can chose other regularization procedures as well.
These differ by the anomalous dimension of the coupling appearing in
$\partial_{k}{\mathcal{R}}_{k}$. For instance, one could adopt
$\Box_{\alpha}\rightarrow
P_{k}(\Box_{\alpha=1})+\left(\Box_{\alpha}-\Box_{\alpha=1}\right)\,.$ (58) In
comparison to (56) the two choices differ by terms proportional to the beta
functions associated with $\alpha_{1}$ and $\alpha_{2}$. These extra
contribution vanish on a fixed point by definition, so that eq. (58) and (56)
lead to the same fixed point structure.

The different dispersion relations listed in Table 1 then lead to the same
profile function $R_{k}$. This function is just a function of the four-
momentum $p^{2}$ and does not include $k$-dependent couplings. In this way,
the argument of $R_{k}$ is covariant and preserves Lorentz symmetry. In this
way, the choice of regulator minimizes the Lorentz-symmetry violating terms
generated by the regularization procedure. Still the non-linearity of the
field decomposition (22) makes the regularization procedure non-covariant.

###  4.2 Projecting and closing the flow equation

At this point, we have all the ingredients for specifying the projection of
the Wetterich equation underlying our computation. Specifically, we project
the flow onto the graviton two-point function singled out in the first line of
Table 1. Substituting this expression into the left-hand side of the flow
equation gives

$32\pi\,\partial_{t}\Gamma^{(hh)}_{k}(p_{0}^{2},\vec{p}^{\,2})=\partial_{t}\left(\frac{\alpha_{1}}{G_{k}}\right)\,p_{0}^{2}\,\Pi_{h}+\partial_{t}\left(\frac{1}{G_{k}}\right)\,\vec{p}^{\,2}\,\Pi_{h}-2\partial_{t}\left(\frac{\Lambda_{k}}{G_{k}}\right)\Pi_{h}\,.$
(59)

This indicates that the projection can track up to three scale-dependent
couplings $\alpha_{1}$, $\Lambda_{k}$, and $G_{k}$.666In principle, the
formalism can also be used to track the scale-dependence of $\alpha_{2}$. This
requires considering one additional two-point function, e.g.,
$\Gamma^{(\Psi\Psi)}_{k}(p_{0}^{2},\vec{p}^{\,2})$ which contains the
corresponding coupling. While this computation is conceptually
straightforward, it essentially doubles the complexity of the underlying
algebra. For this reason, we leave this extension for future work. This
projection then identifies the tensor structures which need to be extracted
from the trace on the right-hand side of the equation. We define

$\begin{split}\frac{1}{2}{\rm
STr}\left[\cdots\right]\simeq&\,T_{p_{0}}(G_{k},\alpha_{1},\alpha_{2},\Lambda_{k},\eta_{N},\partial_{t}\alpha_{1})\,p_{0}^{2}\,\Pi_{h}\\\
&\,+T_{\vec{p}}(G_{k},\alpha_{1},\alpha_{2},\Lambda_{k},\eta_{N},\partial_{t}\alpha_{1})\,\vec{p}^{\,2}\,\Pi_{h}\\\
&\,-T_{0}(G_{k},\alpha_{1},\alpha_{2},\Lambda_{k},\eta_{N},\partial_{t}\alpha_{1})\,\Pi_{h},\end{split}$
(60)

where the $\simeq$ indicates that we neglect terms outside of the projection
subspace and the sign in front of $T_{0}$ has been adjusted for later
convenience. The arguments of the traces highlight the dependence of the
contributions on the couplings and we defined the anomalous dimension of
$G_{k}$ as $\eta_{N}=\partial_{t}G_{k}/G_{k}$. Equating the coefficients
multiplying the independent tensor structures in eqs. (59) and (60) leads to
the following system of first order differential equations

$\begin{split}\frac{1}{32\pi}\partial_{t}\left(\frac{\alpha_{1}}{G_{k}}\right)&\,=T_{p_{0}}(G_{k},\alpha_{1},\alpha_{2},\Lambda_{k},\eta_{N},\partial_{t}\alpha_{1})\,,\\\
\frac{1}{32\pi}\partial_{t}\left(\frac{1}{G_{k}}\right)&\,=T_{\vec{p}}(G_{k},\alpha_{1},\alpha_{2},\Lambda_{k},\eta_{N},\partial_{t}\alpha_{1})\,,\\\
\frac{1}{16\pi}\partial_{t}\left(\frac{\Lambda_{k}}{G_{k}}\right)&\,=T_{0}(G_{k},\alpha_{1},\alpha_{2},\Lambda_{k},\eta_{N},\partial_{t}\alpha_{1})\,.\end{split}$
(61)

As indicated above, the projection does not capture information about the
running of $\alpha_{2}$. So this coupling has to be treated as a dimensionless
parameter.

At this point the computation has reduced to determining the coefficients $T$.
Inspecting eq. (34) shows that this requires information about the
($k$-dependent) $3$\- and $4$-point vertices of the theory. Generically, this
information is not available exactly, and one has to adopt an approximation in
order to close the system of equations. We then generate the relevant $3$\-
and $4$-point vertices by taking additional functional derivatives of
$\Gamma_{k}^{\rm grav}+\Gamma_{k}^{\rm ghost}$ with respect to the collection
of fluctuation fields $\hat{\chi}$ and subsequently specifying to the flat
background. Note that the projection to (59) entails that is is not necessary
to consider all vertices: only $3$-point vertices with at least on leg being
$h_{ij}$ and $4$-point vertices with two legs associated with the field
$h_{ij}$ contribute. Our actual computation of the coefficients $T$ retains
the full momentum dependence of these vertices on the loop-momentum. While the
actual computation is conceptually straightforward, it is technically involved
and we collect the details in App. B.

The $k$-dependence of the couplings is then conveniently encoded in the beta
functions for the dimensionless couplings $(g_{k},\lambda_{k},\alpha_{1})$

$\partial_{t}g_{k}=\beta_{g_{k}}(g_{k},\lambda_{k},\alpha_{1};\alpha_{2})\,,\quad\partial_{t}\lambda_{k}=\beta_{\lambda_{k}}(g_{k},\lambda_{k},\alpha_{1};\alpha_{2})\,,\quad\partial_{t}\alpha_{1}=\beta_{\alpha_{1}}(g_{k},\lambda_{k},\alpha_{1};\alpha_{2})\,,$
(62)

with

$g_{k}\equiv G_{k}\,k^{2}\,,\quad\lambda_{k}\equiv\Lambda_{k}\,k^{-2}\,.$ (63)

The parametric dependence on $\alpha_{2}$ is highlighted by the semicolon.

The explicit expressions for the beta functions (62) are rather lengthy and
can be provided in form of an auxiliary Mathematica notebook upon request.
Here we limit ourselves to giving the results for the foliated Einstein-
Hilbert action ($\alpha_{1}=\alpha_{2}=1$) only. In this case we find

$\begin{split}\beta_{g}=&\,\left(2+\eta_{N}\right)g\,,\\\
\beta_{\lambda}=&\,(\eta_{N}-2)\lambda+\frac{g}{\pi}\left\\{\frac{p^{1}_{\lambda}(\lambda)+\eta_{N}\,\tilde{p}^{1}_{\lambda}(\lambda)}{24\left(1-2\lambda\right)^{2}\left(2-3\lambda\right)^{2}}+\frac{p^{2}_{\lambda}(\lambda)+\eta_{N}\,\tilde{p}^{2}_{\lambda}(\lambda)}{7200\left(1-2\lambda\right)^{3}\left(2-3\lambda\right)^{3}}\right\\}\,.\end{split}$
(64)

The polynomials are tabulated in the first block of Table 2.

$p^{1}_{\lambda}(\lambda)$ | $12\left(18\lambda^{4}-116\lambda^{3}+142\lambda^{2}-61\lambda+8\right)$
---|---
$\tilde{p}^{1}_{\lambda}(\lambda)$ | $-27\lambda^{4}+211\lambda^{3}-259\lambda^{2}+106\lambda-12$
$p^{2}_{\lambda}(\lambda)$ | $10\left(68040\lambda^{6}+439308\lambda^{5}-1156914\lambda^{4}+1012717\lambda^{3}-419962\lambda^{2}+94584\lambda-11760\right)$
$\tilde{p}^{2}_{\lambda}(\lambda)$ | $87480\lambda^{6}-1420428\lambda^{5}+2828930\lambda^{4}-2242429\lambda^{3}+851098\lambda^{2}-166200\lambda+17520$
$p^{3}_{\lambda}(\lambda)$ | $408240\lambda^{8}-2262816\lambda^{7}+6128784\lambda^{6}-10347048\lambda^{5}$
$+10788945\lambda^{4}-6629544\lambda^{3}+2229588\lambda^{2}-344176\lambda+12704$
$\tilde{p}^{3}_{\lambda}(\lambda)$ | $4968\lambda^{4}-67980\lambda^{3}+94335\lambda^{2}-40380\lambda+4148$
$p^{4}_{\lambda}(\lambda)$ | $88248\lambda^{5}-180807\lambda^{4}+125287\lambda^{3}+3744\lambda^{2}-30324\lambda+6376$
$\tilde{p}^{4}_{\lambda}(\lambda)$ | $20055\lambda^{4}-32850\lambda^{3}+19676\lambda^{2}-771\lambda-2158$
Table 2:  Polynomials appearing in the beta functions (64) and the anomalous
dimension (65).

The anomalous dimension takes the form

$\eta_{N}=\frac{gB_{1}(\lambda)}{1-gB_{2}(\lambda)}\,.$ (65)

In the foliated Einstein-Hilbert case, eq. (59) indicates that the functions
$B_{1}(\lambda)$ and $B_{2}(\lambda)$ can either be obtained from the
coefficient of the $p_{0}^{2}$-term ($p_{0}$-projection) or the
$\vec{p}^{\,2}$-term ($\vec{p}$-projection). The $p_{0}$-projection yields

$\begin{split}B_{1}^{p_{0}}(\lambda)=&\frac{48\lambda^{2}-60\lambda+19}{2\pi(1-2\lambda)^{2}(2-3\lambda)^{2}}-\frac{p^{3}_{\lambda}(\lambda)}{360\pi(1-2\lambda)^{4}(2-3\lambda)^{4}}\,,\\\
B_{2}^{p_{0}}(\lambda)=&-\frac{48\lambda^{2}-60\lambda+19}{12\pi(1-2\lambda)^{2}(2-3\lambda)^{2}}+\frac{\tilde{p}^{3}_{\lambda}(\lambda)}{720\pi(1-2\lambda)^{3}(2-3\lambda)^{2}}\,.\\\
\end{split}$ (66)

the projection on the spatial momentum yields an extra contribution to
$B_{1}(\lambda)$ and $B_{2}(\lambda)$:

$\begin{split}B_{1}^{\vec{p}}(\lambda)=&B_{1}^{p_{0}}(\lambda)-\frac{p^{4}_{\lambda}(\lambda)}{315\pi(2-3\lambda)^{2}(1-2\lambda)^{4}}\,,\\\
B_{2}^{\vec{p}}(\lambda)=&B_{2}^{p_{0}}(\lambda)-\frac{\tilde{p}^{4}_{\lambda}(\lambda)}{630\pi(2-3\lambda)^{2}(1-2\lambda)^{3}}\,.\\\
\end{split}$ (67)

The functions $B_{1}$ and $B_{2}$ obtained from the two projections are
compared in Fig. 2. While the results differ quantitatively, the functions
agree on a qualitative level. Together with the generalization of our results
including the beta function for $\alpha_{1}$ and the dimensionless parameter
$\alpha_{2}$ provided in the supplementary material, eqs. (64)-(67) constitute
the main result of this section.

Figure 2:  Comparison between the functions $B_{1}(\lambda)$ (left panel) and
$B_{2}(\lambda)$ (right panel) obtained from reading off the anomalous
dimension $\eta_{N}$ from the time- (blue lines) and spatial component (orange
lines) of the momentum.

We stress that the anomalous dimension $\eta_{N}$, eq. (65), is conceptually
different from the “dynamical anomalous dimension” $\eta_{h}$ considered in
the covariant approach. Determining the latter requires information about the
scale-dependence of the graviton $3$-point vertex in order to disentangle the
running of the graviton self-interaction from the wave function
renormalization of the fluctuation field. This information is not contained in
the present projection though. As a consequence, eq. (64) dictates that
$\eta_{N}(g_{*},\lambda_{*})=-2$ at a NGFP.

We close the section with a conceptual remark. The differences appearing in
the $p_{0}$\- and $\vec{p}$-projection can be traced back to the $3$\- and
$4$-point vertices which depend on $p_{0}$ and $\vec{p}$ in a non-relativistic
way. A prototypical example is the $4$-point vertex
$\Gamma^{(hh\hat{N}\hat{N})}_{k}$. Since the potential terms in the action
(46) are linear in $N$ these do not contribute to this vertex. Considering the
kinetic part of the action, the definition of the extrinsic curvature entails
that this sector gives non-trivial contributions which contain just the time-
derivatives of the spatial metric. Schematically, the vertex then comes with
the momentum structure

$\Gamma^{(hh\hat{N}\hat{N})}_{k}h^{2}\hat{N}^{2}\propto\,p_{0}^{2}\,h^{kl}(-p)h_{kl}(p)\hat{N}(-q)\hat{N}(q),$
(68)

where $p=(p_{0},\vec{p})$ is the external 4-momentum, and
$q=(q_{0},\vec{q})$is the momentum running in the loop. So the vertex (68)
contributes to the $p_{0}$-projection while it does not enter into the
$\vec{p}$-projection.

This illustrates that, in general, the $n$-point functions arising in the ADM-
framework are non-relativistic. At the level of the $2$-point function this
difference has been eliminated by making the specific choice for the gauge-
fixing term (26). The Lorentz invariance breaking effect comes from the non-
linear ADM-decomposition of the metric. It is then tempting to try to dispel
these effects by admitting fluctuation terms of higher order in the gauge-
fixing procedure. An investigation along these lines will be left for the
future work.

##  5 Properties of the renormalization group flow

We proceed by analyzing the fixed point structure and phase diagrams entailed
by the beta functions (62). We start with the foliated Einstein-Hilbert
truncation, where $\alpha_{1}=\alpha_{2}=1$ in Sect. 5.1. The flow of the full
system including the beta function for $\alpha_{1}$ is discussed in Sect. 5.2.
As the main result, we show that all settings exhibit a non-Gaussian fixed
point (NGFP) suitable for providing the high-energy completion of the RG flow.

###  5.1 The foliated Einstein-Hilbert truncation

The beta functions for the foliated Einstein-Hilbert truncation are given in
eq. (64). Depending on whether the anomalous dimension of Newton’s coupling
$\eta_{N}$ is read off from the time or spatial component of the momentum
appearing in the graviton two-point function, the functions $B_{1}(\lambda)$
and $B_{2}(\lambda)$ are given in eqs. (66) and (67), respectively. Following
the terminology introduced in the last section, we refer to these cases as the
$p_{0}$-projection and the $\vec{p}$-projection, respectively.

Work | Projection | Fixed Points | Couplings | Critical Exponents
---|---|---|---|---
|  |  | $g_{*}$ | $\lambda_{*}$ | $\theta_{1}$ | $\theta_{2}$
this article | $p_{0}$-projection | GFP | $0$ | $0$ | $2$ | $-2$
NGFP1 | $1.43$ | $-0.12$ | $4.42\pm 1.38i$
NGFP2 | $-0.40$ | $0.14$ | $1.41\pm 3.84i$
NGFP3 | $5.86$ | $2.24$ | $9.37$ | $4.52$
this article | $\vec{p}$-projection | GFP | $0$ | $0$ | $2$ | $-2$
NGFP1 | $0.78$ | $-0.07$ | $5.01$ | $2.73$
NGFP2 | $-0.23$ | $0.22$ | $-0.80\pm 5.06i$
NGFP3 | $0.36$ | $1.04$ | $27.50$ | $-5.17$
Ref. Manrique:2011jc  | ADM | NGFP | $0.19$ | $0.31$ | $1.07\pm 3.31i$
Ref. Biemans:2017zca  | background level | NGFP | $0.90$ | $0.24$ | $1.06\pm 3.07i$
Ref. Christiansen:2014raa  | covariant | NGFP | $0.61$ | $0.32$ | $1.27\pm 3.01i$
$3$-point function
Table 3:  Summary of the fixed point structure obtained from the foliated
Einstein-Hilbert truncation using the $p_{0}$-projection (top box) and
$\vec{p}$-projection (middle box). The NGFP1 is present in both projection
schemes and serves as a UV-attractor for the RG flow in its vicinity. The
properties of the NGFP obtained within the background computations carried out
within the ADM-framework Manrique:2011jc ; Biemans:2017zca  and the covariant
fluctuation computation of Christiansen:2014raa  are added for comparison.

_Fixed points._ We first identify the fixed points $(g_{*},\lambda_{*})$
where, by definition,
$\beta_{g}(g_{*},\lambda_{*})=0,\beta_{\lambda}(g_{*},\lambda_{*})=0$ and
obtain their stability properties by computing the critical exponents from the
stability matrix (21). We first observe that both projection schemes admit a
GFP

${\rm
GFP:}\qquad(g_{*},\lambda_{*})=(0,0)\,,\qquad\theta_{1}=2\,,\;\;\theta_{2}=-2\,.$
(69)

The critical exponents agree with the ones obtained from canonical power
counting, in agreement with the definition of a GFP advocated in
Reuter:2019byg ; Saueressig:2023irs . The GFP is a saddle point in the
$g$-$\lambda$-plane. The eigenvector associated with the UV-repulsive
eigendirection indicates that this fixed point cannot serve as a UV-completion
for RG-trajectories with $g_{k}>0$.

In addition to the GFP, our numerical search for real roots also revealed
three NGFPs, present in both the $\vec{p}$\- and $p_{0}$-projection. Their
properties are summarized in Table 3. The fixed point of physical interest is
NGFP1. It is located at $g_{*}>0$ and $\lambda_{*}<0$ and serves as a UV-
attractor for the RG flow in its vicinity. The negative sign of $\lambda_{*}$
indicates that the dimensionless graviton mass appearing in the two-point
function at the fixed point is positive. Comparing the results for NGFP2 and
NGFP3, it is evident that the two projections do not necessarily give rise to
the same qualitative behavior. It is therefore remarkable that the NGFP1 found
in both cases share the same qualitative features.

The last three lines summarize properties for NGFPs reported from either ADM-
based computations at the background level Manrique:2011jc ; Biemans:2017zca
or flows of the graviton two-point function studied in the covariant approach
Christiansen:2014raa . Strictly speaking, these results do not lend themselves
to a direct comparison since - from the perspective of the fluctuation field
approach - the couplings $g$ and $\lambda$ listed in the various blocks are
associated with different correlation functions and projection prescriptions.
Nevertheless, the results form a coherent picture in the sense that they all
point towards the existence of a phenomenologically interesting NGFP in
foliation approach. Qualitatively, the features of this fixed point agree with
the ones observed in fluctuation computations carried out in the covariant
framework.

_Phase diagram._ The phase diagram for the foliated Einstein-Hilbert
truncation is shown in Fig. 3. The plots show that the $p_{0}$\- and
$\vec{p}$-projection lead to results which are qualitatively identical.

Figure 3:  Phase diagram obtained from the beta functions of the foliated
Einstein-Hilbert truncation in the $p_{0}$-projection (left diagram) and the
$\vec{p}$-projection (right diagram) with the arrows pointing towards lower
values of the coarse-graining scale $k$. The diagrams focus on the physically
interesting region where $g_{k}\geq 0$. The GFP (69), the NGFP1 given in Table
3, and the IR-FP (70) are marked by the black dots while the red lines mark a
singular locus of the beta functions. The flow is governed by the interplay of
these fixed points. The RG trajectory connecting the NGFP1 in the UV to the
GFP in the IR (Separatrix) is highlighted by the black line. The other black
lines have been added to highlight the boundary of the region where the RG
trajectories interpolate between NGFP1 in the UV and the IR-FP as
$k\rightarrow 0$.

In order to understand the structure, we first note that the line $g=0$
separates the trajectories with positive and negative $g_{k}$ and cannot be
crossed. Moreover, there is a singular locus at $\lambda=1/2$ linked to a
singularity in the beta functions. The beta functions give rise to a second,
non-trivial singular locus which is depicted by the red line. We then limit
the discussion to the physically interesting region $g\geq 0$, $\lambda\leq
1/2$, and the region below the red line. The RG flow in this region is
governed by the interplay of the NGFP1, the GFP, and the IR-fixed point

$\text{IR-FP:}\qquad(g_{*}^{\rm IR},\lambda_{*}^{\rm IR})=(0,1/2)\,.$ (70)

The NGFP2 and NGFP3 are separated from this region by the lines discussed
above and do not influence the flow in this region. As indicated by the
critical exponents, the NGFP1 serves as a UV-attractor for the RG trajectories
in its vicinity. At the IR-FP, the beta functions are ambiguous. Investigating
the scaling properties of RG trajectories in its vicinity, one finds that

$\lambda^{\rm IR}_{*}-\lambda(t)\sim c_{1}\,e^{t}\,,\qquad g(t)-g_{*}^{\rm
IR}\sim c_{2}\,e^{4t}\,,$ (71)

where $c_{1}$ and $c_{2}$ are constants depending on the RG trajectory chosen
and we use the symbol $\sim$ to indicate that the relations are valid in the
scaling regime of the IR-FP.

Among the set of trajectories emanating from NGFP1, two play a distinguished
role in determining the structure of the phase diagram. These are highlighted
by the black solid lines. Firstly, we have one trajectory which connects the
NGFP in the UV to the GFP in the IR. This line has been called separatrix in
Reuter:2001ag . Secondly, there is a “broken” trajectory which connects the
NGFP to a specific point on the singular line (red curve) and subsequently to
the IR-FP in the IR. These black lines bound different phases encountered in
the phase diagram. RG trajectories to the left of the separatrix (and not
terminating in the red, singular line) flow to $(g,\lambda)=(0,-\infty)$ and
lead to negative values $\Lambda_{k}$ in the IR. In this case, the mass
appearing in the two-point function is positive. Trajectories in the triangle
formed by the connecting lines approach the IR-FP in the IR. They all lead to
a zero value of the squared graviton mass $\mu^{2}\equiv-2\Lambda_{0}$.

At this stage, it is interesting to investigate the RG trajectories located
between the solid black lines in more detail. A set of example solutions
situated in this region is shown in Fig. 4. All trajectories are complete: the
high-energy completion is provided by the NGFP1. Lowering $k$ the flow crosses
over to a “classical regime” where the dimensionful couplings $G_{k}$ and
$\Lambda_{k}$ are constant. In terms of the phase diagrams given in Fig. 3,
this corresponds to the regime where the trajectories linger in the vicinity
of the GFP. At even lower values of $k$ the trajectories are captured by the
IR-FP and follow the scaling law (71). This feature has profound consequences
for the mass appearing in the graviton propagator: _based on the flow diagram
the renormalized squared mass $\mu^{2}$ can never be negative_. The scaling
induced by the IR-FP indicates that any positive $\Lambda_{k}$ at $k>0$ is
quenched to $\Lambda_{0}=0$ for trajectories in this region.

Figure 4:  Illustration of typical RG trajectories situated between the solid
black lines shown in Fig. 3 as a function of RG time $t$. The trajectories are
obtained by solving the beta function obtained from the $p_{0}$-projection
numerically with high precision. Starting from large values of $t$ and going
towards lower coarse-graining scales, the trajectories exhibit three distinct
scaling regimes: for $t\gg 1$ the running is controlled by the NGFP1
indicating that the dimensionless couplings $g_{k}$ and $\lambda_{k}$ take
their fixed point values. Lowering $t$, there is a cross-over to the GFP. This
phase is characterized by the dimensionful couplings $G_{k}$ and $\Lambda_{k}$
being constant. For even lower values of $t$ the trjectories leave the scaling
regime of the GFP and approach the IR-FP. Here the scale-dependence of $g_{k}$
and $\lambda_{k}$ follows the relation (71). This implies that
$\lim_{k\rightarrow 0}\Lambda_{k}=0$ for all trajectories situated in this
region.

###  5.2 Fixed points of the non-relativistic system

We proceed by analyzing the full system of beta functions including the flow
of $\alpha_{1}$ and the dependence on the parameter $\alpha_{2}$. Before
delving into the discussion, we make the following observation: if
$\alpha_{1}\not=1$ or $\alpha_{2}\not=1$, the action (16) is no longer
invariant with respect to the full diffeomorphism group Diff($\mathcal{M}$).
Instead the symmetry group is reduced to foliation-preserving diffeomorphisms
Diff($\mathcal{M},\Sigma)\subset$ Diff($\mathcal{M}$). This results in a
mismatch between the dependence of the fields on spacetime points, $N(\tau,y)$
and the time-component of the vector (10) generating infinitesimal foliation-
preserving diffeomorphism which depends on $\tau$ only (see
Rechenberger:2012dt  for detailed discussions).777Projective Hořava-Lifshitz
gravity Horava:2009uw  resolves this mismatch by restricting the lapse
function $N$ to depend on $\tau$ only. The change in the symmetry group
potentially entails consequences for the Faddeev-Popov procedure leading to
the ghost action (27): it is unclear whether the scalar ghost and anti-ghost
should be treated as fully spacetime-dependent fields. Solving this problem is
identical to finding a good gauge-fixing procedure for non-projectable Hořava-
Lifshitz gravity. Resolving this issue is beyond the scope of this work.
Instead we will analyze two versions of the flow equations: the first one
includes the contribution of the scalar ghost sector ($\bar{c}c$) and a second
version where these contributions are not included in the beta functions.

In order to close the beta functions, the analysis also requires an assumption
about the free parameter $\alpha_{2}$. Motivated by the previous section, we
first consider the case where $\alpha_{2}=1$. We briefly comment on other
values for this parameter at the end of this section.

$\alpha_{2}$ | Fixed Points | Ghost Sector | Couplings | Critical Exponents
---|---|---|---|---
|  |  | $g_{*}$ | $\lambda_{*}$ | $\alpha_{1,*}$ | $\theta_{1}$ | $\theta_{2}$ | $\theta_{3}$
1 | NGFP1 | with $\bar{c}c$ | $1.92$ | $-0.25$ | $0.71$ | $6.12$ | $-4.24\pm 5.91i$
1 | NGFP1 | without $\bar{c}c$ | $1.21$ | $-0.14$ | $0.73$ | $5.09$ | $\phantom{-}0.16\pm 4.97i$
$\alpha_{1}$ | NGFP2 | with $\bar{c}c$ | $65.03$ | $-10.96$ | $8.66$ | $10.19$ | $-1.16\pm 2.66i$
$\alpha_{1}$ | NGFP2 | without $\bar{c}c$ | $43.54$ | $-13.71$ | $14.53$ | $13.80$ | $\phantom{-}0.23\pm 3.85i$
Table 4: Fixed point structure of the full beta functions (62) including the
flow of $\alpha_{1}$. For $\alpha_{2}=1$ (top block) and
$\alpha_{1}=\alpha_{2}$ (bottom block), the fixed point without ghost
contributions are connected continuously to the ones with ghost contributions.
The NGFP1 and NGFP2 are not continuously connected though, therefore
justifying their different labels.

_Fixed Points._ We first investigate the generalization of the GFP in the
enlarged truncation. Expanding the beta functions in powers of $g$, one finds

$\begin{split}\beta_{g}&=2g+\mathcal{O}[g^{2}]*f_{g}(\lambda,\alpha_{1})\,,\\\
\beta_{\lambda}&=-2\lambda+\mathcal{O}[g]*f_{\lambda}(\lambda,\alpha_{1})\,,\\\
\beta_{\alpha_{1}}&=\mathcal{O}[g]*f_{\alpha_{1}}(\lambda,\alpha_{1})\,.\\\
\end{split}$ (72)

The first terms in $\beta_{g}$ and $\beta_{\lambda}$ are fixed by the
classical mass dimensions of the couplings and $f_{g}$, $f_{\lambda}$ and
$f_{\alpha_{1}}$ encode the leading corrections to the classical result. From
the expansion (72) one readily verifies that all three beta functions vanish
for $g=\lambda=0$. Hence the system admits _a one-parameter family of GFPs_ :

$\text{GFPs:}\qquad\left(g_{*},\lambda_{*},\alpha_{1,*}\right)=\left(0,0,\alpha_{1,*}\right)\,.$
(73)

The critical exponents of these fixed points are fixed by the mass dimension
of the couplings. Evaluating the stability matrix based on the expansion (72)
one finds $\theta_{1}=2$, $\theta_{2}=-2$, and $\theta_{3}=0$, with the latter
corresponding to a marginal direction.

In addition to the GFPs, the full system also possess a NGFP generalizing the
fixed point NGFP1 of the foliated Einstein-Hilbert truncation. Its position
and stability properties are summarized in Table 4 which gives the results for
both cases where the scalar ghost contribution has been taken into account
(labeled by “with $\bar{c}c$”) and left out (labeled by “without $\bar{c}c$”).
We checked that the two fixed points are related by a continuous deformation
when turning on the scalar ghost contribution in the beta functions. We note
that the fixed point is situated close to but not within the surface
$\alpha_{1}=1$ which is spanned by the foliated Einstein-Hilbert truncation.
The position is rather insensitive to whether the scalar ghost contribution is
included. The latter has a profound consequence on its stability properties
though: in the absence of the $\bar{c}c$-contribution NGFP1 is UV-attractive
in all three directions. Upon including the scalar contribution in the beta
function the fixed point turns into a saddle point with one UV-attractive and
two UV-repulsive directions.

The occurrence of a line of GFPs in truncations of the form (16) has already
been observed in Contillo:2013fua . The non-relativistic viewpoint adopted in
the present work offers a natural explanation of this phenomenon. Let us set
$\Lambda_{k}=0$ for the time being, i.e., all component fields are taken to be
massless. Inspecting the dispersion relations of the component fields, we
encounter the generic form

$\Gamma_{k}^{(ij)}\propto a_{i}\,p_{0}^{2}+b_{i}\,\vec{p}^{\,2}\,,\qquad
c_{i}^{2}\equiv\frac{b_{i}}{a_{i}}\,.$ (74)

The explicit values for the coefficients $a_{i}$ and $b_{i}$ can be read off
from Table 1 and $c_{i}$ is the resulting “speed of light” associated with a
given component field. For instance, the transverse-traceless modes $h_{ij}$
and the scalar $B$ propagate with

$c_{h}^{2}=\frac{1}{\alpha_{1}}\,,\quad\text{and}\qquad
c_{B}^{2}=1+2\alpha_{1}-2\alpha_{2}\,,$ (75)

respectively. 888In the scalar sector, comprising the fields $\Psi$ and
$\hat{N}$ the dispersion relations do not have the simple structure (74) owed
to the off-diagonal terms in the propagator. Generically the eigenvalues of
this matrix involve square-roots. In the special case
$\alpha_{1}=\alpha_{2}=1$, all equations reduce to their relativistic form and
one has $c_{i}=1$ for all component fields. Keeping $\alpha_{2}$ fixed, the
system then admits a one-parameter family of non-interacting theories, where
the component fields satisfy different dispersion relations controlled by the
value of $\alpha_{1}$. This feature then generates the line of GFPs (73). The
fact that differences in propagation speeds are observable quantities suggests
that each point on the line actually corresponds to a physically distinct
theory. This also entails that lines of GFPs should be a rather generic
property of non-relativistic systems where one has several fields with
distinct dispersion relations.

Table 4 also reports a NGFP found within the alternative identification
$\alpha_{1}=\alpha_{2}$. This NGFP comes with identical stability properties
as the $\alpha_{2}=1$ case, albeit at much larger values of the couplings.
Again one readily checks that the NGFPs found with and without the scalar
ghost contributions are connected by analytic continuation. While the
positions and stability coefficients suggest that the fixed points for
$\alpha_{2}=1$ and $\alpha_{1}=\alpha_{2}$ are also continuously connected we
have not been able to construct such a deformation explicitly. Building on the
interpolation $\alpha_{2}=1+\gamma(\alpha_{2}-1)$, $\gamma\in[0,1]$, the lines
followed by the fixed point as a function of $\gamma$ terminate around
$\gamma\approx 0.7$. We conclude that the NGFP1 is robust under a wide range
of values for $\alpha_{2}$. Moreover, it is the scalar ghost sector which has
an decisive effect on the stability properties of the fixed point.

##  6 Summary and discussion

The presence of a foliation on spacetime is essential for transiting from
quantum gravity formulated on Euclidean signature spacetime to its Lorentzian
counterpart. One way to implement this structure is through the Arnowitt-
Deser-Misner (ADM)-decomposition of the metric degrees of freedom. In this
work, we studied the RG flows arising within this setting using non-
perturbative functional renormalization group methods. Our work reports the
first fluctuation field computation in the ADM-framework, studying the RG flow
of the graviton two-point function. It complements earlier investigations
based on the ADM-formalism Manrique2011 ; Rechenberger:2012dt ;
Contillo:2013fua ; Biemans:2016rvp ; Biemans:2017zca ; Houthoff:2017oam  and
covariant computations utilizing a foliation gauge fixing Knorr:2018fdu ;
Eichhorn:2019ybe ; Knorr:2022mvn . It is also closely related to the
Lorentzian signature computations based on the Wetterich equation
Banerjee:2022xvi ; DAngelo:2022vsh ; DAngelo:2023tis . At the technical level,
these computations are significantly more complex than their counterparts in
the covariant setting reviewed in Pawlowski:2020qer . This is owed to the
increased number of component fields carrying the gravitational degrees of
freedom as well as the proliferation of potential vertex structures coming
with independent couplings.

We give a comprehensive introduction to the general setup and structures
featuring in these computations. In particular, we highlight the differences
between fluctuation computations in the covariant and foliated frameworks. The
fact that the latter naturally also comprises non-relativistic theories like
Hořava-Lifshitz gravity Horava:2009uw  introduces new elements. In particular,
one finds that, generically, projections onto tensor structures involving the
spatial and time-parts of the fluctuation field momentum may lead to different
results. Conceptually, this implies that one is dealing with different
coupling constants of the theory which, in the covariant setting, exhibit the
same flow due to the enhanced symmetry.

In order to illustrate the general framework, we performed an explicit
computation of the gravitational RG flow projected onto the graviton two-point
function. The setting is based on a truncation of the effective average action
containing all two-derivative terms compatible with foliation-preserving
diffeomorphisms, cf. eq. (16). This action is used to derive the graviton two-
point function as well as the 3- and 4-point vertices required to close the
flow equation. By projecting the RG equation onto the 2-point function, we
derive the beta functions for Newton’s coupling, the cosmological constant
(featuring as a graviton mass term in the 2-point function) as well as one
coupling $\alpha_{1}$ encoding deviations from general relativity.
Furthermore, the results depend parametrically on a second Lorentz-symmetry
violating coupling $\alpha_{2}$ which does not enter in the 2-point function
under consideration but appears on the right-hand side of the flow equation.

The fixed point structure and phase diagram resulting from the foliated
Einstein-Hilbert action is discussed in detail. As a key result, all
projections studied in this work identify a non-Gaussian fixed point (NGFP)
suitable for asymptotic safety. It serves as a UV-attractor for the two
couplings retained within the projection. This puts the existence of the NGFP
already identified in background field computations on spherical
Manrique:2011jc ; Rechenberger:2012dt , toroidal Houthoff:2017oam , and
cosmological backgrounds Biemans:2016rvp ; Platania:2017djo  on firm grounds.
The NGFP seen in our work is part of an intricate phase diagram, see Fig. 3.
In the physically relevant region, its structure is governed by the interplay
of the GFP, the NGFP, and an IR-FP. The latter controls the IR-behavior of RG-
trajectories which, in principle, could lead to a negative squared graviton
mass $\mu^{2}$. The fixed point quenches this mass to zero. This restricts the
values of the squared graviton mass supported by the phase diagram to
$\mu^{2}\geq 0$.

Remarkably, our phase diagram is qualitatively identical to the one derived
from similar fluctuation field computations in the covariant setting
Christiansen:2012rx ; Christiansen:2014raa .999The phase diagram is also very
similar to the one found for the Einstein-Hilbert truncation at the background
level, first constructed in Reuter:2001ag . In this case, the IR-FP is
replaced by a singular line which leads to a termination of the RG-
trajectories at finite values of the coarse-graining scale though. This is
rather striking, since the underlying computations are based on an entirely
different implementation of the gravitational degrees of freedom. Moreover,
the vertex structures used in the right-hand side of the flow equation are
manifestly different since the relation between the covariant fluctuation
field and the fluctuation fields in the ADM-formalism are non-linear. Taken
together, this indicates a remarkable stability of the phase diagram related
to the graviton 2-point function.

The space of action functionals which are invariant with respect to
diffeomorphisms spans a subspace of all action functionals compatible with
foliation preserving diffeomorphisms. The enhanced symmetry implies that RG
flows on the former are closed Rechenberger:2012dt ; Knorr:2018fdu . The
Wetterich equation adapted to the ADM-formalism allows to study RG flows on
the larger theory space as well. In this work, we did this by deriving the
beta function for the coupling $\alpha_{1}$ encoding different speeds of light
for the component fields in the construction. We show that all projections
within this truncation possess a NGFP which is the analogue of the NGFP seen
in the foliated Einstein-Hilbert case. Based on symmetry arguments, this NGFP
should be located on the surface with enhanced symmetry,
$\alpha_{1}=\alpha_{2}=1$. Since the Wetterich equation for the ADM-formalism
contains diffeomorphism breaking terms the fixed point is slightly shifted
away from this surface though. The stability properties of the NGFP depend on
the implementation of the scalar ghost sector. In case that the scalar ghost
is left out (as suggested by the reduced symmetry of the setup) one finds that
the NGFP is a UV-attractor on the projection space. This is in agreement with
the background computation Contillo:2013fua , which identified a similar UV-
attractive fixed point once diffeomorphism-breaking couplings are included.

Our work also makes a notable step forward in connecting continuum
computations and Monte Carlo simulations of the gravitational path integral.
In particular, the framework introduced in our work gives access to the
spatially averaged correlation functions accessible in the Causal Dynamical
Triangulations (CDT) program Ambjorn:2012jv ; Loll:2019rdj . In this context,
it is important to highlight that the phase diagram obtained in the present
work differs from the one obtained at the background level Reuter:2001ag ;
Manrique:2011jc  by the addition of an IR-fixed point. This feature may be
interesting when interpreting the RG flows Ambjorn:2014gsa ; Ambjorn:2020rcn
obtained from Monte Carlo simulations in the context of CDT. Both settings
share the same foundations in the sense that they build on a foliation
structure and admit anisotropy parameters distinguishing between space and
time. We hope to come back to this intriguing possibility in the future.

Obviously, it will also be interesting to investigate the robustness of the
fixed point structure and phase diagram shown in Fig. 3 once matter degrees of
freedom are included. Specifically, one may want to ad massless gauge fields
supplemented by Lorentz symmetry breaking parameters, similar to $\alpha_{1}$
and $\alpha_{2}$ introduced in the gravitational sector. This setting may
allow to study differences in the light cone structure for the two fields
which can be constrained by multi-messenger astronomy. Moreover, the framework
set up in the present work allows for the direct transition to Lorentzian
signature computations. Results along these lines will be reported in
WangInPrep1  and WangInPrep2 , respectively.

## Acknowledgments

We thank J. Ambjørn, M. Becker, A. Bonanno, G. P. de Brito, A. Ferreiro, B.
Knorr, G. Korver, A. Koshelev, A. Pereira, M. Reuter, and M. Schiffer for
discussion and insightful comments on the manuscript. JW acknowledges the
China Scholarship Council (CSC) for financial support.

##  Appendix A Technical Annex

The derivation of the results presented in Sect. 4 is technically quite
involved. This appendix collects some additional background information and
lengthy formulas underlying these computations. We start by briefly outlining
the implementation of a foliation structure within Mathematica in App. A.1.
App. A.2 collects the propagators for the component fields and the actions
underlying the derivation of the gravity-ghost vertices are given in App. A.3.

###  A.1 Implementation within Mathematica

In the ADM-formalism the relations (8) decompose the spacetime $g_{\mu\nu}$
into the lapse function $N$, a shift vector $N_{i}$, and spatial metric
$\sigma_{ij}$. As a result, the four-dimensional spacetime $\mathcal{M}$
inherits a foliation structure

$\mathcal{M}=\mathbb{R}\times\Sigma_{\tau}\,.$ (76)

The dynamics of the theory is expressed in terms of time-derivative
$\partial_{\tau}$ and covariant derivatives $D_{i}$ constructed from the
metric $\sigma_{ij}$ on the spatial slices.

In order to implement the decomposition (76) in the xAct-package xactref ;
Nutma:2013zea  for Mathematica, we define the three-dimensional manifold
$\Sigma_{\tau}$ and a one-dimensional manifold $\mathbb{R}$. These correspond
to the two factors in (76), respectively. Subsequently, the code defines the
four-dimensional spacetime $\mathcal{M}$ as the product of the two factors.
The field content is then coded as the spatial metric $\sigma_{ij}(\tau,y)$,
one scalar field $N(\tau,y)$, and one vector field $N_{i}(\tau,y)$. The
indices are taken to be covariant with respect to $\Sigma_{\tau}$. The time-
derivative and spatially covariant derivatives are defined as the covariant
derivatives on $\mathbb{R}$ and $\Sigma_{\tau}$, respectively. They are
compatible with the metrics on the two submanifolds, respectively.

Building on this setup, all propagators and vertices needed in our computation
can be generated via the Perturbation command provided by the xAct package.
The expressions for the vertices obtained in this way are very lengthy though.
This holds specifically for the $3$\- and $4$-point vertices which contain
hundreds of contributions. Therefore, we will not give the explicit form of
these contributions within this paper. The mathematica-code generating them is
available on request.

###  A.2 Propagators for the component fields

The $2$-point functions including the contribution from the gauge fixing term
have been summarized in Table 1. The propagators required in the evaluation of
eq. (34) are obtained as the inverse of this matrix. Defining the propagator
matrix including the contributions of the scalar regulator
${\mathcal{R}}_{k}(p^{2})$,

$\mathcal{G}\equiv\left(\Gamma_{k}^{(2)}+{\mathcal{R}}_{k}\right)^{-1}\,,$
(77)

we list the matrix elements in Table 5. In addition to the definition of the
$4$-momentum, $p^{2}=p_{0}^{2}+\vec{p}^{\,2}$, the table uses the shorthand
notations

$\begin{split}f_{a}=\alpha_{1}p_{0}^{2}+\vec{p}^{\,2}\,,\;f_{b}=\frac{1}{\alpha_{1}}p_{0}^{2}+\vec{p}^{\,2}\,,\;f_{d}=\alpha_{2}p_{0}^{2}+\vec{p}^{\,2}\,,\;f_{e}=\frac{1}{\alpha_{2}}p_{0}^{2}+\vec{p}^{\,2}\,,\end{split}$
(78)

for capturing the $\alpha$-dependence of the dispersion relations.

fields $(i,j)$ | $\mathcal{G}^{(ij)}_{k}$
---|---
$h_{ij}h^{kl}$ | $32\pi G_{k}\,\frac{1}{f_{a}-2\Lambda_{k}+R_{k}}\,{\Pi_{h}}^{ij}_{~{}~{}kl}$
$v_{i}v^{j}$ | $16\pi G_{k}\,\frac{1}{f_{a}-2\Lambda_{k}+R_{k}}\,{\Pi_{u}}{}^{i}{}_{j}$
$EE$ | $48\pi G_{k}\,\frac{1}{f_{a}-2\Lambda_{k}+R_{k}}$
$u^{i}u_{j}$ | $\frac{16\pi G_{k}}{\alpha_{1}}\,\frac{1}{f_{b}+R_{k}}\,{\Pi_{u}}{}^{i}{}_{j}$
$BB$ | $16\pi G_{k}\,\frac{1}{p^{2}+2\alpha_{1}f_{b}-2\alpha_{2}f_{e}+(2\alpha_{1}-2\alpha_{2}+1)\,R_{k}}$
$\Psi\Psi$ | $96\pi G_{k}\Big{(}\frac{p^{2}+R_{k}}{f_{a}p^{2}-3p^{2}f_{d}+7p^{2}\Lambda_{k}-6\Lambda_{k}^{2}+(f_{a}-2p^{2}-3f_{d}+7\Lambda_{k})R_{k}-2R_{k}^{2}}\Big{)}$
$\hat{N}\hat{N}$ | $-8\pi G_{k}\Big{(}\frac{6f_{d}-2f_{a}-3p^{2}-2\Lambda_{k}+R_{k}}{f_{a}p^{2}-3p^{2}f_{d}+7p^{2}\Lambda_{k}-6\Lambda_{k}^{2}+(f_{a}-2p^{2}-3f_{d}+7\Lambda_{k})R_{k}-2R_{k}^{2})}\Big{)}$
$\Psi\hat{N}$ | $48\pi G_{k}\Big{(}\frac{p^{2}+R_{k}-2\Lambda_{k}}{f_{a}p^{2}-3p^{2}f_{d}+7p^{2}\Lambda_{k}-6\Lambda_{k}^{2}+(f_{a}-2p^{2}-3f_{d}+7\Lambda_{k})R_{k}-2R_{k}^{2}}\Big{)}$
$\bar{c}c$ | $\frac{1}{\sqrt{2}\,p^{2}}$
$\bar{b}^{i}b_{i}$ | $\frac{1}{\sqrt{2}\,p^{2}}\,\Pi_{u}{}_{j}{}^{i}$
Table 5:  Matrix elements of the regulated propagators required in the
evaluation of (34). For the diagonal blocks given in Table 1, the inversion is
straightforward and the results are given in the first block. For the scalar
sector spanned by the fields $\Psi$ and $\hat{N}$ the propagator is found by
inverting a $2\times 2$-matrix. The determinant arising from this inversion
leads to the rather complicated dispersion relations listed in the second
block. The propagators in the ghost sector are given in the third block.

The projectors $\Pi$ carrying the index structure of the propagators are
defined in eqs. (51) and (52).

###  A.3 Ghost actions on a foliated spacetime

We close the technical annex by giving the ghost action resulting from
evaluating the general formula (27) for the specific choice of gauge fixing
functional (26). The result is used to extract the propagators and interaction
vertices involving the scalar (anti-)ghost $\bar{c},c$ and the vector
(anti-)ghosts $\bar{b}^{i},b_{j}$, respectively. In order to keep the
complexity at a manageable level we chose the background to be flat space.
Moreover, all derivatives act on the right,
$\partial_{\tau}N_{i}c=N_{i}(\partial_{\tau}c)+(\partial_{\tau}N_{i})c$. The
contribution proportional to the scalar anti-ghost $\bar{c}$, arising from the
first term in (27) is

$\begin{split}\Gamma^{\text{scalar}}_{\text{ghost}}=-\sqrt{2}\,\int d\tau
d^{d}y\,\bar{c}\Big{[}&\partial_{\tau}^{2}cN+\partial_{\tau}b^{k}\partial_{k}N-\partial_{\tau}NN^{i}\partial_{i}c+\partial^{i}\partial_{\tau}N_{i}c+\partial^{i}b^{k}\partial_{k}N_{i}\\\
&+\partial^{i}N_{k}\partial_{i}b^{k}+\partial^{i}\sigma_{ki}\partial_{\tau}b^{k}+\partial^{i}N_{k}N^{k}\partial_{i}c+\partial^{i}N^{2}\partial_{i}c\\\
&-\frac{1}{2}\partial_{\tau}\sigma^{ij}c\partial_{\tau}\sigma_{ij}-\frac{1}{2}\partial_{\tau}\sigma^{ij}b^{k}\partial_{k}\sigma_{ij}-\partial_{\tau}\partial_{k}b^{k}-\partial_{\tau}N^{i}\partial_{i}c\Big{]}\,.\end{split}$
(79)

In the vector sector, the terms proportional to the anti-ghost $\bar{b}^{i}$
are

$\begin{split}\Gamma^{\text{vec}}_{\text{ghost}}=-\sqrt{2}\,\int\,d\tau&d^{d}y\,\bar{b}^{i}\Big{[}\partial^{2}_{\tau}N_{i}c+\partial_{\tau}b^{k}\partial_{k}N_{i}+\partial_{\tau}N_{k}\partial_{i}b^{k}+\partial_{\tau}\sigma_{ki}\partial_{\tau}b^{k}\\\
&+\partial_{\tau}N_{k}N^{k}\partial_{i}c+\partial_{\tau}N^{2}\partial_{i}c-\partial_{i}\partial_{\tau}cN-\partial_{i}b^{k}\partial_{k}N\\\
&+\partial_{i}NN^{k}\partial_{k}c-\frac{1}{2}\partial_{i}\sigma^{mn}c\partial_{\tau}\sigma_{mn}-\frac{1}{2}\partial_{i}\sigma^{mn}b^{k}\partial_{k}\sigma_{mn}-\partial_{i}\partial_{k}b^{k}\\\
&-\partial_{i}N^{m}\partial_{m}c+\partial^{j}c\partial_{\tau}\sigma_{ij}+\partial^{j}b^{k}\partial_{k}\sigma_{ij}+\partial^{j}\sigma_{jk}\partial_{i}b^{k}+\partial^{j}\sigma_{ik}\partial_{j}b^{k}\\\
&+\partial^{j}N_{j}\partial_{i}c+\partial^{j}N_{i}\partial_{j}c\,\Big{]}.\end{split}$
(80)

##  Appendix B Trace computations

The beta functions given in Sect. 4.2 are obtained by projecting the Wetterich
equation (17) onto the $2$-point function of the transverse-traceless
fluctuation field $h_{ij}(x)$. Their computation requires evaluating the
traces appearing on the right-hand side of eq. (34). In a flat background,
this is conveniently done by using standard momentum space methods for
evaluating Feynman diagrams. In this appendix, we provide some general
identities useful in these computations in App. B.1. The computation of a
typical diagram built from a $4$-point vertex and $3$-point vertices is
illustrated in App. B.2. We also briefly discusses the analytic properties of
the loop integrals.

###  B.1 Structural aspects of the traces

The right-hand side of the projected flow equation contains one-loop diagrams
built from a pair of $3$-point vertices (bubble diagrams) and a $4$-point
vertex (tadpole diagrams)

$\begin{split}T_{3}=\,&-\frac{1}{2}\text{STr}[\mathcal{G}_{\chi_{a^{i}}\chi_{a^{j}}}{\Gamma}^{(hh\chi_{a^{j}}\chi_{a^{k}})}\mathcal{G}_{\chi_{a^{k}}\chi_{a^{l}}}{\partial_{t}R_{k}}^{\chi_{a^{l}}\chi_{a^{i}}}]\\\
T_{4}=\,&\text{STr}[\mathcal{G}_{\chi_{a^{i}}\chi_{a^{j}}}\Gamma^{({h\chi_{a^{j}}\chi_{a^{k}}})}\mathcal{G}_{\chi_{a^{k}}\chi_{a^{l}}}\Gamma^{(h\chi_{a^{l}}\chi_{a^{m}})}\mathcal{G}_{\chi_{a^{m}}\chi_{a^{n}}}{\partial_{t}R_{k}}^{\chi_{a^{n}}\chi_{a^{i}}}]\,.\end{split}$
(81)

Here the subscript on $T$ refers to the vertices contained in the trace. The
trace contains a sum over all fluctuation fields which can propagate in the
regularized loop and subsequently we will use superscripts to single out the
contributions of specific fields to the complete trace. In addition, the trace
also contains an integral over loop momenta $q_{\mu}=(q_{0},q_{i})$. The
propagators and vertices depend on both the loop momentum and the external
momentum associated with the transverse-traceless field
$h_{ij}(p_{0},\vec{p})$. The integrals over the spatial loop momenta have the
generic form

$I_{n}\equiv\int d^{3}q\,f(\vec{q}^{~{}2})\,T^{i_{1}i_{2}\cdots
i_{n}}q_{i_{1}}q_{i_{2}}\cdots q_{i_{n}}\,.$ (82)

The $T^{i_{1}i_{2}\cdots i_{n}}$ are tensors constructed from the fields and
external momenta allowed by our projection. Canonical building blocks are, for
example, $h^{ij}(p_{0},\vec{p})$ and $p^{i}$. The parts depending on the
magnitude of the spatial loop momentum have been factored out and collected in
the function $f(\vec{q}^{\,2})$.

In the present computation, the maximum number of loop momenta contracted with
$T^{i_{1}i_{2}\cdots i_{n}}$ turns out to be $n=6$. Higher-order terms are
outside of the projection subspace and do not contribute to the computation.
Following Peskin:1995ev , the integrals (82) can be simplified as follows. For
$n$ odd the contributions vanish due to the anti-symmetry in sending
$q_{i}\mapsto-q_{i}$. For even $n$ the integrands can be reduced to functions
depending on $\vec{q}^{\,2}$ only. For the integrals defined in eq. (82), the
relevant identities are

$\begin{split}I_{2}&=\frac{1}{3}\,\int
d^{3}qf(\vec{q}^{\,2})T_{ij}\,\delta^{ij}\vec{q}^{\,2}\,,\\\
I_{4}&=\frac{1}{15}\int
d^{3}qf(\vec{q}^{\,2})T_{ijkl}(\delta^{ij}\delta^{kl}+\delta^{ik}\delta^{jl}+\delta^{il}\delta^{jk})(\vec{q}^{\,2})^{2},\\\
I_{6}&=\frac{1}{105}\int
d^{3}qf(\vec{q}^{\,2})T_{ijklmn}\left(\delta^{ij}\delta^{kl}\delta^{mn}+14\;\text{permutations}\right)(\vec{q}^{\,2})^{3}\,.\end{split}$
(83)

The external legs associated with the transverse-traceless field $h_{ij}$ have
to be attached to the $3$\- and $4$-point vertices. In terms of computational
efficiency, it is convenient to retain these fields in the intermediate steps
in order to deal with scalar instead of tensorial quantities. Upon carrying
out the integration over loop momenta, they can be stripped. The projection
onto the tensor structure $\Pi_{h}$ appearing on the left-hand side is then
evoked by contracting the resulting matrix with the projectors $\Pi_{h}$
defined in eq. (51). The coefficients $T_{p_{0}}$, $T_{\vec{p}}$, and $T_{0}$
defined in (60) are the read off by matching powers of the external momenta.

###  B.2 Computations of selected traces

In general, the evaluation of the tadpole diagrams is simpler than the one of
the bubble diagrams. Hence we start with an example based on the $4$-point
vertex containing four gravitons. Subsequently, we discuss the projection of
the traces associated with bubble-diagrams, expanding the trace arguments in
terms of the external momenta.

####  B.2.1 The tadpole diagram containing the $4$-graviton vertex

The contribution originating from the $4$-graviton vertex has the form

$T_{4}^{hh}=-\frac{1}{2}\text{Tr}[\mathcal{G}_{hh}\Gamma^{(hhhh)}_{k}\mathcal{G}_{hh}{\partial_{t}{\mathcal{R}}_{k}}^{hh}]\,.$
(84)

The propagators are listed in Table 5 and the $4$-point vertex is generated
from the computer algebra. The trace contains an integration over loop-
momentum. The explicit evaluation of these integrals requires choosing a
regulator function $R_{k}$. In practice, we have opted for a Litim-type
regulator Litim:2000ci ; Litim:2001up  where
$R_{k}(q^{2})=(k^{2}-q^{2})\Theta(k^{2}-q^{2})$. In this case, the integration
is restricted to the domain where $q_{0}^{2}+\vec{q}^{\,2}\leq k^{2}$.
Performing all index contractions and using the identities (83), one finds

$\begin{split}T_{4}^{hh}=-\int_{q^{2}\leq
k^{2}}&\frac{dq_{0}d^{3}\vec{q}}{(2\pi)^{4}}\bigg{[}\frac{(k^{2}(\eta_{N}-2)-q^{2}\eta_{N})(5\vec{q}^{~{}2}+9\alpha_{1}q_{0}^{2}-4\alpha_{2}q^{2}_{0}-14k^{2}\lambda_{k})}{5(q_{0}^{2}(\alpha_{1}-1)+k^{2}(1-2\lambda_{k}))^{2}}\\\
&+\frac{(k^{2}(\eta_{N}-2)-q^{2}\eta_{N})}{(q_{0}^{2}(\alpha_{1}-1)+k^{2}(1-2\lambda_{k}))^{2}}\left(\frac{9\alpha_{1}-4\alpha_{2}}{5}p_{0}^{2}+\vec{p}^{\,2}\right)\bigg{]}h^{ij}(-p)h_{ij}(p)\,.\\\
\end{split}$ (85)

This expression illustrates the following feature. We have picked the
regulator to be a function of $q^{2}=q_{0}^{2}+\vec{q}^{\,2}$. The propagators
entering our computation contain the loop momentum $q_{0}^{2}$ and
$\vec{q}^{\,2}$ in different linear combination. In eq. (85) this manifests
itself in form of the terms $q_{0}^{2}(\alpha_{1}-1)$ in the denominators. As
a consequence the integrand may exhibit poles when performing the integral
over $q_{0}$. However, we find that there are regions in the parameter space
of $\alpha_{1}$ and $\lambda_{k}$ where the poles are absent and the integral
is finite. For the integrands in eq. (85) this requires that the following
range of couplings should be excluded:

$hh:\qquad-1\leq\frac{1-2\lambda_{k}}{\alpha_{1}-1}\leq 0\,.$ (86)

Note that this bound is satisfied by the foliated Einstein-Hilbert truncation
where $\alpha_{1}=\alpha_{2}=1$ and $q_{0}$ drops out of the denominators.

We proceed by evaluating (85). Since $\vec{q}$ appears in the numerator only,
performing this integration is rather straightforward. Subsequently, we
integrate over $q_{0}$. Assuming the absence of poles, this integration
results in inverse trigonometric functions. Since the outcome is rather
lengthy, we illustrate the generic structure based on the coefficient
multiplying the tensor structure $h^{ij}h_{ij}~{}\vec{p}^{~{}2}$,

$\begin{split}T_{4}^{hh}=\Big{(}P_{1}(q_{0})+P_{2}(q_{0})~{}f^{hh}_{1}(q_{0})+P_{3}(q_{0})~{}f^{hh}_{2}(q_{0})\left.\Big{)}\right|^{q_{0}=k}_{q_{0}=-k}\times
h^{ij}h_{ij}~{}\vec{p}^{\,2}\,.\end{split}$ (87)

The trigonometric contributions are captured by

$f^{hh}_{1}(q_{0})=\text{arctan}\left(\frac{q_{0}}{\sqrt{k^{2}-q_{0}^{2}}}\right)\,,\quad
f^{hh}_{2}(q_{0})=\text{arctanh}\left(\frac{q_{0}}{\sqrt{k^{2}-q_{0}^{2}}}\sqrt{\frac{\alpha_{1}-2\lambda_{k}}{2\lambda_{k}-1}}\right)\,.$
(88)

The polynomials appearing in this expression read

$\begin{split}P_{1}(q_{0})=&-\frac{q_{0}\sqrt{k^{2}-q_{0}^{2}}}{60\pi^{3}\left(\alpha_{1}-1\right){}^{2}\left(1-2\lambda_{k}\right)\left(k^{2}\left(1-2\lambda_{k}\right)+\left(\alpha_{1}-1\right)q_{0}^{2}\right)}\\\
&\Big{(}\alpha_{1}k^{2}\left(\lambda_{k}\left(10-4\eta_{N}\right)+5\right)+k^{2}\left(-10\lambda_{k}+8\lambda_{k}^{2}\eta_{N}-4\lambda_{k}\eta_{N}+\eta_{N}\right)\\\
&\qquad+\alpha_{1}^{2}k^{2}\left(\eta_{N}-5\right)+\left(2\lambda_{k}-1\right)\eta_{N}q_{0}^{2}+\alpha_{1}\left(1-2\lambda_{k}\right)\eta_{N}q_{0}^{2}\Big{)}\\\
P_{2}(q_{0})=&\frac{k^{2}}{60\pi^{3}\left(\alpha_{1}-1\right){}^{3}}\times\left(8\lambda_{k}\eta_{N}-5\alpha_{1}\left(\eta_{N}-2\right)+\eta_{N}-10\right),\\\
P_{3}(q_{0})=&-\frac{k^{2}\sqrt{\alpha_{1}-2\lambda_{k}}}{60\pi^{3}\left(\alpha_{1}-1\right){}^{3}\left(2\lambda_{k}-1\right){}^{3/2}}\times\Big{(}\alpha_{1}\left(6\lambda_{k}\eta_{N}-5\eta_{N}+20-20\lambda_{k}\right)\\\
&\qquad-16\lambda_{k}^{2}\eta_{N}+10\lambda_{k}\left(\eta_{N}+2\right)+\alpha_{1}^{2}\left(\eta_{N}-5\right)-15\Big{)}.\end{split}$
(89)

Matching the result (87) to the definition (60), one then obtains the
contribution of this specific tadpole diagram to the beta functions.

The remaining tadpole contributions can be computed along the same lines.
Again, one encounters inverse trigonometric functions which put bounds on the
space of admissible couplings. Since the component fields come with different
dispersion relations, one actually finds a set of inequalities. Retaining both
parameters $\alpha_{1}$ and $\alpha_{2}$ we find the following exclusion
regions

$\begin{split}BB:\qquad&-1\leq\frac{2\alpha_{1}-2\alpha_{2}+1}{2(\alpha_{2}-\alpha_{1})}\leq
0\,,\\\
NN:\qquad&-1\leq-\frac{2-7\lambda_{k}+6\lambda_{k}^{2}}{2+\alpha_{1}-3\alpha_{2}}\leq
0\,,\\\ uu:\qquad&-1\leq\frac{\alpha_{1}}{1-\alpha_{1}}\leq 0\,.\end{split}$
(90)

For $\alpha_{2}=1$ the condition for $uu$ is a subset of $BB$, so that one has
to consider only three inequalities in this case. The admissible values for
the couplings leading to convergent loop integrals are most conveniently found
graphically and are illustrated in Fig. 5. We note that the traces found for
the foliated Einstein-Hilbert truncation are always convergent due to our
choice of regulator.

Figure 5:  Illustration of the inequalities (86) and (90) for $\alpha_{2}=1$
(left panel) and $\alpha_{1}=\alpha_{2}$ (right panel). The inequalities are
satisfied in the white regions, implying that the loop integrals do not
converge for this set of couplings. The gray regions are admissible. Note that
the foliated Einstein-Hilbert truncation $\alpha_{1}=\alpha_{2}=1$ is situated
in the gray region and all loop contributions are finite in this case.

####  B.2.2 Bubble diagrams

We complete the appendix with a brief discussion of the contributions
resulting from bubble diagrams, encoded in the first trace of eq. (34). We can
always shift the loop momentum so that the argument of the regulator is
$q^{2}$. The trace contributions then take the form

$T_{3}=\text{STr}\left[\mathcal{G}_{\chi_{a}\chi_{b}}(q)\Gamma^{(h\chi_{b}\chi_{c})}_{k}(p,q)\mathcal{G}_{\chi_{c}\chi_{d}}(p+q)\Gamma^{(h\chi_{d}\chi_{e})}_{k}(p,q)\mathcal{G}_{\chi_{e}\chi_{f}}(q)\,\partial_{t}{\mathcal{R}}_{k}^{\chi_{f}\chi_{a}}(q)\right]\,,$
(91)

where we have highlighted the momentum dependence of the building blocks.

The schematics of (91) shows that there is always one propagator depending on
both the internal and external momentum. This momentum dependence does not
align with the step function contained in the regulator. In general, this
would then lead to a quite complicated integration region. This complication
can be avoided by noting that our computation does not need to track the full
dependence of the trace on the external momentum. It is sufficient to extract
the terms proportional to $\vec{p}^{\,2}$ and $p_{0}^{2}$ to extract the beta
functions. Thus, we can expand the middle propagator in orders of external
momenta and only keep the terms contributing to our projection. The relevant
terms follow from a standard Taylor expansion in multiple variables

$\begin{split}f(|\vec{p}+\vec{q}|,|p_{0}+q_{0}|)=&f(|\vec{q}|,|q_{0}|)+\frac{\partial
f(|\vec{q}|,|q_{0}|)}{\partial|\vec{q}|}\frac{\vec{q}\cdot\vec{p}}{|\vec{q}|}+\frac{\partial
f(|\vec{q}|,|q_{0}|)}{\partial|q_{0}|}\frac{q_{0}\cdot p_{0}}{|q_{0}|}\\\
&+\frac{1}{2}\frac{\partial^{2}f(|\vec{q}|,|q_{0}|)}{\partial|\vec{q}|^{2}}\frac{(\vec{q}\cdot\vec{p})^{2}}{|\vec{q}|^{2}}+\frac{1}{2}\frac{\partial
f(|\vec{q}|,|q_{0}|)}{\partial|\vec{q}|}\left(\frac{\vec{p}\cdot\vec{p}}{|\vec{q}|}-\frac{(\vec{q}\cdot\vec{p})^{2}}{|\vec{q}|^{3}}\right)\\\
&+\frac{1}{2}\frac{\partial^{2}f(|\vec{q}|,|q_{0}|)}{\partial|q_{0}|^{2}}p_{0}^{2}+\frac{\partial^{2}f(|\vec{q}|,|q_{0}|)}{\partial|q_{0}|\partial|\vec{q}|}\frac{\vec{p}\cdot\vec{q}}{|\vec{q}|}\frac{q_{0}\cdot
p_{0}}{|q_{0}|}+\mathcal{O}(p^{3}).\end{split}$ (92)

We can then apply this expansion to $\mathcal{G}_{\chi_{c}\chi_{d}}(p+q)$. As
a result, the propagator and its derivatives depend on the loop momentum only.
The tensor structures related to the external momenta can then be simplified
by applying (83) and the evaluation of the traces proceeds along the same
lines as the computation of the tadpole diagrams described in the previous
subsection.

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