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

# Asymptotically Safe Gravity-Fermion systems
on curved backgrounds

Jesse Daas a Wouter Oosters a Frank Saueressig  a and Jian Wang
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We set up a consistent background field formalism for studying the
renormalization group (RG) flow of gravity coupled to $N_{f}$ Dirac fermions
on maximally symmetric backgrounds. Based on Wetterich’s equation we perform a
detailed study of the resulting fixed point structure in a projection
including the Einstein-Hilbert action, the fermion anomalous dimension, and a
specific coupling of the fermion bilinears to the spacetime curvature. The
latter constitutes a mass-type term which breaks chiral symmetry explicitly.
Our analysis identifies two infinite families of interacting RG fixed points
which are viable candidates to provide a high-energy completion through the
asymptotic safety mechanism. The fixed points exist for all values of $N_{f}$
outside of a small window situated at low values $N_{f}$ and become weakly
coupled in the large $N_{f}$-limit. Symmetry-wise, they correspond to “quasi-
chiral” and “non-chiral” fixed points. The former come with enhanced
predictive power, fixing one of the couplings via the asymptotic safety
condition. Moreover, the interplay of the fixed points allows for cross-overs
from the non-chiral to the chiral fixed point, giving a dynamical mechanism
for restoring the symmetry approximately at intermediate scales. Our
discussion of chiral symmetry breaking effects provides strong indications
that the topology of spacetime plays a crucial role when analyzing whether
quantum gravity admits light chiral fermions.

###### Keywords:

Models of Quantum Gravity, Asymptotic Safety, Renormalisation Group,
Nonperturbative Effects

##  1 Introduction

Asymptotic Safety Percacci:2017fkn ; Reuter:2019byg  constitutes a powerful
mechanism for providing a consistent and predictive high-energy completion of
a quantum field theory. The construction hinges on the presence of a suitable
(interacting) fixed point of the theories renormalization group (RG) flow.
Building on the initial proposal by Weinberg Weinberg:1976xy ; Weinberg:1980gg
, functional renormalization group methods pioneered in Wetterich:1992yh ;
Morris:1993qb ; Reuter:1993kw ; Reuter:1996cp , have provided substantial
evidence supporting the existence of such a fixed point, the so-called Reuter
fixed point, in the context of gravity Souma:1999at ; Reuter:2001ag ;
Lauscher:2001ya ; Litim:2003vp ; Machado:2007ea ; Benedetti:2009rx ;
Machado:2009ph ; Benedetti:2009gn ; Manrique:2009uh ; Manrique:2010am ;
Groh:2010ta ; Eichhorn:2010tb ; Manrique:2010mq ; Manrique:2011jc ;
Benedetti:2012dx ; Rechenberger:2012dt ; Christiansen:2012rx ; Dietz:2012ic ;
Ohta:2013uca ; Falls:2013bv ; Falls:2014tra ; Christiansen:2014raa ;
Becker:2014qya ; Demmel:2014hla ; Christiansen:2015rva ; Morris:2015oca ;
Ohta:2015efa ; Ohta:2015fcu ; Gies:2015tca ; Demmel:2015oqa ; Biemans:2016rvp
; Gies:2016con ; Denz:2016qks ; Platania:2017djo ; Houthoff:2017oam ;
Falls:2017lst ; Knorr:2017fus ; Eichhorn:2017egq ; Christiansen:2017bsy ;
deBrito:2018jxt ; Falls:2018ylp ; Knorr:2018kog ; Kluth:2020bdv ;
Falls:2020qhj ; Knorr:2020ckv  and also a wide range of gravity-matter systems
Narain:2009fy ; Shaposhnikov:2009pv ; Dona:2013qba ; Dona:2014pla ;
Meibohm:2015twa ; Dona:2015tnf ; Oda:2015sma ; Eichhorn:2016vvy ;
Wetterich:2016uxm ; Biemans:2017zca ; Becker:2017tcx ; Christiansen:2017cxa ;
Hamada:2017rvn ; Eichhorn:2017als ; Eichhorn:2017ylw ; Eichhorn:2017sok ;
Alkofer:2018fxj ; Eichhorn:2018akn ; Eichhorn:2018ydy ; Eichhorn:2018nda ;
Pawlowski:2018ixd ; Alkofer:2018baq ; Knorr:2019atm ; Burger:2019upn ;
Eichhorn:2019yzm ; Reichert:2019car ; Daas:2020dyo ; Eichhorn:2020kca ;
Eichhorn:2020sbo . See also Niedermaier:2006wt ; Codello:2008vh ;
Reuter:2012id ; Eichhorn:2018yfc ; Pawlowski:2020qer ; Bonanno:2020bil  for
reviews and Nagy:2012ef ; Pereira:2019dbn ; Reichert:2020mja  for recent
lecture notes. In particular, there has been a significant effort in
developing the form-factor program, analyzing approximations which retain an
arbitrary momentum dependence at the level of the effective action
Knorr:2019atm ; Draper:2020bop , reconstructing the graviton propagator
Bosma:2019aiu ; Bonanno:2021squ ; Knorr:2021niv , and refining the
computational toolbox in arbitrary backgrounds Benedetti:2010nr ;
Codello:2012kq ; Becker:2020mjl ; Knorr:2021slg .

An important step towards a realistic theory of quantum gravity is the
inclusion of matter degrees of freedom. Within the asymptotic safety program
this step is conceptually straightforward. The corresponding matter fields are
simply added to the theory. Subsequently, one analyzes the resulting RG flow,
searching for fixed points which could provide potential high-energy
completions. This opens the perspective on a _new standard model of particle
physics_ in which the matter degrees of freedom mandated by the standard model
of particle physics are supplemented by the graviton as an additional force
carrier. Tentative studies suggest that the RG flow of this setting indeed
supports interacting fixed points which could render the construction
asymptotically safe Dona:2013qba ; Biemans:2017zca ; Alkofer:2018fxj .
Remarkably, these fixed points come with the potential of fixing some of the
free parameters of the standard model as, e.g., the value of the
electromagnetic coupling Harst:2011zx ; Eichhorn:2017lry , the Higgs mass
Shaposhnikov:2009pv ; Pawlowski:2018ixd , or the ratio of quark masses
Eichhorn:2018whv .

An important element in the construction of the _new standard model of
particle physics_ is the inclusion of fermions. Within the asymptotic safety
program, this question has already been looked into in a series of works
Eichhorn:2011pc ; Dona:2012am ; Dona:2013qba ; Meibohm:2016mkp ;
Eichhorn:2017eht ; Eichhorn:2016esv ; Eichhorn:2016vvy ; Biemans:2017zca ;
Alkofer:2018fxj ; Alkofer:2018baq ; Gies:2018jnv ; deBrito:2019epw ;
Daas:2020dyo ; Gies:2021upb . In particular, non-minimal couplings between
fermions and gravity have been considered in Eichhorn:2016vvy ;
Eichhorn:2018nda . This entails the effect of “gravitational catalysis” where
the coupling to the spacetime curvature essentially provides a mass to the
fermions Gies:2018jnv ; Gies:2021upb . Up to now, fermions in Asymptotic
Safety have predominantly been investigated using a hybrid approach where the
beta functions in the matter sector are computed in a flat background. While
this leads to significant technical simplifications and is conceptually close
to computations carried out in the context of particle physics, the specific
background also comes with features and choices that are non-generic when
considering more general curved backgrounds, see, e.g., Dona:2012am  for an
exemplary exposition.

This provides a clear motivation for investigating the RG flow of gravity
coupled to fermionic matter beyond these flat background studies. Building on
our earlier work Daas:2020dyo  we investigate this question in a truncation
comprising the Einstein-Hilbert action, the fermion anomalous dimension and a
coupling between the fermion bilinears and the spacetime curvature. The latter
provides a prototypical example for an interaction breaking chiral symmetry
explicitly. We provide a detailed derivation of the beta functions on a
background-sphere and carry out a thorough analysis of the resulting fixed
point structure. An important feature of our setting is that it can be
restricted to several well-motivated subtruncations, comprising

  * i)

fermions minimally coupled to the Einstein-Hilbert action,

  * ii)

fermions coupled to the Einstein-Hilbert action including the fermion
anomalous dimension,

  * iii)

the setting i) complemented by the coupling of the fermion-bilinear to the
spacetime curvature,

  * iv)

the full system including all couplings and anomalous dimensions.

This feature allows to explicitly test the robustness of a fixed point
structure under a refinement of the approximation. In this way the influence
of various factors (as the matter anomalous dimension) can be investigated
systematically. Moreover, mechanisms leading to new classes of fixed points,
which are not visible in simpler approximations, are readily understood based
on explicit examples.

As a key result, we identify continuous families of interacting
renormalization group fixed points, so-called non-Gaussian fixed points
(NGFPs), which are robust under the refinement strategy. Within our
approximation, the fixed points come with varying degrees of predictive power.
The existence of two of these families are tied to the inclusion of the non-
minimal interaction term coupling the fermion-bilinear to the spacetime
curvature. Moreover, our analysis shows that the role of chiral symmetry in a
curved background is actually more subtle than suggested by flat-background
computations since the change of the background topology unlocks new symmetry-
breaking ingredients, also see Hamada:2020mug  for a related discussion.

Our work is organized as follows. Sect. 2 introduces the Wetterich equation
together with the truncation ansatz studied in this article. In particular,
the fate of chiral symmetry is discussed in detail. The beta functions
encoding the scale-dependence of the couplings and anomalous dimensions
contained in our ansatz are reported in Sect. 3. We perform a detailed
analysis of the fixed point structure encoded in this system in Sect. 4. The
complexity of the approximation is increased from subsection to subsection,
giving control on the robustness of the fixed point structure. The RG flow
controlled by the interplay of these fixed points is investigated in Sect. 5.
Sect. 6 contains our conclusion. The technical details of our investigation,
including a brief introduction to the spin-base formalism employed in our
computation, have been relegated to two appendices.

##  2 Gravitational RG flows including fermions in a curved background

We start by outlining the general framework underlying our computation.
Technical details about the spin-base formalism have been relegated to App. A.

###  2.1 The Wetterich equation and its projection

Starting from the pioneering work Reuter:1996cp , the functional
renormalization group equation (FRGE) for the effective average action
$\Gamma_{k}$ Wetterich:1992yh ; Morris:1993qb ; Reuter:1993kw  has played a
key role in exploring the existence and predictive power of the Reuter fixed
points in the context of gravity and gravity-matter systems. The Wetterich
equation Wetterich:1992yh ; Morris:1993qb ; Reuter:1993kw

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

encodes the change of $\Gamma_{k}$ when integrating out quantum fluctuations
with momenta $p^{2}\approx k^{2}$ where $k$ corresponds to a coarse-graining
scale. Here STr contains a sum over all fluctuation fields, an integral over
loop-momenta, and a minus sign for fermionic degrees of freedom. The Hessian
$\Gamma_{k}^{(2)}$ is the second functional derivative of $\Gamma_{k}$ with
respect to the fluctuations. The regulator $\mathcal{R}_{k}$ suppresses
fluctuations with $p^{2}\lesssim k^{2}$ by a $k$-dependent mass-term. By
construction $\mathcal{R}_{k}$ falls off sufficiently fast for $p^{2}\gg
k^{2}$, ensuring that the trace does not give rise to UV-divergences. The
interplay of the regulator appearing in the numerator and denominator ensures
that the flow of $\Gamma_{k}$ is driven by integrating out momenta close to
the scale $k$ Berges:2002ew ; Dupuis:2020fhh .

In this work, we are interested in studying the RG flow of gravity
supplemented by $N_{f}$ Dirac fermions $\psi$ in a four-dimensional spacetime.
The gravitational degrees of freedom are encoded in the spacetime metric
$g_{\mu\nu}$ which is decomposed into a fixed background metric
$\bar{g}_{\mu\nu}$ and fluctuations $h_{\mu\nu}$ by performing a linear split

$g_{\mu\nu}=\bar{g}_{\mu\nu}+h_{\mu\nu}\,.$ (2)

In the sequel quantities constructed from the background metric will be marked
by a bar. The metric fluctuations are further decompose into their trace $h$
and traceless part $\widehat{h}_{\mu\nu}$,

$h_{\mu\nu}=\widehat{h}_{\mu\nu}+\frac{1}{4}\bar{g}_{\mu\nu}h\,,\qquad\bar{g}^{\mu\nu}\widehat{h}_{\mu\nu}=0\,.$
(3)

Similarly, we decompose the fermions according to

$\psi=\theta+\chi\,,\qquad\bar{\psi}=\bar{\theta}+\bar{\chi}\,,$ (4)

where $\theta$ and $\chi$ are the background and fluctuations, respectively.
We define $\bar{\psi}\equiv\psi^{\dagger}h$ with the internal spinor metric
$h$ satisfying $h^{\dagger}=-h$, c.f. App. A for more details on our spinor
conventions.

We then approximate the effective average action by

$\Gamma_{k}[g,\psi,\bar{\psi},\bar{g}]=\Gamma_{k}^{\rm
grav}[g,\bar{g}]+\Gamma_{k}^{\rm
fermion}[g,\psi,\bar{\psi},\bar{g}]+\Gamma^{\rm gf}_{k}[g,\bar{g}]+S^{\rm
ghost}[g,C,\bar{C},\bar{g}]\,.$ (5)

The gravitational part is approximated by the Einstein-Hilbert action,

$\Gamma_{k}^{\rm grav}[g,\bar{g}]=\frac{1}{16\pi G_{k}}\int
d^{4}x\sqrt{g}\left[-R+2\Lambda_{k}\right]\,,$ (6)

with $G_{k}$ and $\Lambda_{k}$ being the running Newton’s coupling and
cosmological constant, respectively.111We stress that physics should be
extracted from the endpoint of an RG-trajectory at $k=0$ Knorr:2019atm ;
Bonanno:2020bil . This entails that the Newton’s constant and cosmological
constant appearing in physical processes is scale-independent. The
gravitational sector is supplemented by the harmonic gauge condition

$\Gamma_{k}^{\rm gf}[g,\bar{g}]=\frac{1}{32\pi G_{k}}\int
d^{4}x\sqrt{\bar{g}}\,\bar{g}^{\mu\nu}F_{\mu}F_{\nu}\,,\qquad
F_{\mu}=\bar{D}^{\nu}h_{\mu\nu}-\frac{1}{2}\bar{D}_{\mu}\,h\,,$ (7)

and the resulting (classical) ghost action

$S^{\rm ghost}[g,C,\bar{C},\bar{g}]=-\sqrt{2}\int
d^{4}x\sqrt{\bar{g}}\,\bar{C}_{\mu}\,\mathcal{M}^{\mu}{}_{\nu}\,C^{\nu}\,,$
(8)

For $g=\bar{g}$, one has
$\mathcal{M}^{\mu}{}_{\nu}=\delta_{\nu}^{\mu}\bar{D}^{2}+\bar{R}^{\mu}{}_{\nu}$.
The fermionic part of the action is taken to be

$\Gamma_{k}^{\rm fermion}[g,\psi,\bar{\psi};\bar{g}]=\int
d^{4}x\sqrt{g}\,Z^{\psi}_{k}\,\left\\{\,\bar{\psi}\left[i\not{\nabla}+m\gamma_{5}\right]\psi+\tilde{\alpha}_{k}R\,\bar{\psi}\gamma_{5}\psi\right\\}\,.$
(9)

Here $Z^{\psi}_{k}$ denotes the wave function renormalization associated with
the fermionic fields, $\not{\nabla}=\gamma^{\mu}\nabla_{\mu}$ is the covariant
derivative containing the spin connection and $m$ denotes the fermion mass. In
addition, our setting includes the scale-dependent coupling
$\tilde{\alpha}_{k}$ multiplying a mass-type term where the mass of the
fermions is provided by the Ricci curvature $R$. The conventions on the
$\gamma$-matrices are chosen such that the kinetic term squares to the Klein-
Gordon equation while, at the same time, satisfying the identity (72).

We compute the flow of $G_{k}$, $\Lambda_{k}$, $\tilde{\alpha}_{k}$ and the
scale-dependence of $Z^{\psi}_{k}$ in the background field approximation,
evaluating the FRGE (1) at zeroth order in the fluctuation fields. The
information about the $k$-dependence of the couplings is then encoded in the
coefficients multiplying the interaction monomials

$I_{1}=\int d^{4}x\sqrt{\bar{g}}\,,\quad I_{2}=\int
d^{4}x\sqrt{\bar{g}}\bar{R}\,,\quad I_{3}=\int
d^{4}x\sqrt{\bar{g}}\bar{\theta}\,i\not{\nabla}\,\theta\,,\quad I_{4}=\int
d^{4}x\sqrt{\bar{g}}\bar{R}\,\bar{\theta}\gamma_{5}\theta\,.$ (10)

The computation of these coefficients can significantly be simplified by a
clever choice of background fields which allow to disentangle these
contributions. In practice, we work with $\bar{g}_{\mu\nu}$ being a one-
parameter family of metrics on the Euclidean four-sphere, parameterized by the
radius of the spheres. The background curvature tensors then satisfy

$\bar{R}_{\mu\nu}=\frac{1}{4}\bar{g}_{\mu\nu}\,\bar{R}\,,\qquad\bar{R}_{\mu\nu\rho\sigma}=\frac{\bar{R}}{12}\left(\bar{g}_{\mu\rho}\bar{g}_{\nu\sigma}-\bar{g}_{\mu\sigma}\bar{g}_{\nu\rho}\right)\,,\qquad\bar{D}_{\mu}\bar{R}=0\,.$
(11)

In addition, we impose that the background spinor satisfies the generalized
eigenvalue equation

$\nabla_{\mu}\,\theta=i\sqrt{\frac{\bar{R}}{48}}\,\gamma_{\mu}\,\theta\,.$
(12)

Noting that the spectrum of the Dirac operator on a four-sphere is given by
Camporesi:1995fb

${\rm spec}\\{\not{\nabla}\\}=\\{\pm
i\sqrt{\frac{\bar{R}}{12}}\left(l+2\right)\\}\,,\quad l=0,1,2,\cdots\,,$ (13)

it is easily verified that the choice (12) corresponds to the lowest
eigenvalue of $\not{\nabla}$. Notably, this is the only eigenspinor of the
Dirac operator which satisfies a generalized eigenvalue equation of the form
(12), see (76). At this stage, the following important remark is in order.
Structurally, eq. (12) is quite different from a flat background where
eigenspinors of the Dirac operator are obtained in Fourier space,
$i\not{\nabla}\theta=\not{p}\,\theta$. In this case $\not{\nabla}$ has a
continuous spectrum and it is admissible to work with a background spinor
which is covariantly constant, $\nabla_{\mu}\theta=0$. Since the spectrum of
$\not{\nabla}$ on the sphere does not admit a zero mode, it is clear that this
choice is not admissible on spherically symmetric backgrounds for purely
geometrical reasons, though.

The completion of our setup requires specifying the regulator
$\mathcal{R}_{k}$. In the gravitational sector, we generate the matrix
elements via the substitution rule

$\Box\mapsto\Box+R_{k}(\Box)\,.$ (14)

In this sector, we identify the coarse-graining operator to be the Laplacian
constructed from the background metric, $\Box^{\rm
grav}=-\bar{g}^{\mu\nu}\bar{D}_{\mu}\bar{D}_{\nu}$. In the classification
Codello:2008vh , this corresponds to a regulator of type I. Applying (14) to
the graviton propagator (87) yields the matrix elements of $\mathcal{R}_{k}$
associated with the graviton

$\begin{split}\left[\left.\mathcal{R}_{k}\right|_{\widehat{h}\widehat{h}}\right]_{\mu\nu}{}^{\rho\sigma}=&\,\frac{1}{32\pi
G_{k}}\,R_{k}(\Box)\left(\frac{1}{2}\left(\delta_{\mu}^{\rho}\delta_{\nu}^{\sigma}+\delta_{\nu}^{\rho}\delta_{\mu}^{\sigma}\right)-\frac{1}{4}\bar{g}_{\mu\nu}\bar{g}^{\rho\sigma}\right)\,,\\\
\left.\mathcal{R}_{k}\right|_{hh}=&\,-\frac{1}{128\pi
G_{k}}\,R_{k}(\Box)\,.\end{split}$ (15)

The scalar regulator function $R_{k}(\Box)$ is taken to be of Litim-type
Litim:2000ci ; Litim:2001up , $R_{k}^{\rm Litim}(z)=(k^{2}-z)\Theta(k^{2}-z)$,
where $\Theta(x)$ is the Heaviside step function. Following the discussion
Dona:2012am , the fermionic sector admits two canonical choices for $\Box$:

$\Box^{\rm
I}\equiv-\bar{g}^{\mu\nu}\bar{\nabla}_{\mu}\bar{\nabla}_{\nu}\,,\qquad\Box^{\rm
II}\equiv-\bar{\not{\nabla}}^{2}\,.$ (16)

These two choices are related by the Lichnerowicz formula

$-\bar{\not{\nabla}}^{2}=-\bar{g}^{\mu\nu}\bar{\nabla}_{\mu}\bar{\nabla}_{\nu}+\frac{1}{4}\bar{R}\,,$
(17)

i.e., the two coarse-graining operators differ by a covariantly constant
endomorphism only. Hence, they can be combined into a single expression,

$\Box^{\psi}\equiv-\bar{\not{\nabla}}^{2}+\beta\bar{R}\,,$ (18)

where $\beta=0$ and $\beta=-1/4$ corresponds to $\Box^{\psi}$ being the
squared Dirac operator and the Laplacian, respectively. Motivated by the
structure of the mass term appearing in (9), we construct the regulator in the
fermionic sector by replacing $m$ with a $k$-dependent regulator of dimension
one

$\left.\mathcal{R}_{k}\right|_{\bar{\psi}\psi}=k\,Z^{\psi}_{k}\gamma^{5}\left(1-\sqrt{\Box^{\psi}}/k\right)\Theta\left(1-\sqrt{\Box^{\psi}}/k\right)\,.$
(19)

This specific form of the regulator dresses the fermion kinetic term according
to $\pm Z_{k}^{\psi}\sqrt{p^{2}}\rightarrow\pm
Z_{k}^{\psi}\sqrt{p^{2}+k^{2}(r_{k}^{\psi})^{2}}$ with
$r_{k}^{\psi}\equiv\left(1-\sqrt{\Box^{\psi}}/k\right)\Theta\left(1-\sqrt{\Box^{\psi}}/k\right)$
being the dimensionless profile function in the fermionic sector. Hence, it
suppresses fluctuation modes with $p^{2}\ll k^{2}$ irrespective of the
corresponding sign of the eigenvalue. This completes the setup underlying the
RG analysis of our work.

###  2.2 Chiral symmetry on spherically symmetric backgrounds

Before giving the results of our computation, it is instructive to discuss the
role of chiral symmetry breaking in the setting described in the previous
subsection. By definition, chiral transformations rotate the left-handed and
the right-handed components of a spinor by an independent phase,

$\psi\rightarrow
e^{i\theta\gamma_{5}}\,\psi\,,\qquad\bar{\psi}\rightarrow\bar{\psi}\,e^{i\theta\gamma_{5}}\,,$
(20)

with identical transformations for the background and fluctuation fields. The
standard model of particle physics indicates that there are sectors where
chiral symmetry must be unbroken at the typical energy scales associated with
quantum gravity effects, typically assumed to be set by the Planck scale
$M_{\rm Pl}\approx 10^{18}$ GeV. Any phenomenologically viable theory of
quantum gravity must therefore admit “light” chiral fermions Eichhorn:2011pc .
The symmetry may then provide a valuable testbed for possible high-energy
completions of gravity-matter systems in general and via the asymptotic safety
mechanism in particular.

Applying the transformation (20) to our ansatz for $\Gamma_{k}$ shows that the
gravitational sector and the fermion kinetic term is invariant under chiral
transformations. The mass term as well as the interaction coupling the
fermion-bilinear to the spacetime curvature break this symmetry explicitly. On
the basis that gravity couples to Dirac fermions, independently of their
handedness, one expects that there is a consistent solution of the flow
equation where the couplings of the chiral symmetry breaking terms are zero
for all values of the coarse-graining scale $k$.

At this point, we point out though that, besides the chiral symmetry breaking
terms in the action, our setup of the flow equation contains two additional
sources for chiral symmetry breaking. Firstly, the regulator
$\mathcal{R}_{k}^{\psi}$ introduced in eq. (19) is modeled according to a
fermion mass term. As a result it induces a chiral symmetry breaking component
in the flow.222Similarly to the case of split-symmetry breaking, this effect
may be controlled by a modified ward-identity. The flat background computation
recently presented in deBrito:2020dta  followed an alternative route, taking
the regulator to be proportional to the fermion kinetic term, i.e.,
$\mathcal{R}_{k}(p^{2})|_{\bar{\psi}\psi}=Z_{k}^{\psi}\not{p}\left(\sqrt{k^{2}/p^{2}}-1\right)\Theta(1-p^{2}/k^{2})$.
This suggests a chiral symmetry preserving extension to general backgrounds
given by

$\mathcal{R}_{k}(\Box^{\psi})|_{\bar{\psi}\psi}=Z_{k}^{\psi}\,\bar{\not{\nabla}}\,\left(\sqrt{k^{2}/\Box^{\psi}}-1\right)\Theta(1-\Box^{\psi}/k^{2})\,.$
(21)

While it would be interesting to compare our results to a flow computed based
on this choice, this is beyond the present study and may be subject of
subsequent work.

A second source of chiral symmetry breaking emanates from the background
spinor formalism. Explicitly, the generalized eigenvalue equation (12) does
not adhere to chiral symmetry. This may easily be seen by evaluating the
fermion kinetic term for this specific choice of background

$\left.\int
d^{4}x\sqrt{\bar{g}}\,\bar{\psi}\,i\bar{\not{\nabla}}\,\psi\right|_{\bar{\chi}=\chi=0}=-\,\int
d^{4}x\sqrt{\bar{g}}\,\sqrt{\frac{\bar{R}}{3}}\,\bar{\theta}\theta\,.$ (22)

The resulting term has the structure of a mass term, signaling the breaking of
chiral symmetry by the choice of background. This property is actually not
limited to the specific choice of mode made in (12). Any background spinor
which is an eigenvalue of the Dirac operator on the background sphere,
satisfying $\not{\nabla}\theta=\lambda\theta$, induces chiral symmetry
breaking terms based on the argument (22). Since any spinor can be expanded in
this eigenbasis, it is difficult to have a non-trivial fermionic background
field which preserves the transformation law (20) on a spherically symmetric
background.

##  3 Beta functions

We now give the result for the beta-functions encoding the scale-dependence of
the couplings $G_{k}$, $\Lambda_{k}$, and $\tilde{\alpha}_{k}$. The technical
details underlying this computation have been relegated to App. B. Further
details on working with spinors on a spherically symmetric background and the
graviton-fermion vertices can be found in App. A.

###  3.1 Beta functions

The scale-dependence of $G_{k}$, $\Lambda_{k}$ and $\tilde{\alpha}_{k}$
together with the fermion wave-function renormalization $Z_{k}^{\psi}$ is
obtained by substituting the ansatz (5) into the Wetterich equation (1),
compute the resulting operator traces, and comparing the coefficients of the
interaction terms (10) appearing on the left- and right-hand side. The result
is conveniently expressed in terms of the dimensionless couplings

$g_{k}\equiv
G_{k}\,k^{-2}\,,\qquad\lambda_{k}\equiv\Lambda_{k}k^{2}\,,\qquad\alpha_{k}\equiv\tilde{\alpha}_{k}\,k\,,$
(23)

supplemented by the anomalous dimension of Newton’s coupling, $\eta_{N}$ and
the fermionic fields

$\eta_{N}\equiv
G_{k}^{-1}\,k\partial_{k}G_{k}\,,\qquad\eta_{\psi}\equiv-\,k\partial_{k}\ln
Z_{k}^{\psi}\,.$ (24)

The scale-dependence of the couplings (23) is then encoded in an autonomous
system of coupled differential equations

$k\partial_{k}g_{k}=\beta_{g}(g,\lambda,\alpha)\,,\quad
k\partial_{k}\lambda_{k}=\beta_{\lambda}(g,\lambda,\alpha)\,,\quad
k\partial_{k}\alpha_{k}=\beta_{\alpha}(g,\lambda,\alpha)\,.$ (25)

Besides their dependence on the couplings, the beta functions also depend on
the parameter $\beta$ specifying the coarse-graining operator (18). The beta
functions in the gravitational sector are given by333We correct a misprint in
the earlier work Daas:2020dyo  where the factor of $g$ multiplying the
fermionic contribution to $\beta_{\lambda}$ is missing.

$\begin{split}\beta_{g}=&\,(2+\eta_{N})\,g\,,\\\
\beta_{\lambda}=&\left(\eta_{N}-2\right)\lambda+\frac{g}{4\pi}\left[\left(10-\frac{5}{3}\eta_{N}\right)\frac{1}{(1-2\lambda)}-8\right]-\frac{N_{f}\,g}{12}\left(4+(8-3\pi)\eta_{\psi}\right)\,.\end{split}$
(26)

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

$\eta_{N}=\frac{g\left(B_{1}^{\rm grav}+N_{f}(B^{\rm ferm}_{1}+B^{\rm
ferm}_{2}\,\eta_{\psi})\right)}{1-g\,B_{2}^{\rm grav}}\,,$ (27)

where

$\begin{split}B_{1}^{\rm
grav}=&\,-\frac{1}{3\pi}\left[\frac{9}{(1-2\lambda)^{2}}-\frac{5}{(1-2\lambda)}+7\right]\,,\\\
B_{2}^{\rm
grav}=&\,\frac{1}{12\pi}\left[\frac{6}{(1-2\lambda)^{2}}-\frac{5}{(1-2\lambda)}\right]\,\,,\\\
B^{\rm
ferm}_{1}=&\,-\frac{1}{6\pi}\left[(2-\pi)(1+6\beta)+12\alpha\right]\,,\\\
B^{\rm
ferm}_{2}=&\,\frac{1}{12\pi}\left[(2-\pi)-24(3-\pi)\,\alpha+6(8-3\pi)\beta\right]\,.\end{split}$
(28)

The running of $\alpha_{k}$ and the fermion anomalous dimension turn out to be
cubic and quadratic in $\alpha$, respectively, and can be parameterized as

$\beta_{\alpha}=A_{0}+(A_{1}+1+\eta_{\psi})\,\alpha+A_{2}\,\alpha^{2}+A_{3}\,\alpha^{3}\,,$
(29)

and

$\eta_{\psi}=C_{0}+C_{1}\,\alpha+C_{2}\,\alpha^{2}\,.$ (30)

The coefficients $A_{i}$ and $C_{i}$ depend on $g$, $\lambda$ and
$\eta_{\psi}$ and take the form

$A_{i}=\frac{g}{\pi}\Big{[}\frac{A_{i}^{1}+\tilde{A}_{i}^{1}\,\eta_{\psi}}{(1-2\lambda)}+\frac{A_{i}^{2}+\tilde{A}_{i}^{2}\,\eta_{\psi}}{(1-2\lambda)^{2}}+\frac{A_{i}^{3}}{(1-2\lambda)^{3}}\Big{]}\,,\quad
C_{i}=\frac{g}{\pi}\Big{[}\,\frac{C_{i}^{1}+\tilde{C}_{i}^{1}\,\eta_{\psi}}{(1-2\lambda)}+\frac{C_{i}^{2}}{(1-2\lambda)^{2}}\,\Big{]}\,.$
(31)

The explicit computation gives the values tabulated in Table 1. The terms
proportional to $\eta_{\psi}$ can be understood as non-perturbative
contributions resulting from resuming certain perturbative contributions to
the flow. The results for the anomalous dimensions are given in implicit form.
The corresponding system of linear equations is readily solved to obtain the
anomalous dimensions as a function of the scale-dependent couplings
$g,\lambda$ and $\alpha$.

coefficient | numerical value
---|---
$A_{0}^{1}$ | $-\frac{3}{32}$
$A_{1}^{1}$ | $-\frac{7}{6}+\frac{7}{16}\pi-\frac{1}{2}\beta$
$A_{2}^{1}$ | $\frac{47}{12}-\frac{5}{4}\pi+(\frac{45}{4}-\frac{45}{16}\pi)\beta$
$A_{3}^{1}$ | $\frac{9}{10}$
$A_{0}^{2}$ | $\frac{3}{8}-\frac{15}{128}\pi+\frac{1}{32}\eta_{N}$
$A_{1}^{2}$ | $\frac{107}{30}+\frac{1}{32}\pi+(\frac{1}{30}-\frac{13}{64}\pi)\eta_{N}-\frac{1}{4}\beta-(\frac{39}{40}-\frac{21}{64}\pi)\,\beta\,\eta_{N}$
$A_{2}^{2}$ | $\frac{169}{120}-\frac{1}{2}\pi+(\frac{101}{280}-\frac{1}{8}\pi)\eta_{N}+(\frac{9}{2}-\frac{9}{8}\pi)\beta-(\frac{61}{20}-\frac{15}{16}\pi)\,\beta\,\eta_{N}$
$A_{3}^{2}$ | $-\frac{17}{10}+(\frac{79}{28}-\frac{27}{32}\pi)\eta_{N}$
$A_{0}^{3}$ | $\frac{7}{20}-\frac{3}{32}\pi-(\frac{179}{1120}-\frac{3}{64}\pi)\eta_{N}$
$A_{1}^{3}$ | $-\frac{67}{30}-\frac{1}{8}\pi+(\frac{47}{210}+\frac{1}{32}\pi)\eta_{N}$
$A_{2}^{3}$ | $-\frac{17}{105}+(\frac{143}{630}-\frac{1}{16}\pi)\eta_{N}$
$\tilde{A}_{0}^{1}$ | $-\frac{9}{32}+\frac{3}{32}\pi$
$\tilde{A}_{0}^{2}$ | $-\frac{5}{32}+\frac{3}{64}\pi$
$\tilde{A}_{1}^{1}$ | $\frac{21}{16}-\frac{7}{16}\pi-\frac{1}{8}\beta$
$\tilde{A}_{1}^{2}$ | $\frac{1}{12}-\frac{1}{32}\pi$
$\tilde{A}_{2}^{1}$ | $\frac{43}{40}-\frac{7}{4}\pi-\left(\frac{9}{4}-\frac{9}{16}\pi\right)\beta$
$\tilde{A}_{2}^{2}$ | $\frac{49}{60}-\frac{1}{4}\pi$
$\tilde{A}_{3}^{1}$ | $\frac{163}{10}-\frac{21}{4}\pi$
$C_{0}^{1}$ | $\frac{3}{4}-\frac{9}{32}\pi$
$C_{0}^{2}$ | $-\frac{75}{16}-\frac{9}{64}\pi+(\frac{23}{40}+\frac{3}{32}\pi)\,\eta_{N}$
$C_{1}^{1}$ | $\frac{21}{4}-\frac{45}{32}\pi$
$C_{1}^{2}$ | $\frac{21}{10}-\frac{9}{16}\pi-(\frac{277}{280}-\frac{9}{32}\pi)\,\eta_{N}$
$C_{2}^{1}$ | $-\frac{39}{10}+\frac{9}{8}\pi$
$C_{2}^{2}$ | $-\frac{23}{10}+\frac{3}{8}\pi+(\frac{41}{140}-\frac{3}{64}\pi)\,\eta_{N}$
$\tilde{C}_{0}^{1}$ | $-\frac{27}{32}+\frac{9}{32}\pi$
$\tilde{C}_{1}^{1}$ | $-\frac{15}{8}+\frac{9}{16}\pi$
$\tilde{C}_{2}^{1}$ | $-\frac{1}{2}+\frac{3}{16}\pi$
Table 1:  List of the numerical coefficients appearing in the parameterization
(31). The coefficients not listed in the table vanish. The coefficients
$A_{i}^{j}$ agree with the ones published in Daas:2020dyo .

The beta functions (26) and (29) together with the explicit expressions for
the anomalous dimensions (27) and (30) constitute the main result of this
section. They extend the computation presented in Daas:2020dyo  by including
the effect of the wave-function renormalization in the fermionic sector. This
extension gives valuable information about the robustness of the fixed points
and RG flows resulting from the ansatz given in Sect. 2.1.

###  3.2 Structural Properties of the Beta Functions

At this stage, it is instructive to take limits of our beta functions and
compare to earlier results in the literature. Setting the matter contributions
to zero, one recovers the beta functions of the Einstein-Hilbert truncation
Reuter:1996cp ; Reuter:2019byg .

The ambiguity in the coarse-graining operator (18) raises the natural question
if the parameter $\beta$ can affect the RG flow at a qualitative level. This
is conveniently studied by restricting the full setup to the case of minimally
coupled fermions, setting $\alpha_{k}=0$ and switching off the fermion wave-
function renormalization $\eta_{\psi}=0$. The equation $\beta_{g}=0$ then
admits solutions with non-vanishing $g_{*}$ if $\eta_{N}^{*}=-2$. Assuming
that its numerator is positive, eq. (27) then entails that $g_{*}>0$ requires

$B_{1}^{\rm grav}+B^{\rm ferm}<0\,.$ (32)

Substituting the explicit expressions given in (28) entails that $g_{*}$
changes sign if

$\left[\frac{9}{(1-2\lambda)^{2}}-\frac{5}{(1-2\lambda)}+7\right]+\frac{N_{f}}{2}\bigg{[}2-\pi+\left(12-6\pi\right)\beta\bigg{]}=0\,.$
(33)

For $\beta<-\frac{1}{6}$ the straight bracket multiplying $N_{f}$ is positive.
If the first term is positive and finite, this entails that (32) is satisfied
for all values of $N_{f}$. Conversely, $\beta>-\frac{1}{6}$ entails that
(again for $\lambda<1/2$ kept fixed) there is a critical value

$N_{f}^{\rm
crit}=2\left[\frac{9}{(1-2\lambda)^{2}}-\frac{5}{(1-2\lambda)}+7\right]/\bigg{[}\pi-2+\left(6\pi-12\right)\beta\bigg{]}>0\,,$
(34)

where the condition (32) is violated and $g_{*}$ moves to negative values.
This mechanism, leading to a upper bound on the number of fermions for which
$g_{*}>0$, was first discussed in detail in Dona:2012am . In a flat background
one has $\not{p}^{2}=p^{2}$, corresponding to the case $\beta=0$. Computations
on a spherical symmetric background can be based on coarse-graining with the
square of the Dirac operator ($\beta=-1/4)$ which does not lead to a violation
of (32). From the structure of $B^{\rm ferm}$ found in (28), it is clear that
both the inclusion of $\alpha$ and $\eta_{\psi}$ may crucially affect the
condition (32). Following Daas:2020dyo , this observation consititutes one
motivation to study RG flows including the non-minimal fermion interaction
terms (9).

From the viewpoint of chiral symmetry, an intriguing property of
$\beta_{\alpha}$ is the non-vanishing coefficient $A_{0}$. Baring miraculous
cancellations, this entails that any interacting fixed point with
$g_{*}\not=0$ also comes with a non-zero value $\alpha_{*}$. This entails that
any interacting RG fixed point found in the present computation comes with a
(possibly small) chiral symmetry breaking component associated with the
coupling of the spinor bilinear to the spacetime curvature. At this point it
is instructive to scrutinize this property in more detail. For this purpose,
we compute the coefficient $A_{0}$ with the right-hand side of (12)
artificially set to zero

$\begin{split}&\left.A_{0}^{1}\right|_{\nabla_{\mu}\theta=0}=\frac{3}{32}\,,\\\
&\left.A_{0}^{2}\right|_{\nabla_{\mu}\theta=0}=\frac{1}{2}-\frac{15}{128}\pi+\left(\frac{23}{160}-\frac{3}{64}\pi\right)\eta_{N}\,,\\\
&\left.A_{0}^{3}\right|_{\nabla_{\mu}\theta=0}=\frac{7}{20}-\frac{3}{32}\pi-\left(\frac{179}{1120}-\frac{3}{64}\pi\right)\eta_{N}\,.\end{split}$
(35)

In a sense, this may correspond to a hybrid-type of computation where the flow
of the matter couplings is evaluated on a flat background while the
gravitational sector is computed utilizing the background sphere. The
contribution of the non-trivial background background spinor $\Delta
A_{0}^{i}\equiv A_{0}^{i}-\left.A_{0}^{i}\right|_{\nabla_{\mu}\theta=0}$ can
then be isolated by taking the difference between the result (35) and the
coefficients given in Table 1

$\Delta A_{0}^{1}=-\frac{3}{16}\,,\qquad\Delta
A_{0}^{2}=-\frac{1}{8}-\left(\frac{9}{80}-\frac{3}{64}\pi\right)\eta_{N}\,,\qquad\Delta
A_{0}^{3}=0\,.$ (36)

The vanishing of $\Delta A_{0}^{3}$ indicates that this term receives
contributions from the regulator only. In general the $\Delta A_{0}^{i}$ are
non-vanishing though. This establishes that even in the case where no
additional chiral symmetry breaking terms are induces (e.g., by a mass-type
regulator in the fermionic sector), the background spinor equation will induce
a non-trivial coefficient $A_{0}$. This makes the flow qualitatively different
from the one computed for matter sectors on a flat background.

##  4 Fixed Point structure for gravity-fermion systems

The main result of the previous section is the explicit form of the beta
functions encoding the scale-dependence of the dimensionless couplings
$g_{k},\lambda_{k}$, and $\alpha_{k}$ as well as the fermion anomalous
dimension $\eta_{\psi}$. These beta functions depend parameterically on the
number of fermions $N_{f}$ and the parameterization of the coarse-graining
operator $\beta$. The explicit expressions (26) and (29) then serve as our
starting point for studying the fixed point structure of the system. Clearly,
there is a Gaussian fixed point (GFP), situated at $\\{g_{*}^{\rm
GFP},\lambda_{*}^{\rm GFP},\alpha_{*}^{\rm GFP}\\}=\\{0,0,0\\}$ which is
present at all levels of our approximation. Thus our goal is to identify the
non-Gaussian fixed points (NGFPs) which could provide a high-energy completion
of the gravity-fermion system via the asymptotic safety mechanism. The
interplay of the fixed points will be studied in Sect. 5.

Let us start by introducing the general framework. We denote the dimensionless
couplings collectively by $u^{i}=\\{g,\lambda,\alpha\\}$. At a fixed point
$u^{i}_{*}$ all beta functions vanish simultaneously,

$\beta_{u^{j}}(u_{*}^{i})=0\,,\qquad\forall\,i\,.$ (37)

This entails that the dimensionless coupling constants remain finite if the
underlying RG trajectory is dragged into a fixed point as
$k\rightarrow\infty$. The set of these asymptotically safe trajectories spans
the UV-critical hypersurface of the fixed point. Whether a given direction is
UV-attractive (UV-relevant) or UV-repulsive (UV-irrelevant) is conveniently
studied by linearizing the beta functions at the fixed point. Defining the
stability matrix associated with a given fixed point $u_{*}$,

${\bf
B}^{i}{}_{j}\equiv\left.\partial_{u^{j}}\beta_{u^{i}}\right|_{u=u_{*}}\,,$
(38)

eigendirections $V_{I}$ with stability coefficients $\theta_{I}$, ${\bf
B}V_{I}=-\theta_{I}V_{I}$, are UV-attractive if ${\rm Re}\,\theta_{I}>0$ and
UV-repulsive if ${\rm Re}\,\theta_{I}<0$. Since ${\bf B}$ is not necessarily
symmetric, the stability coefficients can be complex. Notably, the
$\theta_{I}$ constitute observable quantities. Thus they provide valuable
probes for judging the robustness of approximations based on truncating the
action functional.

In the following subsections we will gradually build up the analysis, starting
with minimally coupled fermions in Sect. 4.1. The effect of the chiral
symmetry breaking fermion-curvature coupling, initially studied in
Daas:2020dyo  and also Eichhorn:2018nda ; deBrito:2020dta , is reviewed in
Sect. 4.2. We analyze the fixed point structure of the full system in Sect.
4.3 and conclude with a discussion of the chiral symmetry breaking terms in
$\beta_{\alpha}$ in section 4.4. Our main result are the two families of RG
fixed points characterized in Figs. 8 and 9, which are robust under the
extension of the projection space and satisfy the criterion of being almost
Gaussian in the sense that quantum fluctuations do not provide “large
corrections” to the classical values of the critical exponents.

###  4.1 Fermions minimally coupled to gravity

We start by studying the fixed point structure arising from minimally coupled
fermions, restricting ourselves to the subsystem where $\alpha_{k}=0$.

####  4.1.1 Minimally coupled fermions without the fermion anomalous dimension

The first level of the analysis also switches off the fermion wave function
renormalization by setting $\eta_{\psi}=0$. In this case the beta functions
(26) simplify to

$\begin{split}\beta_{g}=&\,(2+\eta_{N})\,g\,,\\\
\beta_{\lambda}=&\left(\eta_{N}-2\right)\lambda+\frac{g}{4\pi}\left[\left(10-\frac{5}{3}\eta_{N}\right)\frac{1}{(1-2\lambda)}-8\right]-\frac{N_{f}\,g}{3}\,.\end{split}$
(39)

The explicit expression for $\eta_{N}$ takes the form

$\eta_{N}=\frac{2g\left(22-36\lambda+56\lambda^{2}-N_{f}\ (\pi-2)\ (1+6\beta)\
(1-2\lambda)^{2}\right)}{g\ (1+10\lambda)-12\pi\ (1-2\lambda)^{2}}.$ (40)

The Einstein-Hilbert truncation without matter fields is recovered by setting
$N_{f}=0$. In this case the flow is independent of $\beta$ and one recovers
the well-known Reuter fixed point

$g_{*}=0.707\,,\quad\lambda_{*}=0.193\,,\quad\theta_{1,2}=1.48\pm 3.04i\,.$
(41)

Figure 1:  The position and critical exponents of the NGFP arising from (39)
as function of $N_{f}$. The blue and orange lines correspond to $\beta=-1/4$
(coarse-graining by the Laplacian) and $\beta=0$ (coarse-graining by the
squared Dirac operator). Notably, the NGFP exists for all values $N_{f}$. For
$\beta=0$ there is a critical number of fermions $N_{f}^{\text{crit}}=12.26$
where the NGFP transition from $g_{*}>0$ to $g_{*}<0$. The real part of the
critical exponents as a function of $N_{f}$ is shown in the bottom row. For
low values $N_{f}$, the stability coefficients $\theta_{1,2}$ are a complex
pair. The critical exponents turn real at $N_{f}\approx 1.33$ ($\beta=-1/4$)
and $N_{f}\approx 1.29$ ($\beta=0$), respectively. This transition is
magnified in the insets shown in the lower row.

The inclusion of fermions results in a deformation of the gravitational fixed
point. Notably, the deformed fixed points exist for all values of $N_{f}$ for
both values of $\beta$. This can be seen in the top line of Fig. 1. A
remarkable feature is that the $\beta=0$-case exhibits a critical number of
fermions $N_{f}^{\text{crit}}=12.26$ where the NGFP transitions from $g_{*}>0$
to $g_{*}<0$. From the perspective of the background field approximation NGFPs
situated at $g_{*}<0$ are problematic for phenomenologically admissible high-
energy completions of gravity: from (39) one concludes that $\beta_{g}=0$ when
$g=0$. As a consequence the RG flow of Newton’s coupling can not change its
sign. At low energy, gravity being attractive requires a positive value
$G_{0}$ which can not be reached from a NGFP situated at $g_{*}<0$.444Also see
our discussion in section 3.2 and ref. Meibohm:2015twa  for arguments
bypassing this logic at the level of fluctuation field computations. On this
basis one could then conclude that not all NGFPs are located such that it may
serve as a building block for a theory with a phenomenologically viable high
energy completion.

The critical exponents of the NGFPs are shown in the lower row of Fig. 1.
Starting from the gravitational fixed point (41) and increasing $N_{f}$, one
observes that they become real at $N_{f}\approx 1.33$ ($\beta=-1/4$) and
$N_{f}\approx 1.29$ ($\beta=0$), respectively. The fact that the critical
exponents vary mildly with $N_{f}$ indicates that the even the inclusion of a
large number of matter fields does not crucially alter the stability
properties of the fixed point. This provides a clear indication that the NGFPs
seen in the present setting are essentially gravity-dominated. Remarkably, the
transition from $g_{*}>0$ to $g_{*}<0$ leaves no imprint on the critical
exponents, justifying the deformation of the gravitational fixed point
persists for all values of $N_{f}$ also in the peculiar case of $\beta=0$.

Intriguingly, the properties of the NGFPs for large numbers of fermion fields
can be studied analytically. Using a large-$N_{f}$-expansion, the position of
the fixed point is given by

$\begin{split}g_{*}\simeq&\,-\frac{12\pi}{N_{f}(\pi-2)\xi}\,,\qquad\lambda_{*}\simeq\,\frac{\pi}{(\pi-2)\xi}+\frac{\lambda^{(1)}_{*}(\xi)}{N_{f}}\,.\end{split}$
(42)

Here we abbreviated $\xi\equiv 1+6\beta$ and the $\simeq$-symbol indicates
that the right-hand side is an expansion in $1/N_{f}$ with terms of order
$\mathcal{O}(1/N_{f}^{2})$ being neglected. The coefficient
$\lambda^{(1)}_{*}(\xi)$ is positive for all values $\beta$. Its explicit form
is tabulated in Table 2. The large-$N_{f}$ expansion of the critical exponents
has a similar structure. Computing the stability matrix, substituting the
general solution for $g_{*}$, and $\lambda_{*}$, and subsequently performing
the large-$N_{f}$ expansion of the critical exponents yields

$\begin{split}\theta_{1}\simeq&\,4+\frac{\theta_{1}^{(1)}}{N_{f}}\,,\qquad\theta_{2}\simeq\,2+\frac{\theta_{2}^{(1)}}{N_{f}}\,.\end{split}$
(43)

The coefficients at subleading order in $N_{f}$ again depend on $\xi$ and are
given in the lower block of Table 2. Evaluating the coefficients for the
specific values of $\beta$, one sees that these asymptotics entailed by eqs.
(42) and (43) matches the one shown in Fig. 1.

A remarkable property of eqs. (42) and (43) is that in the large-$N_{f}$ limit
$g_{*}\propto 1/N_{f}$ becomes perturbatively small. At the same time, the
critical exponents do not match the ones of the free theory, indicating that
we are still dealing with a non-Gaussian fixed point in this limit. This opens
a window for studying the gravity-matter fixed points using standard
perturbative techniques.

####  4.1.2 Minimally coupled fermions including the fermion anomalous
dimension

We now refine the analysis of the previous subsection by including the fermion
anomalous dimension $\eta_{\psi}$ while still keeping $\alpha_{k}=0$. This
extension of the truncation gives important information on the robustness of
the fixed point structure uncovered in the system with $\eta_{\psi}=0$. The
explicit form of the beta functions is

$\begin{split}\beta_{g}=&\,(2+\eta_{N})\,g\,,\\\
\beta_{\lambda}=&\left(\eta_{N}-2\right)\lambda+\frac{g}{4\pi}\left[\left(10-\frac{5}{3}\eta_{N}\right)\frac{1}{(1-2\lambda)}-8\right]-\frac{N_{f}\,g}{12}(4+(8-3\pi)\,\epsilon\,\eta_{\psi})\,.\end{split}$
(44)

The explicit expressions for $\eta_{N}$ and $\eta_{\psi}$ are obtained from
eqs. (27) and (30) by taking $\alpha=0$ and solving the system of linear
equations

$\begin{split}\eta_{N}=&\frac{g\left[\left(B_{1}^{\rm grav}+N_{f}B_{1}^{\rm
ferm}\right)\big{(}\tilde{C}_{0}^{1}g(1-2\lambda)-\pi(1-2\lambda)^{2}\big{)}-g\,N_{f}\,B_{2}^{\rm
ferm}\left(C_{0}^{2a}+C_{0}^{1}(1-2\lambda)\right)\right]}{g^{2}N_{f}B_{2}^{\rm
ferm}C_{0}^{2b}-B_{2}^{\rm
grav}g\big{(}\tilde{C}_{0}^{1}g(1-2\lambda)-\pi(1-2\lambda)^{2}\big{)}+\tilde{C}_{0}^{1}\,g\,(1-2\lambda)-\pi\,(1-2\lambda)^{2}}\,,\\\
\eta_{\psi}=&\,\frac{g\left[g\,B_{1}^{\rm grav}\,C_{0}^{2b}+(1-gB_{2}^{\rm
grav})(C_{0}^{1}(1-2\lambda)+C_{0}^{2a})+N_{f}\,g\,B_{1}^{\rm
ferm}\,C_{0}^{2b}\right]}{(1-gB_{2}^{\rm
grav})(1-2\lambda)(\pi(1-2\lambda)-g\,\tilde{C}_{0}^{1})-N_{f}\,g^{2}\,B_{2}^{\rm
ferm}\,C_{0}^{2b}}\,.\end{split}$ (45)

The constants $C_{0}^{i}$ and $\tilde{C}_{0}^{i}$ are listed in Table 1 and we
decomposed $C_{0}^{2}=C_{0}^{2a}+C_{0}^{2b}\,\eta_{N}$ such that

$C_{0}^{2a}\equiv-\frac{75}{16}-\frac{9}{64}\pi\,,\qquad
C_{0}^{2b}\equiv\frac{23}{40}+\frac{3}{32}\pi\,.$ (46)

The parameter $\epsilon$ introduced in (44) distinguishes between the cases
where the fermion anomalous dimension is included ($\epsilon=1$) or switched
off ($\epsilon=0$).

The impact of the fermion anomalous dimension on the position of the NGFP is
shown in Fig. 2 for both choices of the coarse-graining scheme.

Figure 2:  Corrections to the position of the NGFPs resulting from the
inclusion of the fermion anomalous dimension ($\epsilon=1$, orange lines). The
postion of the NGFP for $\eta_{\psi}=0$ is given by the blue lines. Notably,
the corrections provided by the fermion anomalous dimension are small. They
are most pronounced in small values of $N_{f}$.

Notably, the fixed point positions receive only minor corrections once the
fermionic wave function is included. These corrections are most pronounced in
the range $0.7\lesssim N_{f}\lesssim 1.3$ for $\beta=0$ whereas for
$\beta=-1/4$ the corrections are over a wider range although they get
significantly smaller as $N_{f}$ increases. For $\beta=0$ there is again a
critical number of fermions, albeit at a slightly larger value.

The the critical exponents of the NGFPs together with the corresponding values
for $\eta_{\psi}$ are shown in Fig. 3.

Figure 3:  Illustration of the critical exponents (top line) and fermion
anomalous dimension (bottom line) for $\beta=0$ and $\beta=-1/4$,
respectively. The bifurcation point appearing for low values of $N_{f}$
signals the transition from complex to real critical exponents. The sign
change in $g_{*}$ and $\lambda_{*}$ that occur when $\beta=0$ induces a tiny
discontinuity where the fermion anomalous dimension jumps by
$\Delta\eta_{\psi}=0.03$ without changing its sign.

The comparison between Fig. 1 and 3 shows that the critical exponents obtained
for $\epsilon=0$ and $\epsilon=1$ exhibit the same qualitative behavior. Again
there is a transition from complex to real critical exponents occurring, now
at $N_{f}=0.5$ ($\beta=0$) and $N_{f}=0.55$ ($\beta=-1/4$). Moreover, the
transition from $g_{*}>0$ to $g_{*}<0$ ($\beta=0$) does not reflect in
$\theta_{1,2}$ which are again continuous at $N_{f}^{\rm crit}$. The fermion
anomalous dimension shown in the bottom diagram reveals that $\eta_{\psi}$ is
bounded and vanishes for large values of $N_{f}$. The largest absolute values
are found for small values of $N_{f}$, consistent with the observation that
the corrections arising from the inclusion of the fermionic wave function
renormalization are most pronounced in this regime.

Eqs. (44) and (45) again allow to study the properties of the fixed points in
a large-$N_{f}$ expansion analytically. This expansion takes a form similar to
the one found in the minimally coupled case. The fixed point is located at

$g_{*}\simeq-\frac{12\pi}{(\pi-2)\xi}\frac{1}{N_{f}}\,,\qquad\lambda_{*}\simeq\frac{\pi}{(\pi-2)\xi}+\frac{\lambda_{*}^{(1,\eta_{\psi})}}{N_{f}}\,.$
(47)

The coefficient $\lambda_{*}^{(1,\eta_{\psi})}$ is again a function of $\xi$
and listed in the second row of Table 2. The large-$N_{f}$ behavior of the
stability coefficients is again obtained by constructing the general stability
matrix, substituting the position of the fixed point and then performing the
$N_{f}$-expansion for the resulting eigenvalues. This results in

$\begin{split}\theta_{1}\simeq&\,4+\frac{\theta_{1}^{(1,\eta_{\psi})}}{N_{f}}\,,\qquad\theta_{2}\simeq\,2+\frac{\theta_{2}^{(1,\eta_{\psi})}}{N_{f}}\,.\end{split}$
(48)

Using that $g_{*}\propto 1/N_{f}$ and $\lambda_{*}\propto\text{constant}$, the
asymptotics of $\eta_{\psi}$ follows from (45)

$\eta_{\psi}\simeq\frac{3(\pi-2)(1628+195\pi)\xi+180\pi(8-3\pi)}{80(\pi(\xi-2)-2\xi)^{2}}\frac{1}{N_{f}}\,.$
(49)

The sign of the prefactor depends on $\xi$. For $\xi\geq 0.105$, $\eta_{\psi}$
approaches zero from above while for $\xi\leq 0.105$ it is negative for all
values $N_{f}$. Again the asymptotics (47), (48) and (49) matches the one
found numerically in Figs. 2 and 3.

$\lambda_{*}^{(1)}$ | eq. (42) | $\frac{56\pi^{3}-2(\pi-2)\pi^{2}\xi+19(\pi-2)^{2}\pi\xi^{2}-4(\pi-2)^{3}\xi^{3}}{(\pi-2)^{2}(\pi(\xi-2)-2\xi)^{2}\xi^{2}}$
---|---|---
$\lambda_{*}^{(1,\eta_{\psi})}$ | eq. (47) |  $\frac{1}{320(\pi-2)^{2}(\pi(\xi-2)-2\xi)^{2}\xi^{2}}\Big{[}-1280(\pi-2)^{3}\xi^{3}\\\ -(\pi-2)\pi(\pi(9\pi(328+195\pi)-85040)+168448)\xi^{2}\\\ +4\pi(\pi(\pi(45\pi(9\pi-97)+9401)+9710)-29304)\xi+80\pi^{2}(\pi(71+27\pi)+216)\Big{]}$
$\theta_{1}^{(1)}$ | eq. (43) | $\frac{80(\pi-2)\xi}{(\pi(\xi-2)-2\xi)^{2}}$
$\theta_{2}^{(1)}$ | eq. (43) | $-\frac{2(4\xi^{2}-4\pi\xi(37+\xi)+\pi^{2}(40+74\xi+\xi^{2}))}{(\pi(\xi-2)-2\xi)^{3}}$
$\theta_{1}^{(1,\eta_{\psi})}$ | eq. (48) |  $\frac{800(\pi-2)^{2}\xi^{2}+(\pi-2)\pi(9\pi(674+75\pi)-22576)\xi-45\pi^{2}(8-3\pi)^{2}}{10(\pi(\xi-2)-2\xi)^{3}}$
$\theta_{2}^{(1,\eta_{\psi})}$ | eq. (48) |  $\frac{-1}{80(\pi(\xi-2)-2\xi)^{3}}\Big{[}(\pi-2)(\pi(12004+2295\pi)-48224)\xi^{2}\\\ -2(\pi(\pi(1170\pi-20497)+28250)+35928)\xi+20\pi(\pi(27\pi-805)+3132)\Big{]}$
Table 2:  Summary of the numerical coefficients determining the leading
corrections in a large-$N_{f}$ expansion of the fixed point position (top
rows) and stability coefficients (bottom rows).

_In summary, we find that the gravitational fixed point admits a deformation
by the fermion number $N_{f}$. This deformation exists for all values $N_{f}$.
The critical exponents vary mildly with $N_{f}$, indicating that the fixed
point is “gravity-dominated”. For $\beta>-\frac{1}{6}$ there is a critical
number of fermions for which the value of the background Newton’s coupling
$g_{*}$ transits to negative values. In general, the inclusion of the fermion
anomalous dimension leads to small corrections in the fixed point properties
and in particular to the critical exponents. The one-parameter family of NGFPs
admits a large $N_{f}$ expansion where $g_{*}=0$ to leading order. Moreover,
the leading term in the $N_{f}$-expansion of the stability coefficients are
universal in the sense that they are independent of our choice of coarse-
graining operator._

###  4.2 Including non-minimally coupled gravity-fermion interactions

We now consider the system where $\alpha_{k}\neq 0$ and $\eta_{\psi}=0$. This
subsystem includes the gravity-mediated chiral symmetry breaking term while
neglecting the effect of the fermion wave function renormalization. Since this
case has been discussed in Daas:2020dyo  already, the present exposition will
be brief.

The explicit form of the beta functions for this subsystem is readily obtained
from (26), (27), and (29) and reads

$\begin{split}\beta_{g}=&\,(2+\eta_{N})\,g\,,\\\
\beta_{\lambda}=&\left(\eta_{N}-2\right)\lambda+\frac{g}{4\pi}\left[\left(10-\frac{5}{3}\eta_{N}\right)\frac{1}{(1-2\lambda)}-8\right]-\frac{N_{f}\,g}{3}\,,\\\
\beta_{\alpha}=&A_{0}+(A_{1}+1)\alpha+A_{2}\alpha^{2}+A_{3}\alpha^{3}\,.\end{split}$
(50)

The anomalous dimension $\eta_{N}$ has the explicit form

$\eta_{N}=\frac{2g\left(22-36\lambda+56\lambda^{2}-N_{f}\left(\pi+6\pi\beta-2(1+6\alpha+6\beta)\right)(1-2\lambda)^{2}\right)}{g(1+10\lambda)-12\pi(1-2\lambda)^{2}}\,.$
(51)

The coefficients $A_{i}$ are parametrized as

$A_{i}=\frac{g}{\pi}\left[\frac{A_{i}^{1}}{(1-2\lambda)}+\frac{A_{i}^{2}}{(1-2\lambda)^{2}}+\frac{A_{i}^{3}}{(1-2\lambda)^{2}}\right]\,,$
(52)

with the numerical values $A_{i}^{j}$ are listed in Table 1.

Examining the fixed point structure of this system leads to the following
observations:

  1. 1.

Including the new fermionic term proportional to $\alpha$ gives rise to a new
beta function $\beta_{\alpha}$. This is cubic in $\alpha$ for a fixed
$\lambda$, $g$ and thus guarantees that there is at least one real solution to
the equation $\beta_{\alpha}=0$.

  2. 2.

As already discussed in Sect. 3.2, the coefficient $A_{0}$ in $\beta_{\alpha}$
is generically non-zero. Thus $\alpha=0$ is not a root of $\beta_{\alpha}$. As
a consequence, any fixed point discussed in the previous section has to
generalize to a fixed point where $\alpha_{*}\not=0$.

  3. 3.

When investigating the transition from the case of pure gravity, $N_{f}=0$, to
the inclusion of a small number of fermions, $N_{f}\ll 1$, one finds that the
fixed point from the minimally coupled case splits into 3 families of NGFPs
that are distinguished by their value of $\alpha_{*}$. In addition to these
families there is a fourth one coming in from $\alpha_{*}\to-\infty$.

  4. 4.

When $N_{f}$ is increased to $N_{f}\approx 3$ the NGFPs from
$\alpha_{*}\to-\infty$ and one family emanating from the pure-gravity fixed
point annihilate. The remaining two families of NGFPs extend to arbitrary
values of $N_{f}$. These solutions will be named NGFPA and NGFPB (Family $A$
and Family $B$).

The position and stability properties associated with Family $A$ and Family
$B$ are shown in Figs. 4 and 5, respectively. The two families are
distinguished by their value of $\alpha_{*}$. Family $A$ has the
characteristic feature that $\alpha_{*}^{A}\ll 1$ while Family $B$ is
characterized by $\alpha_{*}^{B}\propto N_{f}$. Family $A$ therefore
corresponds to an (almost) chiral fixed point. Comparing the fixed point
positions shown in Fig. 1 and $g_{*}^{A}$ and $\lambda_{*}^{A}$ reveals that
these exhibit a very similar behavior. Most notably, $g_{*}^{A}$ is again
sensitive in the same way to the choice of $\beta$: there is a critical number
$N_{f}^{\text{crit}}$ where $g_{*}^{A}$ switches sign. The similarity between
these two cases is understood by noting that $\alpha_{*}=0$ is a good
approximation to the case where $\alpha_{*}^{A}\ll 1$. This suggests that the
NGFPs found at minimal coupling should be identified with the fixed points
comprising Family $A$ once the projection is extended to include $\alpha$.

Family $B$ is situated such that $g_{*}^{B}>0$ for all values of $N_{f}$. This
entails that there is no critical number of fermions $N_{f}^{\rm crit}$ for
Family $B$. This can be understood from the fixed point condition
$\eta_{N}^{*}=-2$. Considering the fact that $\alpha_{*}^{B}>0$ continuously
grows proportional to $N_{f}$ one sees that the contribution of $\alpha$
dominates over the contribution from the regulator, $\beta$, as
$\alpha_{*}^{B}\gg\beta$. With this, one finds that $g_{*}^{B}$ is always
positive. The large values for $\alpha_{*}^{B}$ indicates that Family $B$
constitutes a new class of NGFPs where chiral symmetry is broken by the
coupling to gravity. Owed to the fact that this class comes with
$\alpha_{*}^{B}\not=0$ it is clear that this family may not be seen in
computations which do not include the coupling $\alpha$ in the projection.

The critical exponents for both families are shown in Fig. 5. This shows that
Family $A$ and Family $B$ come with two and three relevant directions,
respectively. This result is independent of the values for $N_{f}$ and
$\beta$. Consequently, a high-energy completion based on NGFPA has the power
to predict the value of $\alpha$ appearing in the effective action
$\Gamma_{k=0}$. Conversely, a high-energy completion based on NGFPB has this
coupling as a free parameter which needs to be fixed by experimental
observations. Besides this, one can see that the critical exponents become
real at $N_{f}^{A}\approx 1.34$ and $N_{f}^{B}\approx 14$ in the case that
$\beta=-1/4$ whereas they become real at $N_{f}^{A}\approx 1.29$ and
$N_{f}^{B}\approx 8.6$ in the case that $\beta=0$.

Figure 4:  The position of the fixed points as a function of $N_{f}$ is shown
for the system (50). The blue and yellow lines indicate the difference in
$\beta$, and therefore in the coarse-graining operator. Family $A$ is
characterized by $\alpha_{*}^{A}\approx 0$ and structurally resembles the
fixed points found at minimal coupling, cf. Fig. 1. For $\beta=0$ there is a
critical number of fermions $N_{f}^{\text{crit}}$ that changes the sign of
$g_{*}^{A}$. Family $B$ on the other hand is characterized by $\alpha_{*}^{B}$
that grows proportional to $N_{f}$. This in turn makes it so that
$\alpha_{*}^{B}\gg\beta$ and $g_{*}^{B}$ is always positive.

Figure 5:  The critical exponents associated with the NGFPs comprising Family
$A$ and Family $B$ for both values of $\beta$. The families come with two and
three relevant directions, respectively. This is independent of the choice of
coarse-graining operator and $N_{f}$. Furthermore, the critical exponents
become real at $N_{f}^{A}\approx 1.34$ and $N_{f}^{B}\approx 14$ in the case
that $\beta=-1/4$ whereas they become real at $N_{f}^{A}\approx 1.29$ and
$N_{f}^{B}\approx 8.6$ in the case that $\beta=0$.

The position of the NGFPs for large $N_{f}$ can again be obtained analytically
by performing a scaling analysis within the fixed point equations. For NGFPA
the leading terms in the expansion are given by

$\begin{split}\quad
g_{*}^{A}&\,\simeq-\tfrac{12\pi}{(\pi-2)\,\xi\,N_{f}}\,,\;\,\lambda_{*}^{A}\,\simeq\tfrac{\pi}{(\pi-2)\xi}\,,\;\,\alpha_{*}^{A}\,\simeq-\tfrac{1008\pi^{2}+42\pi(\pi-2)(16-15\pi)\xi-3(\pi-2)^{2}(796-273\pi)\xi^{2}}{224\,(\pi(\xi-2)-2\xi)^{3}\,N_{f}}\,,\end{split}$
(53)

while for NGFPB

$\begin{split}\quad
g_{*}^{B}&\,\simeq\frac{\pi(3608-945\pi)}{560\,N_{f}^{2}}\,,\quad\lambda_{*}^{B}\,\simeq-\frac{\pi}{12}\,\frac{3608-945\pi}{560}\,\frac{1}{N_{f}}\,,\quad\alpha_{*}^{B}\,\simeq\frac{560}{3608-945\pi}N_{f}\,.\end{split}$
(54)

These asymptotic formulas again match the behavior displayed in Fig. 4,
obtained by numerical methods. Thus both Family $A$ and Family $B$ exist for
arbitrary positive values $N_{f}$. Moreover, the system becomes weakly coupled
as $N_{f}\rightarrow\infty$.

_In summary, the inclusion of $\alpha$ leads to new fixed points. The fixed
points NGFPA comes with $\alpha_{*}\ll 1$ and corresponds to an (almost)
chiral fixed point. In addition there is a family of non-chiral fixed points
NGFPB. The later family is not resolved by the projections analyzed in Sect.
4.1, since these do not track the chiral symmetry breaking coupling $\alpha$
discriminating the two families. The large-$N_{f}$ expansion shows that both
families admit a weak-coupling limit, $\lim_{N_{f}\rightarrow\infty}g_{*}=0$._

###  4.3 The complete system

We now investigate the fixed point-structure of the complete set of beta
functions (26) and (29). Our main finding is the existence of three continuous
families of NGFPs (denoted as NGFPA, NGFPB, and NGFPC) whose properties are
summarized in Figs. 8, 9, and 10, respectively. Families $A$ and $B$ exhibit
the same qualitative features as the fixed points found in Sect. 4.2. Family
$C$ is novel and ows its existence to the inclusion of $\eta_{\psi}$. While
this new family of fixed points passes several non-trivial tests for its
validity, the evidence supporting this class of fixed points is not on the
same footing as the one for the other families which are observed in “simpler”
truncations of the system already.

Before reporting on the results, two technical remarks are in order. First,
eqs. (27) and (30) are implicit equations determining the anomalous
dimensions. Since they are linear in $\eta_{\psi}$ and $\eta_{N}$, it is
straightforward to solve them leading to explicit expressions for
$\eta_{\psi}$ and $\eta_{N}$ depending on the couplings
$\\{g,\lambda,\alpha\\}$ and parameters ${\beta,N_{f}}$. The corresponding
expressions are quite lengthy and little illuminating, so we refrain from
reproducing them at this point. Secondly, the inclusion of $\eta_{\psi}$
complicates the structure of the beta functions such that the NSolve-algorithm
provided by Mathematica is unable to provide a complete list of fixed points
for fixed values of ${\beta,N_{f}}$. Consequently, we adapted our search
strategy as follows. In the first step, we performed a detailed search for
fixed points at $N_{f}=10^{-4}$ and $N_{f}=20$, systematically varying the
seeds of Mathematica’s FindRoot-routine on a three-dimensional cube spanned by
$\\{g,\lambda,\alpha\\}$. This resulted in two lists of solutions which served
as seeds in the subsequent analysis. In order to reduce the numerical
complexity of the search, the list obtained for $N_{f}=10^{-4}$ was reduced to
the fixed points emanating from the Reuter fixed point (41), but coming with
arbitrary values for $\alpha_{*}$. The seeds at $N_{f}=10^{-4}$ ($N_{f}=20$)
were then extended towards increasing (decreasing) values of $N_{f}$. In this
way our search algorithm covered the parameter space $N_{f}\in[0,20]$ twice.
This allowed to eliminate outliers and spurious solutions appearing in the
numerical search and resulted in the robust picture of the fixed point
structure detailed below.

Figure 6:  In the presence of the the fermion anomalous dimension
$\eta_{\psi}$, the fixed point condition $\eta_{N}^{*}=-2$ constitutes a
quadratic equation for $g_{*}$ whose solution includes a square root. The
figure displays the sign of its argument as a function of $\lambda$ and
$\alpha$ for fixed parameters $N_{f}=0.05$ and $\beta=0$. Negative values are
marked in blue while a positive argument is obtained in the sandy region. The
purple dot represents the fixed point from Family $B$. As the value of $N_{f}$
increases, the fixed point moves into the blue region where the argument of
the square root is negative.

_Merging fixed points - an algebraic consideration_
The analysis of the fixed point structure conveniently starts with the
following observation. The condition $\beta_{g}=0$ together with the
requirement $g_{*}\not=0$ requires $\eta_{N}^{*}=-2$. The latter relation can
be solved explicitly for $g_{*}$ which then becomes a function of the
remaining couplings $\\{\lambda_{*},\alpha_{*}\\}$ and the parameters
$\\{N_{f},\beta\\}$. For $\eta_{\psi}=0$, the condition $\eta_{N}^{*}=-2$ is
linear in $g_{*}$ so that there is a unique solution. Including $\eta_{\psi}$
turns this relation into a quadratic equation for $g_{*}$. While this equation
can still be solved algebraically, one now obtains two branches of solutions.
The coordinates $\\{g_{*},\lambda_{*}\\}$ obtained from the first branch
connect continuously to the Reuter fixed point (41). It is this branch that is
traced in the sequel. The novel feature in the fixed point function
$g_{*}(\lambda_{*},\alpha_{*};N_{f},\beta)$ is the appearance of a square-root
whose argument depends on $\\{\lambda_{*},\alpha_{*},N_{f},\beta\\}$.
Demanding that $g_{*}(\lambda_{*},\alpha_{*};N_{f},\beta)$ is real then
requires that the argument of the square-root is positive. This in turn leads
to an additional constraint on the position of the NGFPs. The explicit form of
this inequality is rather bulky and will not be given here. Instead we
illustrate its content for a specific choice of parameters
$\\{\beta=0,N_{f}=0.05\\}$ in Fig. 6. The collision of NGFPB with this region
shifts the fixed points into the complex plane from which they re-emerge at
larger values $N_{f}$.

Figure 7:  Illustration of the interplay between the NGFPs and the region of
instability exemplified in Fig. 6 ($\beta=0$). The figure gives the $g_{*}$
coordinate of various fixed point solutions for values $N_{f}$ below the
window (55).

In addition, a detailed numerical investigation of the fixed point structure
for $\beta=0$ and values $N_{f}<0.55$ shows an intricate pattern of fixed
points moving into and out of the complex plane. This pattern is illustrated
in Fig. 7. Reading the figure from left to right, we first encounter three
fixed points branching off the Reuter fixed point for $N_{f}\ll 1$. At
$N_{f}\approx 0.06$ a new pair of fixed points emerges from the complex plane.
At $N_{f}\approx 0.08$ the lower branch of this pair merges with the branch
NGFPB and both fixed points vanish into the complex plane. The upper branch of
the new pair merges with Family $A$ at the lower boundary given in (55),
$N_{f}\approx 0.55$. These fixed points eventually re-emerge at $N_{f}\approx
1.05$. The precise values of $N_{f}$ where these fixed points take a detour
through the complex plane depends on the choice for $\beta$. For the two cases
investigated in the present work this window is found at

$\begin{split}\beta=-1/4:&\qquad 0.65<N_{f}<1.25\,,\\\ \beta=0:&\qquad
0.55<N_{f}<1.05\,.\end{split}$ (55)

Family $C$ is unaffected by the region of instability and exists for arbitrary
values $N_{f}$. The comparison with the results obtained when setting
$A_{0}=0$ (see the next section) where the window (55) is absent suggests that
it is the chiral symmetry breaking contribution $A_{0}$ which is responsible
for creating this “region of instability”.

| $N_{f}$ | $\beta$ |  $\theta_{1}$ |  $\theta_{2}$ |  $\theta_{3}$ |  $g_{*}$ |  $\lambda_{*}$ |  $\alpha_{*}$ |  $\eta_{\psi}$
---|---|---|---|---|---|---|---|---|---
NGFPA | $3$ | $0$ | $4.21$ | $2.05$ | $-0.84$ | $4.18$ | $-1.65$ | $0.04$ | $-0.53$
| $3$ | $-1/4$ | $4.22$ | $1.78$ | $-0.76$ | $2.35$ | $-0.89$ | $0.04$ | $-0.70$
| $20$ | $0$ | $3.96$ | $2.28$ | $-1.01$ | $-6.32$ | $11.6$ | $0.0081$ | $0.017$
| $20$ | $-1/4$ | $4.11$ | $2.01$ | $-0.99$ | $1.43$ | $-2.63$ | $0.0075$ | $-0.089$
NGFPB | $3$ | $0$ | $4.29$ | $2.52$ | $0.68$ | $0.43$ | $-0.12$ | $1.86$ | $-0.95$
| $3$ | $-1/4$ | $4.14$ | $2.53$ | $0.61$ | $0.44$ | $-0.12$ | $1.62$ | $-0.86$
| $20$ | $0$ | $4.40$ | $3.46$ | $0.58$ | $0.016$ | $-0.032$ | $8.95$ | $-0.65$
| $20$ | $-1/4$ | $4.38$ | $3.45$ | $0.58$ | $0.016$ | $-0.032$ | $8.74$ | $-0.62$
NGFPC | $3$ | $0$ | $4.07$ | $1.31$ | $-67.4$ | $0.0025$ | $-0.0061$ | $91.0$ | $-25.9$
| $3$ | $-1/4$ | $4.07$ | $1.33$ | $-68.9$ | $0.0024$ | $-0.0059$ | $93.0$ | $-26.2$
| $20$ | $0$ | $4.01$ | $1.33$ | $-68.7$ | $5.45\cdot 10^{-5}$ | $-9.40\cdot 10^{-4}$ | $608$ | $-26.4$
| $20$ | $-1/4$ | $4.01$ | $1.34$ | $-69.0$ | $5.42\cdot 10^{-5}$ | $-9.36\cdot 10^{-4}$ | $610$ | $-26.5$
Table 3:  Examples illustrating the generic properties associated with the
three families of fixed points for selected values $N_{f}$ and $\beta$. The
NGFP’s of Family $A$ and $C$ constitute saddle points for the RG flow while
NGFPB act as UV-attractors for the RG flow in the
$g$-$\lambda$-$\alpha-$hyperplane. In contrast to the Reuter fixed point (41),
all examples come with real stability coefficients.

_Fixed points - numerical results_
We now summarize the key properties of the three continuous families of NGFPs
identified by our numerical search algorithm. Explicit examples corresponding
to specific choices of the parameters $N_{f}$ and $\beta$ are listed in Table
3. When plotting the $N_{f}$-dependence of the solutions, the blue and yellow
lines appearing in the position plots correspond to the two choices of coarse-
graining operators, $\beta=-1/4$ and $\beta=0$, respectively. Gray bands mark
the position for $\beta=0$ of region (55) where NGFPA and NGFPB have been
shifted to complex positions.

NGFPA: The properties of the non-Gaussian fixed points comprising Family $A$
are shown in Fig. 8.

Figure 8:  Results from the numerical investigation of the NGFPs belonging to
Family $A$. Reading the diagrams from top-left to bottom right, the first
three plots show the position of the fixed points as a function of $N_{f}$.
The blue and yellow lines correspond to the choice in coarse-graining
operator, $\beta=-1/4$ and $\beta=0$, respectively. The fermion anomalous
dimension $\eta_{\psi}^{*}$ is shown in the fourth diagram. The stability
coefficients are plotted in the bottom row of the diagram indicating that the
fixed points come with two UV-attractive and one UV-repulsive eigendirection
once $N_{f}$ exceeds the upper bound of the window (55).

Figure 9:  Results from the numerical investigation of the fixed points
belonging to Family $B$. The first three diagrams show the position of the
fixed points as a function of $N_{f}$ for $\beta=-1/4$ (blue line) and
$\beta=0$ (orange line). The fixed points exist for all values $N_{f}$ outside
the window (55). The fermion anomalous dimension evaluated at the fixed point
is depicted in the fourth diagram, indicating that $\eta_{\psi}^{*}<0$ in all
cases. Notably, small values of $N_{f}$ lead to rather large absolute values
$|\eta_{\psi}^{*}|$ while values $N_{f}$ to the right of the region of
instability have $\eta_{N}^{*}\approx-1$ (cf. Table 3). The stability
coefficients show that the NGFPB are saddle points with two UV-relevant
directions (three UV-attractors) for values $N_{f}$ below (above) the bounds
(55). The inserts magnify the fermion anomalous dimension and the stability
coefficients obtained for the NGFPB along the orange line in Fig. 7.

Figure 10:  Characteristic properties of Family $C$ obtained from
investigating the system eqs. (26) and (29) numerically. The first three
diagrams show the position of the fixed points as a function of $N_{f}$ for
$\beta=-1/4$ (blue line) and $\beta=0$ (orange line). NGFPC exists for all
values $N_{f}$ and is fairly insensitive to the choice of $\beta$. It is
unaffected by the window (55). The fermion anomalous dimension at the fixed
points is depicted in the fourth diagram, showing that it is negative and
rather large. The stability analysis identifies two UV-attractive and one UV-
repulsive stability coefficient, indicating that NGFPC is a saddle point in
the $\\{g,\lambda,\alpha\\}$-plane.

The fixed points exist for all values $N_{f}$ outside the window (55). The
effect of the fermion anomalous dimension on the system is most pronounced for
small fermion numbers $N_{f}\lesssim 2$ where $|\eta_{\psi}^{*}|>1$. For
values $N_{f}$ located to the left of the gray region, the fixed points act as
UV-attractors and are characterized by a complex pair of stability
coefficients. For $N_{f}$ larger than the bound (55), the NGFPs are
characterized by two real positive and one negative stability coefficient,
indicating that they are saddle points for the RG flow. In this region the
effect of including $\eta_{\psi}$ is rather small. As a consequence, the
properties of the fixed points are in qualitative agreement with the findings
in Sect. 4.2. In particular, the case $\beta=0$ still exhibits a critical
number of fermions $N_{f}^{\text{crit}}$ at which the sign of $g_{*}$ changes
and $\alpha_{*}\ll 1$ so that the fixed points remain (approximately) chiral.

NGFPB: The properties of the NGFPs comprising Family $B$ are shown in Fig. 9.
We first focus on the region where $N_{f}$ is larger than the values indicated
in (55). Here the NGFPs exist for all $N_{f}$. The signature feature of Family
$B$ in this region is the linear increase of $\alpha_{*}^{B}$ with $N_{f}$.
This feature is accompanied by a negative fermion anomalous dimension
$\eta^{B}_{\psi}\simeq-1$. Moreover, the stability analysis reveals that the
three stability coefficients are real and positive, indicating that the fixed
points serve as UV-attractors for all three couplings
$\\{\lambda,g,\alpha\\}$. This picture matches the one established without
considering the effect of the fermion wave function renormalization (cf. Figs.
4 and 5). This indicates that the small negative value of $\eta_{\psi}^{B}$
does not affect the fixed point structure of Family $B$ in a qualitative way.

For values $N_{f}$ to the left of the gray vertical band, $\eta_{\psi}^{B}$ is
negative with relatively large absolute values. As a consequence, the
inclusion of $\eta_{\psi}^{B}$ affects the fixed point structure rather
drastically. This includes the rather intricate pattern of fixed points being
created and annihilated as a function of $N_{f}$ which is detailed in Fig. 7.
Moreover, $\eta_{\psi}^{B}$ turns the real part of one of the stability
coefficient from positive to negative so that the NGFPB are saddle points for
low values $N_{f}$.

NGFPC: The genuinely novel feature arising from the inclusion of the fermion
anomalous dimension is a new family of NGFPs which we label Family $C$. Its
properties are summarized in Fig. 10. Notably, Family $C$ exists for all
values $N_{f}$: it is unaffected by the instability window (55). We find that,
again, $\alpha_{*}^{C}$ increases linearly with $N_{f}$. As a consequence, the
family is rather robust against changes in the coarse-graining operator
encoded in $\beta$. As its characteristic feature, Family $C$ comes with a
negative and rather large fermion anomalous dimension, approaching
$\eta_{\psi}^{C}\approx-26$ for large values $N_{f}$. The stability analysis
shows that NGFPC is a saddle point with two (real) UV-attractive and one UV-
repulsive eigendirection. Clearly, it is the interplay between $\alpha$ and
the large absolute value of $\eta_{\psi}^{C}$ which gives existence to this
family of fixed points.

_In summary, the inclusion of $\eta_{\psi}$ leads to three continuous families
of NGFPs: Family $A$ (saddle point), Family $B$ (UV-attractor), and Family $C$
(saddle point). Families $A$ and $B$ are qualitatively similar to the fixed
points identified in the previous section while Family $C$ is novel. All
families posses a weak coupling limit $\lim_{N_{f}\rightarrow\infty}g_{*}=0$.
The effect of $\eta_{\psi}$ is most notable for small values $N_{f}\lesssim
2$. In this region the fermion anomalous dimension leads to an intricate
pattern of fixed points moving in and out of the complex plane, see Fig. 7.
Family $C$ is unaffected by these transitions and exists continuously for all
values $N_{f}$._

###  4.4 The complete system excluding chiral symmetry breaking contributions

We conclude our analysis of the fixed point structure by studying the beta
functions (26) and (29) with $A_{0}$ set to zero by hand. This approximation
mimics the structure of $\beta_{\alpha}$ found in flat-background computations
Eichhorn:2016vvy . The main effect is the disappearance of the bounds (55) for
Family $A$ and $B$. Family $C$ is structurally left unaltered. Our analysis
follows the one of the previous subsection. In particular, the technical
remarks made at the beginning of section 4.3 again apply.

Merging fixed points - algebraic considerations
We start by analyzing the condition $\eta_{N}^{*}=-2$. Notably, $A_{0}$ does
not enter into this equation. As a consequence, one again obtains a quadratic
equation fixing $g_{*}$ as in terms of
$\\{\lambda_{*},\alpha_{*},N_{f},\beta\\}$. One branch is connected
continuously to the Reuter fixed point (41). The equation for $g_{*}$ again
contains square roots whose arguments depend on
$\\{\lambda_{*},\alpha_{*},N_{f},\beta\\}$. The condition that the arguments
should be positive in order to ensure that $g_{*}$ is real again leads to the
constraints illustrated in Fig. 6.

For $N_{f}<0.2$ this property again leads to an intricate interplay of fixed
points moving in and out of the complex plane. For $\beta=0$, the resulting
pattern is illustrated in Fig. 11.

Figure 11:  Illustration of the interplay between the fixed points for
$\beta=0$ and small values $N_{f}<0.2$. Family B (orange line) crosses over
into the complex plane upon colliding with a short-lived fixed point given by
the purple line. The Family B’ (green line) extends to arbitrary values of
$N_{f}$. On this basis, we identify the Family B’ as the “continuation” of
Family B to arbitrary large values $N_{f}$.

The result is very similar to the case where $A_{0}\neq 0$. However, the
region (55) has disappeared and Family B is unable to return from the complex
plane. Instead one has a new branch, dubbed Family B’, which extends to
arbritrary values of $N_{f}$.

Fixed points - numerical results
We now illustrate the key properties of the three continuous families of NGFPs
identified by our numerical search algorithm. When plotting the
$N_{f}$-dependence of the solutions, the blue and yellow lines appearing in
the position plots correspond to the two choices of coarse-graining operators,
$\beta=-1/4$ and $\beta=0$, respectively.

NGFPA: The properties of the fixed points comprising Family A are shown in
Fig. 12. It is then suggestive to call this family of fixed points “quasi-
chiral”. The fixed points exist for all values of $N_{f}$ as the window (55)
is now absent. As a consequence of excluding the chiral symmetry breaking term
$A_{0}$, Family A is now situated at $\alpha_{*}^{A}=0$ for all values of
$N_{f}$. The stability coefficients are structurally equal to the ones from
the previous section: for low values $N_{f}$ the fixed points act as UV-
attractors while they turn into saddle points once $N_{f}$ exceeds one.

Figure 12:  Characteristic properties of Family $A$ obtained from
investigating the system eqs. (26) and (29) with $A_{0}=0$. The first three
diagrams show the position of the fixed points as a function of $N_{f}$ for
$\beta=-1/4$ (blue line) and $\beta=0$ (orange line). NGFPA exists for all
values $N_{f}$ and is characterized by $\alpha_{*}^{A}=0$. The fermion
anomalous dimension at the fixed points is depicted in the fourth diagram,
showing no structural differences with respect to the previous section apart
from the disappearence of the bounds (55). The stability analysis identifies
two UV-attractive and one UV-repulsive stability coefficient for values of
$N_{f}$ extending beyond the window (55), indicating that NGFPA is a saddle
point in the $\\{g,\lambda,\alpha\\}$-plane. For smaller values of $N_{f}$ the
stability analysis identies three UV-attractive stability coefficients.

Figure 13:  Characteristic properties of Family $B$ obtained from
investigating the system eqs. (26) and (29) with $A_{0}=0$. The first three
diagrams show the position of the fixed points as a function of $N_{f}$ for
$\beta=-1/4$ (blue line) and $\beta=0$ (orange line). NGFPB exists for all
values $N_{f}$ for $\beta=-1/4$. For $\beta=0$ family B moves into the complex
plane at $N_{f}\approx 0.6$ and does not re-emerge due to the absence of the
window (55). Family B’ continues and is shown as a green line. The fermion
anomalous dimension at the fixed points is depicted in the fourth diagram,
showing that it is negative and rather large. The stability analysis
identifies two UV-attractive and one UV-repulsive stability coefficient for
values below the outer edge of (55), indicating that NGFPB is a saddle point
in this region. For values beyond this edge, the fixed points function as UV-
attractors.

Figure 14:  Characteristic properties of Family $C$ obtained from
investigating the system eqs. (26) and (29) with $A_{0}=0$. The first three
diagrams show the position of the fixed points as a function of $N_{f}$ for
$\beta=-1/4$ (blue line) and $\beta=0$ (orange line). NGFPC exists for all
values $N_{f}$ and is fairly insensitive to the choice of $\beta$. The fermion
anomalous dimension at the fixed points is depicted in the fourth diagram,
showing that it is negative and rather large. The stability analysis
identifies two UV-attractive and one UV-repulsive stability coefficient,
indicating that NGFPC is a saddle point in the $\\{g,\lambda,\alpha\\}$-plane.

NGFPB and NGFP${}^{\rm B^{\prime}}$: The properties of the NGFPs comprising
Families B and B’ are shown in Fig. 13. Their position and fermion anomalous
dimension resembles the one found for NGFPB shown in Fig. 9. Apart from the
disappearance of the forbidden region (55) there are no structural changes
with respect to the full system. Moreover, for $\beta=-1/4$ we find that
Family B is UV-attractive, in agreement with the previous section. The fixed
point collisions resolved in Fig. 11 have a drastic effect on the stability
coefficients of the system though: the Family B’ (located at $N_{f}\gtrsim
0.06$) comes with one UV-relevant and two UV-irrelevant directions. This
indicates that the inclusion of $A_{0}$ has a significant stabilization effect
on the fixed point structure. The direct comparison to the case with
$A_{0}\neq 0$ shows that the interplay of $A_{0}$ with the other terms in the
beta functions leads to a fixed point structure which is less sensitive to
varying $\beta$ in the sense that Family B can be recovered for both $\beta=0$
and $\beta=-1/4$.

NGFPC: Structurally, this family of fixed points is left unaltered. For
completeness the properties of the NGFPs comprising Family C are shown in Fig.
14.

_In summary, switching of the chiral symmetry breaking contribution in
$\beta_{\alpha}$ by hand has little effect on the solutions NGFPA and NGFPC.
NGFPA is now situated at $\alpha_{*}=0$, warranting the label “quasi-chiral”
fixed point. In this approximation, Family B develops a significant
$\beta$-dependence though. Only for $\beta=0$ one recovers the qualitative
features exhibited by the full system, indicating that $A_{0}$ plays an
intriguing role in stabilizing the fixed point structure._

##  5 Bounding chiral symmetry through asymptotic safety

In Sect. 4.3 we presented an in-depth analysis of the fixed points encoded in
the beta functions (26) and (29). In this course, we identified four fixed
points: the GFP, NGFPA, NGFPB and NGFPC. The goal of this section is to
illustrate the RG flow resulting from the interplay of these fixed points and
its consequences for the chiral symmetry breaking coupling $\alpha$. For
concreteness, we set $\beta=-1/4$ and $N_{f}=3$. The position and stability
coefficients for this case are listed in Table 3.

Ultimately, physics should be extracted from the effective average action
$\Gamma_{k}$ at $k=0$ where all quantum fluctuations have been integrated out
Knorr:2019atm ; Bonanno:2020bil . Asking for a (semi-)classical regime then
requires that there is a cross-over in the RG flow from the NGFP providing the
UV-completion and the GFP controlling the low-energy behavior of the flow
Reuter:2001ag ; Reuter:2004nx . This restricts the discussion to the interplay
of the GFP, the quasi-chiral fixed point NGFPA and the non-chiral fixed point
NGFPB. The NGFPC is distached from the semi-classical regime and will not play
a role in the subsequent discussion.

As a starting point, we give a more detailed analysis of the stability matrix
(38) associated with these fixed points. For the GFP situated at
$\\{g^{\text{GFP}}_{*},\lambda^{\text{GFP}}_{*},\alpha^{\text{GFP}}_{*}\\}=\\{0,0,0\\}$
the stability coefficients are fixed by canonical power counting and the
corresponding eigenvectors in the $g$-$\lambda$-$\alpha$–plane are555We give
all results rounded to two decimal digests. The notation $0.$ implies that the
component is non-zero but rounds to zero in this representation.

$\begin{array}[]{ll}\theta_{1}=2,&\qquad V_{1}=\\{0,1,0\\}\,,\\\
\theta_{2}=-2,&\qquad V_{2}=\\{0.98,0.21,0.01\\}\,,\\\ \theta_{3}=-1,&\qquad
V_{3}=\\{0,0,1\\}\,.\end{array}$ (56)

Thus, we have two IR-attractive eigendirections which are essentially aligned
with the $g$\- and $\alpha$-axis, while the fixed point is IR-repulsive along
the $\lambda$-direction. The quasi-chiral NGFPA is located at
$\\{g^{A}_{*},\lambda^{A}_{*},\alpha^{A}_{*}\\}=\\{2.35,-0.89,0.04\\}$ and
possesses the eigensystem

$\begin{array}[]{ll}\theta_{1}=4.22,&\qquad V_{1}=\\{0.33,0.941,0.01\\}\,,\\\
\theta_{2}=1.78,&\qquad V_{2}=\\{0.92,-0.40,0.\\}\,,\\\
\theta_{3}=-0.76,&\qquad V_{3}=\\{0.90,-0.38,-0.21\\}\,.\end{array}$ (57)

Thus, we are dealing with a saddle-point whose UV-attractive directions
essentially lie within the $g$-$\lambda$-plane while the $\alpha$-direction is
UV-repulsive. Finally, the non-chiral NGFPB sits at
$\\{g^{B}_{*},\lambda^{B}_{*},\alpha^{B}_{*}\\}=\\{0.44,-0.12,1.62\\}$ and
acts as a UV-attractor

$\begin{array}[]{ll}\theta_{1}=4.14,&\qquad V_{1}=\\{0.29,0.19,0.94\\}\,,\\\
\theta_{2}=2.53,&\qquad V_{2}=\\{0.61,-0.28,0.74\\}\,,\\\
\theta_{3}=0.61,&\qquad V_{3}=\\{0.31,-0.09,-0.95\\}\,.\end{array}$ (58)

An intriguing feature of these stability properties is that they admit RG
trajectories which emanate from NGFPB in the UV, cross over to NGFPA, and
subsequently obtain their low-energy completion from the GFP. This process is
illustrated in Fig. 15.

Figure 15:  Illustration of the interplay between the three fixed points (56),
(57), and (58) for $N_{f}=3$ and $\beta=-1/4$. The left and right panel show
$g_{k}$ and (the logarithm of) $\alpha_{k}$-component obtained from solving
the full system of beta functions numerically as a function of $t\equiv\log
k$. The green lines correspond to the UV-safe trajectories emanating from
NGFPB. The thick green line marks the crossover from NGFPB to NGFPA and
constitutes the barrier between the safe and unsafe trajectories depicted in
orange. For small values $t$, the flow undergoes a second cross-over from
NGFPA to the GFP. The linear decrease of $\log\alpha_{k}$ in this regime is
characteristic for the dimensionful coupling obtaining a constant
($k$-independent) IR-value.

The interplay among the fixed points furthermore suggests that there is a
window for the IR values of $\alpha_{k}$ provided by the asymptotic safety
condition. In order to obtain a qualitative idea for the admissible values
$\alpha_{0}$, we consider a two-parameter family of initial conditions
$\\{g_{\rm init},\lambda_{\rm init},\alpha_{\rm init}\\}$ placed on a two-
sphere centered on the UV-attractor NGFPB, i.e.,

$g_{\rm init}=g_{*}^{B}+R\ \text{sin}\,\theta\
\text{cos}\,\phi,\quad\;\lambda_{\rm init}=\lambda_{*}^{B}+R\
\text{sin}\,\theta\ \text{sin}\,\phi,\quad\;\alpha_{\rm
init}=\alpha_{*}^{B}+R\ \text{cos}\,\theta.$ (59)

Provided that $R$ is taken sufficiently small, this construction guarantees
that all trajectories constructed from these initial conditions are pulled
into NGFPB as $k\rightarrow\infty$. Thus one can construct the UV-critical
hypersurface of the fixed point by varying $\theta,\phi$ and integrating the
RG flow towards $k\to 0$ numerically. Subsequently, one can extract the
scaling of $\alpha_{k}$ in the vicinity of the GFP and reconstruct the value
of the dimensionful coupling $\tilde{\alpha}_{k}$ via the relation (23).
Concretely, our sampling algorithm considered two different values for the
radius, $R=0.05$ and $R=0.005$, and sampled the two-sphere with angle
differences of $0.1$ and $0.3$ degrees, respectively. In the IR, this
algorithm identified the following maximal and minimal values for
$\tilde{\alpha}_{0}$:

$\begin{split}R=0.005:&\qquad 0.020\leq\tilde{\alpha}_{0}\leq 0.19\,,\\\
R=0.05:&\qquad 0.014\leq\tilde{\alpha}_{0}\leq 0.16\,.\end{split}$ (60)

The fact that different values for $R$ give slightly different limits results
from the fact that the set of RG trajectories created by sampling the spheres
at different radius is not identical: the initial conditions set at $R=0.05$
and $R=0.005$ are not positioned on the same RG trajectory and thus provide
two different samplings of the UV-critical hypersurface. Our result then gives
a good indication for the values of $\tilde{\alpha}_{0}$ which are compatible
with an asymptotically safe UV completion by the non-chiral fixed point NGFPB.
In particular, they show that the couplings are not exponentially large, so
that (60) could be in agreement with the phenomenological requirements of
having light chiral fermions. In a flat background where NGFPA may be exactly
chiral, this cross-over may provide an elegant mechanism for starting from a
non-chiral theory at high energy (NGFPB) and restoring the symmetry (at least
approximately) by spending a sufficient amount of RG time in the vicinity of
NGFPA before flowing to the GFP.

##  6 Summary and Conclusions

Our work provides a detailed study of the asymptotic safety mechanism for
gravity coupled to $N_{f}$ Dirac fermions within the background field
formalism. Besides a fermion-kinetic term (and the associated fermion
anomalous dimension), our study includes a non-minimal interaction coupling
the fermion bilinears to the spacetime curvature. The extension beyond minimal
coupling is interesting for two reasons. Firstly, the new term gives a
contribution to the anomalous dimension of Newton’s coupling which is
proportional to the number of fermionic fields. Depending on the sign of the
corresponding coupling, this leads to a novel screening/anti-screening
contribution in the gravitational RG flow which works towards either
stabilizing or destabilizing the asymptotic safety mechanism. Secondly, the
non-minimal interaction explicitly breaks chiral symmetry, as it provides a
fermion mass set by the spacetime curvature. Thus the analysis has a natural
connection to interesting questions including the existence of light chiral
fermions Eichhorn:2011pc  and the role of global symmetries within the
asymptotic safety program Ali:2020znq . While the effect of the non-minimal
coupling has already been explored in Eichhorn:2016vvy , our work (and its
companion Daas:2020dyo ) is the first where the role of the explicit symmetry
breaking terms is studied _in a non-flat background_. Together with the flat-
space results, our findings establish that the topology of spacetime has a
crucial effect on symmetries supported by the matter sector (also see
Gies:2018jnv ; Hamada:2020mug  for related discussions).

The renormalization group flow of the system projected onto the subspace
spanned by the ansatz (10) comes with an intricate fixed point structure. At
the same time the ansatz is sufficiently sophisticated so that it admits
several non-trivial subsystems. Thus it allows to test the robustness of the
fixed point structure under the inclusion of further interaction monomials. In
the absence of the non-minimal coupling, the systems exhibits a one-parameter
family of interacting fixed points, called NGFPA, existing for all values
$N_{f}$. Their structure resembles the pure gravity fixed point. In
particular, they come with two UV-relevant directions associated with Newton’s
coupling and the cosmological constant while the fermion anomalous dimension
is negative and of order unity. Notably, this family admits a
large-$N_{f}$-expansion showing that the theory becomes weakly coupled in the
large $N_{f}$-limit.

Extending the projection subspace by the non-minimal coupling has two
significant effects. Firstly, the “chiral” fixed points NGFPA persist but are
shifted towards a non-zero value of the non-minimal coupling. The coupling
behaves as a “shifted” Gaussian matter coupling, i.e., the cubic beta function
(29) for the new coupling has a non-vanishing constant term. One source for
this shift is the contribution of the background fermion fields, eq. (12),
which violates chiral symmetry.666This is reminiscent of the discussion of
gravitational catalysis occuring in fermionic systems in hyperbolic spacetimes
Ebert:2008pc , where it is also the spectrum of the Dirac operator which is
responsible for chiral symmetry breaking. The new direction of the
renormalization group flow is irrelevant, so that the symmetry breaking dies
off towards low energy and does not provide an obstruction to the existence of
light (standard-model like) chiral fermions. Secondly, the inclusion of the
non-minimal coupling reveals the existence of a family of “non-chiral” fixed
points, called NGFPB, characterized by the new coupling taking values
$\alpha_{*}^{B}\geq 1$. These come with three UV-relevant directions. The
specific values for the stability coefficients and anomalous dimensions
suggest that this class of fixed points also behaves “almost gaussian” in the
sense that quantum corrections do not overrule the canonical power counting.
Supplementing the full system with the fermion anomalous dimension induces a
“region of instability” in the $N_{f}$-dependence of the fixed point
structure. This region is situated at small values $N_{f}\sim 2$ and
characterized by fixed points shifting into and emerging from the complex
plane (see Fig. 7). In addition, the anomalous dimension creates one more
branch of fixed point solutions, NGFPC. In this case the fermion anomalous
dimension and eigenvalues deviate significantly from canonical power counting.

These results fit well with the fixed point structure found on flat
backgrounds. The properties of NGFPA are in qualitative agreement with the
chiral fixed point investigated in Dona:2013qba ; Dona:2014pla ;
Meibohm:2015twa  while non-chiral fixed points similar to the class NGFPB have
been reported in Eichhorn:2016vvy . In particular the critical exponents and
fermion anomalous dimensions are in qualitative agreement when the values of
$N_{f}$ allow for such a comparison. Our work then adds two important new
insights to our understanding of the asymptotic safety mechanism of gravity
coupled to fermionic matter: firstly, there is an intriguing interplay between
the topology of the background and the possibility of realizing global
symmetries at an interacting fixed point. Our analysis provides an explicit
example where the choice of background structure induces a non-zero value for
a coupling responsible for the explicit breaking of a global symmetry.
Secondly, we were able to trace the gravity-matter fixed points to large
values $N_{f}$. This feature is in agreement with the fixed point structure
observed in fluctuation computations Meibohm:2016mkp . Intriguingly, for large
values of $N_{f}$, the fixed points become weakly coupled in the background
Newton coupling. This opens the exciting possibility that this regime may
actually be accessible via perturbative methods.

Conceptually, it would be interesting to enhance the projection subspace of
our work by promoting the fermionic wave-function renormalization to a
(matrix-valued) momentum-dependent function $Z_{k}^{\psi}\rightarrow
Z_{k}^{\psi}(\not{\nabla})$. When expanding $Z_{k}^{\psi}(x)$ in a power
series at $x=0$, the lowest order terms correspond to a mass-term, the fermion
kinetic term, and the invariant $\int
d^{4}x\sqrt{g}\,\bar{\psi}{\not{\nabla}}^{2}\psi$.777Based on a flat-
background study, partial results on the fixed point structure associated with
this coupling have been reported in Eichhorn:2016vvy . This class of
interactions is singled out by the observation that, besides the structures
included in (10), they are the only invariants which may still contribute to
the anomalous dimension of Newton’s coupling at the background level. With the
present choice of background fermions, it is not possible to disentangle this
coupling from an $R\bar{\psi}\psi$ term though. While it is clear that the
methods developed in this work can be used to analyze this coupling in an
approximation excluding the $R\bar{\psi}\psi$-term, we leave this
investigation to future work.

##  Appendix A Fermions on curved backgrounds

Our evaluation of the Wetterich equation (1) in the fermionic sector utilizes
the spin-base formalism developed in Gies:2013noa ; Gies:2015cka ;
Lippoldt:2015cea  and summarized in Lippoldt:2016ayw . The most important
formulas resulting from this approach together with our conventions for the
spinors are summarized in this appendix.

###  A.1 Spinor conventions

Let us start by considering a flat spacetime $\mathbb{R}^{4}$ with Euclidean
metric $\delta_{ab}$, $a,b=1,2,3,4$. The spacetime admits a set of complex
$4\times 4$-matrices $\bar{\gamma}^{a}$, the Dirac matrices, satisfying the
Clifford algebra

$\left\\{\bar{\gamma}^{a},\bar{\gamma}^{b}\right\\}=2\,\delta^{ab}\,{\bf
1}\,,$ (61)

where ${\bf 1}$ is the $4\times 4$-dimensional unit matrix. In addition there
is the fifth $\gamma$-matrix ,
$\gamma^{5}\equiv-\bar{\gamma}^{1}\bar{\gamma}^{2}\bar{\gamma}^{3}\bar{\gamma}^{4}$,
satisfying $\bar{\gamma}^{a}\,\gamma^{5}=-\gamma^{5}\,\bar{\gamma}^{a}$ and
$(\gamma^{5})^{2}={\bf 1}$. The $\gamma$-matrices can be used to build a
convenient basis for Dirac space. Explicitly, the elements are

$\big{\\{}\Gamma^{n}\big{\\}}_{n=1,...,16}=\big{\\{}{\bf
1},\bar{\gamma}^{a},\frac{i}{2}\big{[}\bar{\gamma}^{a},\bar{\gamma}^{b}\big{]},\gamma^{5},i\,\bar{\gamma}^{a}\gamma^{5}\big{\\}},$
(62)

and satisfy

${\rm tr}(\Gamma^{n}\Gamma^{m})=4\,\delta^{nm}.$ (63)

One can then project the structure $\bar{\psi}A\psi$, with $A$ an arbitrary
matrix in Dirac space onto $\bar{\psi}\Gamma^{a}\psi$ using

$\bar{\psi}A\psi=\frac{1}{4}\sum_{n}\operatorname{tr}\big{(}A\Gamma^{n})\,\bar{\psi}\Gamma^{n}\psi\,.$
(64)

By making use of the explicit basis (62) basis one can derive the Fierz
reordering formula

$\big{(}\bar{u}_{1}\Gamma^{m}u_{2}\big{)}\big{(}\bar{u}_{3}\Gamma^{n}u_{4}\big{)}=\,-\frac{1}{16}\sum_{o,p}{\rm
tr}(\Gamma^{m}\Gamma^{n}\Gamma^{o}\Gamma^{p})\big{(}\bar{u}_{1}\Gamma^{o}u_{4}\big{)}\big{(}\bar{u}_{3}\Gamma^{p}u_{2}\big{)},$
(65)

where the $u_{i}$ are arbitrary Dirac spinors. The minus sign is incorporated
since we work with Grassmann-valued spinors.

We will now discuss the generalization to Euclidean curved spacetimes with
metric $g_{\mu\nu}$, admitting a suitable spin-structure. Here it is
convenient to introduce vierbeins $e^{a}_{\,\mu}$ satisfying

$g_{\mu\nu}=e^{a}_{\,\mu}e^{b}_{\,\nu}\,\delta_{ab}.$ (66)

Notably, these vierbeins are not unique. They are fixed up to local
SO$(4)$-transformations only. Requiring invariance under these transformations
serves as an important guiding principle for the construction of various
objects. The Dirac matrices $\gamma^{\mu}$ satisfying the generalized Clifford
algebra

$\big{\\{}\gamma^{\mu},\gamma^{\nu}\big{\\}}=2\,g^{\mu\nu}\,{\bf 1},$ (67)

can be constructed from the flat space gamma matrices $\bar{\gamma}^{a}$ as

$\gamma^{\mu}\equiv e^{\mu}_{\,a}\,\bar{\gamma}^{a}.$ (68)

Note that in general the Dirac matrices $\gamma^{\mu}$ will now depend on the
spacetime coordinate.

In order to construct fermion bilinears and in particular the fermion kinetic
term two additional objects are required, the spin metric $h$ and the spin
connection $\Gamma_{\mu}$ providing the connection piece in the spin covariant
derivative $\nabla_{\mu}$. The spin metric is used to construct the Dirac-
adjoint

$\bar{\psi}\equiv\psi^{\dagger}h\,.$ (69)

The spin metric has to satisfy $|\det(h)|=1$ in order to ensure no new scale
is introduced. In addition the reality of the spinor bilinear forces
$h^{\dagger}=-h$ and $\gamma_{\mu}^{\dagger}=-h\gamma_{\mu}h^{-1}$. Starting
from some basic assumptions for both $h$ and $\nabla_{\mu}$ (e.g. linearity,
product rule, metric compatibility and covariance) one can show that for an
arbitrary spinor $\psi$ containing components of arbitrary spacetime rank we
have

$\nabla_{\mu}\psi=D_{\mu}\psi+\Gamma_{\mu}\psi,\quad\nabla_{\mu}\bar{\psi}=D_{\mu}\bar{\psi}-\bar{\psi}\Gamma_{\mu}.$
(70)

Here $D_{\mu}$ is the Levi-Civita connection and $\Gamma_{\mu}$ is a spin
connection piece. Throughout this work we assume a torsion free connection:
$\operatorname{tr}(\Gamma_{\mu})=0$. Through covariance of the vierbeins it
can be shown that

$\nabla_{\mu}\gamma^{\nu}=0\,.$ (71)

From the properties of $\Gamma_{\mu}$ one can also derive the important
commutator identity

$\big{[}\nabla_{\mu},\nabla_{\nu}\big{]}=\frac{1}{8}R_{\mu\nu\alpha\beta}\big{[}\gamma^{\alpha},\gamma^{\beta}\big{]}.$
(72)

This relation allows to derive the well-known Lichnerowicz formula

$-\not{\nabla}^{2}\psi=\big{(}\Delta_{\psi}+\frac{R}{4}-\frac{1}{2}\,\gamma^{\mu}\gamma^{\nu}\,\big{[}D_{\mu},D_{\nu}\big{]}\big{)}\psi,$
(73)

where $\Delta_{\psi}\coloneqq-\nabla_{\mu}\nabla^{\mu}$. At this stage an
action giving rise to the dispersion relation $(\Delta_{\psi}+m^{2})\psi=0$
can be constructed as

$S^{\rm ferm}=\int
d^{4}x\sqrt{g}\,\bar{\psi}\big{(}i\not{\nabla}+m\gamma^{5}\big{)}\psi.$ (74)

The reality condition $h^{\dagger}=-h$ guarantees that the action is real.

We close this section by showing that, on a background given by a four-sphere,
an eigenmode of the Dirac operator satisfying (12) must necessarily be the
lowest eigenmode. Let the spinor $\psi$ satisfy the relation
$\nabla_{\mu}\psi=ic\,\gamma_{\mu}\psi$, for some real number $c$. Then we
have

$\big{[}\nabla_{\mu},\nabla_{\nu}\big{]}\psi=-c^{2}\big{[}\gamma_{\nu},\gamma_{\mu}\big{]}\psi.$
(75)

By rewriting the right-hand side via (72) and exploiting (11) one then
concludes that $c$ must take the value

$c=\sqrt{\frac{R}{48}}\,.$ (76)

This corresponds to the lowest eigenvalue of the Dirac operator
Camporesi:1995fb . Hence $\psi$ must the lowest momentum mode of the Dirac
operator. Notably, this does not mean that the lowest momentum mode
necessarily satisfies relation (12). Working in an explicit basis however (see
for instance Camporesi:1995fb ), it can be verified that it does.

###  A.2 Variations with respect to the metric field

The computation of the flow in the fermion sector requires the explicit form
of the Hessian $\Gamma_{k}^{(2)}$ entering (1). For this one needs the
expansion of the EAA in terms of the fluctuation fields at quadratic order.
Only the fermion fields $\psi$ can give rise to the fermion-fluctuating fields
$\chi$ whereas all other objects present in the EAA produce terms containing
the fluctuating metric $h_{\mu\nu}$. The variations of the gamma matrices can
be determined by equating the Clifford algebra (67) order by order in the
metric perturbation. For our purpose, the first two orders in this expansion
are sufficient

$\delta\gamma^{\mu}=-\frac{1}{2}h^{\mu\nu}\gamma_{\nu},\qquad\delta^{2}\gamma^{\mu}=\frac{3}{8}\gamma^{\alpha}h_{\beta\alpha}h^{\mu\beta}.$
(77)

Here indices are raised/lowered by the background metric $\bar{g}$. The matrix
$\gamma^{5}$ can be constructed such that the variations with respect to the
metric vanish. For the variations of $\nabla_{\mu}$ we have Lippoldt:2016ayw

$\begin{split}&\delta\nabla_{\mu}=\frac{1}{8}\big{[}\gamma^{\alpha},\gamma^{\beta}\big{]}D_{\beta}h_{\mu\alpha},\\\
&\delta^{2}\nabla_{\mu}=\frac{1}{8}\big{[}\gamma^{\alpha},\gamma^{\beta}\big{]}\big{(}h^{\lambda}_{\,\alpha}D_{\beta}h_{\mu\lambda}+h^{\lambda}_{\,\beta}D_{\lambda}h_{\mu\alpha}+\frac{1}{2}h^{\lambda}_{\alpha}D_{\mu}h_{\beta\lambda}\big{)}.\end{split}$
(78)

###  A.3 Vertices and Propagators

We close the discussion by giving the explicit form of the vertices entering
into $\Gamma_{k}^{(2)}$. Given the linear split (4) we adopt

$\frac{\delta\psi^{i}(x)}{\delta\chi^{j}(y)}=\delta^{i}_{\,j}\,\delta(x-y),\qquad\frac{\delta\bar{\psi}_{i}(x)}{\delta\bar{\chi}_{j}(y)}=\delta^{\,j}_{i}\,\delta(x-y)\,.$
(79)

All the other variations are taken to be zero, which corresponds to treating
$\psi$ and $\bar{\psi}$ as independent fields. Introducing a multiplet
$\Upsilon=\big{\\{}\hat{h}_{\mu\nu},h,\chi^{a},\bar{\chi}_{a}\big{\\}}$ the
Hessian $\Gamma_{k}^{(2)}$ is defined as

$\Gamma_{k}^{(2)}={\frac{\overrightarrow{\delta}}{\delta\Upsilon}}\,\Gamma_{k}\,\frac{\overleftarrow{\delta}}{\delta\Upsilon^{T}}.$
(80)

The variations from the right are taken with respect to the transpose in
spinor space. Denoting the entries of this matrix in field space by
$\Gamma_{f_{i}f_{j}}\equiv\frac{\overrightarrow{\delta}}{\delta
f_{i}}\,\Gamma_{k}\,\overleftarrow{\frac{\overleftarrow{\delta}}{\delta
f_{j}^{T}}}$, and employing the identity
$\big{(}\bar{\psi}A\psi\big{)}^{T}=-\psi^{T}A^{T}\psi^{\dagger}$, the off-
diagonal entries are given by

$\begin{split}\Gamma_{\hat{h}\chi}/Z_{k}^{\psi}&=\frac{i}{4}\,\overleftarrow{D}^{\mu}\,\bar{\theta}\gamma^{\nu}+\bar{\alpha}_{k}\overleftarrow{D}^{\mu}\overleftarrow{D}^{\nu}\,\bar{\theta}\gamma^{5}\\\
\Gamma_{h\chi}/Z_{k}^{\psi}&=-\frac{3i}{16}\,\overleftarrow{D}^{\mu}\,\bar{\theta}\gamma_{\mu}+\frac{\bar{\alpha}_{k}}{4}\,(3\Delta+R)\,\bar{\theta}\gamma^{5}-\frac{3}{2}\,c\,\bar{\theta}\\\
\Gamma_{\hat{h}\bar{\chi}}/Z_{k}^{\psi}&=\frac{i}{4}\,\overleftarrow{D}^{\mu}\,\gamma^{\nu}\theta-\bar{\alpha}_{k}\overleftarrow{D}^{\mu}\overleftarrow{D}^{\nu}\,\gamma^{5}\theta\\\
\Gamma_{h\bar{\chi}}/Z_{k}^{\psi}&=-\frac{3i}{16}\,\overleftarrow{D}^{\mu}\,\gamma_{\mu}\theta-\frac{\bar{\alpha}_{k}}{4}\,(3\Delta+R)\,\gamma^{5}\theta+\frac{3}{2}\,c\,\theta\\\\[20.0pt]
\Gamma_{\chi\hat{h}}/Z_{k}^{\psi}&=-\frac{i}{4}\,\overleftarrow{D}^{\mu}\,\bar{\theta}\gamma^{\nu}-\bar{\alpha}_{k}\overleftarrow{D}^{\mu}\overleftarrow{D}^{\nu}\,\bar{\theta}\gamma^{5}\\\
\Gamma_{\chi
h}/Z_{k}^{\psi}&=\frac{3i}{16}\,\bar{\theta}\gamma^{\mu}D_{\mu}-\frac{\bar{\alpha}_{k}}{4}\,\bar{\theta}\gamma^{5}\,(3\Delta+R)+\frac{3}{2}\,c\,\bar{\theta}\\\
\Gamma_{\bar{\chi}\hat{h}}/Z_{k}^{\psi}&=-\frac{i}{4}\,\gamma^{\nu}\theta
D^{\mu}+\bar{\alpha}_{k}\,\gamma^{5}\theta D^{\mu}D^{\nu}\\\
\Gamma_{\bar{\chi}h}/Z_{k}^{\psi}&=\frac{3i}{16}\,\gamma^{\mu}\theta
D_{\mu}+\frac{\bar{\alpha}_{k}}{4}\,\gamma^{5}\theta\,(3\Delta+R)-\frac{3}{2}\,c\,\theta.\end{split}$
(81)

Following (76), we have $c=\sqrt{\frac{R}{48}}$ and all objects are
constructed from the background metric only. The left arrow over the
derivatives arrives from partial integration and is there to signal the
derivative acts on the gravitational degrees of freedom. The boson-boson
variations, ignoring the contributions of the kinetic term, are given by

$\begin{split}\Gamma_{\hat{h}\hat{h}}/Z_{k}^{\psi}&=\frac{5}{2}c\,\bar{\theta}\theta\,-\bar{\alpha}_{k}\,\bar{\theta}\gamma^{5}\theta\,\,(\frac{1}{2}\Delta+\frac{1}{3}R)\big{(}\mathbb{1}-P\big{)}^{\mu\nu}_{\alpha\beta}\\\
&\qquad\qquad-
D_{\sigma}D^{\rho}\big{(}\mathbb{1}-P\big{)}^{\mu\nu}_{\lambda\rho}\big{(}\mathbb{1}-P\big{)}^{\lambda\sigma}_{\alpha\beta}+\mathcal{O}(D_{\mu})\\\
\Gamma_{hh}/Z_{k}^{\psi}&=-\frac{3}{8}c\,\bar{\theta}\theta+\frac{3}{16}\bar{\alpha}_{k}\,\bar{\theta}\gamma^{5}\theta\,\Delta+\mathcal{O}(D_{\mu})\,.\end{split}$
(82)

Here $\mathbb{1}$ denotes the identity on the space of symmetric two-tensors
and $P$ projects out the trace, meaning

$\big{(}\mathbb{1}-P\big{)}^{\mu\nu}_{\alpha\beta}=\frac{1}{2}(\delta^{\mu}_{\alpha}\delta^{\nu}_{\beta}+\delta^{\nu}_{\alpha}\delta^{\mu}_{\beta})-\frac{1}{4}g_{\alpha\beta}g^{\mu\nu}\,.$
(83)

The explicit form of the $\mathcal{O}(D_{\mu})$-terms is irrelevant since
their contribution vanishes after they are traced over. Finally, the fermion-
fermion variations result in

$\begin{split}\Gamma_{\bar{\psi}\psi}/Z_{k}^{\psi}&=i{\not{\nabla}}+\bar{\alpha}_{k}R\gamma^{5}\\\
\Gamma_{\psi\bar{\psi}}/Z_{k}^{\psi}&=-\big{(}i{\not{\nabla}}+\bar{\alpha}_{k}R\gamma^{5}\big{)}^{T}.\end{split}$
(84)

These are all the variations that enter into the calculation for the beta
function encoding the running of couplings in the fermionic sector. Combining
(84) with the regulator (19) and inverting the resulting expression gives the
explicit form of the fermionic propagator

$G_{\psi}^{-1}=i{\not{\nabla}}\,\widetilde{W}_{\chi}({\not{\nabla}}^{2})+\gamma^{5}\,W_{\chi}({\not{\nabla}}^{2})\,.$
(85)

The functions $\widetilde{W}_{\chi}({\not{\nabla}}^{2})$ and
$W_{\chi}({\not{\nabla}}^{2})$ are defined as

$\begin{split}&W_{\chi}({\not{\nabla}}^{2})\equiv\frac{R_{k}^{\psi}+\bar{\alpha}R}{-{\not{\nabla}}^{2}+(R_{k}^{\psi})^{2}+2\bar{\alpha}R\,R_{k}^{\psi}}\,,\qquad\widetilde{W}_{\chi}({\not{\nabla}}^{2})\equiv\frac{1}{-{\not{\nabla}}^{2}+(R_{k}^{\psi})^{2}+2\bar{\alpha}R\,R_{k}^{\psi}}\,.\end{split}$
(86)

Here the terms $-{\not{\nabla}}^{2}$ are short-hand for the right-hand side of
(17). The graviton propagator resulting from the Einstein-Hilbert action in
harmonic gauge is readily obtained from Reuter:1996cp ; Reuter:2019byg  and
reads

$\begin{split}G_{h}^{-1}=&\,-128\pi G_{k}\,(\Delta-2\Lambda_{k})^{-1}\,,\\\
G_{\hat{h}}^{-1}=&\,32\pi
G_{k}\,(\Delta+\tfrac{2}{3}R-2\Lambda_{k})^{-1}\,\big{(}\mathbb{1}-P\big{)}^{\mu\nu}_{\alpha\beta}\,.\end{split}$
(87)

##  Appendix B Functional traces involving fermionic background fields

In this appendix, we illustrate the techniques used for evaluating operator
traces containing background spinor fields. In practise these computations
amount to evaluating the Feynman diagrams shown in Fig. 16 for the Feynman
rules obtained from (5) on a background four-sphere.

Figure 16:  Feynman diagrams encoding the contributions (88) to the RG flow.
Solid and double internal lines correspond to the fermion and gravity
propagators while the crossed circle marks the insertion of the corresponding
regulator. The external straight lines denote background fermionic fields. The
explicit expressions for the vertices are collected in section A.2.

For the trace-mode $h$ of the graviton fluctuations, the explicit expressions
corresponding to these diagrams (from left to right) are

$\begin{split}D_{3}^{hh\chi}&={\rm
Tr}_{0}\big{[}G_{h}^{-1}\,\Gamma_{h\chi}\,G_{\psi}^{-1}\,\Gamma_{\bar{\chi}h}\,G_{h}^{-1}\partial_{t}\mathcal{R}_{k}^{h}\Big{]}\\\
D_{3}^{\chi\chi h}&=-{\rm
Tr}_{\psi}\Big{[}G_{\psi}^{-1}\,\Gamma_{\bar{\chi}h}\,G_{h}^{-1}\,\Gamma_{h\chi}\,G_{\psi}^{-1}\partial_{t}\mathcal{R}_{k}^{\psi}\Big{]}\\\
D^{h}_{\rm Tad}&=-\frac{1}{2}{\rm
Tr}_{0}\Big{[}G_{h}^{-1}\,\Gamma_{hh}\,G_{h}^{-1}\,\partial_{t}\mathcal{R}_{k}^{h}\Big{]}\,.\end{split}$
(88)

The explicit form of the vertices is given in eqs. (81) and (82) while the
graviton and fermion propagators takes the form (87) and (85), respectively.
In addition, there is an identical set of expressions where $h$ is replaced by
the traceless fluctuations $\widehat{h}$ and Tr0 substituted by the trace over
traceless, symmetric matrices.

###  B.1 Useful identities for background spinor computations

Our goal is to project (88) onto the invariants $I_{3}$ and $I_{4}$ listed in
eq. (10). For the specific choice of background spinor (12), this corresponds
to extracting the contributions proportional to

$\int d^{4}x\sqrt{\bar{g}}\,\sqrt{\bar{R}}\,\bar{\theta}\theta\,,\quad\int
d^{4}x\sqrt{\bar{g}}\bar{R}\,\bar{\theta}\gamma_{5}\theta\,.$ (89)

These structures carry the information about the fermion anomalous dimension
and the scale-dependence of $\alpha_{k}$, respectively. The projection then
entails that we can ignore all terms of order $\bar{R}^{3/2}$ and higher.
Moreover, the trace arguments in (88) typically contain products of
$\gamma$-matrices which need to be mapped to the corresponding basis elements
$\Gamma^{n}$ appearing in the fermion bilinears. This mapping is carried out
using the formula (64), for $\Gamma^{n}={\bf 1}$ and $\Gamma^{n}=\gamma^{5}$,
respectively. A corrollary of this projection is that all terms containing an
odd number of gamma matrices between the background spinors will not
contribute to the set (64).

In order to keep track of the order of the background curvature, all
derivatives acting on the background spinors are eliminated. To accomplish
this, we will list three identities. The first identity shows how to commute
an arbitrary function of the squared Dirac operator through a gamma matrix and
reads

${\not{\nabla}}^{2}\gamma^{\alpha}\psi_{\alpha..}=\gamma^{\alpha}{\not{\nabla}}^{2}\psi_{\alpha..}+2\gamma^{\nu}\big{[}D_{\nu},D_{\alpha}\big{]}\psi^{\alpha}_{\,\,..}+\frac{R}{4}\gamma^{\alpha}\psi_{\alpha..}.$
(90)

In addition to the index $\alpha$, the (generalized) spinor $\psi$ may carry
an arbitrary set of spacetime indices (which is indicated by the dots).
Secondly, we give an identity allowing us to commute a function of
${\not{\nabla}}^{2}$ and a background spinor (12)

$\begin{split}f({\not{\nabla}}^{2})\,\theta\,T=&\,\theta\,f(-\Delta)\,T-16c^{2}\,\theta\,f^{\prime}(-\Delta)\,T+\frac{1}{2}\gamma^{\mu}\gamma^{\nu}\theta\,f^{\prime}(-\Delta)\,\big{[}D_{\mu},D_{\nu}\big{]}\,T\\\
&+2c^{2}\,\theta\,\Delta
f^{\prime\prime}(-\Delta)\,T+2ic\,\gamma^{\mu}\theta\,D_{\mu}\,f^{\prime}(-\Delta)\,T.\end{split}$
(91)

Here $T$ is a tensor of arbitrary rank with the indices suppressed. This
formula can be derived by repeatedly acting with ${\not{\nabla}}^{2}$ onto the
structure $\theta\,T$ and is valid up to terms of order $\bar{R}^{3/2}$. The
final identity allows to pull out a spinor bilinear $\bar{\theta}\,A\,\theta$,
with $A$ being an arbitrary (not necessarily constant) matrix in Dirac space
out of the functional trace

${\rm Tr}\Big{[}(\Delta\phi)\,g(\Delta)\Big{]}=0,$ (92)

where $\phi$ is a scalar function. The reason this identity holds is because
the Laplacian $\Delta$ projects out the zero mode of the spherical harmonics.
Off-diagonal heat kernel methods (see Benedetti:2010nr ) then show that the
functional trace vanishes. We now have all ingredients to evaluate the traces
(88).

For reference, we further note the explicit commutation relations appearing in
explicit applications of (90) and (91). On a background four-sphere and up to
terms of order $\mathcal{O}(R^{2})$ we have

$\big{[}D_{\mu},W(\Delta)\big{]}\,\phi_{\alpha_{1}...\alpha_{n}}=W^{\prime}(\Delta)\big{[}D_{\mu},\Delta\big{]}\,\phi_{\alpha_{1}...\alpha_{n}}\,.$
(93)

For tensors of rank one and symmetric traceless tensors $\hat{\phi}^{\mu\nu}$
the commutators evaluate to the explicit expressions

$\begin{split}&\big{[}D_{\mu},\Delta\big{]}\,\phi^{\mu}=-\frac{1}{4}\,R\,D_{\mu}\,\phi^{\mu}\,,\qquad\big{[}D_{\mu},\Delta\big{]}\,\hat{\phi}^{\mu\nu}=-\frac{5}{12}\,R\,D_{\mu}\,\hat{\phi}^{\mu\nu}\,.\end{split}$
(94)

###  B.2 Results for selected traces

We exemplify our computational strategy by evaluating $D_{3}^{hh\chi}$
explicitly, before giving some concluding remarks on the evaluation of the
other trace expressions. In order to ease the notation, we focus on extracting
the contributions proportional to $\bar{R}\,\bar{\theta}\gamma_{5}\theta$
only. We also set $Z_{k}=1$ for simplicity. The terms proportional to
$\sqrt{\bar{R}}$ are obtained along the same lines. Inserting the explicit
expressions for the propagators and vertices one has

$\begin{split}D_{3}^{hh\chi}={\rm
Tr}_{0}\Big{[}&W_{h}(\Delta)\,\big{(}\frac{3}{16}\,i\,D_{\mu}\bar{\theta}\gamma^{\mu}+\frac{\bar{\alpha}}{4}\,(3\Delta+R)\bar{\theta}\gamma^{5}-\frac{3}{2}ic\,\bar{\theta}\big{)}\\\
&\frac{i{\not{\nabla}}+\gamma^{5}R_{k}^{\psi}+\gamma^{5}\,\bar{\alpha}R}{-{\not{\nabla}}^{2}+(R_{k}^{\psi})^{2}+2\bar{\alpha}R\,R_{k}^{\psi}}\big{(}\frac{3}{16}\,i\,\gamma^{\nu}\theta\,D_{\nu}+\gamma^{5}\,\frac{\bar{\alpha}}{4}\theta\,(3\Delta+R)-\frac{3}{2}ic\,\theta\big{)}\Big{]},\end{split}$
(95)

where the function $W_{h}(\Delta)$ is given by

$W_{h}(\Delta)\coloneqq
G_{h}^{-2}(\Delta)\,\partial_{t}\mathcal{R}_{k}^{h}(\Delta).$ (96)

The strategy for evaluating the right-hand side of (95) is then the following

  1. 1.

Factor out the vertex terms in order to obtain traces with a specific
combination of $\gamma$-matrices sandwiched between the background fermions.

  2. 2.

For each of these combinations, use the representation (85) to identify the
part of the fermion propagator which will give the non-vanishing contribution
to the trace (recalling that traces with odd numbers of $\gamma$-functions
will vanish under the projection and keeping terms up to order $\bar{R}$).

  3. 3.

Commute the gamma matrix with the function of ${\not{\nabla}}^{2}$ with
identity (90).

  4. 4.

Use identity (91) to deal with the derivatives acting on the background
spinor.

  5. 5.

Take the spinorial scalar out of the functional trace (allowed by identity
(92)) and project onto correct spinor structure using (64).

  6. 6.

Compute all the commutators and contract the indices.

  7. 7.

Add all the terms together.

This algorithm results in the following set of scalar traces

$\begin{split}D_{3}^{hh\chi}\Big{|}_{\bar{\theta}\gamma^{5}\theta}=&-\frac{9}{256}{\rm
Tr}_{0}\big{[}W_{h}(\Delta)\,\Delta\,W_{\chi}(-\Delta)\big{]}+\frac{3}{64}{\rm
Tr}_{0}\big{[}W_{h}(\Delta)\,R\,W_{\chi}(-\Delta)\big{]}\\\
&+\frac{3}{1024}{\rm
Tr}_{0}\big{[}W_{h}(\Delta)\,\Delta\,R\,W^{\prime}_{\chi}(-\Delta)\big{]}-\frac{3}{2048}{\rm
Tr}_{0}\big{[}W_{h}(\Delta)\,\Delta^{2}\,R\,W^{\prime\prime}_{\chi}(-\Delta)\big{]}\\\
&+\frac{9}{32}\bar{\alpha}\,{\rm
Tr}_{0}\big{[}W_{h}(\Delta)\,\Delta^{2}\,\widetilde{W}_{\chi}(-\Delta)\big{]}+\frac{3}{32}\bar{\alpha}\,{\rm
Tr}_{0}\big{[}W_{h}(\Delta)\,\Delta\,R\,\widetilde{W}_{\chi}(-\Delta)\big{]}\\\
&-\frac{9}{128}\bar{\alpha}\,{\rm
Tr}_{0}\big{[}W_{h}(\Delta)\,\Delta^{2}\,R\,\widetilde{W}^{\prime}_{\chi}(-\Delta)\big{]}+\frac{3}{256}\bar{\alpha}\,{\rm
Tr}_{0}\big{[}W_{h}(\Delta)\,\Delta^{3}\,R\,\widetilde{W}^{\prime\prime}_{\chi}(-\Delta)\big{]}\\\
&+\frac{9}{16}\bar{\alpha}^{2}\,{\rm
Tr}_{0}\big{[}W_{h}(\Delta)\,\Delta^{2}\,W_{\chi}(-\Delta)\big{]}+\frac{3}{8}\bar{\alpha}^{2}\,{\rm
Tr}_{0}\big{[}W_{h}(\Delta)\,\Delta\,R\,W_{\chi}(-\Delta)\big{]}\\\
&-\frac{3}{16}\bar{\alpha}^{2}\,{\rm
Tr}_{0}\big{[}W_{h}(\Delta)\,\Delta^{2}\,R\,W^{\prime}_{\chi}(-\Delta)\big{]}+\frac{3}{128}\bar{\alpha}^{2}\,{\rm
Tr}_{0}\big{[}W_{h}(\Delta)\,\Delta^{3}\,R\,W^{\prime\prime}_{\chi}(-\Delta)\big{]}\,,\end{split}$
(97)

where the subscript on the right-hand side indicates that the diagram has been
projected onto the fermion-bilinear ${\bar{\theta}\gamma^{5}\theta}$. At this
stage the argument of all scalar traces has been reduced to functions of the
Laplacian $\Delta$. These expressions are then readily evaluated using the
early-time expansion of the heat-kernel Reuter:1996cp ; Codello:2008vh ;
Reuter:2019byg , in $d=4$ dimensions and up to terms quadratic in the
curvature

${\rm Tr}_{s}[W(\Delta)]=\frac{1}{(4\pi)^{2}}\,{\rm tr}_{s}[{\bf 1}]\,\int
d^{4}x\sqrt{g}\,\left(Q_{2}[W]+\frac{1}{6}R\,Q_{1}[W]\right)+\mathcal{O}(R^{2})\,,$
(98)

with

$Q_{n}[W]\equiv\frac{1}{\Gamma(n)}\int_{0}^{\infty}dzz^{n-1}W(z)\,,\qquad
n>0\,.$ (99)

Applying these formulas to (97) gives the final expression for
$D_{3}^{hh\chi}\Big{|}_{R\,\bar{\theta}\gamma^{5}\theta}$

$\begin{split}D_{3}^{hh\chi}\Big{|}_{R\,\bar{\theta}\gamma^{5}\theta}=&\frac{G\,k}{(1-2\lambda)^{2}\pi}\Bigg{\\{}-\frac{29}{256}+(-\frac{89}{1280}+\frac{33}{1024}\pi)\eta_{N}+(-\frac{9}{16}+\frac{9}{64}\pi)\beta+(\frac{9}{128}-\frac{9}{512}\pi)\beta\,\eta_{N}\\\
&+\alpha\,\Big{[}\frac{3}{16}-\frac{27}{128}\pi+(-\frac{13}{128}+\frac{15}{256}\pi)\eta_{N}-\frac{3}{8}\beta+(-\frac{117}{80}+\frac{63}{128}\pi)\beta\eta_{N}\Big{]}\\\
&+\alpha^{2}\Big{[}\frac{371}{80}-\frac{21}{16}\pi+(-\frac{5}{14}+\frac{3}{32}\pi)\eta_{N}+(\frac{27}{4}-\frac{27}{16}\pi)\beta+(\frac{183}{40}+\frac{45}{32}\pi)\beta\,\eta_{N}\Big{]}\\\
&+\alpha^{3}\Big{[}-\frac{51}{20}+(\frac{237}{56}-\frac{81}{64}\pi)\eta_{N}\Big{]}\Bigg{\\}}.\end{split}$
(100)

We now discuss the contribution of $D_{3}^{\chi\chi h}$. A priori,
$D_{3}^{\chi\chi h}$ constitutes a trace in spinor-space,

$D_{3}^{\chi\chi
h}=-\sum_{n}\langle\bar{\psi}_{n}|\Gamma_{\bar{\chi}h}\,G_{h}^{-1}\Gamma_{h\chi}G_{\psi}^{-1}\partial_{t}\mathcal{R}_{k}^{\psi}\,G_{\psi}^{-1}|\psi_{n}\rangle\,,$
(101)

where the sum is over a complete basis of spinors on the four-sphere. One can
then exploit that the vertices $\Gamma_{h\chi}$ act as “intertwiners”
converting from a spinorial to a scalar expression. This allows to rewrite
(101) as a scalar trace by inserting a complete basis of scalar functions
satisfying $\sum_{m}|\phi_{m}\rangle\langle\phi_{m}|={\bf 1}$

$\begin{split}D_{3}^{\chi\chi
h}&=-\sum_{n,m}\langle\bar{\psi}_{n}|\Gamma_{\bar{\chi}h}\,G_{h}^{-1}\,|\phi_{m}\rangle\langle\phi_{m}|\,\Gamma_{h\chi}G_{\psi}^{-1}\partial_{t}\mathcal{R}_{k}^{\psi}\,G_{\psi}^{-1}|\psi_{n}\rangle\\\
&=-\sum_{n,m}\langle\phi_{m}|\,\Gamma_{h\chi}G_{\psi}^{-1}\partial_{t}\mathcal{R}_{k}^{\psi}\,G_{\psi}^{-1}|\psi_{n}\rangle\langle\bar{\psi}_{n}|\Gamma_{\bar{\chi}h}\,G_{h}^{-1}\,|\phi_{m}\rangle\\\
&=-\sum_{m}\langle\phi_{m}|\,G_{h}^{-1}\,\Gamma_{h\chi}G_{\psi}^{-1}\partial_{t}\mathcal{R}_{k}^{\psi}\,G_{\psi}^{-1}\Gamma_{\bar{\chi}h}\,|\phi_{m}\rangle\,.\end{split}$
(102)

After this reordering the structure of $D_{3}^{\chi\chi h}$ becomes identical
to the one found in $D_{3}^{hh\chi}$. The evaluation of $D_{3}^{\chi\chi h}$
then proceeds along the same lines as the one of $D_{3}^{hh\chi}$, replacing

$\begin{array}[]{ll}W_{h}(\Delta)\mapsto G_{h}^{-1}(\Delta)\,,&\qquad
G_{\psi}^{-1}({\not{\nabla}})\mapsto
G_{\psi}^{-2}({\not{\nabla}})\partial_{t}\mathcal{R}_{k}^{\psi}({\not{\nabla}})\,,\\\\[6.45831pt]
W_{\chi}({\not{\nabla}}^{2})\mapsto
V_{\chi}({\not{\nabla}}^{2})\,,&\qquad\widetilde{W}_{\chi}({\not{\nabla}}^{2})\mapsto\widetilde{V}_{\chi}({\not{\nabla}}^{2})\,,\end{array}$
(103)

where

$\begin{split}V_{\chi}({\not{\nabla}}^{2})\equiv\frac{{\not{\nabla}}^{2}+\big{(}R_{k}^{\psi}\big{)}^{2}+2\bar{\alpha}R\,R_{k}^{\psi}}{\big{[}-{\not{\nabla}}^{2}+\big{(}R_{k}^{\psi}\big{)}^{2}+2\bar{\alpha}R\,R_{k}^{\psi}\big{]}^{2}}\partial_{t}R_{k}^{\psi},\\\
\widetilde{V}_{\chi}({\not{\nabla}}^{2})\equiv\frac{2(\bar{\alpha}R+R_{k}^{\psi})}{\big{[}-{\not{\nabla}}^{2}+\big{(}R_{k}^{\psi}\big{)}^{2}+2\bar{\alpha}R\,R_{k}^{\psi}\big{]}^{2}}\partial_{t}R_{k}^{\psi}.\end{split}$
(104)

Making these substitutions in (97) and taking into account the relative minus
sign then yields the contribution of the fermion loop diagram

$\begin{split}D_{3}^{\chi\chi
h}\Big{|}_{R\,\bar{\theta}\gamma^{5}\theta}=&\frac{G\,k}{(1-2\lambda)\pi}\Bigg{\\{}-\frac{39}{256}+(-\frac{27}{32}+\frac{27}{128}\pi)\beta\\\
&+\alpha\,\Big{[}\frac{21}{16}-\frac{27}{64}\pi-\frac{3}{4}\beta\Big{]}+\alpha^{2}\Big{[}\frac{89}{8}-\frac{105}{32}\pi+(\frac{135}{8}-\frac{135}{32}\pi)\beta\Big{]}+\alpha^{3}\frac{27}{20}\Bigg{\\}}.\end{split}$
(105)

The evaluation of the Feynman diagrams including the traceless part of the
graviton fluctuations proceeds along the same lines. In order to simplify the
computation we perform a transverse-traceless decomposition York:1973ia ;
Lauscher:2002sq ; Benedetti:2010nr ; Groh:2011dw  of $\widehat{h}_{\mu\nu}$ on
the spherically symmetric background, setting

$\widehat{h}_{\mu\nu}=h^{TT}_{\mu\nu}+D_{\mu}\xi_{\nu}+D_{\nu}\xi_{\mu}+\big{(}D_{\mu}D_{\nu}+\frac{1}{4}g_{\mu\nu}\,\Delta\big{)}\sigma\,,$
(106)

followed by the field redefinition

$\xi_{\mu}\mapsto\frac{1}{\sqrt{2}}\big{[}\Delta-\frac{1}{4}R\big{]}^{-1/2}\,\xi_{\mu},\quad\sigma\mapsto\big{[}\frac{3}{4}\Delta^{2}-\frac{1}{4}R\Delta\big{]}^{-1/2}\,\sigma\,.$
(107)

The component fields, given by a transverse-traceless symmetric tensor
$h^{TT}_{\mu\nu}$, a transverse vector $\xi_{\mu}$, and a scalar $\sigma$,
satisfy the differential constraints

$g^{\mu\nu}h^{TT}_{\mu\nu}=0,\quad D^{\mu}h^{TT}_{\mu\nu}=0,\quad
D^{\mu}\xi_{\mu}=0.$ (108)

The redefinition (107) ensures that the decomposition does not give rise to
non-trivial (operator-dependent) Jacobians.

Following the algorithm described above allows to reduce the operator traces
appearing in $D_{3}^{\hat{h}\hat{h}\chi}$ to traces whose arguments depend on
the Laplacian $\Delta$ only. The expression analogous to (97) reads

$\begin{split}&D_{3}^{\hat{h}\hat{h}\chi}\Big{|}_{\bar{\theta}\gamma^{5}\theta}=\\\
&\;\;-\frac{1}{32}\,{\rm
Tr}_{TV}\big{[}W_{\hat{h}}\,\Delta\,W_{\chi}(-\Delta)\big{]}+\frac{1}{128}\,{\rm
Tr}_{TV}\big{[}W_{\hat{h}}\,R\,W_{\chi}(-\Delta)\big{]}+\frac{1}{96}\,{\rm
Tr}_{TV}\big{[}W_{\hat{h}}\,\Delta\,R\,W^{\prime}_{\chi}(-\Delta)\big{]}\\\
&\;\;-\frac{1}{768}\,{\rm
Tr}_{TV}\big{[}W_{\hat{h}}\,\Delta^{2}\,R\,W^{\prime\prime}_{\chi}(-\Delta)\big{]}+\frac{5}{384}\,{\rm
Tr}_{TV}\big{[}W^{\prime}_{\hat{h}}\,\Delta\,R\,W_{\chi}(-\Delta)\big{]}-\frac{3}{64}\,{\rm
Tr}_{0}\big{[}W_{\hat{h}}\,\Delta\,W_{\chi}(-\Delta)\big{]}\\\
&\;\;+\frac{1}{64}\,{\rm
Tr}_{0}\big{[}W_{\hat{h}}\,R\,W_{\chi}(-\Delta)\big{]}+\frac{1}{256}\,{\rm
Tr}_{0}\big{[}W_{\hat{h}}\,\Delta\,R\,W^{\prime}_{\chi}(-\Delta)\big{]}-\frac{1}{512}\,{\rm
Tr}_{0}\big{[}W_{\hat{h}}\,\Delta^{2}\,R\,W^{\prime\prime}_{\chi}(-\Delta)\big{]}\\\
&\;\;+\frac{1}{32}\,{\rm
Tr}_{0}\big{[}W^{\prime}_{\hat{h}}\,\Delta\,R\,W_{\chi}(-\Delta)\big{]}+\frac{3}{8}\bar{\alpha}\,{\rm
Tr}_{0}\big{[}W_{\hat{h}}\,\Delta^{2}\,\widetilde{W}_{\chi}(-\Delta)\big{]}-\frac{1}{8}\bar{\alpha}\,{\rm
Tr}_{0}\big{[}W_{\hat{h}}\,\Delta\,R\,\widetilde{W}_{\chi}(-\Delta)\big{]}\\\
&\;\;-\frac{3}{32}\bar{\alpha}\,{\rm
Tr}_{0}\big{[}W_{\hat{h}}\,\Delta^{2}\,R\widetilde{W}^{\prime}_{\chi}(-\Delta)\big{]}+\frac{1}{64}\bar{\alpha}\,{\rm
Tr}_{0}\big{[}W_{\hat{h}}\,\Delta^{3}\,R\,\widetilde{W}^{\prime\prime}_{\chi}(-\Delta)\big{]}-\frac{1}{4}\bar{\alpha}\,\,{\rm
Tr}_{0}\big{[}W^{\prime}_{\hat{h}}\,\Delta^{2}\,R\,\widetilde{W}_{\chi}(-\Delta)\big{]}\\\
&\;\;+\frac{3}{4}\bar{\alpha}^{2}\,{\rm
Tr}_{0}\big{[}W_{\hat{h}}\,\Delta^{2}\,W_{\chi}(-\Delta)\big{]}-\frac{1}{4}\bar{\alpha}^{2}\,{\rm
Tr}_{0}\big{[}W_{\hat{h}}\,\Delta\,R\,W_{\chi}(-\Delta)\big{]}-\frac{1}{4}\bar{\alpha}^{2}\,{\rm
Tr}_{0}\big{[}W_{\hat{h}}\,\Delta^{2}\,R\,W^{\prime}_{\chi}(-\Delta)\big{]}\\\
&\;\;+\frac{1}{32}\bar{\alpha}^{2}\,{\rm
Tr}_{0}\big{[}W_{\hat{h}}\,\Delta^{3}\,R\,W^{\prime\prime}_{\chi}(-\Delta)\big{]}-\frac{1}{2}\bar{\alpha}^{2}\,{\rm
Tr}_{0}\big{[}W^{\prime}_{\hat{h}}\,\Delta^{2}\,R\,W_{\chi}(-\Delta)\big{]}.\end{split}$
(109)

Here the subscripts $TV$ and $0$ indicate that the corresponding traces are
over the space of transverse vectors and scalars, respectively. Applying (98),
one then finds

$\begin{split}D_{3}^{\hat{h}\hat{h}\chi}\Big{|}_{R\,\bar{\theta}\gamma^{5}\theta}=&\frac{G\,k}{(1-2\lambda)^{2}\pi}\Bigg{\\{}\frac{13}{256}+(\frac{129}{1280}-\frac{33}{1024}\pi)\eta_{N}+\frac{1}{1-2\lambda}\big{[}\frac{7}{20}-\frac{3}{32}\pi+(-\frac{179}{1120}+\frac{3}{64}\pi)\eta_{N}\big{]}\\\
&\qquad\qquad\quad+(\frac{9}{16}-\frac{9}{64}\pi)\beta+(-\frac{9}{128}+\frac{9}{256}\pi)\beta\,\eta_{N}\Big{]}\\\
&+\alpha\,\Big{[}-\frac{37}{48}+\frac{55}{128}\pi+(\frac{1379}{1920}-\frac{67}{256}\pi)\eta_{N}+\frac{1}{1-2\lambda}\big{[}-\frac{7}{30}-\frac{1}{8}\pi+(-\frac{11}{420}+\frac{1}{32}\pi)\eta_{N})\big{]}\\\
&\qquad\qquad\quad+\frac{1}{8}\beta+(\frac{39}{80}-\frac{21}{128}\pi)\beta\,\eta_{N}\Big{]}\\\
&+\alpha^{2}\Big{[}-\frac{707}{240}+\frac{13}{16}\pi+(\frac{201}{280}-\frac{7}{32}\pi)\eta_{N}+\frac{1}{1-2\lambda}\big{[}-\frac{17}{105}+(\frac{143}{630}-\frac{1}{16}\pi)\eta_{N}\big{]}\\\
&\qquad\qquad\quad+(-\frac{9}{4}+\frac{9}{16}\pi)\beta+(\frac{61}{40}-\frac{15}{32}\pi)\beta\,\eta_{N}\Big{]}\\\
&+\alpha^{3}\Big{[}\frac{17}{20}+(-\frac{79}{56}+\frac{27}{64}\pi)\eta_{N}\Big{]}\Bigg{\\}}.\end{split}$
(110)

Finally, one can compute
$D_{3}^{\chi\chi\hat{h}}\Big{|}_{\bar{\theta}\gamma^{5}\theta}$ by plugging
the substitutions (103) into (109), giving