#  OU-HET/873-2015 KANAZAWA-15-15
Non-minimal coupling in Higgs-Yukawa model
with asymptotically safe gravity

Kin-ya Oda and Masatoshi Yamada

∗Department of Physics, Osaka University, Osaka 560-0043, Japan
†Institute for Theoretical Physics, Kanazawa University, Kanazawa 920-1192,
Japan

E-mail<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We study the fixed point structure of the Higgs-Yukawa model, with its scalar
being non-minimally coupled to the asymptotically safe gravity, using the
functional renormalization group. We have obtained the renormalization group
equations for the cosmological and Newton constants, the scalar mass-squared
and quartic coupling constant, and the Yukawa and non-minimal coupling
constants, taking into account all the scalar, fermion, and graviton loops. We
find that switching on the fermionic quantum fluctuations makes the non-
minimal coupling constant irrelevant around the Gaussian-matter fixed point
with the asymptotically safe gravity.

##  1 Introduction

Construction of quantum gravity is one of the most important, and challenging,
subjects in physics. The general relativity is derived from the Einstein-
Hilbert action

$\displaystyle S_{\rm EH}=\int\text{d}^{D}x\,\sqrt{g}\left(-{\frac{1}{16\pi
G}R}+\lambda\right),$ (1)

where $G$ and $\lambda$ are the Newton and cosmological constants and we work
with the Euclidean action throughout this paper. The Einstein gravity (1) can
accurately account for the macroscopic phenomena such as the perihelion
precession of Mercury and the gravitational lensing. Therefore we believe that
the action (1) correctly describes the dynamics of gravity in the long range.
On the other hand, its quantization is quite difficult because of the non-
renormalizability. The asymptotically safe quantum gravity, suggested by
Weinberg [1], is one of the possible candidates of quantum gravity.

It is essential for the scenario of asymptotic safety that there exists a non-
trivial ultraviolet (UV) fixed point.111 In general we call a fixed point “UV”
if it has a relevant direction. (Throughout this paper, we call an operator
“relevant” if its coupling constant departs from the fixed point in the flow
from UV to IR.) Sometimes “IR fixed point” is defined by the condition that
all the directions become either irrelevant or marginal around it. Here
instead we call a fixed point “IR”, even when there exists a relevant
direction around it, if the RG flow from the UV fixed point is attracted
toward it, as is the case for the Wilson-Fisher fixed point. See Fig. 1.
Around the UV fixed point, two hypersurfaces are defined: the UV and infrared
(IR) critical surfaces.222 Usually, “critical surface” refers to what is
called the “IR critical surface” here. In the literature on the asymptotic
safety, the wording “UV critical surface” is used frequently, and we put “IR”
on what is usually called “critical surface”, in order to distinguish it from
the other.  The UV critical surface consists of the renormalized trajectories
that are flowing out of the UV fixed point and is in general finite
dimensional, whereas the IR critical surface is its orthogonal complement and
is infinite dimensional in general. See Fig. 1.

Figure 1: Schematic figure for the RG flow in the theory space. The arrows
indicate the direction from UV to IR. The left (red) and right (purple)
points, labelled IRFP and UVFP, are the IR and UV fixed points, respectively.
The UV critical surface (green) is the finite-dimensional subspace spanned by
the renormalized trajectories that are flowing out of the UV fixed point.
Under the asymptotic safety, our low energy effective theory is one of the
points on the renormalized trajectory. The right (blue) surface is the IR
critical surface, which is generally infinite dimensional; see footnote 2. The
other (orange) generic flows cannot be used to construct an asymptotically
safe theory. The left (yellow) surface is a finite-dimensional subspace
spanned by the relevant directions around the IR fixed point.

The renormalization group (RG) flow of the renormalized trajectory on the UV
critical surface takes infinite steps of renormalization transformations near
the UV fixed point. If the IR physics is realized as a point on the UV
critical surface, then the continuum limit $\Lambda\to\infty$ can be taken,
and the theory is free from UV divergences. Furthermore, when the dimension of
the UV critical surface is finite, the theory is non-perturbatively
renormalizable even if it is non-renormalizable in perturbation theory; see
e.g. Refs. [2, 3, 4, 5, 6]. The idea of the asymptotic safety has been applied
not only to gravity but also to the extra-dimensional model [7, 8, 9, 10] and
to the Higgs-Yukawa model in flat spacetime [11, 12, 13, 14, 15]. The quantum
Einstein gravity theory is asymptotically safe if there exists the UV critical
surface including the Newton constant. We will further review the concept of
the asymptotic safety in section 2.

In earlier study the exsistence of UV fixed point of the Newton constant $G$
has been studied by an $\epsilon$-expansion in $2+\epsilon$ dimensions [1,
16]. The fixed point of the dimensionless rescaled Newton constant
$\tilde{G}:=G\Lambda^{\epsilon}$ is found as $\tilde{G}^{*}=3\epsilon/38$ when
ignoring the cosmological constant $\lambda$. The dimensionful Newton constant
$G$ vanishes asymptotically around the fixed point if $\epsilon>0$, that is,
$G\simeq\tilde{G}^{*}/\Lambda^{\epsilon}\to 0$ for $\Lambda\to\infty$. Then
the theory is asymptotically free.

The $\epsilon$ expansion method has difficulties in applying to arbitrary
space-time dimensions and in analyzing the theory in detail. The functional
renormalization group (FRG) [17, 18, 19, 20, 21, 22] is useful for such
purposes. After its pioneering application to the quantum Einstein gravity
given by Reuter [23], the UV fixed point and the RG flow structure of the
Einstein gravity have been investigated in Refs. [24, 25, 26, 27, 28, 29, 30,
31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47], and its
extended models with matters are studied in Refs. [48, 49, 50, 51, 52, 53, 54,
55, 56, 57, 58, 59, 60, 61, 62, 63, 64]; see also Refs. [65, 66, 67, 68, 69,
70] for reviews.333 In Refs. [71, 72, 47], the issue of gauge dependence has
been discussed.  The existence of the UV fixed point and the stability of the
dimension of the UV critical surface when extending the theory space have been
studied in Refs. [73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87,
88, 89, 90]. For example in Ref. [87], the $f(R)$ gravity that has powers of
the Ricci scalar $R$ up to order $R^{34}$ is studied, and it has been shown
that the number of dimensions of the UV critical surface is stable to be
three, namely, the relevant operators are $\lambda$, $R$ and $R^{2}$.
Furthermore, the Higgs mass was predicted to be $\simeq 126\,\text{GeV}$
before the Higgs discovery by requiring the Higgs quartic coupling to vanish
around the Planck scale in the context of the asymptotically safe gravity
[91]. These results encourage the asymptotic safety scenario for the quantum
gravity.

The purpose of this paper is to contribute to the investigation whether the
asymptotically safe gravity can have a large non-minimal coupling $\xi$
between $R$ and a scalar field in the IR limit. Such a large non-minimal
coupling plays a crucial role in the Higgs inflation scenario [92, 93, 94, 95,
96, 97, 98, 99, 100, 101, 102, 103]; see Ref. [104] for a phenomenological
study in a concrete model of the Higgs inflation under the asymptotically safe
gravity.444 In Refs. [105, 106], the asymptotically safe gravity has been
applied to the Starobinsky $R^{2}$ inflation model. In Ref. [107], it has been
claimed that the Higgs potential becomes flat above a certain transition scale
under the asymptotic safety. See Refs. [108, 109, 110, 111] for attempts of
the so-called asymptotically safe inflation, and also Refs. [112, 113].  In
the first attempts of Higgs inflation [92, 93], it was necessary to have an
extremely large value of $\xi$ of order $10^{4}$–$10^{5}$ to account for the
cosmological data. Later it has been pointed out [96, 97] that it is possible
to have a successful Higgs inflation with smaller $\xi\sim 10$, given the
criticality of the Higgs potential, i.e., the fact that both the Higgs quartic
coupling and its beta function can vanish at around the Planck scale
$\sim\left(32\pi G\right)^{-1/2}\simeq 10^{18}\,\text{GeV}$; see also Ref.
[95]. For the criticality of the Higgs potential in the Standard Model, it is
essential that the Higgs field has the large Yukawa coupling to the top quark.
Therefore, it is important to understand the asymptotic safety in a Higgs-
Yukawa system which is non-minimally coupled to the gravity.

In Refs. [49, 51, 52, 114, 115], the authors have analyzed a simplified
scalar-gravity system without fermions, taking into account the non-minimal
coupling $\xi$ between a neutral scalar and the Ricci scalar, under the local
potential approximation (LPA) in FRG: The non-minimal coupling is shown to be
_relevant_ around the UV fixed point. In Ref. [53], the authors have analyzed
a simplified Higgs-Yukawa system with the same neutral scalar and an
additional fermion and without $\xi$, in the flat spacetime. We combine these
two approaches and analyze the running of $\xi$ and the Yukawa coupling under
the influence of the fermionic quantum fluctuations, in the simplified Higgs-
Yukawa system that is non-minimally coupled to gravity. We find that $\xi$
becomes _irrelevant_ by inclusion of the fermions.

This paper is organized as follow: We briefly review the concept of asymptotic
safety in the next section. In section 3, we introduce the Higgs-Yukawa model
which is non-minimally coupled to gravity. In Sec. 4, we show explicitly the
RG equations of the model. In section 5, we present the methods and results of
the numerical analysis. In Sec. 6, we give summary and discussions. In
appendix A, we briefly sketch how the Wetterich equation is derived from the
cutoff dependence of the effective action equipped with the cutoff function.
In appendix B, we list the formulae for supermatrix. In appendix C, we rewrite
the Wetterich equation into suitable form to be used in our application, using
the supermatrix formula. In appendix D, we review the heat kernel expansion
techniques which are used to sum up the eigenvalues of the differential
operators. In appendix E, we show the explicit derivations of the beta
functions in our system.

##  2 Asymptotic safety

In this section we explain the basic idea of the asymptotic safety. We start
from a system described by an effective action

$\displaystyle\Gamma_{\Lambda}=\int\text{d}^{D}x\displaystyle\sum_{i}^{\infty}\frac{g_{i,\Lambda}}{\Lambda^{D_{\mathcal{O}_{i}}-D}}\mathcal{O}_{i},$
(2)

where $g_{i,\Lambda}$ are the dimensionless coupling constants,
$\mathcal{O}_{i}$ are operator bases, and $D_{\mathcal{O}_{i}}$ is the
dimension of $\mathcal{O}_{i}$. Let us write the RGE for the coupling constant
$g_{i,\Lambda}$

$\displaystyle-\Lambda\frac{\partial
g_{i,\Lambda}}{\partial\Lambda}=\beta_{i}\\!\left(g_{\Lambda}\right),$ (3)

where $\beta_{i}\\!\left(g_{\Lambda}\right)$ is the abbreviation for
$\beta_{i}\\!\left(g_{1,\Lambda},\,g_{2,\Lambda},\,\dots\right)$. The fixed
point $g^{*}$ is given by the solution to the vanishing beta functions:

$\displaystyle\beta_{i}\\!\left(g^{*}\right)=0.$ (4)

In many cases, there exists the trivial (Gaussian) fixed point: $g_{i}^{*}=0$
for all $i$.

Here we consider the case that the coupled equation (4) has a non-trivial
fixed point with $g_{j}^{*}\neq 0$ for some $j$ and that this is the UV fixed
point, namely, there exists a relevant direction flowing out of this point.
The resultant RG flows, given by Eq. (3), are as schematically shown in Fig.
1. As said in Introduction, the IR critical surface separates the theory space
(the space of coupling constants) into two phases, and is spanned by an
infinite number of irrelevant operators. On the other hand, the UV critical
surface (the green renormalized trajectories in Fig. 1) controls the IR
physics. That is, an arbitrary RG flow from the neighborhood of UV fixed point
approaches this hypersurface at IR scales. In other words, when we fix the
physics at IR scales and take the continuum limit $\Lambda\to\infty$, the
theory on the renormalized trajectory approaches the UV non-trivial fixed
point and dose not diverge from it.

To conclude, such an RG flow can be a candidate of the UV complete theory.
Furthermore, if the dimension of the UV critical surface is finite, the theory
is renormalizable: The finite number of parameters spanning the UV critical
surface determine all other parameters.555The renormalizability in low energy
region is guaranteed by an existence of the stable hypersurface with finite
dimention. It is known as the Polchinski theorem; see [116, 117, 118] for the
scalar theory and [119] for QED. Note that a perturbative expansion can be
done only at the vicinity of the trivial fixed point $g_{i,\Lambda}^{*}=0$ and
that we need a non-perturbative method to analyze the whole structure the RG
flows to find out the fixed points.

To see the renormalizability of the theory, we evaluate the critical exponents
$\theta_{i}$ of the coupling constants by linearizing the RGEs (3) around the
fixed point $g_{i,\Lambda}^{*}$:

$\displaystyle
g_{i,\Lambda}=g_{i,\Lambda}^{*}+\displaystyle\sum_{j=0}^{\infty}\zeta_{ij}\left(\frac{\Lambda_{0}}{\Lambda}\right)^{\theta_{j}},$
(5)

where $\Lambda_{0}$ is a UV cutoff scale.666 An explicit derivation is shown
in Sec. 5.  The RG flow going away from the fixed point has a critical
exponent with positive real part,777 The imaginary part of the critical
exponent corresponds to the mixing with other couplings when flowing out of
the UV fixed point.  and is on the UV critical surface. Therefore, we can
examine whether the UV critical surface is finite dimensional or not, by
investigating the number of positive critical exponents. We will investigate
the fixed point structure and the critical exponent of our theory in Sec. 5.

Let us illustrate the situation with the case where the UV critical surface is
spanned by a single operator $\mathcal{O}_{1}$. The dimensionful parameter
$G_{1,\Lambda}$ reads

$\displaystyle
g_{1,\Lambda}^{*}=\Lambda^{D_{\mathcal{O}_{1}}-D}G_{1,\Lambda}.$ (6)

When $D_{\mathcal{O}_{1}}-D>0$, the dimensionful parameter $G_{1,\Lambda}$ at
vicinity of the UV fixed point goes to zero in the UV limit
$\Lambda\to\infty$. That is, the theory becomes asymptotically free. When
$D_{\mathcal{O}_{1}}=D$, the coupling constant $G_{1,\Lambda}$ is
dimensionless and the theory becomes asymptotically non-free.888 If the fixed
point $g_{1,\Lambda}^{*}$ is a trivial UV fixed point $g_{1,\Lambda}^{*}=0$
with $D_{\mathcal{O}_{1}}=D$, the coupling constant $G_{1,\Lambda}$ in the UV
limit, and the theory becomes asymptotically free. This is the case for the
quantum chromodynamics.  To summarize, the asymptotic safety is a
generalization of the asymptotic freedom. We have
$\zeta_{1j}=g_{1,\Lambda}\delta_{1j}$ with a positive critical exponent $\rm
Re(\theta_{1})>0$, while the others are negative, and $g_{1,\Lambda}$ becomes
a single physical free parameter of the theory.

We comment on the dimension of an operator and its coupling constant. The beta
function of $g_{1,\Lambda}$ is typically given as999 The FRG is one-loop exact
and the term of $\mathcal{O}\\!\left(g_{1,\Lambda}^{3}\right)$ does not appear
in general.

$\displaystyle\beta_{g_{1,\Lambda}}=-\left(D_{\mathcal{O}_{1}}-D\right)g_{1,\Lambda}+Lg_{1,\Lambda}^{2},$
(7)

where $L$ is a loop factor and we have ignored the contributions from other
couplings. Note that the anomalous dimension from the field renormalization is
ignored throughout this paper as we are taking the LPA. Around the trivial
fixed point $g_{1,\Lambda}^{*}=0$, the first term (the so-called canonical
scaling term) in Eq. (7) becomes dominant. The coefficient
$-\left(D_{\mathcal{O}_{1}}-D\right)$ is the dimension of $g_{1,\Lambda}$, and
the dimension of the operator is simply $D_{\mathcal{O}_{1}}$. On the other
hand, at the non-trivial fixed point, we get
$g_{1,\Lambda}^{*}=\left(D_{\mathcal{O}_{1}}-D\right)/L$, and the beta
function is rewritten as

$\displaystyle\beta_{g_{1,\Lambda}}=\left(D_{\mathcal{O}_{1}}-D\right)\left(g_{1,\Lambda}-g_{1,\Lambda}^{*}\right)+L\left(g_{1,\Lambda}-g_{1,\Lambda}^{*}\right)^{2}.$
(8)

The dimension of operator is effectively changed to
$D-\left(D_{\mathcal{O}_{1}}-D\right)=2D-D_{\mathcal{O}_{1}}$.

Let us consider the case of the gravity in $D=4$. The operator
${\mathcal{O}_{1}}=R$ and its coupling constant $g_{1,\Lambda}=1/16\pi G$ have
the canonical dimension $D_{\mathcal{O}_{1}}=2$ and
$-\left(D_{\mathcal{O}_{1}}-D\right)=2$, respectively. At the non-trivial
fixed point, the effective dimension of the operator becomes
$2D-D_{\mathcal{O}_{1}}=6$, and that of the gravitational coupling constant
$g_{1,\Lambda}$ has been changed from the canonical dimension
$-\left(D_{\mathcal{O}_{1}}-D\right)=2$ to the value
$D_{\mathcal{O}_{1}}-D=-2$. The critical exponent $D_{\mathcal{O}_{1}}-D$ in
Eq. (8) is physically the effective dimension of the coupling around the non-
trivial fixed point.

There are attempts to read off the number of effective degrees of freedom
(namely the spectral dimension) from the RG flows of the theory, in order to
test whether the asymptotically safe gravity can be achieved or not: Such
attempts have been made in Refs. [120, 121, 122, 69, 123] using the FRG and in
Refs. [124, 125] using the lattice simulation; see also Refs. [126, 127, 128,
129, 130] for related studies.

##  3 Non-minimal Higgs-Yukawa model

###  3.1 The model

As a toy model for the Higgs inflation scenario under the asymptotically safe
gravity, we study a Higgs-Yukawa model with a real scalar field
$\widehat{\phi}$ and with $N_{\text{f}}$-flavors of Dirac fermions
$\widehat{\psi}$, where its flavor index is suppressed. We write the metric
$\widehat{g}_{\mu\nu}$ and the volume element $\sqrt{\widehat{g}}$. We
decompose the integration variables $\widehat{g}_{\mu\nu}$, $\widehat{\phi}$
and $\widehat{\psi}$ in the functional integral over all field configurations
according to

$\displaystyle\widehat{g}_{\mu\nu}$ $\displaystyle=g_{\mu\nu}+h_{\mu\nu},$
$\displaystyle\widehat{\phi}$ $\displaystyle=\phi+\varphi,$
$\displaystyle\widehat{\psi}$ $\displaystyle=\psi+\chi,$ (9)

where $g_{\mu\nu}$, $\phi$ and $\psi$ are fixed background fields so that the
integration over $\widehat{g}_{\mu\nu}$, $\widehat{\phi}$ and $\widehat{\psi}$
may be replaced by an integration over $h_{\mu\nu}$, $\varphi$ and $\chi$,
respectively.

We write the truncated effective action in Euclidean space:

$\displaystyle\Gamma_{\Lambda}[g_{\mu\nu},\phi,\psi;\,h_{\mu\nu},\varphi,\chi]$
$\displaystyle=\int\text{d}^{4}x\sqrt{\widehat{g}}\Bigg{\\{}V_{\Lambda}\\!\left(\widehat{\phi}^{2}\right)-F_{\Lambda}\\!\left(\widehat{\phi}^{2}\right)\widehat{R}+\frac{1}{2}\widehat{g}^{\mu\nu}\,\partial_{\mu}{\widehat{\phi}}\,\partial_{\nu}\widehat{\phi}+\overline{\widehat{\psi}}\widehat{{\ooalign{\hfil/\hfil\crcr$D$}}}\widehat{\psi}+y_{\Lambda}\widehat{\phi}\overline{\widehat{\psi}}\widehat{\psi}\Bigg{\\}}$
(12) $\displaystyle\quad+S_{\rm GF}+S_{\rm gh},$ (13)

where $\partial_{\mu}$ and $\widehat{{\ooalign{\hfil/\hfil\crcr$D$}}}$ are the
general covariant derivatives on the scalar fields and on spinor fields that
includes the spin connection; $S_{\text{GF}}$ and $S_{\text{gh}}$ are the
gauge fixing and ghost terms, respectively, shown below; and the widehat
symbol $\widehat{\mbox{ }}$ denotes that the corresponding quantity is made of
the metric $\widehat{g}_{\mu\nu}$ and veirbein $\widehat{e}_{\mu}^{a}$. We
have imposed the $Z_{2}$ symmetry: $\widehat{\phi}\to-\widehat{\phi}$ and
$\widehat{\psi}\to\gamma_{5}\widehat{\psi}$.101010 A background $\phi\neq 0$
breaks this $Z_{2}$ symmetry. In this paper, we restrict our attention to the
case $\phi=0$.

We expand the scalar potential and the non-minimal coupling of
$\widehat{\phi}$ to the gravity:

$\displaystyle V_{\Lambda}\\!\left(\widehat{\phi}^{2}\right)$
$\displaystyle=\hat{\lambda}_{0}\\!\left(\Lambda\right)+\hat{\lambda}_{2}\\!\left(\Lambda\right)\,\widehat{\phi}^{2}+\hat{\lambda}_{4}\\!\left(\Lambda\right)\,\widehat{\phi}^{4}+\cdots,$
(14) $\displaystyle F_{\Lambda}\\!\left(\widehat{\phi}^{2}\right)$
$\displaystyle=\hat{\xi}_{0}\\!\left(\Lambda\right)+\hat{\xi}_{2}\\!\left(\Lambda\right)\,\widehat{\phi}^{2}+\hat{\xi}_{4}\\!\left(\Lambda\right)\,\widehat{\phi}^{4}+\cdots.$
(15)

In more conventional language, $\hat{\lambda}_{0}$ is the cosmological
constant; $\hat{\lambda}_{2}=m^{2}/2$ gives the mass parameter of the scalar
field; and $\hat{\xi}_{0}=1/16\pi G$ the Newton constant. The non-minimal
coupling $\hat{\xi}_{2}$ plays a crucial role in the Higgs inflation scenario
[92]; see also Ref. [104].

We employ the following gauge-fixing and ghost actions for the diffeomorphisms
[23, 49, 51, 52]

$\displaystyle S_{\rm GF}$
$\displaystyle=\frac{1}{2\alpha}\int\text{d}^{D}x\sqrt{g}\,F\\!\left(\phi^{2}\right){g}^{\mu\nu}\Sigma_{\mu}\Sigma_{\nu},$
(16) $\displaystyle S_{\rm gh}$
$\displaystyle=-\int\text{d}^{D}x\sqrt{g}\,\bar{C}_{\mu}\left[g^{\mu\rho}{\partial}^{2}+\frac{1-\beta}{2}{\partial}^{\mu}{\partial}^{\rho}+{R}^{\mu\rho}\right]C_{\rho},$
(17)

where $C_{\mu}$ and $\bar{C}_{\mu}$ are the ghost and anti-ghost fields for
the diffeomorphisms, respectively; $\alpha$ and $\beta$ are gauge parameters;
and

$\displaystyle\Sigma_{\mu}$
$\displaystyle:=\partial^{\nu}h_{\nu\mu}-\frac{\beta+1}{D}{\partial}_{\mu}h,$
(18)

with $h$ being the trace part of the fluctuation $g^{\mu\nu}h_{\mu\nu}$.
Throughout this paper, the expression without the widehat symbol
$\widehat{\mbox{ }}$ indicates that the indices are raised and lowered by the
background metric $g_{\mu\nu}$, and expressions such as $R$ and  / $D$ are
written in terms of $g_{\mu\nu}$ and the background vierbein $e_{\mu}^{a}$.

###  3.2 Two-point functions

We collectively write the background fields
$\Phi:=\left(g_{\mu\nu},\phi,\psi\right)$ and the fluctuations
$\Upsilon:=\left(h_{\mu\nu},\varphi,\chi,C_{\mu},\bar{C}_{\mu}\right)$.111111
The ghost $C_{\mu}$ and anti-ghost $\bar{C}_{\mu}$ are treated as fluctuations
only.  The effective action is written as $\Gamma_{\Lambda}[\Phi;\Upsilon]$,
which is expanded as

$\displaystyle\Gamma_{\Lambda}[\Phi;\Upsilon]=\Gamma_{\Lambda}[\Phi]+\Gamma_{\Lambda}^{(1)}[\Phi;\Upsilon]+\Gamma_{\Lambda}^{(2)}[\Phi;\Upsilon]+\mathcal{O}\\!\left(\Upsilon^{3}\right),$
(19)

where $\Gamma_{\Lambda}^{(n)}[\Phi;\Upsilon]$ contains the terms of order
$\Upsilon^{n}$.

To derive the beta functions for the Higgs-Yukawa model, we need to evaluate
the $\Gamma_{\Lambda}^{(2)}$ terms. After some computations, we obtain

$\displaystyle\Gamma_{\Lambda}^{(2)}[\Phi;\Upsilon]$
$\displaystyle=\frac{1}{2}\int\text{d}^{4}x\sqrt{g}\bigg{[}-\frac{1}{2}F\\!\left(\phi^{2}\right)h^{\mu\nu}\partial^{2}h_{\mu\nu}+\frac{1}{2}F\\!\left(\phi^{2}\right)h\partial^{2}h-F\\!\left(\phi^{2}\right)h\partial_{\mu}\partial_{\nu}h^{\mu\nu}+F\\!\left(\phi^{2}\right)h^{\mu\nu}\partial_{\mu}\partial_{\rho}h^{\rho}_{~{}\nu}$
$\displaystyle\phantom{=\frac{1}{2}\int\text{d}^{4}x\sqrt{g}\bigg{[}}+\bigg{(}\frac{1}{4}h^{2}-\frac{1}{2}h_{\mu\nu}h^{\mu\nu}\bigg{)}\left(V\\!\left(\phi^{2}\right)+y\phi\overline{\psi}\psi-F\\!\left(\phi^{2}\right)R\right)$
$\displaystyle\phantom{=\frac{1}{2}\int\text{d}^{4}x\sqrt{g}\bigg{[}}+F\\!\left(\phi^{2}\right)hh^{\mu\nu}R_{\mu\nu}-F\\!\left(\phi^{2}\right)h_{\rho}^{~{}\nu}h^{\mu\rho}R_{\mu\nu}-F\\!\left(\phi^{2}\right)h^{\mu\nu}R_{\rho\mu\sigma\nu}h^{\rho\sigma}$
$\displaystyle\phantom{=\frac{1}{2}\int\text{d}^{4}x\sqrt{g}\bigg{[}}-\frac{1}{16}h^{~{}\mu}_{\rho}\partial_{\nu}h_{\sigma\mu}\overline{\psi}\gamma^{\nu}[\gamma^{\rho},\gamma^{\sigma}]\psi\bigg{]}$
$\displaystyle\quad+\int\text{d}^{4}x\sqrt{g}\,\varphi\bigg{[}-2\phi
F^{\prime}\\!\left(\phi^{2}\right)\left\\{\partial_{\mu}\partial_{\nu}-\partial^{2}g_{\mu\nu}\right\\}h^{\mu\nu}$
$\displaystyle\phantom{\quad+\int\text{d}^{4}x\sqrt{g}\,\varphi\bigg{[}}+h\left\\{\phi
V^{\prime}\\!\left(\phi^{2}\right)+\frac{1}{2}y\overline{\psi}\psi-\phi
F^{\prime}\\!\left(\phi^{2}\right)R\right\\}+h^{\mu\nu}\left\\{2\phi
F^{\prime}\\!\left(\phi^{2}\right)+R_{\mu\nu}\right\\}\bigg{]}$
$\displaystyle\quad+\int\text{d}^{4}x\sqrt{g}\,h\bigg{[}\frac{1}{2}y\phi\left(\overline{\psi}\chi+\overline{\chi}\psi\right)\bigg{]}$
$\displaystyle\quad+\frac{1}{2}\int\text{d}^{4}x\sqrt{g}\,\varphi\bigg{[}\left\\{-\partial^{2}+2V^{\prime}\\!\left(\phi^{2}\right)+4\phi^{2}V^{\prime\prime}\\!\left(\phi^{2}\right)\right\\}-R\left\\{2F^{\prime}\\!\left(\phi^{2}\right)+4\phi^{2}F^{\prime\prime}\\!\left(\phi^{2}\right)\right\\}\bigg{]}\varphi$
$\displaystyle\quad+\int\text{d}^{4}x\sqrt{g}\bigg{[}\frac{1}{4}\left(-\partial_{\mu}h+\partial_{\nu}h^{\nu}_{~{}\mu}\right)\left(\overline{\psi}\gamma^{\mu}\chi-\overline{\chi}\gamma^{\mu}\psi\right)\bigg{]}$
$\displaystyle\quad+\int\text{d}^{4}x\sqrt{g}\,\varphi\bigg{[}y\left(\overline{\psi}\chi+\overline{\chi}\psi\right)\bigg{]}+\int\text{d}^{4}x\sqrt{g}\,\overline{\chi}\bigg{[}{\ooalign{\hfil/\hfil\crcr$\partial$}}+y\phi\bigg{]}\chi+S_{\rm
GF}+S_{\rm gh},$ (22)

where $S_{\rm GF}$ and $S_{\rm gh}$ are given in Eqs. (16) and (17),
respectively,121212 They are already bilinear terms of the fluctuations.  and
the prime symbol ′ denotes a derivative with respect to $\phi^{2}$ so that

$\displaystyle V_{\phi}$ $\displaystyle=2\phi V^{\prime},$ $\displaystyle
V_{\phi\phi}$ $\displaystyle=2V^{\prime}+4\phi^{2}V^{\prime\prime},$
$\displaystyle F_{\phi}$ $\displaystyle=2\phi F^{\prime},$ $\displaystyle
F_{\phi\phi}$ $\displaystyle=2F^{\prime}+4\phi^{2}F^{\prime\prime}.$ (23)

Inserting (24) into $h_{\mu\nu}$ of (22), we get the two-point functions for
each field. We write down their explicit forms below.

###  3.3 York decomposition

We decompose the graviton fluctuation as [131]

$\displaystyle
h_{\mu\nu}=h_{\mu\nu}^{\perp}+\partial_{\mu}\tilde{\xi}_{\nu}+\partial_{\nu}\tilde{\xi}_{\mu}+\left(\partial_{\mu}\partial_{\nu}-\frac{1}{D}g_{\mu\nu}\partial^{2}\right)\tilde{\sigma}+\frac{1}{D}g_{\mu\nu}h,$
(24)

where $\partial^{2}:=g^{\mu\nu}\partial_{\mu}\partial_{\nu}$;
$h^{\perp}_{\mu\nu}$ is the transverse and traceless tensor field with spin 2;
$\tilde{\xi}_{\mu}$ is the transverse vector field with spin 1; and
$\tilde{\sigma}$ and $h:={g}^{\mu\nu}h_{\mu\nu}$ are the scalar fields with
spin 0. These fields satisfy the following conditions:
${g}^{\mu\nu}h^{\perp}_{\mu\nu}=0$, $\partial^{\nu}h^{\perp}_{\mu\nu}=0$, and
$\partial^{\mu}\tilde{\xi}_{\mu}=0$.

We decompose the ghosts into the transverse and scalar components:

$\displaystyle C_{\mu}$
$\displaystyle=C_{\mu}^{\perp}+\partial_{\mu}\tilde{C},$
$\displaystyle\tilde{\bar{C}}_{\mu}$
$\displaystyle=\tilde{\bar{C}}_{\mu}^{\perp}+\partial_{\mu}{\bar{C}},$ (25)

where $\tilde{C}$, $\tilde{\bar{C}}$ are spin-0 scalar fields and
$C^{\perp}_{\mu}$, $\bar{C}_{\mu}^{\perp}$ are spin-1 transverse vector fields
that satisfy
$\partial^{\mu}C^{\perp}_{\mu}=\partial^{\mu}{\bar{C}}^{\perp}_{\mu}=0$.

In order to absorb the Jacobean of the path integral measure from the above
decompositions, we redefine several components of the fluctuations as follows:

$\displaystyle\xi_{\mu}$
$\displaystyle=\sqrt{-\partial^{2}-\frac{R}{D}}\,\tilde{\xi}_{\mu},$
$\displaystyle{\sigma}$
$\displaystyle=\sqrt{-\partial^{2}-\frac{R}{D-1}}\sqrt{-\partial^{2}}\,\tilde{\sigma},$
$\displaystyle{C}$ $\displaystyle=\sqrt{-\partial^{2}}\,\tilde{C},$
$\displaystyle{\bar{C}}$ $\displaystyle=\tilde{\bar{C}}\sqrt{-\partial^{2}};$
(26)

see e.g. Ref. [70].

To summarize, the degrees of freedom in our system are the spin two
$h^{\perp}_{\mu\nu}$, the spin one $\xi_{\mu}$, $C^{\perp}_{\mu}$,
$\bar{C}^{\perp}_{\mu}$ the spin half $\psi$, and the spin zero $\sigma$, $h$,
$\phi$, $C$, $\bar{C}$.

###  3.4 Explicit form of two-point functions

For bosonic fields, we obtain

$\displaystyle\Gamma_{\text{BB}}$
$\displaystyle=\begin{bmatrix}\Gamma_{h^{\perp}_{\mu\nu}h^{\perp}_{\rho\sigma}}&0&0\\\
0&\Gamma_{\xi_{\mu}\xi_{\nu}}&0\\\ 0&0&\Gamma_{\text{SS}}\end{bmatrix},$ (27)

with

$\displaystyle\Gamma_{h^{\perp}_{\mu\nu}h^{\perp}_{\rho\sigma}}$
$\displaystyle=\left(g^{\mu\rho}g^{\nu\sigma}\right)_{\text{sym}}\left[{F\over
2}\left(p^{2}+{2R\over 3}\right)-{V+Y\over 2}\right]+(\text{spin connection
term}),$ (28) $\displaystyle\Gamma_{\xi_{\mu}\xi_{\nu}}$
$\displaystyle=g^{\mu\nu}\left[{F\over\alpha}\left(p^{2}+{2\alpha-1\over
4}R\right)-V-Y\right]+(\text{spin connection term}),$ (29)
$\displaystyle\Gamma_{\text{SS}}$
$\displaystyle=\bordermatrix{&\sigma&h&\varphi\cr\sigma&{\frac{3F}{16}\left(\frac{3-\alpha}{\alpha}p^{2}+\frac{\alpha-1}{\alpha}R\right)-\frac{3\left(V+Y\right)}{8}}&{\frac{3F}{16}\frac{\beta-\alpha}{\alpha}\sqrt{p^{2}-\frac{R}{3}}\sqrt{p^{2}}}&{-\frac{3F_{\varphi}}{4}\sqrt{p^{2}-\frac{R}{3}}\sqrt{p^{2}}}\cr
h&{\frac{3F}{16}\frac{\beta-\alpha}{\alpha}\sqrt{p^{2}-\frac{R}{3}}\sqrt{p^{2}}}&{-\frac{F}{16}\frac{3\alpha-\beta^{2}}{\alpha}p^{2}+\frac{V+Y}{8}}&{-\frac{3F_{\varphi}}{4}\left(p^{2}+\frac{R}{3}\right)+{V_{\varphi}\over
2}+{Y_{\varphi}\over
2}}\cr\varphi&{-\frac{3F_{\varphi}}{4}\sqrt{p^{2}-\frac{R}{3}}\sqrt{p^{2}}}&{-\frac{3F_{\varphi}}{4}\left(p^{2}+\frac{R}{3}\right)+{V_{\varphi}\over
2}+{Y_{\varphi}\over 2}}&{p^{2}+V_{\varphi\varphi}-RF_{\varphi\varphi}}},$
(30)

where $\left(\cdots\right)_{\text{sym}}$ indicates that the indices inside
parentheses are properly symmetrized;131313 Explicitly,
$\left(g^{\mu\rho}g^{\nu\sigma}\right)_{\text{sym}}={1\over
4}\left(g^{\mu\rho}g^{\nu\sigma}+g^{\nu\rho}g^{\mu\sigma}+g^{\nu\sigma}g^{\mu\rho}+g^{\mu\sigma}g^{\nu\rho}\right)$.
we write a minus of the d’Alembertian in the de Sitter space
$p^{2}:=-\partial^{2}$; the over-left-arrow
$\overleftarrow{\phantom{\partial}}$ denotes that the differential operator
acts on the left; and $Y:=y\phi\overline{\psi}\psi$ and
$Y_{\phi}:=y\overline{\psi}\psi$ are the Yukawa interaction and its derivative
with respect to $\phi$, respectively. The “spin connection term” in Eqs. (28)
and (29) is coming from the derivatives of the spin connection with respect to
the metric, which only affect operators involving higher powers of $\psi$ and
$\overline{\psi}$; such operators are truncated in our effective action.

Other parts are given by

$\displaystyle\Gamma_{\text{BF}}$
$\displaystyle=\bordermatrix{&\chi&\smash{\overline{\chi}^{\text{T}}}\cr
h^{\perp}_{\mu\nu}&0&0\cr\xi_{\mu}&-\frac{1}{4}\sqrt{\overleftarrow{p}^{2}}\left(\overline{\psi}\gamma^{\mu}\right)&-\frac{1}{4}\sqrt{\overleftarrow{p}^{2}}\left(\gamma^{\mu}\psi\right)^{\text{T}}\cr\sigma&-\frac{3}{16}\left(\overleftarrow{\partial_{\mu}}\right)\left(\overline{\psi}\gamma^{\mu}\right)&-\frac{3}{16}\left(\overleftarrow{\partial_{\mu}}\right)\left(\gamma^{\mu}\psi\right)^{\text{T}}\cr
h&\frac{y}{2}\phi{\overline{\psi}}-\frac{3}{16}\left(\overleftarrow{\partial_{\mu}}\right)\left(\overline{\psi}\gamma^{\mu}\right)&-\frac{y}{2}\phi\psi^{\text{T}}-\frac{3}{16}\left(\overleftarrow{\partial_{\mu}}\right)\left(\gamma^{\mu}\psi\right)^{\text{T}}\cr\varphi&y\overline{\psi}&-y\psi^{\text{T}}},$
(31) $\displaystyle\Gamma_{\text{FB}}$
$\displaystyle=\bordermatrix{&h^{\perp}_{\mu\nu}&\xi_{\mu}&\sigma&h&\varphi\cr\chi^{\text{T}}&0&\frac{1}{4}\left({\overline{\psi}}\gamma^{\mu}\right)^{\text{T}}\sqrt{p^{2}}&\frac{3}{16}\left({\overline{\psi}}\gamma^{\mu}\right)^{\text{T}}\partial_{\mu}&-\frac{y}{2}\phi{\overline{\psi}}^{\text{T}}+\frac{3}{16}\left({\overline{\psi}}\gamma^{\mu}\right)^{\text{T}}\partial_{\mu}&-y{\overline{\psi}}^{\text{T}}\cr\overline{\chi}&0&\frac{1}{4}\left(\gamma^{\mu}\psi\right)\sqrt{p^{2}}&\frac{3}{16}\left(\gamma^{\mu}\psi\right)\partial_{\mu}&\frac{y}{2}\phi\psi+\frac{3}{16}\left(\gamma^{\mu}\psi\right)\partial_{\mu}&y\psi},$
(32) $\displaystyle\Gamma_{\text{FF}}$
$\displaystyle=\begin{bmatrix}\Gamma^{\text{phys}}_{\text{FF}}&0\\\
0&\Gamma^{\text{ghost}}_{\text{FF}}\end{bmatrix},$ (33)

where ${}^{\text{T}}$ denotes the transposition of the spinor indices and

$\displaystyle\Gamma^{\text{phys}}_{\text{FF}}$
$\displaystyle=\bordermatrix{&\chi&\smash{\overline{\chi}^{\text{T}}}\cr\chi^{\text{T}}&0&-\left(\overleftarrow{{\ooalign{\hfil/\hfil\crcr$D$}}}^{\text{T}}+y\phi\right)\cr\overline{\chi}&{{\ooalign{\hfil/\hfil\crcr$D$}}}+y\phi&0},$
(34) $\displaystyle\Gamma^{\text{ghost}}_{\text{FF}}$
$\displaystyle=\bordermatrix{&\smash{C^{\perp}_{\nu}}&\smash{\bar{C}^{\perp}_{\nu}}&C&\bar{C}\cr
C^{\perp}_{\mu}&0&-g^{\mu\nu}\left(p^{2}-{R\over
4}\right)&0&0\cr\bar{C}^{\perp}_{\mu}&g^{\mu\nu}\left(p^{2}-{R\over
4}\right)&0&0&0\cr C&0&0&0&-\left[\left(2-{1+\beta\over 2}\right)p^{2}-{R\over
2}\right]\cr\bar{C}&0&0&\left(2-{1+\beta\over 2}\right)p^{2}-{R\over 2}&0}.$
(35)

###  3.5 Explicit form of cutoff functions

We write down the cutoff function for the bosons in the non-minimal Higgs-
Yukawa model:

$\displaystyle\mathcal{R}_{\text{BB}}$
$\displaystyle=\begin{bmatrix}\mathcal{R}_{h^{\perp}_{\mu\nu}h^{\perp}_{\rho\sigma}}&0&0\\\
0&\mathcal{R}_{\xi_{\mu}\xi_{\nu}}&0\\\
0&0&\mathcal{R}_{\text{SS}}\end{bmatrix},$ (36)

with

$\displaystyle\mathcal{R}_{h^{\perp}_{\mu\nu}h^{\perp}_{\rho\sigma}}$
$\displaystyle=\left(g^{\mu\rho}g^{\nu\sigma}\right)_{\text{sym}}{F\over
2}R_{\Lambda}\\!\left(p^{2}\right),$ (37)
$\displaystyle\mathcal{R}_{\xi_{\mu}\xi_{\nu}}$
$\displaystyle=g^{\mu\nu}{F\over\alpha}R_{\Lambda}\\!\left(p^{2}\right),$ (38)
$\displaystyle\mathcal{R}_{\text{SS}}$
$\displaystyle=\bordermatrix{&\sigma&h&\varphi\cr\sigma&\frac{3F}{16}\frac{3-\alpha}{\alpha}R_{\Lambda}\\!\left(p^{2}\right)&\frac{3F}{16}\frac{\beta-\alpha}{\alpha}K_{\Lambda}\\!\left(p^{2}\right)&-\frac{3F_{\phi}}{4}K_{\Lambda}\\!\left(p^{2}\right)\cr
h&\frac{3F}{16}\frac{\beta-\alpha}{\alpha}K_{\Lambda}\\!\left(p^{2}\right)&-\frac{F}{16}\frac{3\alpha-\beta^{2}}{\alpha}R_{\Lambda}\\!\left(p^{2}\right)&-\frac{3F_{\phi}}{4}R_{\Lambda}\\!\left(p^{2}\right)\cr\varphi&-\frac{3F_{\phi}}{4}K_{\Lambda}\\!\left(p^{2}\right)&-\frac{3F_{\phi}}{4}R_{\Lambda}\\!\left(p^{2}\right)&R_{\Lambda}\\!\left(p^{2}\right)},$
(39)

where we employ the optimized cutoff function [132]:

$\displaystyle R_{\Lambda}\\!\left(p^{2}\right)$
$\displaystyle:=\left(\Lambda^{2}-p^{2}\right)\theta\left(\Lambda^{2}-p^{2}\right),$
(40) $\displaystyle K_{\Lambda}\\!\left(p^{2}\right)$
$\displaystyle:=\sqrt{p^{2}+R_{\Lambda}\\!\left(p^{2}\right)-{R\over
3}}\,\sqrt{p^{2}+R_{\Lambda}\\!\left(p^{2}\right)}-\sqrt{p^{2}-{R\over
3}}\sqrt{p^{2}}.$ (41)

Note that

$\displaystyle{\partial R_{\Lambda}\\!\left(p^{2}\right)\over\partial\Lambda}$
$\displaystyle=2\Lambda\,\theta\\!\left(\Lambda^{2}-p^{2}\right),$ (42)
$\displaystyle{\partial K_{\Lambda}\\!\left(p^{2}\right)\over\partial\Lambda}$
$\displaystyle={\Lambda\left(2\Lambda^{2}-{R\over
3}\right)\theta\\!\left(\Lambda^{2}-p^{2}\right)\over\sqrt{p^{2}+\left(\Lambda^{2}-p^{2}\right)\theta\\!\left(\Lambda^{2}-p^{2}\right)}\sqrt{p^{2}-{R\over
3}+\left(\Lambda^{2}-p^{2}\right)\theta\\!\left(\Lambda^{2}-p^{2}\right)}}.$
(43)

More explicitly,

$\displaystyle R_{\Lambda}\\!\left(p^{2}\right)$
$\displaystyle=\begin{cases}\Lambda^{2}-p^{2},&\\\ 0,&\end{cases}$
$\displaystyle{\partial R_{\Lambda}\\!\left(p^{2}\right)\over\partial\Lambda}$
$\displaystyle=\begin{cases}2\Lambda&\quad\text{for $p^{2}<\Lambda^{2}$,}\\\
0&\quad\text{for $p^{2}\geq\Lambda^{2}$,}\end{cases}$ (44) $\displaystyle
K_{\Lambda}\\!\left(p^{2}\right)$
$\displaystyle=\begin{cases}\sqrt{\Lambda^{2}-{R\over
3}}\,\sqrt{\Lambda^{2}}-\sqrt{p^{2}-{R\over 3}}\sqrt{p^{2}},&\\\
0,&\end{cases}$ $\displaystyle{\partial
K_{\Lambda}\\!\left(p^{2}\right)\over\partial\Lambda}$
$\displaystyle=\begin{cases}{2\Lambda^{2}-{R\over
3}\over\sqrt{\Lambda^{2}-{R\over 3}}}&\quad\text{for $p^{2}<\Lambda^{2}$,}\\\
0&\quad\text{for $p^{2}\geq\Lambda^{2}$.}\end{cases}$ (45)

The cutoff function for fermions is given by

$\displaystyle\mathcal{R}_{\text{FF}}$
$\displaystyle=\begin{bmatrix}\mathcal{R}_{\text{physical}}&0\\\
0&\mathcal{R}_{\text{ghost}}\end{bmatrix},$ (46)

where

$\displaystyle\mathcal{R}_{\text{physical}}$
$\displaystyle=\bordermatrix{&\chi&\smash{\overline{\chi}^{\text{T}}}\cr\chi^{\text{T}}&0&-\overleftarrow{{\ooalign{\hfil/\hfil\crcr$D$}}}^{\text{T}}\left(\sqrt{1+{R_{\Lambda}\\!\left(p^{2}+{R\over
4}\right)\over p^{2}+{R\over
4}}}-1\right)\cr\overline{\chi}&\left(\sqrt{p^{2}+{R\over
4}+R_{\Lambda}\\!\left(p^{2}+{R\over 4}\right)\over p^{2}+{R\over
4}}-1\right){\ooalign{\hfil/\hfil\crcr$D$}}&0\cr},$ (47)
$\displaystyle\mathcal{R}_{\text{ghost}}$
$\displaystyle=\bordermatrix{&\smash{C^{\perp}_{\nu}}&\smash{\bar{C}^{\perp}_{\nu}}&C&\bar{C}\cr
C^{\perp}_{\mu}&0&-g^{\mu\nu}R_{\Lambda}\\!\left(p^{2}\right)&0&0\cr\bar{C}^{\perp}_{\mu}&g^{\mu\nu}R_{\Lambda}\\!\left(p^{2}\right)&0&0&0\cr
C&0&0&0&-\left(2-{1+\beta\over
2}\right)R_{\Lambda}\\!\left(p^{2}\right)\cr\bar{C}&0&0&\left(2-{1+\beta\over
2}\right)R_{\Lambda}\\!\left(p^{2}\right)&0}.$ (48)

For $\mathcal{R}_{\rm physical}$, we have employed the so-called type II
cutoff function [43] in order to give the correct sign of the fermionic
quantum corrections for the non-minimal potential $F(\phi^{2})$.

Figure 2: The propagators in the truncated effective action

We have spelled out the two-point and cutoff functions. From them, we can
construct the inverse propagators as in Fig. 2. The vertex structures included
in $\Gamma_{\text{FB}}$ and $\Gamma_{\text{BF}}$ are shown in Fig. 3.

Figure 3: The vertices with fermion in external or internal line

##  4 RG equations

###  4.1 Computational methods

Figure 4: One loop contribution to $V$ and $F$. The gray circle denotes the
mixing of scalar fields.

There are two methods to compute the beta functions for $V_{\Lambda}$,
$F_{\Lambda}$ and $Y_{\Lambda}$. One is a direct evaluation of the Wetterich
equation by algebraic matrix manipulation from the expression

$\displaystyle{\partial\over\partial\Lambda}\Gamma_{\Lambda}$
$\displaystyle={1\over
2}\operatorname{Tr}\bigg{[}{\mathcal{M}}_{\text{BB}}^{-1}{\partial\mathcal{R}_{\text{BB}}\over\partial\Lambda}\bigg{]}$
$\displaystyle\quad-{1\over
2}\operatorname{Tr}\bigg{[}\Big{(}{\mathcal{M}}_{\text{FF}}-{\mathcal{M}}_{\text{FB}}{\mathcal{M}}_{\text{BB}}^{-1}{\mathcal{M}}_{\text{BF}}\Big{)}^{-1}\left({\partial\mathcal{R}_{\text{FF}}\over\partial\Lambda}+{\mathcal{M}}_{\text{FB}}{\mathcal{M}}_{\text{BB}}^{-1}\,{\partial\mathcal{R}_{\text{BB}}\over\partial\Lambda}\,{\mathcal{M}}_{\text{BB}}^{-1}{\mathcal{M}}_{\text{BF}}\right)\bigg{]},$
(49)

where
$\mathcal{M}_{\Upsilon\Upsilon}=\Gamma_{\Upsilon\Upsilon}+\mathcal{R}_{\Upsilon\Upsilon}$
as in Eq. (124); explicit forms of the cutoff functions
$\mathcal{R}_{\Upsilon\Upsilon}$ are given in Sec. 3.5; and
$\Gamma_{\Upsilon\Upsilon}$ can be read off as the coefficients of the
quadratic terms of the fluctuations $\Upsilon$ in Eq. (22). For the detailed
derivation of Eq. (49), see Appendix C. We evaluate this expression employing
the de-Donder gauge $\alpha=0$, $\beta=1$ after taking the inverse and the
trace.

The other is a one-loop diagrammatic computation, obtained by rewriting the
Wetterich equation as

$\displaystyle\partial_{t}\Gamma_{\Lambda}$ $\displaystyle=\frac{1}{2}{\rm
Tr}\left.\frac{\partial_{t}{\mathcal{R}}_{\Lambda}}{\Gamma_{\Lambda}^{(1,1)}+{\mathcal{R}}_{\Lambda}}\right|_{h^{\perp}h^{\perp}}+\frac{1}{2}{\rm
Tr}^{\prime}\left.\frac{\partial_{t}{\mathcal{R}}_{\Lambda}}{\Gamma_{\Lambda}^{(1,1)}+{\mathcal{R}}_{\Lambda}}\right|_{\xi\xi}+\frac{1}{2}{\rm
Tr}^{\prime\prime}\left.\frac{\partial_{t}{\mathcal{R}}_{\Lambda}}{\Gamma_{\Lambda}^{(1,1)}+{\mathcal{R}}_{\Lambda}}\right|_{\rm
SS}$
$\displaystyle\quad-\left.\operatorname{Tr}\frac{\partial_{t}{\mathcal{R}}_{\Lambda}}{\Gamma_{\Lambda}^{(1,1)}+{\mathcal{R}}_{\Lambda}}\right|_{\overline{\chi}\chi}-\left.\operatorname{Tr}\frac{\partial_{t}{\mathcal{R}}_{\Lambda}}{\Gamma_{\Lambda}^{(1,1)}+{\mathcal{R}}_{\Lambda}}\right|_{{\bar{C}}^{\perp}C}-\left.\operatorname{Tr}\frac{\partial_{t}{\mathcal{R}}_{\Lambda}}{\Gamma_{\Lambda}^{(1,1)}+{\mathcal{R}}_{\Lambda}}\right|_{\bar{C}C},$
(50)

where we write

$\displaystyle\partial_{t}$
$\displaystyle:=-\Lambda{\partial\over\partial\Lambda}$ (51)

and each prime symbol ′ on the trace denotes a subtraction of a negative
eigenvalue of the differential operator from the trace.141414 Since the
negative eigenvalues arise from order $R^{2}$, we ignore it hereafter; see
appendix of Ref. [70] for details.  We see that each term in Eq. (50) can be
represented by the corresponding diagram in Fig. 4. Detailed computations are
shown in Appendix E. In Appendix D, we summarizes the values of the heat
kernel coefficients used in Appendix E.

###  4.2 Running of $V$ and $F$

As explained above, we compute the beta function for $V$ and $F$. Its
diagrammatic and algebraic derivations are shown in Appendices E and C,
respectively. The final results for the beta functions are

$\displaystyle\partial_{t}V$
$\displaystyle=\frac{\Lambda^{4}}{192\pi^{2}}\bigg{[}-6-\frac{30V}{\Psi}-\frac{6(\Lambda^{2}\Psi+24\phi^{2}\Lambda^{2}F^{\prime}\Psi^{\prime}+F\Lambda^{2}\Sigma_{1})}{\Delta}$
$\displaystyle\phantom{=\frac{\Lambda^{4}}{192\pi^{2}}\bigg{[}}+\partial_{t}F\left(\frac{4}{F}+\frac{5\Lambda^{2}}{\Psi}+\frac{\Lambda^{2}\Sigma_{1}}{\Delta}\right)+\partial_{t}F^{\prime}\frac{24\phi^{2}\Lambda^{2}\Psi^{\prime}}{\Delta}\bigg{]}+\frac{N_{\text{f}}}{8\pi^{2}}\frac{\Lambda^{6}}{\Sigma_{3}},$
(52) $\displaystyle\partial_{t}F$
$\displaystyle=-\frac{\Lambda^{2}}{2304\pi^{2}}\bigg{\\{}150+\frac{120\Lambda^{2}F(3\Lambda^{2}F-V)}{\Psi^{2}}$
$\displaystyle\phantom{=\frac{\Lambda^{2}}{2304\pi^{2}}\bigg{\\{}}-\frac{24}{\Delta}(\Lambda^{2}\Psi+24\phi^{2}\Lambda^{2}F^{\prime}\Psi^{\prime}+F\Lambda^{2}\Sigma_{1})$
$\displaystyle\phantom{=\frac{\Lambda^{2}}{2304\pi^{2}}\bigg{\\{}}-\frac{36}{\Delta^{2}}\bigg{[}-4\phi^{2}(6\Lambda^{4}F^{\prime
2}+\Psi^{\prime 2})\Delta$
$\displaystyle\phantom{=\frac{\Lambda^{2}}{2304\pi^{2}}\bigg{\\{}}\phantom{-\frac{36}{\Delta^{2}}\bigg{[}}+4\phi^{2}\Psi\Psi^{\prime}\left(7\Lambda^{2}F^{\prime}-V^{\prime}\right)\left(\Sigma_{1}-\Lambda^{2}\right)$
$\displaystyle\phantom{=\frac{\Lambda^{2}}{2304\pi^{2}}\bigg{\\{}}\phantom{-\frac{36}{\Delta^{2}}\bigg{[}}+4\phi^{2}\Sigma_{1}\left(7\Lambda^{2}F^{\prime}-V^{\prime}\right)\left(2\Psi
V^{\prime}-V\Psi^{\prime}\right)+(2\Lambda^{4}\Psi^{2}+48\Lambda^{4}F^{\prime}\phi^{2}\Psi\Psi^{\prime}-24\Lambda^{4}F\phi^{2}\Psi^{\prime
2})\Sigma_{2}\bigg{]}$
$\displaystyle\phantom{=\frac{\Lambda^{2}}{2304\pi^{2}}\bigg{\\{}}+\frac{\partial_{t}F}{F}\bigg{[}30-\frac{10\Lambda^{2}F(7\Psi+4V)}{\Psi^{2}}$
$\displaystyle\phantom{=\frac{\Lambda^{2}}{2304\pi^{2}}\bigg{\\{}}\phantom{+\frac{\partial_{t}F}{F}\bigg{[}}+\frac{6}{\Delta^{2}}\left(\Lambda^{2}F\Sigma_{1}\Delta+4\phi^{2}V^{\prime}\Psi^{\prime}\Delta-24\Lambda^{4}F\phi^{2}\Psi^{\prime
2}\Sigma_{2}-4\phi^{2}\Lambda^{2}F\Psi^{\prime}\Sigma_{1}\left(7\Lambda^{2}F^{\prime}-V^{\prime}\right)\right)\bigg{]}$
$\displaystyle\phantom{=\frac{\Lambda^{2}}{2304\pi^{2}}\bigg{\\{}}-\partial_{t}F^{\prime}\frac{24\Lambda^{2}\phi^{2}}{\Delta^{2}}\left(\left(\Lambda^{2}F^{\prime}+5V^{\prime}\right)\Delta-2\left(7F^{\prime}\Lambda^{2}-V^{\prime}\right)\Psi\Sigma_{1}-12\Lambda^{2}\Psi\Psi^{\prime}\Sigma_{2}\right)\bigg{\\}}$
$\displaystyle\quad+\frac{N_{\text{f}}}{48\pi^{2}}\frac{\Lambda^{4}}{\Sigma_{3}},$
(53)

where we employ the notations

$\displaystyle\Psi$ $\displaystyle:=F\Lambda^{2}-V,$ $\displaystyle\Sigma_{1}$
$\displaystyle:=\Lambda^{2}+2V^{\prime}+4\phi^{2}V^{\prime\prime},$
$\displaystyle\Sigma_{2}$
$\displaystyle:=2F^{\prime}+4\phi^{2}F^{\prime\prime},$
$\displaystyle\Sigma_{3}$ $\displaystyle:=\Lambda^{2}+y^{2}\phi^{2},$
$\displaystyle\Delta$ $\displaystyle:=12\phi^{2}\Psi^{\prime
2}+\Psi\Sigma_{1}.$ (54)

In the case of $N_{\text{f}}=0$, these results (52) and (53) agree with those
in Ref. [51]. On the other hand, we also reproduce the result in Ref. [53]
when we put the vanishing non-minimal coupling, i.e. $F=\hat{\xi}_{0}$ in Eq.
(55). We get the RGE for each coupling constant by expanding $V(\phi^{2})$ and
$F(\phi^{2})$ into polynomials of the squared scalar field $\phi^{2}$:

$\displaystyle V\\!\left(\phi^{2}\right)$
$\displaystyle=\displaystyle\sum_{n=0}^{\infty}\hat{\lambda}_{2n}\phi^{2n},$
$\displaystyle F\\!\left(\phi^{2}\right)$
$\displaystyle=\displaystyle\sum_{n=0}^{\infty}\hat{\xi}_{2n}\phi^{2n}.$ (55)

To investigate the fixed point structure, we define the rescaled dimensionless
coupling constants:

$\displaystyle\lambda_{2n}$ $\displaystyle:=\hat{\lambda}_{2n}\Lambda^{2n-4},$
$\displaystyle\xi_{2n}$ $\displaystyle:=\hat{\xi}_{2n}\Lambda^{2n-2}.$ (56)

The cutoff $\Lambda$ disappears from RGEs for these dimensionless coupling
constants, and there remain the so-called canonical scaling terms:

$\displaystyle\partial_{t}\lambda_{2n}$
$\displaystyle=-\left(2n-4\right)\lambda_{2n}+\text{fluctuations},$
$\displaystyle\partial_{t}\xi_{2n}$
$\displaystyle=-\left(2n-2\right)\xi_{2n}+\text{fluctuations},$ (57)

where “fluctuations” indicate the loop contributions, which are one-loop
exact. Note that the coefficient of the canonical scaling term becomes the
dimension of the coupling constant in the LPA.

###  4.3 Running of scalar and gravitational coupling constants

We have considered the truncation of full system by restricting to the
functional form (13). Now we truncate the series in Eq. (55) up to
$\hat{\lambda}_{4}$ and to $\hat{\xi}_{2}$. We can read off the beta functions
for $\xi_{0}$, $\lambda_{0}$, $\xi_{2}$, $\lambda_{2}$, and $\lambda_{4}$ from
Eqs. (52) and (53). We show the results in the symmetric phase $\phi=0$:

$\displaystyle\partial_{t}\xi_{0}$
$\displaystyle=2\xi_{0}-\frac{1}{384\pi^{2}}\bigg{[}25-\frac{4}{1+2\lambda_{2}}-\frac{24\xi_{2}}{\left(1+2\lambda_{2}\right)^{2}}+\frac{8\xi_{0}\left(7\xi_{0}-2\lambda_{0}\right)}{\left(\xi_{0}-\lambda_{0}\right)^{2}}\bigg{]}$
$\displaystyle\quad+\frac{1}{1152\pi^{2}}\frac{\partial_{t}\xi_{0}-2{\xi_{0}}}{\xi_{0}}\frac{17\xi_{0}^{2}+18\lambda_{0}\xi_{0}-15\lambda_{0}^{2}}{\left(\xi_{0}-\lambda_{0}\right)^{2}}+\frac{N_{\text{f}}}{48\pi^{2}},$
(58) $\displaystyle\partial_{t}\lambda_{0}$
$\displaystyle=4\lambda_{0}-\frac{1}{32\pi^{2}}\bigg{[}2+\frac{1}{1+2\lambda_{2}}+\frac{6\lambda_{0}}{\xi_{0}-\lambda_{0}}\bigg{]}+\frac{\partial_{t}\xi_{0}-2{\xi_{0}}}{96\pi^{2}\xi_{0}}\frac{5\xi_{0}-2\lambda_{0}}{\xi_{0}-\lambda_{0}}+\frac{N_{\text{f}}}{8\pi^{2}},$
(59) $\displaystyle\partial_{t}\xi_{2}$
$\displaystyle=-\frac{1}{576\pi^{2}}\bigg{[}\frac{1+2\lambda_{2}}{\xi_{0}-\lambda_{0}}\left(9+\frac{39\xi_{0}}{\xi_{0}-\lambda_{0}}+\frac{60\xi_{0}^{2}}{\left(\xi_{0}-\lambda_{0}\right)^{2}}\right)+\frac{3\left(3+32\xi_{2}\right)}{\xi_{0}-\lambda_{0}}-\frac{6\xi_{0}\left(11+2\xi_{2}\right)}{\left(\xi_{0}-\lambda_{0}\right)^{2}}$
$\displaystyle\phantom{=-\frac{1}{576\pi^{2}}\bigg{[}}-\frac{60\xi_{0}^{2}\left(1+2\xi_{2}\right)}{\left(\xi_{0}-\lambda_{0}\right)^{3}}+\frac{216\xi_{2}\left(1+2\xi_{2}\right)^{2}}{\left(1+2\lambda_{2}\right)^{3}\left(\xi_{0}-\lambda_{0}\right)}+\frac{9\left[\lambda_{0}\left(5-2\xi_{2}\right)-2\xi_{0}\left(1+2\xi_{2}\right)\right]\left(1+2\xi_{2}\right)}{\left(1+2\lambda_{2}\right)\left(\xi_{0}-\lambda_{0}\right)^{2}}$
$\displaystyle\phantom{=-\frac{1}{576\pi^{2}}\bigg{[}}+\frac{27\left(1+2\xi_{2}\right)\left(1-10\xi_{2}-16\xi_{2}^{2}\right)}{\left(1+2\lambda_{2}\right)^{2}\left(\xi_{0}-\lambda_{0}\right)}+\frac{108\xi_{0}\xi_{2}\left(1+2\xi_{2}\right)^{2}}{\left(1+2\lambda_{2}\right)^{2}\left(\xi_{0}-\lambda_{0}\right)^{2}}+\frac{72{\lambda}_{4}}{\left(1+2\lambda_{2}\right)^{2}}\frac{1+12\xi_{2}+2\lambda_{2}}{1+2\lambda_{2}}\bigg{]}$
$\displaystyle\quad+\frac{\partial_{t}\xi_{0}-2\xi_{0}}{1152\pi^{2}\xi_{0}}\bigg{[}\frac{1+2\lambda_{2}}{\xi_{0}-\lambda_{0}}\bigg{(}3+\frac{18\xi_{0}}{\xi_{0}-\lambda_{0}}+\frac{20\xi_{0}^{2}}{\left(\xi_{0}-\lambda_{0}\right)^{2}}\bigg{)}+\frac{15\xi_{2}}{\xi_{0}}-\frac{6\left(1+\xi_{2}\right)}{\xi_{0}-\lambda_{0}}-\frac{10\xi_{0}\left(3+4\xi_{2}\right)}{\left(\xi_{0}-\lambda_{0}\right)^{2}}$
$\displaystyle\phantom{\quad+\frac{\partial_{t}\xi_{0}-2\xi_{0}}{1152\pi^{2}\xi_{0}}\bigg{[}}-\frac{20\xi_{0}^{2}\left(1+2\xi_{2}\right)}{\left(\xi_{0}-\lambda_{0}\right)^{3}}-\frac{3\left[\lambda_{0}-\xi_{0}\left(5-4\xi_{2}\right)\right]\left(1+2\xi_{2}\right)}{\left(1+2\lambda_{2}\right)\left(\xi_{0}-\lambda_{0}\right)^{2}}+\frac{36\xi_{0}\xi_{2}\left(1+2\xi_{2}\right)^{2}}{\left(1+2\lambda_{2}\right)^{2}\left(\xi_{0}-\lambda_{0}\right)^{2}}\bigg{]}$
$\displaystyle\quad+\frac{\partial_{t}\xi_{2}}{1152\pi^{2}\xi_{0}}\bigg{[}-15+\frac{54\xi_{0}}{\xi_{0}-\lambda_{0}}+\frac{20\xi_{0}^{2}}{\left(\xi_{0}-\lambda_{0}\right)^{2}}-\frac{6\xi_{0}\left(7+2\xi_{2}\right)}{\left(1+2\lambda_{2}\right)\left(\xi_{0}-\lambda_{0}\right)}-\frac{144\xi_{0}\xi_{2}\left(1+2\xi_{2}\right)}{\left(1+2\lambda_{2}\right)\left(\xi_{0}-\lambda_{0}\right)}\bigg{]}$
$\displaystyle\quad-\frac{N_{\text{f}}y^{2}}{48\pi^{2}},$ (60)
$\displaystyle\partial_{t}\lambda_{2}$
$\displaystyle=2\lambda_{2}-\frac{1}{48\pi^{2}}\bigg{[}\frac{9\lambda_{0}\left(1+2\xi_{2}\right)}{2\left(\xi_{0}-\lambda_{0}\right)^{2}}-\frac{9\left(2\lambda_{0}-\xi_{0}\right)\left(1+2\xi_{2}\right)^{2}}{2\left(1+2\lambda_{2}\right)\left(\xi_{0}-\lambda_{0}\right)^{2}}-\frac{9\left(1+2\xi_{2}\right)^{2}}{2\left(1+2\lambda_{2}\right)^{2}\left(\xi_{0}-\lambda_{0}\right)}-\frac{18\lambda_{4}}{\left(1+2\lambda_{2}\right)^{2}}\bigg{]}$
$\displaystyle\quad+\frac{\partial_{t}\xi_{0}-2\xi_{0}}{96\pi^{2}\xi_{0}}\bigg{[}-\frac{2\xi_{2}}{\xi_{0}}+\frac{3\xi_{0}\left(1+2\xi_{2}\right)}{2\left(\xi_{0}-\lambda_{0}\right)^{2}}-\frac{3\xi_{0}\left(1+2\xi_{2}\right)^{2}}{2\left(1+2\lambda_{2}\right)\left(\xi_{0}-\lambda_{0}\right)^{2}}\bigg{]}$
$\displaystyle\quad+\frac{1}{96\pi^{2}}\frac{\partial_{t}\xi_{2}}{\xi_{0}}\bigg{[}2-\frac{3\xi_{0}}{\xi_{0}-\lambda_{0}}+\frac{6\xi_{0}\left(1+2\xi_{2}\right)}{\left(1+2\lambda_{2}\right)\left(\xi_{0}-\lambda_{0}\right)}\bigg{]}-\frac{N_{\text{f}}y^{2}}{8\pi^{2}},$
(61) $\displaystyle\partial_{t}\lambda_{4}$
$\displaystyle=-\frac{1}{48\pi^{2}}\Bigg{[}\frac{9}{4\left(\xi_{0}-\lambda_{0}\right)^{2}}\Bigg{(}5\left(1+2\lambda_{2}\right)\left(1+4\xi_{2}\right)-\left(1+2\xi_{2}\right)\left(21+62\xi_{2}\right)+\frac{33\left(1+2\xi_{2}\right)^{3}}{1+2\lambda_{2}}$
$\displaystyle\phantom{=-\frac{1}{48\pi^{2}}\Bigg{[}}\phantom{\frac{9}{4\left(\xi_{0}-\lambda_{0}\right)^{2}}\Bigg{(}}-\frac{\left(1+2\xi_{2}\right)^{3}\left(23+24\xi_{2}\right)}{\left(1+2\lambda_{2}\right)^{2}}+\frac{6\left(1+2\xi_{2}\right)^{4}}{\left(1+2\lambda_{2}\right)^{3}}\Bigg{)}$
$\displaystyle\phantom{=-\frac{1}{48\pi^{2}}\Bigg{[}}+\frac{9\xi_{0}\left(\xi_{2}-\lambda_{2}\right)^{2}}{\left(\xi_{0}-\lambda_{0}\right)^{3}}\left(6\frac{\left(1+2\xi_{2}\right)^{2}}{\left(1+2\lambda_{2}\right)^{2}}-10\frac{1+2\xi_{2}}{1+2\lambda_{2}}+5\right)-\frac{72\lambda_{2}\lambda_{4}\left(1+2\xi_{2}\right)\left(1-4\lambda_{2}+6\xi_{2}\right)}{\left(\xi_{0}-\lambda_{0}\right)\left(1+2\lambda_{2}\right)^{3}}$
$\displaystyle\phantom{=-\frac{1}{48\pi^{2}}\Bigg{[}}+\frac{9\xi_{0}\lambda_{4}}{\left(\xi_{0}-\lambda_{0}\right)^{2}}\left(6\frac{\left(1+2\xi_{2}\right)^{2}}{\left(1+2\lambda_{2}\right)^{2}}-8\frac{1+2\xi_{2}}{1+2\lambda_{2}}+3\right)+\frac{216\lambda_{4}^{2}}{\left(1+2\lambda_{2}\right)^{3}}\Bigg{]}$
$\displaystyle\phantom{=-\frac{1}{48\pi^{2}}\Bigg{[}}+\frac{\partial_{t}\xi_{0}-2\xi_{0}}{96\pi^{2}\xi_{0}}\Bigg{[}\frac{2\xi_{2}^{2}}{\xi_{0}^{2}}+\frac{3\xi_{0}\left(\xi_{2}-\lambda_{2}\right)^{2}}{\left(\xi_{0}-\lambda_{0}\right)^{3}}\left(6\frac{\left(1+2\xi_{2}\right)^{2}}{\left(1+2\lambda_{2}\right)^{2}}-10\frac{1+2\xi_{2}}{1+2\lambda_{2}}+5\right)$
$\displaystyle\phantom{=-\frac{1}{48\pi^{2}}\Bigg{[}}\phantom{+\frac{\partial_{t}\xi_{0}-2\xi_{0}}{96\pi^{2}\xi_{0}}\Bigg{[}}+\frac{3\xi_{0}\lambda_{4}}{\left(\xi_{0}-\lambda_{0}\right)^{2}}\left(6\frac{\left(1+2\xi_{2}\right)^{2}}{\left(1+2\lambda_{2}\right)^{2}}-8\frac{1+2\xi_{2}}{1+2\lambda_{2}}+3\right)\Bigg{]}$
$\displaystyle\quad+\frac{1}{96\pi^{2}}\frac{\partial_{t}\xi_{2}}{\xi_{0}}\Bigg{[}-\frac{2\xi_{2}}{\xi_{0}}-\frac{24\xi_{0}\lambda_{4}\left(1-4\lambda_{2}+6\xi_{2}\right)}{\left(1+2\lambda_{2}\right)^{2}\left(\xi_{0}-\lambda_{0}\right)}$
$\displaystyle\phantom{\quad+\frac{1}{96\pi^{2}}\frac{\partial_{t}\xi_{2}}{\xi_{0}}\Bigg{[}}-\frac{3\xi_{0}\left(\xi_{2}-\lambda_{2}\right)}{\left(\xi_{0}-\lambda_{0}\right)^{2}}\left(12\frac{\left(1+2\xi_{2}\right)^{2}}{\left(1+2\lambda_{2}\right)^{2}}-21\frac{1+2\xi_{2}}{1+2\lambda_{2}}+10\right)\Bigg{]}+\frac{N_{\text{f}}y^{4}}{8\pi^{2}}.$
(62)

The last term of each beta function is coming from the fermionic fluctuation.
The others agree with the results in Ref. [51]. When $y=0$, we see that the
loop of $\psi$ contributes only to the beta functions of $\xi_{0}$ and
$\lambda_{0}$.

In Ref. [53], the authors have studied the Higgs-Yukawa model without the non-
minimal coupling $\xi_{2}=0$. We have checked that when $\xi_{2}=0$, our RG
equation for the scalar potential (52) reduces to theirs, namely the first
line of Eq. (4) in Ref. [53], if we impose that the dimension _ful_
gravitational coupling constant $\hat{\xi}_{0}$ does not run,
$\partial_{t}\hat{\xi}_{0}=\partial_{t}\xi_{0}-2\xi_{0}=0$, in the right hand
side of the RG equation. (We write $\xi_{0}=1/16\pi\tilde{G}$ where
$\tilde{G}$ is the dimensionless Newton constant.) Similarly, we can see that
the RG equations for $\lambda_{0}$ (59), for $\lambda_{2}$ (61), and for
$\lambda_{4}$ (62) reduce to Eq. (6) in Ref. [53] if we put
$\lambda_{0}=\xi_{2}=0$ and $\partial_{t}\hat{\xi}_{0}=0$.

###  4.4 Running of Yukawa coupling

(I)(II)(III)(IV)(V)(VI)(VII)(VIII)(IX)(X)(XI)(XII)(XIII)

Figure 5: The corrections to the Yukawa coupling constant in our truncation.
The black dot and the gray circle denote the Yukawa interaction vertex and the
mixing of scalar fields, respectively.

By the same two methods described in Sec. 4.1, we obtain the RGE for $Y$, and
read off the beta function for the Yukawa coupling constant $y$ in the
symmetric phase $\phi=0$:

$\displaystyle\partial_{t}y$
$\displaystyle=\frac{5y\Lambda^{6}}{32\pi^{2}}\left(\frac{\partial_{t}\hat{\xi}_{0}}{6}-\hat{\xi}_{0}\right)I[2,0,0]$
$\displaystyle\quad+\frac{y\Lambda^{6}}{32\pi^{2}}\bigg{[}24\left(\hat{\xi}_{2}-\frac{\partial_{t}\hat{\xi}_{2}}{6}\right)I[1,1,0]-\left(\hat{\xi}_{0}-\frac{\partial_{t}\hat{\xi}_{0}}{6}\right)I[2,0,0]-{12}\,C\left(\hat{\xi}_{0}-\frac{\partial_{t}\hat{\xi}_{0}}{6}\right)I[2,1,0]$
$\displaystyle\phantom{\quad+\frac{y\Lambda^{6}}{32\pi^{2}}\bigg{(}}-{12}\,CI[1,2,0]\bigg{]}$
$\displaystyle\quad-\frac{y^{3}\Lambda^{6}}{16\pi^{2}}\left(I[0,1,2]+I[0,2,1]\right)-\frac{y\Lambda^{8}}{128\pi^{2}}\left[I[1,0,2]+\left(\hat{\xi}_{0}-\frac{\partial_{t}\hat{\xi}_{0}}{8}\right)I[2,0,1]\right]$
$\displaystyle\quad+\frac{3y\Lambda^{8}}{40\pi^{2}}\bigg{[}I[1,0,2]+\left(\hat{\xi}_{0}-\frac{\partial_{t}\hat{\xi}_{0}}{7}\right)I[2,0,1]-\frac{1}{2\Lambda^{2}}I[1,0,1]\bigg{]}$
$\displaystyle\quad-\frac{3y\Lambda^{8}}{20\pi^{2}}\bigg{[}-\left(\hat{\xi}_{0}-\frac{\partial_{t}\hat{\xi}_{0}}{7}\right)CI[2,1,1]+\left(\hat{\xi}_{2}-\frac{\partial_{t}\hat{\xi}_{2}}{7}\right)I[1,1,1]$
$\displaystyle\phantom{\quad-\frac{y\Lambda^{8}}{16\pi^{2}}\bigg{(}}-C\left(I[1,2,1]+I[1,1,2]-\frac{1}{2\Lambda^{2}}I[1,1,1]\right)\bigg{]},$
(63)

where $C=\hat{\xi}_{2}\Lambda^{2}-\hat{\lambda}_{2}$ and

$\displaystyle
I[n_{g},n_{b},n_{f}]:=\left.\frac{1}{\Psi^{n_{g}}\Sigma_{1}^{n_{b}}\Sigma_{3}^{n_{f}}}\right|_{\phi=0}=\frac{1}{(\hat{\xi}_{0}\Lambda^{2}-\hat{\lambda}_{0})^{n_{g}}(\Lambda^{2}+2\hat{\lambda}_{2})^{n_{b}}(\Lambda^{2})^{n_{f}}}.$
(64)

This is one of our main results.

The first term in the beta function corresponds to the diagram (I) in Fig. 5.
The second term includes the diagrams (III), (IV) and (V). The term in
proportion to $y^{3}$ corresponds to the diagram (VI). The terms in fourth and
fifth lines in the beta function correspond to (VIII)(IX) and (X)(XI),
respectively. The last terms correspond to (XII) and (XIII).

Using the dimensionless rescaled coupling constants $\lambda_{2n}$, $\xi_{2n}$
introduced in Eq. (56), we can easily rewrite the beta function of $y$ by the
replacements

$\displaystyle\partial_{t}\hat{\lambda}_{2n}$
$\displaystyle={1\over\Lambda^{2n-4}}\left[\partial_{t}\lambda_{2n}+\left(2n-4\right)\lambda_{2n}\right],$
$\displaystyle\partial_{t}\hat{\xi}_{2n}$
$\displaystyle={1\over\Lambda^{2n-2}}\left[\partial_{t}\xi_{2n}+\left(2n-2\right)\xi_{2n}\right].$
(65)

Since the Yukawa coupling constant is dimensionless and its canonical scaling
term vanishes in its beta function in the LPA, we have been omitting the hat
$\hat{\mbox{ }}$ for $y$.

Let us try to put $\lambda_{0}=\xi_{2}=0$ as in the end of Sec. 4.3. If we
impose that the dimensionful constant does not run,
${\partial_{t}\hat{\xi}_{0}=\Lambda^{2}(\,}\partial_{t}\xi_{0}-2\xi_{0}{)}=0$,
we obtain151515The difference of overall sign in the beta function is due to
the sign convention for the dimensionless scale $t$. Recall that in our
notation, $t=-\log\\!\left(\Lambda/\Lambda_{0}\right)$.

$\displaystyle\dot{y}$
$\displaystyle:=-\beta_{y}=\frac{y^{3}(1+\lambda_{2})}{8\pi^{2}\left(1+2\lambda_{2}\right)^{2}}+\tilde{G}y{29-4\lambda_{2}\left(1-5\lambda_{2}\right)\over
20\pi\left(1+2\lambda_{2}\right)^{2}}.$ (66)

This is to compare with Eq. (6) in Ref. [53]. We see that the first term,
which corresponds to the diagram (VI) in Fig. 5, agrees each other, while the
second term does not. To study the fixed-point structure, we are rather
interested in the limit where dimensionless coupling constant does not run
$\partial_{t}{\xi_{0}}\to 0$, which results in

$\displaystyle\dot{y}$
$\displaystyle:=-\beta_{y}=\frac{y^{3}\left(1+\lambda_{2}\right)}{8\pi^{2}\left(1+2\lambda_{2}\right)^{2}}+\tilde{G}y\frac{2395+4\lambda_{2}\left({347}+{315}\lambda_{2}\right)}{1120\pi\left(1+2\lambda_{2}\right)^{2}}+\mathcal{O}\\!\left({\tilde{G}}^{2}\right).$
(67)

However, the dimensionless cosmological constant $\lambda_{0}$ is not
vanishing at the UV fixed point, and we will rely on the numerical computation
in the next section.

##  5 Numerical Analysis

###  5.1 Fixed Point structure

The fixed points are defined by vanishing beta functions
$\beta_{i}\\!\left(g^{*}\right)=0$ at which RG flows completely stop. To study
the behavior of the RG flow near the fixed point $g^{*}$, let us consider the
linearized flow equations. Let $N$ be the dimension of our (truncated)
coupling space. We expand the beta function around $g^{*}$,

$\displaystyle\beta_{i}\\!\left(g\right)=\beta_{i}\\!\left(g^{*}\right)+\sum_{j=1}^{N}\left.\frac{\partial\beta_{i}}{\partial
g_{j}}\right|_{g=g^{*}}\left(g_{j}-g_{j}^{*}\right)+\cdots.$ (68)

Using $\beta_{i}\\!\left(g^{*}\right)=0$ and neglecting the higher order terms
in $v:=g-g^{*}$, we obtain the linearlized RGE

$\displaystyle\partial_{t}v_{i}=\sum_{j=1}^{N}\left.\frac{\partial\beta_{i}}{\partial
g_{j}}\right|_{g=g^{*}}v_{j}.$ (69)

Let us diagonalize the matrix

$\displaystyle M_{ij}:=\left.{\partial\beta_{i}\over\partial
g_{j}}\right|_{g=g^{*}}$ (70)

by a constant matrix $V$ so that

$\displaystyle\sum_{i,j=1}^{N}\left(V^{-1}\right)_{li}\left.{\partial\beta_{i}\over\partial
g_{j}}\right|_{g=g^{*}}V_{jk}$ $\displaystyle=\theta_{k}\delta_{lk},$ (71)

where $k$ is not summed. That is, the $k$th eigenvalue of $M$ is $\theta_{k}$,
and the corresponding eigenvector is
$V^{(k)}=\left(V_{jk}\right)_{j=1,\dots,N}$:

$\displaystyle MV^{(k)}=\theta_{k}V^{(k)}.$ (72)

Now Eq. (69) reduces to

$\displaystyle\partial_{t}\kappa_{i}=\theta_{i}\kappa_{i},$ (73)

where the index $i$ is not summed and we have written
$v_{i}=\sum_{j=1}^{N}V_{ij}\kappa_{j}$. The solutions to Eq. (73) are

$\displaystyle\kappa_{i}(t)=C_{i}e^{\theta_{i}t},$ (74)

where $C_{i}$ are constants. When we recover the original dimensionless
coupling constants $g_{i}$, Eq. (74) reads

$\displaystyle
g_{i}(t)=g_{i}^{*}+\displaystyle\sum_{j=1}^{N}V_{ij}C_{j}\left(\frac{\Lambda_{0}}{\Lambda}\right)^{\theta_{j}},$
(75)

which becomes Eq. (5) with $\zeta_{ij}=C_{j}V_{ij}$. In general, a non-zero
$\operatorname{Im}\\!\left(\theta_{i}\right)$ implies that the corresponding
coupling $g_{i}$ is mixed with other couplings in the RG flow from UV to IR,
$\Lambda\to 0$, i.e. $t\to\infty$. Let us see three cases in turn:

  * •

For the directions with $\operatorname{Re}\\!\left(\theta_{i}\right)>0$, we
see that $\kappa_{i}$ grow when we increase $t$ in the flow from UV to IR.
Then $g_{i}$ become the couplings of the relevant operators, and the factor
$C_{i}$ become physical free parameters. When we vary ratios of $C_{i}$, the
direction of the flow to IR changes, and we get a different IR physics.

  * •

For the directions with $\operatorname{Re}(\theta_{i})=0$, the solutions (74)
generally become oscillatory, and the corresponding operators are marginal.

  * •

For the directions with $\operatorname{Re}(\theta_{i})\,{<}\,0$, the solutions
shrink to the UV fixed point, and hence they are the coupling of the
irrelevant operators.

The relevant operators span the hypersurface called the renormalized
trajectory or the UV critical surface, and the number of such operators gives
its dimension.

###  5.2 Pure gravity

We first revisit the pure gravity case obtained in Refs. [49, 51, 52]. The
beta functions for the dimensionless gravitational coupling $\xi_{0}$ and the
dimensionless cosmological constant $\lambda_{0}$ become

$\displaystyle\beta_{\xi_{0}}$
$\displaystyle=2\xi_{0}-\frac{1}{384\pi^{2}}\left[21+\frac{8\xi_{0}(7\xi_{0}-2\lambda_{0})}{\left(\xi_{0}-\lambda_{0}\right)^{2}}\right]+\frac{\partial_{t}\xi_{0}-2\xi_{0}}{1152\pi^{2}\xi_{0}}\frac{17\xi_{0}^{2}+18\xi_{0}\lambda_{0}-15\lambda_{0}^{2}}{\left(\xi_{0}-\lambda_{0}\right)^{2}},$
(76) $\displaystyle\beta_{\lambda_{0}}$
$\displaystyle=4\lambda_{0}-\frac{1}{32\pi^{2}}\left[3+\frac{6\lambda_{0}}{\xi_{0}-\lambda_{0}}\right]+\frac{\partial_{t}\xi_{0}-2\xi_{0}}{96\pi^{2}\xi_{0}}\frac{5\xi_{0}-2\lambda_{0}}{\xi_{0}-\lambda_{0}}.$
(77)

Solving the coupled equation $\beta_{\xi_{0}}=0$ and $\beta_{\lambda_{0}}=0$,
we find the non-trivial fixed point:

$\displaystyle\xi_{0}^{*}$ $\displaystyle=2.38\times 10^{-2},$
$\displaystyle\lambda_{0}^{*}$ $\displaystyle=8.62\times 10^{-3}.$ (78)

Around the fixed point, the matrix (70) becomes:

$\displaystyle
M=\left.\begin{pmatrix}\displaystyle\frac{\partial\beta_{\xi_{0}}}{\partial\xi_{0}}&\displaystyle\frac{\partial\beta_{\xi_{0}}}{\partial\lambda_{0}}\\\\[10.0pt]
\displaystyle\frac{\partial\beta_{\lambda_{0}}}{\partial\xi_{0}}&\displaystyle\frac{\partial\beta_{\lambda_{0}}}{\partial\lambda_{0}}\end{pmatrix}\right|_{\xi_{0}=\xi_{0}^{*}\atop\lambda_{0}=\lambda_{0}^{*}}=\begin{pmatrix}4.79429&-6.42602\\\
2.33613&-0.505996\end{pmatrix}.$ (79)

The eigenvalues for this matrix are $\theta_{1,2}=2.14414\pm 2.82644i$. We see
that $\xi_{0}$ and $\lambda_{0}$ are the relevant coupling constants around
the UV fixed point. The corresponding eigenvectors are

$\displaystyle V^{(1)},V^{(2)}$ $\displaystyle=\begin{pmatrix}0.856378\\\
0.353177\pm 0.376673i\end{pmatrix},$ $\displaystyle V$
$\displaystyle=\begin{pmatrix}0.856378&0.856378\\\
0.353177+0.376673i&0.353177-0.376673i\end{pmatrix}.$ (80)

That is,

$\displaystyle\xi_{0}(t)$
$\displaystyle=\xi_{0}^{*}+0.856378\left(\Lambda_{0}\over\Lambda\right)^{2.14414}\left[A\cos\\!\left(\ln\left(\Lambda_{0}\over\Lambda\right)^{2.82644}\right)+B\sin\\!\left(\ln\left(\Lambda_{0}\over\Lambda\right)^{2.82644}\right)\right],$
(81) $\displaystyle\lambda_{0}(t)$
$\displaystyle=\lambda_{0}^{*}+\left(\Lambda_{0}\over\Lambda\right)^{2.14414}\Bigg{[}\left(0.353177A+0.376673B\right)\cos\\!\left(\ln\left(\Lambda_{0}\over\Lambda\right)^{2.82644}\right)$
$\displaystyle\phantom{=\lambda_{0}^{*}+\left(\Lambda_{0}\over\Lambda\right)^{2.14414}\Bigg{[}}+\left(-0.376673A+0.353177B\right)\sin\\!\left(\ln\left(\Lambda_{0}\over\Lambda\right)^{2.82644}\right)\Bigg{]},$
(82)

where $A:=C_{1}+C_{2}$ and $B:=i\left(C_{1}-C_{2}\right)$ are real constants
that are free parameters of the asymptotically safe theory. We see that the
two relevant couplings mix with each other in the RG flow to IR scales.

###  5.3 Scalar-gravity model

Next we turn to the extension of the system with the neutral scalar field [49,
51, 52]. This section is still a review. In truncated theory space
$g_{i}=\\{\xi_{0},\lambda_{0},\xi_{2},\lambda_{2},\lambda_{4}\\}$, we find the
fixed point

$\displaystyle\xi_{0}^{*}$ $\displaystyle=2.38\times 10^{-2},$
$\displaystyle\lambda_{0}^{*}$ $\displaystyle=8.62\times 10^{-3},$
$\displaystyle\xi_{2}^{*}$ $\displaystyle=0,$ $\displaystyle\lambda_{2}^{*}$
$\displaystyle=0,$ $\displaystyle\lambda_{4}^{*}$ $\displaystyle=0.$ (83)

The gravitational coupling constants have the same non-trivial fixed point,
while the matter fixed point is trivial, i.e. Gaussian. The fixed point (83)
is called the Gaussian-matter fixed point.161616 It could be that the
Gaussian-matter fixed point is a special property of the present truncation
with LPA: If we take into account the higher-derivative matter self-
interactions, which are induced by the gravitational fluctuations, then the
matter self interaction might become non-vanishing at the fixed point [62].
The matrix (70) becomes

$\displaystyle M=\begin{pmatrix}4.85544&-6.51993&0.00766245&-0.00262748&0\\\
2.40051&-0.570309&0.00234951&0.0055494&0\\\
0&0&2.85544&{-6.51993}&-0.0157649\\\ 0&0&2.40051&-2.57031&0.0332964\\\
0&0&0&0&-2.62692\end{pmatrix},$ (84)

and the eigenvalues and the corresponding eigenvectors are

$\displaystyle\theta_{1,2}$ $\displaystyle=2.143\pm 2.879i,$ $\displaystyle
V^{(1)},V^{(2)}$ $\displaystyle=\begin{pmatrix}0.8549\\\ 0.3557\pm 0.3776i\\\
0\\\ 0\\\ 0\end{pmatrix},$ (85) $\displaystyle\theta_{3,4}$
$\displaystyle=0.143\pm 2.879i,$ $\displaystyle V^{(3)},V^{(4)}$
$\displaystyle=\begin{pmatrix}\left(-1.8059\pm 0.731i\right)\times 10^{-3}\\\
\left(3.0723\pm 1.0763i\right)\times 10^{-3}\\\ 0.3557\pm 0.3776i\\\ 0.8549\\\
0\end{pmatrix},$ (86) $\displaystyle\theta_{5}$ $\displaystyle=-2.627,$
$\displaystyle V^{(5)}$ $\displaystyle=\begin{pmatrix}2.0687\times 10^{-5}\\\
2.7445\times 10^{-5}\\\ -1.3805\times 10^{-2}\\\ -1.3542\times 10^{-2}\\\
0.999813\end{pmatrix}.$ (87)

Several comments are in order:

  * •

Since the vectors $V^{(1)}$ and $V^{(2)}$ have the values at only first and
second rows, these vectors correspond to the mixing between $\xi_{0}$ and
$\lambda_{0}$. These coupling constants are relevant as their critical
exponents are positive: $\operatorname{Re}\theta_{1}>0$ and
$\operatorname{Re}\theta_{2}>0$. We see that the impact of scalar fluctuations
to the gravitational couplings is not large since the values of the critical
exponents $\theta_{1,2}$ hardly change by its inclusion.

  * •

Although the vectors $V^{(3)}$ and $V^{(4)}$ include the mixing of $\xi_{0}$,
$\lambda_{0}$, $\xi_{2}$ and $\lambda_{2}$, the contributions from the
gravitational couplings $\xi_{0}$ and $\lambda_{0}$ are smaller than those
from $\xi_{2}$ and $\lambda_{2}$. Therefore they are mainly $\xi_{2}$ and
$\lambda_{2}$. These coupling constants are relevant as their critical
exponent is positive: $\operatorname{Re}\theta_{3}>0$ and
$\operatorname{Re}\theta_{4}>0$. Note that the non-minimal coupling constant
$\xi_{2}$ is marginal at the trivial fixed point $g^{*}=0$, and hence, the
gravitational effects have made it relevant.

  * •

The scalar quartic coupling $\lambda_{4}$ is irrelevant as its critical
exponent is negative: $\operatorname{Re}\theta_{5}<0$. Although $\lambda_{4}$
is marginal at the trivial fixed point, the gravitational effects make it
irrelevant at the UV fixed point.

In this truncated theory space, the UV critical surface is spanned by the
operators with $\xi_{0}$, $\lambda_{0}$, $\xi_{2}$ and $\lambda_{2}$. Hence,
these coupling constants are physical free parameters [49, 51]. In next part
we will see that the fermionic fluctuation makes $\xi_{2}$ and $\lambda_{2}$
irrelevant so that these couplings cannot be physical free parameters anymore.

###  5.4 Inclusion of a fermion

Now let us extend the theory space to
$\Set{g_{i}}_{i=1,\dots,6}=\Set{\xi_{0},\lambda_{0},\xi_{2},\lambda_{2},\lambda_{4},y}$
with $N_{\text{f}}=1$. Solving the coupled equation $\beta_{g_{i}}=0$ with
Eqs. (58)–(62), we again obtain the Gaussian-matter fixed point:

$\displaystyle\xi_{0}^{*}$ $\displaystyle=1.63\times 10^{-2},$
$\displaystyle\lambda_{0}^{*}$ $\displaystyle=3.72\times 10^{-3},$
$\displaystyle\xi_{2}^{*}$ $\displaystyle=0,$ $\displaystyle\lambda_{2}^{*}$
$\displaystyle=0,$ $\displaystyle\lambda_{4}^{*}$ $\displaystyle=0,$
$\displaystyle y^{*}$ $\displaystyle=0.$ (88)

Around this fixed point, the matrix (70) becomes

$\displaystyle M=\begin{pmatrix}3.6814&-5.39674&0.00776027&-0.00258676&0&0\\\
1.99718&-0.663341&0.00295698&0.00534691&0&0\\\
0&0&1.6814&-5.39674&-0.0155205&0\\\ 0&0&1.99718&-2.66334&0.0320815&0\\\
0&0&0&0&-2.60696&0\\\ 0&0&0&0&0&-1.46426\end{pmatrix}.$ (89)

The eigenvalues $\theta_{i}$ and eigenvectors $V^{(i)}$ of the matrix (89) are

$\displaystyle\theta_{1,2}$ $\displaystyle=1.50903\pm 2.46151i,$
$\displaystyle V^{(1)},V^{(2)}$ $\displaystyle=\begin{pmatrix}0.854336\\\
0.343899\pm 0.389672i\\\ 0\\\ 0\\\ 0\\\ 0\end{pmatrix},$ (90)
$\displaystyle\theta_{3,4}$ $\displaystyle=-0.490968\pm 2.46151i,$
$\displaystyle V^{(3)},V^{(4)}$ $\displaystyle=\begin{pmatrix}(-3.11425\mp
1.27885i)\times 10^{-3}\\\ \ (-1.92736\pm 0.618506i)\times 10^{-3}\\\
0.854329\\\ 0.343897\mp 0.389669i\\\ 0\\\ 0\end{pmatrix},$ (91)
$\displaystyle\theta_{5}$ $\displaystyle=-2.60696,$ $\displaystyle V^{(5)}$
$\displaystyle=\begin{pmatrix}3.88533\times 10^{-5}\\\ 2.91974\times
10^{-5}\\\ -1.65107\times 10^{-2}\\\ -1.59949\times 10^{-2}\\\ 0.999736\\\
0\end{pmatrix},$ (92) $\displaystyle\theta_{6}$ $\displaystyle=-1.46426,$
$\displaystyle V^{(6)}$ $\displaystyle=\begin{pmatrix}0\\\ 0\\\ 0\\\ 0\\\ 0\\\
1\end{pmatrix}.$ (93)

Several comments are in order:

  * •

We see from Eq. (89) that there is no mixing between the Yukawa coupling and
the others at this fixed point. As the critical exponent of $y$ is negative,
the Yukawa interaction is irrelevant up to this truncation.

  * •

The critical exponents $\operatorname{Re}\theta_{1,2}$ for the gravitational
constants, namely the Newton constant $g_{1}$ ($=\xi_{0}=1/16\pi\tilde{G}$)
and the cosmological constant $g_{2}$ ($=\lambda_{0}$), are substantially
reduced from those in Sec. 5.3 by the inclusion of fermions, even without the
Yukawa coupling. This is due to the last terms in Eqs. (58) and (59).

  * •

We see from Eqs. (60)–(62) that when $y=0$, the RG equations of the non-
minimal coupling $g_{3}$ ($=\xi_{2}$), the scalar mass-squared $g_{4}$
($=\lambda_{2}$), and the scalar quartic coupling $g_{5}$ ($=\lambda_{4}$) do
not differ from those in the scalar gravity model in Sec 5.3. However, even
without the Yukawa coupling, the fermion loops do affect the gravitational
constants $g_{1,2}$ as above. As a result, the critical exponents
$\operatorname{Re}\theta_{3,4}$ of $g_{3,4}$ turn to negative from the
positive values in the scalar-gravity model. The non-minimal coupling $g_{3}$
and the scalar mass-squared $g_{4}$ are both made irrelevant. These coupling
constants are not on the UV critical surface anymore.

On the last point, we note that the matter couplings $\lambda_{2}$,
$\lambda_{4}$, $\xi_{2}$, and $y$ vanish at the Gaussian-matter fixed point
and hence that they do not affect the critical exponents
$\operatorname{Re}\theta_{3,4}$ of the non-minimal coupling $g_{3}$
($=\xi_{2}$) and the mass-squared $g_{4}$ ($=\lambda_{2}$). What is important
for flipping the sign of the critical exponents is the fact that the fixed-
point values for the gravitational sector, $g_{1}^{*}$ ($=\xi_{0}^{*}$) and
$g_{2}^{*}$ ($=\lambda_{0}^{*}$), are reduced by the fermion loops. Indeed,
even if we put $N_{\text{f}}=0$ with the values (88), we still get
$\operatorname{Re}\theta_{3,4}=-0.508$. Also, even if we put $N_{\text{f}}=1$
for the values (83) without fermion loop, we still obtain
$\operatorname{Re}\theta_{3,4}=0.144$, which is very close to the true value
0.143 for $N_{\text{f}}=0$. Finally for illustration, we show analytic
formulae for the submatrix of $M$ in Eq. (89):

$\displaystyle M_{33}$
$\displaystyle={51\left(77-8N_{\text{f}}\right)\over\left(1152\pi^{2}\xi_{0}^{*}-17\right)^{2}}-{27648\pi^{2}\left(13824\pi^{2}\xi_{0}^{*}-104N_{\text{f}}+797\right)\over\left(1152\pi^{2}\xi_{0}^{*}-17\right)^{3}}\lambda_{0}^{*}+\mathcal{O}\\!\left(\lambda_{0}^{*2}\right),$
(94) $\displaystyle M_{34}$
$\displaystyle=-{12\left(27648\pi^{2}\xi_{0}^{*}-104N_{\text{f}}+593\right)\over\left(1152\pi^{2}\xi_{0}^{*}-17\right)^{2}}$
$\displaystyle\quad+{96\left(576\pi^{2}\xi_{0}^{*}\left(19584\pi^{2}\xi_{0}^{*}+739-72N_{\text{f}}\right)+4275-740N_{\text{f}}\right)\over\xi_{0}^{*}\left(1152\pi^{2}\xi_{0}^{*}-17\right)^{3}}\lambda_{0}^{*}+\mathcal{O}\\!\left(\lambda_{0}^{*2}\right),$
(95) $\displaystyle M_{43}$
$\displaystyle={180\left(77-8N_{\text{f}}\right)\over\left(1152\pi^{2}\xi_{0}^{*}-17\right)^{2}}$
$\displaystyle\quad-{17\left(-9667+1672N_{\text{f}}\right)+3456\pi^{2}\xi_{0}^{*}\left(9667-1672N_{\text{f}}+4608\pi^{2}\xi_{0}^{*}\left(93-4N_{\text{f}}+576\pi^{2}\xi_{0}^{*}\right)\right)\over
32\pi^{2}\xi_{0}^{*2}\left(1152\pi^{2}\xi_{0}^{*}-17\right)^{3}}\lambda_{0}^{*}+\mathcal{O}\\!\left(\lambda_{0}^{*2}\right),$
(96) $\displaystyle M_{44}$
$\displaystyle={-9667+1672N_{\text{f}}+64\pi^{2}\xi_{0}^{*}\left(-8837+432N_{\text{f}}+2304\pi^{2}\xi_{0}^{*}\left(576\pi^{2}\xi_{0}^{*}-71\right)\right)\over
32\pi^{2}\xi_{0}^{*2}\left(1152\pi^{2}\xi_{0}^{*}-17\right)^{2}}$
$\displaystyle\quad+{258309-44920N_{\text{f}}-9216\pi^{2}\xi_{0}^{*}\left(-3641+414N_{\text{f}}-144\pi^{2}\xi_{0}^{*}\left(6912\pi^{2}\xi_{0}^{*}+893-24N_{\text{f}}\right)\right)\over
16\pi^{2}\xi_{0}^{*}\left(1152\pi^{2}\xi_{0}^{*}-17\right)^{3}}\lambda_{0}^{*}+\mathcal{O}\\!\left(\lambda_{0}^{*2}\right).$
(97)

##  6 Summary and discussions

In this paper we have investigated the fixed point structure of the Higgs-
Yukawa model that is non-minimally coupled to gravity, using the FRG. The full
set of RG equations of this system are obtained for the first time. We find a
Gaussian-matter fixed point which is a non-trivial UV fixed point for the
gravitational coupling constants, namely the Newton and cosmological
constants, and is a trivial one for the other coupling constants among
matters. It has been known that the Gaussian-matter fixed point for the
scalar-gravity system without fermion has the non-minimal coupling $\xi_{2}$
as relevant direction, together with the scalar mass-squared ($=\lambda_{2}$)
[49, 51, 52]. We have found that the inclusion of fermion to this scalar-
gravity system makes both of them irrelevant, no matter whether the Yukawa
coupling is turned on or not. Therefore both of them in this toy model cannot
be on the UV critical surface, and hence cannot be the free parameters of the
theory in the asymptotic safety scenario.

It is important to investigate whether the non-minimal coupling of the Higgs
to the Ricci scalar in the SM (and its extensions), $\xi\left|H\right|^{2}R$,
becomes relevant or not when we take into account the large degrees of
freedom, both bosonic and fermionic, that couple to the Higgs. The large non-
minimal coupling constant plays crucial role in the Higgs inflation scenario
[92, 96]. If the non-minimal coupling becomes a free parameter in the
asymptotic safety scenario in the above sense, then we can use it to account
for the cosmological data by the Higgs inflation. In this toy model, we have
found that the non-minimal coupling cannot be such a free parameter. If this
is the case for the SM too, then the Higgs inflation model is a cutoff theory
and cannot be a UV complete model within the asymptotically safe gravity
scenario.

In this paper, we have have studied the asymptotic safety of the simple Higgs-
Yukawa model with non-minimal coupling, in the symmetric phase
$\langle\phi\rangle=0$. In the Higgs inflation using the SM criticality, the
typical value of the Higgs field becomes close to the Planck scale [99]. For
such an application, it is important to extend our analysis to the broken
phase $\langle\phi\rangle\neq 0$.

We comment on the unitarity of gravity. The earlier studies indicate that the
asymptotically safe quantum gravity would be described by the three
dimensional UV critical surface, spanned by the cosmological constant, $R$,
and $R^{2}$; see e.g. [69, 87]. It is worth studying whether this remains the
case or not if we include other forms of higher dimensional operators. For
example, the operators $R_{\mu\nu}R^{\mu\nu}$ and
$R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}$ have not been taken into account in
the literature although they give the same order of contribution as $R^{2}$,
due to technical difficulties in distinguishing these three in the heat kernel
expansions around the $S^{4}$ background; see e.g. Refs. [27, 29, 73]. It is
very important to include these terms beyond the current truncation. If they
turn out to take part in the UV critical surface then the UV gravity is not
unitary anymore in general.171717 The standard line of reasoning that higher-
derivative gravity leads to ghost poles in the propagator and thus violates
unitarity is not necessarily applicable in the asymptotic safety context:
Towards the UV, the FRG propagators are still regularized and thus there are
no ghost poles by construction, though this does not mean that the theory is
automatically unitary. The issue of unitarity can only be clarified once all
fluctuations are integrated out and the resulting vacuum state turns out to be
stable (with Minkowski signature). These aspects are discussed in more detail
in Ref. [65]. We thank H. Gies for clarifying this point.  It might also be
interesting if the theory still remains meaningful under such a situation.

In this paper, we have limited ourselves within the LPA where we neglect the
field renormalization and see only the local couplings without external
momenta. LPA has been a useful tool to investigate e.g. the vacuum structure
of the quantum chromodynamics. Although we expect that the LPA is applicable
for sufficiently homogeneous field configurations, it is not clear how good an
approximation the LPA is for the analysis of the asymptotically safe gravity.
It would be useful to go beyond the LPA by taking into account the anomalous
dimensions from the field renormalization.

If the quartic scalar coupling $\lambda_{4}$ has a non-trivial UV fixed point,
namely, $\lambda_{4}^{*}\neq 0$ and $\lambda_{4}\phi^{4}$ is relevant around
the fixed point, then it becomes a solution to the triviality of the scalar
$\phi^{4}$ theory; see e.g. [15, 133]. It would be interesting to study the
triviality under the presence of gravity extending the study in Refs. [58], by
including e.g. the non-minimal coupling.

We comment on the so-called hierarchy problem of the Higgs mass-squared. Let
us write the dimensionful mass-squared
$m^{2}\\!\left(\Lambda\right):=2\lambda_{2}\\!\left(\Lambda\right)\Lambda^{2}$
at the scale $\Lambda$. This reduces to the bare mass at the UV cutoff scale:
$m^{2}\\!\left(\Lambda_{0}\right)=m_{0}^{2}$. For illustration purpose, let us
switch off all the coupling constants except for $\lambda_{2}$ and $y$ in the
RG equation for the mass-squared (61), namely, we take the limits
$\lambda_{0}\to 0$, $\xi_{0}={1\over 16\pi\tilde{G}}\to\infty$, $\xi_{2}\to
0$, and $\lambda_{4}\to 0$:

$\displaystyle-\Lambda{\partial\over\partial\Lambda}m^{2}$
$\displaystyle=-{N_{\text{f}}y^{2}\over 4\pi^{2}}\Lambda^{2}.$ (98)

If we neglect the running of $y$, we get

$\displaystyle m^{2}\\!\left(\Lambda^{2}\right)$ $\displaystyle\simeq
m_{0}^{2}-{N_{\text{f}}y^{2}\over
8\pi^{2}}\Lambda_{0}^{2}+{N_{\text{f}}y^{2}\over 8\pi^{2}}\Lambda^{2}.$ (99)

At very low scales $\Lambda\ll\Lambda_{0}$, the mass-sqaured becomes
$m^{2}\\!\left(\Lambda^{2}\right)\to m_{0}^{2}-{N_{\text{f}}y^{2}\over
8\pi^{2}}\Lambda_{0}^{2}$ and we need the fine-tuning between the bare mass-
squared and the loop correction if we want
$m^{2}\\!\left(\Lambda^{2}\right)\ll\Lambda_{0}^{2}$ ($\sim m_{0}^{2}$). This
is the fine-tuning problem.

This problem still remains in the SM, in principle, even under the
asymptotically safe gravity, e.g. considered in Ref [91]: Suppose we start
from the UV cutoff scale much larger than the Planck scale, $\Lambda_{0}\gg
1/\sqrt{32\pi G}$. Naively, if the dimensionless mass-squared
$\lambda_{2}\\!\left(\Lambda\right)$ turns to be irrelevant around the UV
fixed point as in our result, one might expect that it could be a solution to
the hierarchy problem. However, even if we start from small
$\lambda_{2}\\!\left(\Lambda\right)$ near the UV fixed point
$\Lambda\sim\Lambda_{0}$, and further gets the exponential suppression due to
its irrelevance in the RG evolution departing from the UV fixed point along
the UV critical surface, eventually $\lambda_{2}\\!\left(\Lambda\right)$ will
mix with other relevant operators in the coupled non-linear evolution down to
the Planck scale, and the resultant mass will be of the order of the Planck
scale in general, $\lambda_{2}\\!\left(\Lambda\right)\Lambda^{2}\sim 1/32\pi
G$. It would be interesting to look for a mechanism to keep
$\lambda_{2}\\!\left(\Lambda\right)$ tiny for the scales down to the Planck
scale. Then this sets the boundary condition at the Planck scale for the
subsequent RG evolution further down to the electroweak scale, in which
$\lambda_{2}\\!\left(\Lambda\right)\Lambda^{2}$ and $1/\sqrt{32\pi G}$
correspond to $m_{0}^{2}$ and $\Lambda_{0}$ in Eq. (99), respectively. If we
further manage to find a mechanism to make the sum of SM loop corrections,
corresponding to the second term in the right hand side of Eq. (99), to
vanish, as is speculated by Veltman [134], then the fine-tuning problem is
solved. Note that the observed Higgs mass allows the Veltman condition to be
satisfied at the Planck scale and that the two loop correction to the Veltman
condition is negligibly small [135], although the theoretical explanation why
it holds is still missing. Similarly the cosmological constant problem is yet
to be solved.

### Acknowledgement

We thank Ken-Ichi Aoki, Holger Gies, Yuta Hamada, Aaron Held, Hikaru Kawai,
Jan M. Pawlowski, Roberto Percacci, Gian Paulo Vacca, Satoshi Yamaguchi, and
Ryo Yamamura for useful discussions and comments. We are grateful to the
referees for careful reading and providing the constructive comments. M.Y.
thanks the hospitality of the Particle Physics Theory Group at Osaka
University during his stays. We thank Yuyake Jet at which this collaboration
has started. The work of K.O. is in part supported by the Grant-in-Aid for
Scientific Research Nos. 23104009 and 15K05053. The work of M.Y. is supported
by a Grant-in-Aid for JSPS Fellows (No. 25-5332).

## Appendix

##  Appendix A Wetterich equation

We briefly sketch out the derivation of the Wetterich equation using a simple
scalar theory in flat spacetime without employing the background field method.
Physically, we will derive the effective action $\Gamma_{\Lambda}$ with the
cutoff $\Lambda$, from the bare action $S_{0}$ defined at the UV cutoff scale
$\Lambda_{0}$. Note that in the asymptotic safety scenario, the UV finite
theory is defined on the finite dimensional UV critical surface that consists
of the renormalized trajectories flowing out of the UV fixed point: We define
the bare theory at a point on one of such renormalized trajectories. The
choice of this point and the scale $\Lambda_{0}$ assigned to it are more or
less arbitrary, given the point is right on the renormalized trajectory.181818
The asymptotic safety is somewhat contrary to the ordinary low-energy
effective field theory picture in the sense that we must define the theory
right on the UV critical surface and that even an infinitesimal displacement
from it results in the divergence from it when we track back the
renormalization flow toward UV direction. That is, if we write down all the
possible operators at IR scales allowed by symmetry, then there is
infinitesimally small chance to reach the asymptotically safe theory when we
trace back the renormalization group flow toward UV direction.

We write the partition function

$\displaystyle
Z_{\Lambda}[J]=e^{W_{\Lambda}[J]}:=\int\left[\mathcal{D}\varphi\right]e^{-S_{0}[\varphi]-\Delta
S_{\Lambda}[\varphi]+J\cdot\varphi}$ (100)

with the regulator term

$\displaystyle\Delta S_{\Lambda}[\varphi]$ $\displaystyle:={1\over
2}\int\frac{\text{d}^{4}p}{(2\pi)^{4}}\varphi\\!\left(-p\right)^{\text{T}}R_{\Lambda}\\!\left(p\right)\varphi\\!\left(p\right),$
(101)

where $W_{\Lambda}[J]$ is the generating functional of connected diagrams; we
write $\int_{x}:=\int\text{d}^{4}x$; and we introduce the cutoff profile
function

$\displaystyle R_{\Lambda}\\!\left(p\right)$
$\displaystyle\sim\begin{cases}\Lambda^{2}&\text{for $p<\Lambda$},\\\
0&\text{for $p>\Lambda$},\end{cases}$ (102)

which suppresses the lower momentum modes with $p<\Lambda$ and leaves the
higher ones with $\Lambda<p<\Lambda_{0}$. That is, the low momentum modes with
$p<\Lambda$ are given the extra mass $\Lambda$ in the path integral (100) and
are not effectively path-integrated in the partition function (100).
Therefore, $\Lambda$ can be interpreted as a new UV cutoff scale in
$Z_{\Lambda}[J]$ in which the high momentum modes with $\Lambda<p<\Lambda_{0}$
are integrated out. Namely, $\Lambda$ is the IR cutoff scale for the
integrated high momentum modes, and the UV cutoff scale for the unintegrated
low momentum modes. We also introduce the position-space cutoff function
$R_{\Lambda}\\!\left(x,y\right)=R_{\Lambda}\\!\left(y,x\right)$ by

$\displaystyle\Delta S_{\Lambda}[\varphi]={1\over
2}\int_{x}\int_{y}\,\varphi\\!\left(x\right)^{\text{T}}R_{\Lambda}\\!\left(x,y\right)\varphi\\!\left(y\right).$
(103)

The effective action $\Gamma_{\Lambda}$ is given by the Legendre
transformation of $W_{\Lambda}$:

$\displaystyle\Gamma_{\Lambda}[\Phi]$
$\displaystyle:=J^{\Phi}_{\Lambda}\cdot\Phi-
W_{\Lambda}{[J^{\Phi}_{\Lambda}]}-\Delta S_{\Lambda}{[\Phi]},$ (104)

where $J^{\Phi}_{\Lambda}$ is defined by

$\displaystyle{\delta W_{\Lambda}\over\delta
J\\!\left(x\right)}[J^{\Phi}_{\Lambda}]$
$\displaystyle=\Phi\\!\left(x\right).$ (105)

Then we get

$\displaystyle{\delta\Gamma_{\Lambda}\over\delta\Phi\\!\left(x\right)}[\Phi]$
$\displaystyle=\int_{y}{\delta
J^{\Phi}_{\Lambda}\\!\left(y\right)\over\delta\Phi\\!\left(x\right)}\Phi\\!\left(y\right)+J^{\Phi}_{\Lambda}\\!\left(x\right)-\int_{y}{\delta
W_{\Lambda}\over\delta J\\!\left(y\right)}[J^{\Phi}_{\Lambda}]\,{\delta
J^{\Phi}_{\Lambda}\\!\left(y\right)\over\delta\Phi\\!\left(x\right)}-{\delta\Delta
S_{\Lambda}\over\delta\Phi\\!\left(x\right)}[\Phi]$
$\displaystyle=J^{\Phi}_{\Lambda}\\!\left(x\right)-\int_{y}R_{\Lambda}\\!\left(x,y\right)\Phi\\!\left(y\right),$
(106)

and further

$\displaystyle{\delta^{2}\Gamma_{\Lambda}\over\delta\Phi\\!\left(x\right)\delta\Phi\\!\left(y\right)}[\Phi]$
$\displaystyle={\delta
J^{\Phi}_{\Lambda}\\!\left(x\right)\over\delta\Phi\\!\left(y\right)}-R_{\Lambda}\\!\left(x,y\right).$
(107)

We similarly define $\Phi^{J}_{\Lambda}$ by

$\displaystyle{\delta W_{\Lambda}\over\delta J\\!\left(x\right)}[J]$
$\displaystyle=\Phi^{J}_{\Lambda}\\!\left(x\right)$ (108)

so that $\Phi^{J}_{\Lambda}=\Phi$ if $J=J^{\Phi}_{\Lambda}$. Taking a
functional derivative of Eq. (108), we get

$\displaystyle{\delta^{2}W_{\Lambda}\over\delta J\\!\left(x\right)\delta
J\\!\left(y\right)}[J]$
$\displaystyle={\delta\Phi^{J}_{\Lambda}\\!\left(x\right)\over\delta
J\\!\left(y\right)}[J],$ (109)

and hence

$\displaystyle{\delta^{2}W_{\Lambda}\over\delta J\\!\left(x\right)\delta
J\\!\left(y\right)}[J_{\Lambda}^{\Phi}]$
$\displaystyle={\delta\Phi^{J}_{\Lambda}\\!\left(x\right)\over\delta
J\\!\left(y\right)}[J_{\Phi}^{\Lambda}]=\left({\delta
J^{\Phi}_{\Lambda}\\!\left(x\right)\over\delta\Phi\\!\left(y\right)}[\Phi]\right)^{-1}=\left({\delta^{2}\Gamma_{\Lambda}\over\delta\Phi\\!\left(x\right)\delta\Phi\\!\left(y\right)}[\Phi]+R_{\Lambda}\\!\left(x,y\right)\right)^{-1},$
(110)

where the inverse is in the functional space spanned by $x$ and $y$ and we
have used Eq. (107) in the last step.

We want to evaluate

$\displaystyle{\text{d}\Gamma_{\Lambda}[\Phi]\over\text{d}\Lambda}$
$\displaystyle=\int_{x}{\text{d}J^{\Phi}_{\Lambda}\\!\left(x\right)\over\text{d}\Lambda}\Phi\\!\left(x\right)-{\text{d}W_{\Lambda}[J^{\Phi}_{\Lambda}]\over\text{d}\Lambda}-{\text{d}\Delta
S_{\Lambda}[\Phi]\over\text{d}\Lambda}.$ (111)

After some computation, the second term in Eq. (111) becomes191919 Concretely,
$\displaystyle-{\text{d}W_{\Lambda}[J^{\Phi}_{\Lambda}]\over\text{d}\Lambda}$
$\displaystyle={1\over
Z[J^{\Phi}_{\Lambda}]}\int\left[\mathcal{D}\varphi\right]e^{-S_{\Lambda_{0}}[\varphi]-\Delta
S_{\Lambda}[\varphi]+J^{\Phi}_{\Lambda}\cdot\varphi}\left({\text{d}\Delta
S_{\Lambda}[\varphi]\over\text{d}\Lambda}-{\text{d}J^{\Phi}_{\Lambda}\over\text{d}\Lambda}\cdot\varphi\right)$
$\displaystyle={1\over
Z[J^{\Phi}_{\Lambda}]}\int\left[\mathcal{D}\varphi\right]e^{-S_{\Lambda_{0}}[\varphi]-\Delta
S_{\Lambda}[\varphi]+J^{\Phi}_{\Lambda}\cdot\varphi}\left(\int_{x}\int_{y}{1\over
2}\varphi\\!\left(x\right){\text{d}R_{\Lambda}\\!\left(x,y\right)\over\text{d}\Lambda}\varphi\\!\left(y\right)-\int_{x}{\text{d}J^{\Phi}_{\Lambda}\\!\left(x\right)\over\text{d}\Lambda}\varphi\\!\left(x\right)\right)$
$\displaystyle=\left({1\over
Z[J^{\Phi}_{\Lambda}]}\int\left[\mathcal{D}\varphi\right]e^{-S_{\Lambda_{0}}[\varphi]-\Delta
S_{\Lambda}[\varphi]+J^{\Phi}_{\Lambda}\cdot\varphi}\int_{x}\int_{y}{1\over
2}\varphi\\!\left(x\right){\text{d}R_{\Lambda}\\!\left(x,y\right)\over\text{d}\Lambda}\varphi\\!\left(y\right)\right)-\int_{x}{\text{d}J^{\Phi}_{\Lambda}\\!\left(x\right)\over\text{d}\Lambda}{\delta
W_{\Lambda}\over\delta J\\!\left(x\right)}[J^{\Phi}_{\Lambda}]$
$\displaystyle=\left({1\over
Z[J^{\Phi}_{\Lambda}]}\int\left[\mathcal{D}\varphi\right]e^{-S_{\Lambda_{0}}[\varphi]-\Delta
S_{\Lambda}[\varphi]+J^{\Phi}_{\Lambda}\cdot\varphi}\,{1\over
2}\int_{x}\int_{y}\varphi\\!\left(x\right){\text{d}R_{\Lambda}\\!\left(x,y\right)\over\text{d}\Lambda}\varphi\\!\left(y\right)\right)-\int_{x}{\text{d}J^{\Phi}_{\Lambda}\\!\left(x\right)\over\text{d}\Lambda}\Phi\\!\left(x\right)$
$\displaystyle={1\over
2Z[J^{\Phi}_{\Lambda}]}\int_{x}\int_{y}{\delta^{2}Z\over\delta
J\\!\left(x\right)\delta
J\\!\left(y\right)}[J^{\Phi}_{\Lambda}]\,{\text{d}R_{\Lambda}\\!\left(x,y\right)\over\text{d}\Lambda}-\int_{x}{\text{d}J^{\Phi}_{\Lambda}\\!\left(x\right)\over\text{d}\Lambda}\Phi\\!\left(x\right)$
$\displaystyle={1\over
2}\int_{x}\int_{y}\left({\delta^{2}W_{\Lambda}\over\delta
J\\!\left(x\right)\delta
J\\!\left(y\right)}[J^{\Phi}_{\Lambda}]+\Phi\\!\left(x\right)\Phi\\!\left(y\right)\right){\text{d}R_{\Lambda}\\!\left(x,y\right)\over\text{d}\Lambda}-\int_{x}{\text{d}J^{\Phi}_{\Lambda}\\!\left(x\right)\over\text{d}\Lambda}\Phi\\!\left(x\right)$
$\displaystyle={1\over 2}\int_{x}\int_{y}{\delta^{2}W_{\Lambda}\over\delta
J\\!\left(x\right)\delta
J\\!\left(y\right)}[J^{\Phi}_{\Lambda}]\,{\text{d}R_{\Lambda}\\!\left(x,y\right)\over\text{d}\Lambda}+{\text{d}\Delta
S_{\Lambda}[\Phi]\over\text{d}\Lambda}-\int_{x}{\text{d}J^{\Phi}_{\Lambda}\\!\left(x\right)\over\text{d}\Lambda}\Phi\\!\left(x\right),$
where we have used  $\displaystyle{\delta W_{\Lambda}\over\delta
J\\!\left(x\right)}[J]$ $\displaystyle={1\over Z[J]}{\delta
Z_{\Lambda}\over\delta J\\!\left(x\right)}[J],$
$\displaystyle{\delta^{2}W_{\Lambda}\over\delta J\\!\left(x\right)\delta
J\\!\left(y\right)}[J]$ $\displaystyle={1\over
Z[J]}{\delta^{2}Z_{\Lambda}\over\delta J\\!\left(x\right)\delta
J\\!\left(y\right)}[J]-\Phi^{J}_{\Lambda}\\!\left(x\right)\Phi^{J}_{\Lambda}\\!\left(y\right).$

$\displaystyle-{\text{d}W_{\Lambda}[J^{\Phi}_{\Lambda}]\over\text{d}\Lambda}$
$\displaystyle={1\over 2}\int_{x}\int_{y}{\delta^{2}W_{\Lambda}\over\delta
J\\!\left(x\right)\delta
J\\!\left(y\right)}[J^{\Phi}_{\Lambda}]\,{\text{d}R_{\Lambda}\\!\left(x,y\right)\over\text{d}\Lambda}+{\text{d}\Delta
S_{\Lambda}[\Phi]\over\text{d}\Lambda}-\int_{x}{\text{d}J^{\Phi}_{\Lambda}\\!\left(x\right)\over\text{d}\Lambda}\Phi\\!\left(x\right).$
(112)

Therefore,

$\displaystyle{\text{d}\Gamma_{\Lambda}[\Phi]\over\text{d}\Lambda}$
$\displaystyle={1\over 2}\int_{x}\int_{y}{\delta^{2}W_{\Lambda}\over\delta
J\\!\left(x\right)\delta
J\\!\left(y\right)}[J^{\Phi}_{\Lambda}]\,{\text{d}R_{\Lambda}\\!\left(x,y\right)\over\text{d}\Lambda}.$
(113)

Putting Eq. (110) into Eq. (113), we get the Wetterich equation

$\displaystyle{\text{d}\Gamma_{\Lambda}[\Phi]\over\text{d}\Lambda}$
$\displaystyle={1\over
2}\int_{x}\int_{y}\left({\delta^{2}\Gamma_{\Lambda}\over\delta\Phi\\!\left(x\right)\delta\Phi\\!\left(y\right)}[\Phi]+R_{\Lambda}\\!\left(x,y\right)\right)^{-1}\,{\text{d}R_{\Lambda}\\!\left(x,y\right)\over\text{d}\Lambda}.$
(114)

For general case including fermions, this expression becomes Eq. (122). We see
that the Wetterich equation is one-loop exact from its derivation.

##  Appendix B Supertrace

A supertrace of a supermatrix

$\displaystyle M$ $\displaystyle=\begin{bmatrix}M_{\text{BB}}&M_{\text{BF}}\\\
M_{\text{FB}}&M_{\text{FF}}\end{bmatrix}$ (115)

is defined by

$\displaystyle\operatorname{str}M$
$\displaystyle=\operatorname{tr}M_{\text{BB}}-\operatorname{tr}M_{\text{FF}},$
(116)

which satisfies
$\operatorname{str}\\!\left(MN\right)=\operatorname{str}\\!\left(NM\right)$. A
superdeterminant is defined by

$\displaystyle\operatorname{sdet}M$
$\displaystyle=\exp\\!\left(\operatorname{str}\ln M\right),$ (117)

satisfying

$\displaystyle\operatorname{sdet}\\!\left(MN\right)$
$\displaystyle=\operatorname{sdet}M\,\operatorname{sdet}N.$ (118)

We see

$\displaystyle\operatorname{sdet}M$
$\displaystyle=\operatorname{sdet}\begin{bmatrix}M_{\text{BB}}&0\\\
M_{\text{FB}}&1\end{bmatrix}\,\operatorname{sdet}\begin{bmatrix}1&M_{\text{BB}}^{-1}M_{\text{BF}}\\\
0&M_{\text{FF}}-M_{\text{FB}}M_{\text{BB}}^{-1}M_{\text{BF}}\end{bmatrix}$
$\displaystyle={\det
M_{\text{BB}}\over\det\\!\left(M_{\text{FF}}-M_{\text{FB}}M_{\text{BB}}^{-1}M_{\text{BF}}\right)}.$
(119)

If we decompose the matrix as

$\displaystyle
M=\begin{bmatrix}M_{\text{BB}}-M_{\text{BF}}M_{\text{FF}}^{{-1}}M_{\text{FB}}&M_{\text{BF}}M_{\text{FF}}^{-1}\\\
0&1\end{bmatrix}\begin{bmatrix}1&0\\\
M_{\text{FB}}&M_{\text{FF}}\end{bmatrix},$ (120)

we see

$\displaystyle\operatorname{sdet}M$
$\displaystyle={\det\\!\left(M_{\text{BB}}-M_{\text{BF}}M_{\text{FF}}^{{-1}}M_{\text{FB}}\right)\over\det
M_{\text{FF}}}.$ (121)

In this paper we use (119).

##  Appendix C Functional renormalization group for the effective action

In Appendix A, we have briefly reviewed the derivation of the Wetterich
equation for a simple scalar case. For general case including fermions, the
Wetterich equation reads [136, 137]

$\displaystyle{\partial\over\partial\Lambda}\Gamma_{\Lambda}$
$\displaystyle=\frac{1}{2}\operatorname{STr}_{x,y}\left[\left(\frac{\overrightarrow{\delta}}{\delta\Phi\\!\left(x\right)}\Gamma_{\Lambda}\frac{\overleftarrow{\delta}}{\delta\Phi\\!\left(y\right)}+\mathcal{R}_{\Lambda}\\!\left(x,y\right)\right)^{-1}\cdot{\partial\over\partial\Lambda}\mathcal{R}_{\Lambda}\\!\left(y,x\right)\right].$
(122)

For later convenience, let us briefly review how to treat the supermatrix in
the Wetterich equation.

We separate the two-point function and the cutoff function
$\mathcal{R}_{\Lambda}$ into bosonic and fermionic parts, respectively,

$\displaystyle\begin{bmatrix}\Gamma_{\text{BB}}&\Gamma_{\text{BF}}\\\
\Gamma_{\text{FB}}&\Gamma_{\text{FF}}\end{bmatrix}$
$\displaystyle:=\frac{\overrightarrow{\delta}}{\delta\Phi\\!\left(x\right)}\Gamma_{\Lambda}\frac{\overleftarrow{\delta}}{\delta\Phi\\!\left(y\right)},$
$\displaystyle\begin{bmatrix}\mathcal{R}_{\text{BB}}&0\\\
0&\mathcal{R}_{\text{FF}}\end{bmatrix}$
$\displaystyle:=\mathcal{R}_{\Lambda}\\!\left(x,y\right).$ (123)

We also define

$\displaystyle\begin{bmatrix}{\mathcal{M}}_{\text{BB}}&{\mathcal{M}}_{\text{BF}}\\\
{\mathcal{M}}_{\text{FB}}&{\mathcal{M}}_{\text{FF}}\end{bmatrix}$
$\displaystyle:=\begin{bmatrix}\Gamma_{\text{BB}}&\Gamma_{\text{BF}}\\\
\Gamma_{\text{FB}}&\Gamma_{\text{FF}}\end{bmatrix}+\begin{bmatrix}\mathcal{R}_{\text{BB}}&0\\\
0&\mathcal{R}_{\text{FF}}\end{bmatrix}.$ (124)

Then we rewrite the Wetterich equation as

$\displaystyle{\partial\over\partial\Lambda}\Gamma_{\Lambda}$
$\displaystyle=\widetilde{\partial\over\partial\Lambda}{1\over
2}\ln\\!\left(\operatorname{SDet}\begin{bmatrix}{\mathcal{M}}_{\text{BB}}&{\mathcal{M}}_{\text{BF}}\\\
{\mathcal{M}}_{\text{FB}}&{\mathcal{M}}_{\text{FF}}\end{bmatrix}\right)$
$\displaystyle={1\over
2}\widetilde{\partial\over\partial\Lambda}\left(\ln\bigg{[}\operatorname{Det}{\mathcal{M}}_{\text{BB}}\bigg{]}-\ln\bigg{[}\operatorname{Det}\Big{(}{\mathcal{M}}_{\text{FF}}-{\mathcal{M}}_{\text{FB}}{\mathcal{M}}_{\text{BB}}^{-1}{\mathcal{M}}_{\text{BF}}\Big{)}\bigg{]}\right)$
$\displaystyle={1\over
2}\widetilde{\partial\over\partial\Lambda}\operatorname{Tr}\bigg{[}\ln{\mathcal{M}}_{\text{BB}}\bigg{]}-{1\over
2}\widetilde{\partial\over\partial\Lambda}\operatorname{Tr}\bigg{[}\ln\Big{(}{\mathcal{M}}_{\text{FF}}-{\mathcal{M}}_{\text{FB}}{\mathcal{M}}_{\text{BB}}^{-1}{\mathcal{M}}_{\text{BF}}\Big{)}\bigg{]},$
(125)

where $\widetilde{\partial/\partial\Lambda}$ acts only on
$\mathcal{R}_{\text{BB}}$ and $\mathcal{R}_{\text{FF}}$ and we used the
formulation for supermatrix summarized in Appendix B. Performing the
derivative, we obtain

$\displaystyle{\partial\over\partial\Lambda}\Gamma_{\Lambda}$
$\displaystyle={1\over
2}\operatorname{Tr}\bigg{[}{\mathcal{M}}_{\text{BB}}^{-1}{\partial\mathcal{R}_{\text{BB}}\over\partial\Lambda}\bigg{]}$
$\displaystyle\quad-{1\over
2}\operatorname{Tr}\bigg{[}\Big{(}{\mathcal{M}}_{\text{FF}}-{\mathcal{M}}_{\text{FB}}{\mathcal{M}}_{\text{BB}}^{-1}{\mathcal{M}}_{\text{BF}}\Big{)}^{-1}\left({\partial\mathcal{R}_{\text{FF}}\over\partial\Lambda}+{\mathcal{M}}_{\text{FB}}{\mathcal{M}}_{\text{BB}}^{-1}\,{\partial\mathcal{R}_{\text{BB}}\over\partial\Lambda}\,{\mathcal{M}}_{\text{BB}}^{-1}{\mathcal{M}}_{\text{BF}}\right)\bigg{]}.$
(126)

The first term in RHS of (126) is the fluctuations of bosonic fields. The
second term includes not only the fluctuations of fermionic fields but also
the mixing of fermion and boson. We have directly computed the algebraic
expression in the right hand side of Eq. (126) to cross-check the results in
Sec. 3 that is obtained diagrammatically.

We may also expand the second term in Eq. (125) as

$\displaystyle-{1\over
2}\widetilde{\partial\over\partial\Lambda}\operatorname{Tr}\bigg{[}\ln\Big{(}{\mathcal{M}}_{\text{FF}}-{\mathcal{M}}_{\text{FB}}{\mathcal{M}}_{\text{BB}}^{-1}{\mathcal{M}}_{\text{BF}}\Big{)}\bigg{]}$
$\displaystyle=-{1\over
2}\widetilde{\partial\over\partial\Lambda}\operatorname{Tr}\bigg{[}\ln{\mathcal{M}}_{\text{FF}}+\ln\Big{(}1-{\mathcal{M}}_{\text{FF}}^{-1}{\mathcal{M}}_{\text{FB}}{\mathcal{M}}_{\text{BB}}^{-1}{\mathcal{M}}_{\text{BF}}\Big{)}\bigg{]}$
$\displaystyle=-{1\over
2}\widetilde{\partial\over\partial\Lambda}\left(\operatorname{Tr}\bigg{[}\ln{\mathcal{M}}_{\text{FF}}\bigg{]}-\operatorname{Tr}\bigg{[}{\mathcal{M}}_{\text{FF}}^{-1}{\mathcal{M}}_{\text{FB}}{\mathcal{M}}_{\text{BB}}^{-1}{\mathcal{M}}_{\text{BF}}\bigg{]}+\cdots\right)$
$\displaystyle=-{1\over
2}\operatorname{Tr}\bigg{[}{\mathcal{M}}_{\text{FF}}^{-1}\,{\partial\mathcal{R}_{\text{FF}}\over\partial\Lambda}\bigg{]}$
$\displaystyle\quad-{1\over
2}\operatorname{Tr}\bigg{[}{\mathcal{M}}_{\text{FF}}^{-1}\,{\partial\mathcal{R}_{\text{FF}}\over\partial\Lambda}\,{\mathcal{M}}_{\text{FF}}^{-1}{\mathcal{M}}_{\text{FB}}{\mathcal{M}}_{\text{BB}}^{-1}{\mathcal{M}}_{\text{BF}}\bigg{]}$
$\displaystyle\quad-{1\over
2}\operatorname{Tr}\bigg{[}{\mathcal{M}}_{\text{FF}}^{-1}{\mathcal{M}}_{\text{FB}}{\mathcal{M}}_{\text{BB}}^{-1}\,{\partial\mathcal{R}_{\text{BB}}\over\partial\Lambda}\,{\mathcal{M}}_{\text{BB}}^{-1}{\mathcal{M}}_{\text{BF}}\bigg{]}+\cdots{,}$
(127)

where the higher order terms, represented by dots, are all higher-dimensional
operators being already truncated in Eq. (13), and hence we neglect them in
this paper. This expression (127) is useful to compare with the Feynman
diagramatic computation since the vertex structure is clearer. It is
especially useful when evaluating the beta function of the Yukawa coupling
constant. We have also used this expression to further cross-check the results
in Sec. 3.

##  Appendix D Heat kernel trace

In this section we briefly review how to take the trace in the heat kernel
expansion; see e.g. Refs. [70, 138, 139] for more detailed reviews. Consider
an arbitrary function $W\\!\left(p^{2}\right)$ and its trace
$\operatorname{Tr}\left[W\\!\left(p^{2}\right)\right]$. Using the Laplace
transformation, we get

$\displaystyle\operatorname{Tr}\left[W\\!\left(p^{2}\right)\right]=\int^{\infty}_{-\infty}\text{d}s\,{\tilde{W}}\\!\left(s\right)\operatorname{Tr}\left[e^{-s\,p^{2}}\right].$
(128)

The trace in the right-hand side can be expanded as

$\displaystyle\operatorname{Tr}\left[e^{-s\,p^{2}}\right]=\sum_{n=0}B_{n}\\!\left(p^{2}\right)s^{-\left(D-n\right)/2},$
(129)

where

$\displaystyle
B_{n}\\!\left(p^{2}\right)=\frac{1}{(4\pi)^{D/2}}\int\text{d}^{D}x\sqrt{g}\,\operatorname{tr}\left[{\bf
b}_{n}\right].$ (130)

The heat kernel coefficients ${\bf b}_{n}$ are given by ${\bf b}_{0}={\bf 1}$,
${\bf b}_{2}={R\over 6}\,{\bf 1}$, etc., where ${\bf 1}$ is the identity on
the spin representation of the field. Their explicit values are shown in Table
1. For higher order ($n>2$), see e.g., the appendix of [70]. By inserting
(130) into the right-hand side of (128), we obtain

$\displaystyle\operatorname{Tr}\,[W(p^{2})]=\frac{1}{(4\pi)^{D/2}}\bigg{\\{}Q_{\frac{D}{2}}[W]\int\text{d}^{D}x\sqrt{g}\operatorname{tr}[\bar{\bf
b}_{0}]+Q_{\frac{D}{2}-1}[W]\int\text{d}^{D}x\sqrt{g}R\operatorname{tr}[\bar{\bf
b}_{2}]+{\mathcal{O}}(R^{2})\bigg{\\}},$ (131)

where $\bar{\bf b}_{0}:={\bf 1}$, $\bar{\bf b}_{2}:={\bf 1}/6$, and

$\displaystyle
Q_{n}[W]:=\int^{\infty}_{-\infty}\text{d}s(-s)^{-n}{\tilde{W}}(s).$ (132)

Its Mellin transformation yields

$\displaystyle Q_{0}[W]$ $\displaystyle=W(0),$ $\displaystyle Q_{n}[W]$
$\displaystyle=\frac{1}{\Gamma[n]}\int^{\infty}_{0}\text{d}z\,z^{n-1}W[z],$
(133)

where $\Gamma[n]$ is the Gamma function. Thanks to above relations (131) and
(133), the trace for the eigenvalues of the derivative operator can be
evaluated in curved space.

Table 1: heat kernel coefficients for the individual fields in $D=4$ |  $h^{\perp}_{\mu\nu}$ (spin 2) |  $\xi_{\mu}$, $C^{\perp}_{\mu}$ (spin 1) |  $\psi,{\overline{\psi}}$ (spin 1/2) |  $h,\sigma,\phi,C,{\bar{C}}$ (spin 0)
---|---|---|---|---
$\operatorname{tr}[\bar{\bf b}_{0}]=:b_{0}$ | $5$ | $3$ | 4 | 1
$\operatorname{tr}[\bar{\bf b}_{2}]=:b_{2}$ | $\displaystyle-\frac{5}{6}$ | $\displaystyle\frac{1}{4}$ | $\displaystyle\frac{2}{3}$ | $\displaystyle\frac{1}{6}$

##  Appendix E Derivation of the beta functions

In this appendix we show the diagrammatic derivation of the beta functions.

###  E.1 Bosonic contributions

We evaluate the first term in Eq. (126) corresponding to the boson loops:

$\displaystyle{1\over
2}\operatorname{Tr}\bigg{[}{\mathcal{M}}_{\text{BB}}^{-1}{\partial\mathcal{R}_{\text{BB}}\over\partial\Lambda}\bigg{]}=\frac{1}{2}{\rm
Tr}\left.\frac{\partial_{t}{\mathcal{R}}_{\Lambda}}{\Gamma_{\Lambda}^{(1,1)}+{\mathcal{R}}_{\Lambda}}\right|_{h^{\perp}h^{\perp}}+\frac{1}{2}{\rm
Tr}^{\prime}\left.\frac{\partial_{t}{\mathcal{R}}_{\Lambda}}{\Gamma_{\Lambda}^{(1,1)}+{\mathcal{R}}_{\Lambda}}\right|_{\xi\xi}+\frac{1}{2}{\rm
Tr}^{\prime\prime}\left.\frac{\partial_{t}{\mathcal{R}}_{\Lambda}}{\Gamma_{\Lambda}^{(1,1)}+{\mathcal{R}}_{\Lambda}}\right|_{\rm
SS}.$ (134)

We evaluate these contributions part by part using the explicit form of two-
point functions and cutoff functions exhibited in subsection 3.2.

####  E.1.1 The loop contribution of the transverse gravity field

The loop contribution of $h^{\perp}$ field is evaluated:

$\displaystyle\frac{1}{2}{\rm
Tr}\left.\frac{\partial_{t}{\mathcal{R}}_{\Lambda}}{\Gamma_{\Lambda}^{(1,1)}+{\mathcal{R}}_{\Lambda}}\right|_{h^{\perp}h^{\perp}}$
$\displaystyle=\frac{1}{2}{\rm
Tr}~{}\frac{\frac{1}{2}\left(\partial_{t}F\right)R_{\Lambda}+\frac{1}{2}F\left(\partial_{t}R_{\Lambda}\right)}{\frac{1}{2}F(P_{\Lambda}+\frac{2}{3}{R})-\frac{1}{2}V-\frac{1}{2}Y}$
$\displaystyle=\frac{1}{2}{\rm
Tr}~{}\frac{\left(\partial_{t}F\right)R_{\Lambda}+F\left(\partial_{t}R_{\Lambda}\right)}{FP_{\Lambda}-V-Y}-\frac{1}{3}{\rm
Tr}~{}\frac{\left(\partial_{t}F\right)R_{\Lambda}+F\left(\partial_{t}R_{\Lambda}\right)}{(FP_{\Lambda}-V-Y)^{2}}\,F{R}+\mathcal{O}({R}^{2}).$
(135)

Using the heat kernel expansion given in Appendix D, we evaluate the trace for
$\mathcal{O}(R^{0})$,

$\displaystyle\frac{1}{2}{\rm Tr}[W_{h^{\perp}}^{1}]$
$\displaystyle:=\frac{1}{2}{\rm
Tr}\,\frac{\left(\partial_{t}F\right)R_{\Lambda}+F\left(\partial_{t}R_{\Lambda}\right)}{FP_{\Lambda}-V-Y}$
$\displaystyle=\frac{1}{2}\frac{1}{\left(4\pi\right)^{2}}\left\\{b_{0}^{h^{\perp}}Q_{2}[W_{h^{\perp}}^{1}]\int\text{d}^{4}x\sqrt{g}+b_{2}^{h^{\perp}}Q_{1}[W_{h^{\perp}}^{1}]\int\text{d}^{4}x\sqrt{g}{R}\right\\}+\mathcal{O}({R}^{2})$
(136)

with the functions

$\displaystyle Q_{2}[W_{h^{\perp}}^{1}]$
$\displaystyle=\frac{1}{\Gamma(2)}\int_{0}^{\infty}\text{d}z\,zW_{h^{\perp}}^{1}(z)=\int_{0}^{\infty}\text{d}z\,z\frac{\left(\partial_{t}F\right)R_{\Lambda}+F\left(\partial_{t}R_{\Lambda}\right)}{FP_{\Lambda}-V-Y}$
$\displaystyle=\frac{\Lambda^{6}}{F\Lambda^{2}-V}\left[\frac{1}{6}\left(\partial_{t}F\right)-F\right]+\frac{\Lambda^{6}}{\left(F\Lambda^{2}-V\right)^{2}}\left[\frac{1}{6}\left(\partial_{t}F\right)-F\right]\,Y+\mathcal{O}(Y^{2}),$
(137) $\displaystyle Q_{1}[W_{h^{\perp}}^{1}]$
$\displaystyle=\frac{1}{\Gamma(1)}\int^{\infty}_{0}\text{d}z~{}W_{T1}(z)=\int^{\infty}_{0}\text{d}z~{}\frac{\left(\partial_{t}F\right)R_{\Lambda}+F\left(\partial_{t}R_{\Lambda}\right)}{FP_{\Lambda}-V-Y}$
$\displaystyle=\frac{\Lambda^{4}}{F\Lambda^{2}-V}\left[\frac{1}{2}\left(\partial_{t}F\right)-2F\right]+\mathcal{O}(Y),$
(138)

and for $\mathcal{O}(R)$,

$\displaystyle-\frac{1}{3}{\rm Tr}[W_{h^{\perp}}^{2}]{R}$
$\displaystyle:=-\frac{1}{3}{\rm
Tr}\,\frac{\left(\partial_{t}F\right)R_{\Lambda}+F\left(\partial_{t}R_{\Lambda}\right)}{(FP_{\Lambda}-V-Y)^{2}}\,F{R}$
$\displaystyle=\frac{1}{3}\frac{1}{\left(4\pi\right)^{2}}\left\\{b_{0}^{h^{\perp}}Q_{2}[W_{h^{\perp}}^{2}]\int\text{d}^{4}x\sqrt{g}{R}\right\\}+\mathcal{O}(Y,{R}^{2})$
(139)

with

$\displaystyle Q_{2}[W_{h^{\perp}}^{2}]$
$\displaystyle=\frac{1}{\Gamma(2)}\int^{\infty}_{0}\text{d}z\,z\frac{\left(\partial_{t}F\right)R_{\Lambda}+F\left(\partial_{t}R_{\Lambda}\right)}{(FP_{\Lambda}-V)^{2}}F$
$\displaystyle=\frac{\Lambda^{6}F}{\left(F\Lambda^{2}-V\right)^{2}}\left[\frac{1}{6}\left(\partial_{t}F\right)-F\right].$
(140)

To summarize, we get the loop contributions of $h^{\perp}$ field:

$\displaystyle\frac{1}{2}{\rm
Tr}\left.\frac{\partial_{t}{\mathcal{R}}_{\Lambda}}{\Gamma_{\Lambda}^{(1,1)}+{\mathcal{R}}_{\Lambda}}\right|_{h^{\perp}h^{\perp}}$
$\displaystyle=\frac{b_{0}^{h^{\perp}}}{2\left(4\pi\right)^{2}}\frac{\Lambda^{6}}{F\Lambda^{2}-V}\left[\frac{1}{6}(\partial_{t}F)-F\right]\int\text{d}^{4}x\sqrt{g}$
$\displaystyle\quad+\frac{1}{\left(4\pi\right)^{2}}\left\\{\frac{b_{2}^{h^{\perp}}}{2}\frac{\Lambda^{4}}{F\Lambda^{2}-V}\left[\frac{1}{2}\left(\partial_{t}F\right)-2F\right]-\frac{b_{0}^{h^{\perp}}}{3}\frac{\Lambda^{6}F}{\left(F\Lambda^{2}-V\right)^{2}}\left[\frac{1}{6}\left(\partial_{t}F\right)-F\right]\right\\}$
$\displaystyle\qquad\times\int\text{d}^{4}x\sqrt{g}{R}$
$\displaystyle\quad+\frac{b_{0}^{h^{\perp}}}{\left(4\pi\right)^{2}}\frac{\Lambda^{6}}{\left(F\Lambda^{2}-V\right)^{2}}\left[\frac{1}{6}\left(\partial_{t}F\right)-F\right]\int\text{d}^{4}x\sqrt{g}Y+\mathcal{O}(Y^{2},{R}^{2}).$
(141)

The first, second and third term are the contribution to $V$, $F$ and
$Y=y\phi\overline{\psi}\psi$, respectively. The correction to the Yukawa
interaction corresponds to the diagram (I) in Fig. 5.

####  E.1.2 The loop contribution of the gravity field with spin 1

We evaluate the loop contribution of $\xi$ field:

$\displaystyle\frac{1}{2}{\rm
Tr}^{\prime}\left.\frac{\partial_{t}{\mathcal{R}}_{\Lambda}}{\Gamma_{\Lambda}^{(1,1)}+{\mathcal{R}}_{\Lambda}}\right|_{\xi\xi}$
$\displaystyle=\frac{1}{2}{\rm
Tr}^{\prime}\frac{\left(\partial_{t}F\right)R_{\Lambda}+F\left(\partial_{t}R_{\Lambda}\right)}{F(P_{\Lambda}+\frac{2\alpha-1}{4}{R})-\alpha(V+Y)}$
$\displaystyle=\frac{1}{2}{\rm
Tr}^{\prime}\frac{\left(\partial_{t}F\right)R_{\Lambda}+F\left(\partial_{t}R_{\Lambda}\right)}{FP_{\Lambda}-\alpha(V+Y)}-\frac{1}{8}{\rm
Tr}^{\prime}\frac{\left(\partial_{t}F\right)R_{\Lambda}+F\left(\partial_{t}R_{\Lambda}\right)}{(FP_{\Lambda}-\alpha(V+Y))^{2}}F(2\alpha-1){R}+{\mathcal{O}}({R}^{2}).$
(142)

We obtain

$\displaystyle\frac{1}{2}{\rm Tr}[W_{\xi}^{1}]$
$\displaystyle:=\frac{1}{2}{\rm
Tr}\frac{\left(\partial_{t}F\right)R_{\Lambda}+F\left(\partial_{t}R_{\Lambda}\right)}{FP_{\Lambda}-\alpha(V+Y)}$
$\displaystyle=\frac{1}{2}\frac{1}{\left(4\pi\right)^{2}}\left\\{b^{\xi}_{0}Q_{2}[W_{\xi}^{1}]\int\text{d}^{4}x\sqrt{g}+b^{\xi}_{2}Q_{1}[W_{\xi}^{1}]\int\text{d}^{4}x\sqrt{{g}}{R}\right\\}+{\mathcal{O}}(R^{2}),$
(143)

where

$\displaystyle Q_{2}[W_{\xi}^{1}]$
$\displaystyle=\frac{1}{\Gamma(2)}\int^{\infty}_{0}\text{d}z\,z\frac{\left(\partial_{t}F\right)R_{\Lambda}+F\left(\partial_{t}R_{\Lambda}\right)}{FP_{\Lambda}-\alpha(V+Y)}=\frac{\Lambda^{6}}{F\Lambda^{2}-\alpha(V+Y)}\left[\frac{1}{6}\left(\partial_{t}F\right)-F\right],$
(144) $\displaystyle Q_{1}[W_{\xi}^{1}]$
$\displaystyle=\frac{1}{\Gamma(1)}\int^{\infty}_{0}\text{d}z\,\frac{\left(\partial_{t}F\right)R_{\Lambda}+F\left(\partial_{t}R_{\Lambda}\right)}{FP_{\Lambda}-\alpha(V+Y)}=\frac{\Lambda^{4}}{F\Lambda^{2}-\alpha
V}\left[\frac{1}{2}\left(\partial_{t}F\right)-2F\right],$ (145)

and

$\displaystyle-\frac{1}{8}{\rm Tr}[W_{\xi}^{2}]{R}$
$\displaystyle:=-\frac{1}{8}{\rm
Tr}\frac{\left(\partial_{t}F\right)R_{\Lambda}+F\left(\partial_{t}R_{\Lambda}\right)}{(FP_{\Lambda}-\alpha(V+Y))^{2}}F(2\alpha-1){R}$
$\displaystyle=-\frac{1}{8}\frac{1}{\left(4\pi\right)^{2}}\left\\{b^{\xi}_{0}Q_{2}[W_{\xi}^{2}]\int\text{d}^{4}x\sqrt{g}{R}\right\\}+{\mathcal{O}}({R}^{2}),$

with

$\displaystyle Q_{2}[W_{\xi}^{2}]$
$\displaystyle=\frac{1}{\Gamma(2)}\int^{\infty}_{0}\text{d}z\,z\frac{\left(\partial_{t}F\right)R_{\Lambda}+F\left(\partial_{t}R_{\Lambda}\right)}{(FP_{\Lambda}-\alpha(V+Y))^{2}}F(2\alpha-1)$
$\displaystyle=\frac{\Lambda^{6}F(2\alpha-1)}{(F\Lambda^{2}-\alpha(V+Y))^{2}}\left[\frac{1}{6}(\partial_{t}F)-F\right].$
(147)

We get the loop contributions of $\xi$ field:

$\displaystyle\frac{1}{2}{\rm
Tr}^{\prime}\left.\frac{\partial_{t}{\mathcal{R}}_{\Lambda}}{\Gamma_{\Lambda}^{(1,1)}+{\mathcal{R}}_{\Lambda}}\right|_{\xi\xi}$
$\displaystyle=\frac{b_{0}^{\xi}}{2\left(4\pi\right)^{2}}\frac{\Lambda^{6}}{F\Lambda^{2}-\alpha(V+Y)}\bigg{[}\frac{1}{6}\left(\partial_{t}F\right)-F\bigg{]}\int\text{d}^{4}x\sqrt{g}$
$\displaystyle\quad+\frac{1}{\left(4\pi\right)^{2}}\bigg{\\{}\frac{b_{2}^{\xi}}{2}\frac{\Lambda^{4}}{F\Lambda^{2}-\alpha
V}\bigg{[}\frac{1}{2}\left(\partial_{t}F\right)-2F\bigg{]}$
$\displaystyle\quad-\frac{b_{0}^{\xi}}{8}\frac{\Lambda^{6}F(2\alpha-1)}{(F\Lambda^{2}-\alpha(V+Y))^{2}}\bigg{[}\frac{1}{6}(\partial_{t}F)-F\bigg{]}\bigg{\\}}\int\text{d}^{4}x\sqrt{g}{R}.$
(148)

Obviously, when employing the de-Donder gauge $\alpha=0$, the terms with
$\alpha$ vanish. Thus, the correction exhibited as the diagram (II) in Fig. 5
does not contribute to the beta functions in the de-Donder gauge.

####  E.1.3 The loop contribution of the gravity fields with spin 0 and the
scalar field

Let us evaluate the spin 0 field loop contribution:

$\displaystyle\frac{1}{2}{\rm
Tr}^{\prime\prime}\left.\frac{\partial_{t}{\mathcal{R}}_{\Lambda}}{\Gamma_{\Lambda}^{(1,1)}+{\mathcal{R}}_{\Lambda}}\right|_{\rm
SS}.$ (149)

We calculate the inverse matrix of
$\Gamma_{\Lambda}^{(1,1)}+{\mathcal{R}}_{\Lambda}$, multiply as
$(\Gamma_{\Lambda}^{(1,1)}+{\mathcal{R}}_{\Lambda})^{-1}\,\partial_{t}{\mathcal{R}}_{\Lambda}$,
and take the de-Donder gauge to obtain

$\displaystyle\frac{1}{2}{\rm
Tr}^{\prime\prime}\left.\frac{\partial_{t}{\mathcal{R}}_{\Lambda}}{\Gamma_{\Lambda}^{(1,1)}+{\mathcal{R}}_{\Lambda}}\right|_{\rm
SS}$ $\displaystyle=\frac{b_{0}^{\rm
S}}{2\left(4\pi\right)^{2}}Q_{2}[A]\int\text{d}^{4}x\sqrt{g}+\frac{1}{2\left(4\pi\right)^{2}}\left\\{b_{2}^{\rm
S}Q_{1}[A]+b_{0}^{\rm S}Q_{2}[B]\right\\}\int\text{d}^{4}x\sqrt{g}{R}$
$\displaystyle\quad+\frac{b_{0}^{\rm
S}}{2(4\pi)^{2}}Q_{2}[C]\int\text{d}^{4}x\sqrt{g}Y+\mathcal{O}({R}^{2},Y^{2}),$
(150)

where

$\displaystyle\frac{b_{0}^{\rm S}}{2}\frac{1}{\left(4\pi\right)^{2}}Q_{2}[A]$
$\displaystyle=\frac{b_{0}^{\rm
S}}{\left(4\pi\right)^{2}}\frac{1}{\Gamma(2)}\int\text{d}z\,z\,\theta\\!\left(\Lambda^{2}-z\right)$
$\displaystyle\quad\times\bigg{[}-\frac{2}{2}\frac{F\Lambda^{2}(\Lambda^{2}\Psi+24\phi^{2}\Lambda^{2}F^{\prime
2}\Psi^{\prime}+F\Lambda^{2}\Sigma_{1}+(\Psi\Sigma_{1}+12\phi^{2}\Psi^{\prime
2}))}{F\Lambda^{2}(\Psi\Sigma_{1}+12\Psi^{\prime 2}\phi^{2})}$
$\displaystyle\phantom{\quad\times\bigg{[}}+\frac{1}{2}(\Lambda^{2}-z)\left(\frac{F\Lambda^{2}\Sigma_{1}+(\Psi\Sigma_{1}+12\Psi^{\prime
2}\phi^{2})}{F\Lambda^{2}(\Psi\Sigma_{1}+12\Psi^{\prime
2}\phi^{2})}\right)\partial_{t}F$
$\displaystyle\phantom{\quad\times\bigg{[}}+\frac{24}{2}(\Lambda^{2}-z)\left(\frac{F\Lambda^{2}\phi^{2}\Psi^{\prime}}{F\Lambda^{2}(\Psi\Sigma_{1}+12\Psi^{\prime
2}\phi^{2})}\right)\partial_{t}F^{\prime}\bigg{]}$
$\displaystyle=\frac{b_{0}^{\rm
S}\Lambda^{4}}{192\pi^{2}}\left[-6-\frac{6(\Lambda^{2}\Psi+24\phi^{2}\Lambda^{2}F^{\prime}\Psi^{\prime}+F\Lambda^{2}\Sigma_{1})}{\Delta}+\left(\frac{\Lambda^{2}\Sigma_{1}}{\Delta}+\frac{1}{F}\right)\partial_{t}F+\left(\frac{24\phi^{2}\Lambda^{2}\Psi^{\prime}}{\Delta}\right)\partial_{t}F^{\prime}\right],$
(151) $\displaystyle\frac{b_{2}^{\rm S}}{2}\frac{1}{(4\pi)^{2}}Q_{1}[A]$
$\displaystyle=\frac{b_{2}^{\rm
S}}{\left(4\pi\right)^{2}}\frac{1}{\Gamma(1)}\int\text{d}z~{}\theta\\!\left(\Lambda^{2}-z\right)$
$\displaystyle\quad\times\bigg{[}-1-\frac{\Lambda^{2}\Psi+24\phi^{2}\Lambda^{2}F^{\prime}\Psi^{\prime}+F\Lambda^{2}\Sigma_{1}}{\Delta}$
$\displaystyle\phantom{\quad\times\bigg{[}}+\frac{1}{2}(\Lambda^{2}-z)\left(\frac{\Sigma_{1}}{\Delta}+\frac{1}{F\Lambda^{2}}\right)\partial_{t}F+\frac{24}{2}(\Lambda^{2}-z)\left(\frac{\phi^{2}\Psi^{\prime}}{\Delta}\right)\partial_{t}F^{\prime}\bigg{]}$
$\displaystyle=\frac{b_{2}^{\rm
S}\Lambda^{2}}{16\pi^{2}}\left[-1-\frac{\Lambda^{2}\Psi+24\phi^{2}\Lambda^{2}F^{\prime}\Psi^{\prime}+F\Lambda^{2}\Sigma_{1}}{\Delta}+\frac{1}{4}\left(\frac{\Lambda^{2}\Sigma_{1}}{\Delta}+\frac{1}{F}\right)\partial_{t}F+\frac{24}{4}\left(\frac{\Lambda^{2}\phi^{2}\Psi^{\prime}}{\Delta}\right)\partial_{t}F^{\prime}\right],$
(152)

and

$\displaystyle\frac{b_{0}^{\rm S}}{2}\frac{1}{\left(4\pi\right)^{2}}Q_{2}[B]$
$\displaystyle=\frac{b_{0}^{\rm
S}}{(4\pi)^{2}}\frac{1}{\Gamma(2)}\int\text{d}z\,z\theta\\!\left(\Lambda^{2}-z\right)$
$\displaystyle\quad\times\bigg{[}\frac{-\Delta^{2}+4\phi^{2}(6\Lambda^{4}F^{\prime
2}+\Psi^{\prime
2})\Delta-4\phi^{2}\Psi\Psi^{\prime}\left(7\Lambda^{2}F^{\prime}-V^{\prime}\right)(\Sigma_{1}-\Lambda^{2})}{2\Lambda^{2}\Delta^{2}}$
$\displaystyle\phantom{\quad\times\bigg{[}}+\frac{-4\phi^{2}\Sigma_{1}\left(7\Lambda^{2}F^{\prime}-V^{\prime}\right)(2\Psi
V^{\prime}-V\Psi^{\prime})+(-2\Lambda^{4}\Psi^{2}-48\Lambda^{4}F^{\prime}\phi^{2}\Psi\Psi^{\prime}+24\Lambda^{4}F\phi^{2}\Psi^{\prime
2})\Sigma_{2}}{2\Lambda^{2}\Delta^{2}}$
$\displaystyle\phantom{\quad\times\bigg{[}}+(\Lambda^{2}-z)\left(\frac{\Delta^{2}+4\phi^{2}V^{\prime}\Psi^{\prime}\Delta-24\Lambda^{4}F\phi^{2}\Psi^{\prime
2}\Sigma_{2}-4\phi^{2}\Lambda^{2}F\Psi^{\prime}\Sigma_{1}\left(7\Lambda^{2}F^{\prime}-V^{\prime}\right)}{4F\Lambda^{4}\Delta^{2}}\right)\partial_{t}F$
$\displaystyle\phantom{\quad\times\bigg{[}}+4F\Lambda^{2}(\Lambda^{2}-z)\phi^{2}\left(\frac{-12\Psi^{\prime}(7F^{\prime}\Lambda^{2}-V^{\prime})\phi^{2}+(7F^{\prime}\Lambda^{2}-V^{\prime})\Psi\Sigma_{1}+12\Lambda^{2}\Psi\Psi^{\prime}\Sigma_{2}}{4F\Lambda^{4}\Delta^{2}}\right)\partial_{t}F^{\prime}\bigg{]}$
$\displaystyle=\frac{b_{0}^{\rm
S}}{\left(4\pi\right)^{2}}\bigg{[}\frac{\Lambda^{2}}{4}\frac{-\Delta^{2}+4\phi^{2}(6\Lambda^{4}F^{\prime
2}+\Psi^{\prime
2})\Delta-4\phi^{2}\Psi\Psi^{\prime}\left(7\Lambda^{2}F^{\prime}-V^{\prime}\right)(\Sigma_{1}-\Lambda^{2})}{\Delta^{2}}$
$\displaystyle\phantom{=\frac{b_{0}^{\rm
S}}{\left(4\pi\right)^{2}}\bigg{[}}+\frac{\Lambda^{2}}{4}\frac{-4\phi^{2}\Sigma_{1}\left(7\Lambda^{2}F^{\prime}-V^{\prime}\right)(2\Psi
V^{\prime}-V\Psi^{\prime})}{\Delta^{2}}$
$\displaystyle\phantom{=\frac{b_{0}^{\rm
S}}{\left(4\pi\right)^{2}}\bigg{[}}+\frac{\Lambda^{2}}{4}\frac{(-2\Lambda^{4}\Psi^{2}-48\Lambda^{4}F^{\prime}\phi^{2}\Psi\Psi^{\prime}+24\Lambda^{4}F\phi^{2}\Psi^{\prime
2})\Sigma_{2}}{\Delta^{2}}$ $\displaystyle\phantom{=\frac{b_{0}^{\rm
S}}{\left(4\pi\right)^{2}}\bigg{[}}+\frac{\Lambda^{2}}{24}\left(\frac{\Delta^{2}+4\phi^{2}V^{\prime}\Psi^{\prime}\Delta-24\Lambda^{4}F\phi^{2}\Psi^{\prime
2}\Sigma_{2}-4\phi^{2}\Lambda^{2}F\Psi^{\prime}\Sigma_{1}\left(7\Lambda^{2}F^{\prime}-V^{\prime}\right)}{F\Delta^{2}}\right)\partial_{t}F$
$\displaystyle\phantom{=\frac{b_{0}^{\rm
S}}{\left(4\pi\right)^{2}}\bigg{[}}+\frac{\Lambda^{4}}{6}\phi^{2}\left(\frac{-12\Psi^{\prime}(7F^{\prime}\Lambda^{2}-V^{\prime})\phi^{2}+(7F^{\prime}\Lambda^{2}-V^{\prime})\Psi\Sigma_{1}+12\Lambda^{2}\Psi\Psi^{\prime}\Sigma_{2}}{\Delta^{2}}\right)\partial_{t}F^{\prime}\bigg{]},$
(153)

where $\Psi$, $\Sigma_{1}$, $\Sigma_{2}$ and $\Delta$ are given in (4.2).

The order $Y$ is given by

$\displaystyle\frac{b_{0}^{\rm S}}{2(4\pi)^{2}}Q_{2}[C]=$
$\displaystyle\frac{\Lambda^{6}}{32\pi^{2}}\bigg{[}24\left(\xi_{2}-\frac{\partial_{t}\xi_{2}}{6}\right)I[1,1,0]-\left(\xi_{0}-\frac{\partial_{t}\xi_{0}}{6}\right)I[2,0,0]$
$\displaystyle-{12\,}C\left(\xi_{0}-\frac{\partial_{t}\xi_{0}}{6}\right)I[2,1,0]-{12\,}CI[1,2,0]\bigg{]},$
(154)

where we have introduced

$\displaystyle
I[n_{g},n_{b},b_{f}]=\left.\frac{1}{\left(F\Lambda^{2}-V\right)^{n_{g}}(\Lambda^{2}+2V^{\prime}+4\phi^{2}V^{\prime\prime})^{n_{b}}(\Lambda^{2}+y^{2}\phi^{2})^{n_{f}}}\right|_{\phi=0},$
(155)

$C=F^{\prime}\Lambda^{2}-V^{\prime}|_{\phi=0}=\xi_{2}\Lambda^{2}-\lambda_{2}$,
and omitted the hat on the dimensionful coupling constants. These terms
include the corrections from the diagrams (III), (IV) and (V) in Fig. 5.

###  E.2 Fermionic contributions

We evaluate the fermionic loop contributions. From (127), we have

$\displaystyle-{1\over
2}\operatorname{Tr}\bigg{[}{\mathcal{M}}_{\text{FF}}^{-1}\,{\partial\mathcal{R}_{\text{FF}}\over\partial\Lambda}\bigg{]}$
$\displaystyle-{1\over
2}\operatorname{Tr}\bigg{[}{\mathcal{M}}_{\text{FF}}^{-1}\,{\partial\mathcal{R}_{\text{FF}}\over\partial\Lambda}\,{\mathcal{M}}_{\text{FF}}^{-1}{\mathcal{M}}_{\text{FB}}{\mathcal{M}}_{\text{BB}}^{-1}{\mathcal{M}}_{\text{BF}}\bigg{]}-{1\over
2}\operatorname{Tr}\bigg{[}{\mathcal{M}}_{\text{FF}}^{-1}{\mathcal{M}}_{\text{FB}}{\mathcal{M}}_{\text{BB}}^{-1}\,{\partial\mathcal{R}_{\text{BB}}\over\partial\Lambda}\,{\mathcal{M}}_{\text{BB}}^{-1}{\mathcal{M}}_{\text{BF}}\bigg{]}.$
(156)

The first term contributes to the beta functions of $V$ and $F$ and the second
and third terms contribute to the beta function of Yukawa coupling constant.

First we evaluate the first term:

$\displaystyle-{1\over
2}\operatorname{Tr}\bigg{[}{\mathcal{M}}_{\text{FF}}^{-1}\,{\partial\mathcal{R}_{\text{FF}}\over\partial
t}\bigg{]}=-{1\over 2}\operatorname{Tr}\bigg{[}{\mathcal{M}}_{\text{FF}}^{\rm
phys}{}^{-1}\,{\partial\mathcal{R}_{\text{FF}}^{\rm phys}\over\partial
t}\bigg{]}-{1\over 2}\operatorname{Tr}\bigg{[}{\mathcal{M}}_{\text{FF}}^{\rm
ghost}{}^{-1}\,{\partial\mathcal{R}_{\text{FF}}^{\rm ghost}\over\partial
t}\bigg{]}.$

####  E.2.1 The physical fermion contributions

The physical part is calculated as

$\displaystyle-{1\over 2}\operatorname{Tr}\left[{\mathcal{M}}_{\text{FF}}^{\rm
phys}{}^{-1}\,{\partial\mathcal{R}_{\text{FF}}^{\rm phys}\over\partial
t}\right]$ $\displaystyle=-{1\over 2}\widetilde{\partial\over\partial
t}\operatorname{Tr}\left[\ln\\!\left({\mathcal{M}}_{\text{FF}}^{\rm
phys}\right)\right]$ $\displaystyle=-{1\over
2}\widetilde{\partial\over\partial
t}\ln\operatorname{Det}\begin{pmatrix}0&-{{\ooalign{\hfil/\hfil\crcr$D$}}}^{\text{T}}-R_{\Lambda}\\!\left({{\ooalign{\hfil/\hfil\crcr$D$}}}^{\text{T}}\right)-y\phi\\\
{{\ooalign{\hfil/\hfil\crcr$D$}}}+R_{\Lambda}\\!\left({{\ooalign{\hfil/\hfil\crcr$D$}}}\right)+y\phi&0\end{pmatrix}$
$\displaystyle=-{\rm
Tr}\left.\frac{\partial_{t}{\mathcal{R}}_{\Lambda}}{\Gamma_{\Lambda}^{(1,1)}+{\mathcal{R}}_{\Lambda}}\right|_{{\overline{\chi}}\chi}$
$\displaystyle=-{\rm
Tr}\frac{\left(\partial_{t}{\mathcal{R}}^{(2,1)}_{\Lambda}\right)\left(-\frac{\sqrt{P_{\Lambda}\\!\left(p^{2}+{R\over
4}\right)}}{\sqrt{p^{2}+\frac{R}{4}}}{{\ooalign{\hfil/\hfil\crcr$D$}}}+y\phi\right)}{\left(\frac{\sqrt{P_{\Lambda}\\!\left(p^{2}+{R\over
4}\right)}}{\sqrt{p^{2}+\frac{R}{4}}}{{\ooalign{\hfil/\hfil\crcr$D$}}}+y\phi\right)\left(-\frac{\sqrt{P_{\Lambda}\\!\left(p^{2}+{R\over
4}\right)}}{\sqrt{p^{2}+\frac{R}{4}}}{{\ooalign{\hfil/\hfil\crcr$D$}}}+y\phi\right)}$
(163) $\displaystyle=-{\rm
Tr}\frac{\left(\frac{\frac{1}{2}\partial_{t}P_{\Lambda}\\!\left(p^{2}+{R\over
4}\right)}{\sqrt{P_{\Lambda}\\!\left(p^{2}+{R\over
4}\right)}\sqrt{p^{2}+\frac{R}{4}}}{{\ooalign{\hfil/\hfil\crcr$D$}}}\right)\left(-\frac{\sqrt{P_{\Lambda}\\!\left(p^{2}+{R\over
4}\right)}}{\sqrt{p^{2}+\frac{R}{4}}}{{\ooalign{\hfil/\hfil\crcr$D$}}}+y\phi\right)}{\left(\frac{P_{\Lambda}\\!\left(p^{2}+{R\over
4}\right)}{p^{2}+\frac{R}{4}}\left(p^{2}+\frac{R}{4}\right)+y^{2}\phi^{2}\right)}$
(168) $\displaystyle=-{\rm
Tr}\frac{\frac{1}{2}\partial_{t}\,P_{\Lambda}\\!\left(p^{2}+{R\over
4}\right)}{P_{\Lambda}\\!\left(p^{2}+{R\over
4}\right)+y^{2}\phi^{2}}=-\frac{1}{2}{\rm
Tr}\frac{\partial_{t}R_{\Lambda}}{P_{\Lambda}\\!\left(p^{2}+{R\over
4}\right)+y^{2}\phi^{2}}=:-\frac{1}{2}{\rm Tr}[W_{\rm f}]$
$\displaystyle=-\frac{1}{2}\frac{1}{\left(4\pi\right)^{2}}\left\\{b_{0}^{\rm
f}\,Q_{2}[W_{\rm f}]\int\text{d}^{d}x\sqrt{g}+b_{2}^{\rm f}\,Q_{1}[W_{\rm
f}]\int\text{d}^{d}x\sqrt{g}{R}\right\\}+{\mathcal{O}}({R}^{2}),$ (169)

where we have used $-{{\ooalign{\hfil/\hfil\crcr$D$}}}^{2}=p^{2}+{R}/4$ and

$\displaystyle Q_{2}[W_{\rm f}]$
$\displaystyle=\frac{1}{\Gamma(2)}\int^{\infty}_{0}\text{d}z\,z\frac{\partial_{t}P_{\Lambda}}{P_{\Lambda}+y^{2}\phi^{2}}=\int^{\Lambda^{2}}_{0}\text{d}z\,z\frac{-2\Lambda^{2}}{\Lambda^{2}+y^{2}\phi^{2}}=\frac{-\Lambda^{6}}{\Lambda^{2}+y^{2}\phi^{2}},$
$\displaystyle Q_{1}[W_{\rm f}]$
$\displaystyle=\frac{1}{\Gamma(1)}\int^{\infty}_{0}\text{d}z\frac{\partial_{t}P_{\Lambda}}{P_{\Lambda}+y^{2}\phi^{2}}=\int^{\Lambda^{2}}_{0}\text{d}z\frac{-2\Lambda^{2}}{\Lambda^{2}+y^{2}\phi^{2}}=\frac{-2\Lambda^{4}}{\Lambda^{2}+y^{2}\phi^{2}}.$

We obtain

$\displaystyle-{1\over
2}\operatorname{Tr}\bigg{[}{\mathcal{M}}_{\text{FF}}^{\rm
phys}{}^{-1}\,{\partial\mathcal{R}_{\text{FF}}^{\rm phys}\over\partial
t}\bigg{]}$ $\displaystyle=-\frac{b_{0}^{\rm
f}}{2\left(4\pi\right)^{2}}\frac{-\Lambda^{6}}{\Lambda^{2}+y^{2}\phi^{2}}\int\text{d}^{d}x\sqrt{g}$
$\displaystyle\quad-\frac{b_{2}^{\rm
f}}{2\left(4\pi\right)^{2}}\frac{-2\Lambda^{4}}{\Lambda^{2}+y^{2}\phi^{2}}\int\text{d}^{d}x\sqrt{g}{R}+\mathcal{O}({R}^{2}).$
(170)

####  E.2.2 The ghost fields contributions

We evaluate the ghost field contributions:

$\displaystyle-{1\over
2}\operatorname{Tr}\bigg{[}{\mathcal{M}}_{\text{FF}}^{\rm
ghost}{}^{-1}\,{\partial\mathcal{R}^{\rm ghost}_{\text{FF}}\over\partial
t}\bigg{]}$ $\displaystyle=-{1\over 2}\widetilde{\partial\over\partial
t}\operatorname{Tr}\left[\ln\\!\left({\mathcal{M}}_{\text{FF}}^{\rm
ghost}\right)\right]=-{1\over 2}\widetilde{\partial\over\partial
t}\ln\\!\left(\operatorname{Det}\left[{\mathcal{M}}_{\text{FF}}^{\rm
ghost}\right]\right)$ $\displaystyle=-{\rm
Tr}\left.\frac{\partial_{t}{\mathcal{R}}_{\Lambda}}{\Gamma_{\Lambda}^{(1,1)}+{\mathcal{R}}_{\Lambda}}\right|_{{\bar{C}}^{\perp}C^{\perp}}-{\rm
Tr}^{\prime}\left.\frac{\partial_{t}{\mathcal{R}}_{\Lambda}}{\Gamma_{\Lambda}^{(1,1)}+{\mathcal{R}}_{\Lambda}}\right|_{\bar{C}C}.$

For spin 1 ghost field, we have

$\displaystyle-{\rm
Tr}\left.\frac{\partial_{t}{\mathcal{R}}_{\Lambda}}{\Gamma_{\Lambda}^{(1,1)}+{\mathcal{R}}_{\Lambda}}\right|_{C^{\perp}C^{\perp}}=-{\rm
Tr}\frac{\partial_{t}R_{\Lambda}}{P_{\Lambda}-\frac{R}{4}}=-{\rm
Tr}\frac{\partial_{t}R_{\Lambda}}{P_{\Lambda}}-{\rm
Tr}\frac{\partial_{t}R_{\Lambda}}{P_{\Lambda}^{2}}\frac{R}{4}+{\mathcal{O}}({R}^{2}),$
(171)

with

$\displaystyle-{\rm Tr}\frac{\partial_{t}R_{\Lambda}}{P_{\Lambda}}$
$\displaystyle=-\frac{1}{\left(4\pi\right)^{2}}\left\\{b^{C^{\perp}}_{0}Q_{2}\left[\frac{\partial_{t}R_{\Lambda}}{P_{\Lambda}}\right]\int\text{d}^{4}x\sqrt{g}+b^{C^{\perp}}_{2}Q_{1}\left[\frac{\partial_{t}R_{\Lambda}}{P_{\Lambda}}\right]\int\text{d}^{4}x\sqrt{g}{R}\right\\}+{\mathcal{O}}({R}^{2}),$
(172) $\displaystyle
Q_{2}\left[\frac{\partial_{t}R_{\Lambda}}{P_{\Lambda}}\right]$
$\displaystyle=\frac{1}{\Gamma(2)}\int^{\infty}_{0}\text{d}z\,z\frac{\partial_{t}R_{\Lambda}}{P_{\Lambda}}=\int^{\Lambda^{2}}_{0}\text{d}z\,z\frac{-2\Lambda^{2}}{\Lambda^{2}}=-\Lambda^{4},$
(173) $\displaystyle
Q_{1}\left[\frac{\partial_{t}R_{\Lambda}}{P_{\Lambda}}\right]$
$\displaystyle=\frac{1}{\Gamma(1)}\int^{\infty}_{0}\text{d}z\,\frac{\partial_{t}R_{\Lambda}}{P_{\Lambda}}=\int^{\Lambda^{2}}_{0}\text{d}z\,\frac{-2\Lambda^{2}}{\Lambda^{2}}=-2\Lambda^{2},$
(174)

and

$\displaystyle-{\rm
Tr}\frac{\partial_{t}R_{\Lambda}}{P_{\Lambda}^{2}}\frac{R}{4}$
$\displaystyle=-\frac{1}{\left(4\pi\right)^{2}}\left\\{\frac{b^{C^{\perp}}_{0}}{4}Q_{2}\left[\frac{\partial_{t}R_{\Lambda}}{P_{\Lambda}^{2}}\right]\int\text{d}^{4}x\sqrt{g}{R}\right\\}+{\mathcal{O}}({R}^{2}),$
(175) $\displaystyle
Q_{2}\left[\frac{\partial_{t}R_{\Lambda}}{P_{\Lambda}^{2}}\right]$
$\displaystyle=\frac{1}{\Gamma(2)}\int^{\infty}_{0}\text{d}z\,z\frac{\partial_{t}R_{\Lambda}}{P_{\Lambda}^{2}}=\int^{\Lambda^{2}}_{0}\text{d}z~{}z\frac{-2\Lambda^{2}}{\Lambda^{4}}=-\Lambda^{2}.$
(176)

We obtain the contributions of the ghost field with spin 1:

$\displaystyle-{\rm
Tr}\left.\frac{\partial_{t}{\mathcal{R}}_{\Lambda}}{\Gamma_{\Lambda}^{(1,1)}+{\mathcal{R}}_{\Lambda}}\right|_{C^{\perp}C^{\perp}}$
$\displaystyle=-\frac{b_{0}^{C^{\perp}}(-\Lambda^{4})}{\left(4\pi\right)^{2}}\int\text{d}^{4}x\sqrt{g}$
$\displaystyle-\frac{1}{\left(4\pi\right)^{2}}\left\\{b_{2}^{C^{\perp}}(-2\Lambda^{2})+\frac{b_{0}^{C^{\perp}}}{4}\left(-\Lambda^{2}\right)\right\\}\int\text{d}^{4}x\sqrt{g}{R}.$
(177)

For spin 0 ghost field, we have

$\displaystyle-{\rm
Tr}^{\prime}\left.\frac{\partial_{t}{\mathcal{R}}_{\Lambda}}{\Gamma_{\Lambda}^{(1,1)}+{\mathcal{R}}_{\Lambda}}\right|_{{\bar{C}}{C}}$
$\displaystyle=-{\rm
Tr}^{\prime}\frac{(2-\frac{1+\beta}{2})\partial_{t}R_{\Lambda}}{(2-\frac{1+\beta}{2})P_{\Lambda}-\frac{R}{2}}=-{\rm
Tr}\frac{\partial_{t}R_{\Lambda}}{P_{\Lambda}}-{\rm
Tr}\frac{\partial_{t}R_{\Lambda}}{P_{\Lambda}^{2}}\frac{R}{3-\beta}+{\mathcal{O}}({R}^{2}),$
(178)

with

$\displaystyle-{\rm Tr}\frac{\partial_{t}R_{\Lambda}}{P_{\Lambda}}$
$\displaystyle=-\frac{1}{\left(4\pi\right)^{2}}\left\\{b^{C}_{0}Q_{2}\left[\frac{\partial_{t}R_{\Lambda}}{P_{\Lambda}}\right]\int\text{d}^{4}x\sqrt{g}+b^{C}_{2}Q_{1}\left[\frac{\partial_{t}R_{\Lambda}}{P_{\Lambda}}\right]\int\text{d}^{4}x\sqrt{g}{R}\right\\}+{\mathcal{O}}({R}^{2}),$
(179) $\displaystyle