11institutetext: Nobuyoshi Ohta 22institutetext: Department of Physics,
National Central University, Zhongli, Taoyuan 320317, Taiwan and Research
Institute for Science and Technology, Kindai University, Higashi-Osaka, Osaka
577-8502, Japan, 22email<EMAIL_ADDRESS>
# One-loop divergences in higher-derivative gravity

Nobuyoshi Ohta

###### Abstract

We give a review of the one-loop divergences in higher derivative gravity
theories. We first make the bilinear expansion in the quantum fluctuation on
arbitrary backgrounds, introduce a higher-derivative gauge fixing and show
that higher-derivative gauge fixing must have ghosts in addition to those
naively expected. We give general formulae for the one-loop divergences in
such theories, and give explicit results for theories with quadratic curvature
terms. In this calculation, we need the heat kernel coefficients for the four-
derivative minimal operators and two-derivative nonminimal vector operators,
which are summarized. We also discuss the beta functions in the
renormalization group, and show that the dimensionless couplings are
asymptotically free. The calculation is also extended to the theories with
arbitrary functions of $R$ and $R_{\mu\nu}^{2}$. We show that the result is
independent of metric parametrization and gauge on shell.

## Keywords

Quantum gravity, Quadratic gravity, Higher-derivative gravity, Perturbation
theory, One-loop divergences, Background method, Heat kernel expansion,
Effective action, Renormalization group, Asymptotic freedom

##  1 Introduction

The study of quantum effects in gravity started with the seminal work of ’t
Hooft and Veltmann TV , in which one-loop divergences were first studied. It
was shown that there exist divergences in the quadratic curvature terms, but
these counterterms were shown to be transformed away by a field redefinition.
However it was shown later the Einstein gravity is nonrenormalizable at two
loops GS . It was further shown that gravity theory containing quadratic
curvature terms is a renormalizable theory Stelle . Since this theory contains
curvature squares and four derivatives, we refer to such theories as
quadratic, four-derivative or higher-derivative gravity. Unfortunately the
theory is probably nonunitary in perturbation theory, but see D for
discussions.

The actual calculation of one-loop divergences in the four-derivative quantum
gravity involves some complicated techniques. We use the technique of
background field method by separating the quantum fields into backgrounds and
fluctuations, and integrate over the fluctuations. To really do this at one
loop, we have to know the bilinear forms in the fluctuation fields of all
relevant terms, such as Weyl curvature square, scalar curvature square (or
other combinations, like Ricci curvature square and Riemann curvature square),
and Einstein-Hilbert term.

The next step is to gauge fix the theory. Typically this also involves
introduction of the Faddeev-Popov (FP) ghosts which were first discovered by
Feynman F , and introduced in more formal way by DeWitt DeWitt , and later
elegantly formulated by Faddeev and Popov FP . When higher-derivative gauge
fixing is chosen, this leads to a complication; in addition to those naively
expected JT , we have to include additional ghost contribution [denoted as
${\rm Tr}\log Y$ in eq. (1)] first noticed in FT1 ; FT2 , and derived at the
one-loop level in BC . Here instead of examining the one-loop result, we give
a more elegant derivation of all the ghosts based on the symmetry principle
Ohta:2020 .

The general formula for the divergences is then schematically given as

$\displaystyle\Gamma_{\rm div}=\frac{1}{2}{\rm Tr}\log{\cal H}-{\rm
Tr}\log\Delta_{gh}-\frac{1}{2}{\rm Tr}\log Y,$ (1)

where the trace is meant to sum over the spectra of the operators, and the
first, second and last terms are the contributions from the graviton, FP
ghosts and additional ghosts, respectively. To calculate this, it is most
convenient to use heat kernel technique, which is explained here in some
details. The calculation involves fourth-order minimal operators in the tensor
sector and second-order nonminimal operators for the ghosts. We give the
necessary formulae to evaluate these G ; GK . Then we sum all the
contributions to find the one-loop divergences in the quadratic gravity
theory. We mention that there is also the technique called Schwinger-DeWitt
method, which is basically the same as the heat kernel technique BV . There
were some mistakes in the early calculations of the logarithmic divergences JT
; FT1 ; FT2 , and the correct results were given in AB .

As a simple application of the technique, we also study general higher-
derivative theory including an arbitrary function of Ricci tensor squared and
Ricci scalar curvature in a very general parametrization with arbitrary gauge
fixing. We will find the general result for this case, and show that on shell,
it depends on neither the parametrizations nor gauge parameters.

##  2 Bilinear expansion of quadratic terms

We will consider the Euclidean actions of the general form

$\displaystyle S=\int
d^{d}x\sqrt{-g}\Big{[}\frac{1}{\kappa^{2}}(2\Lambda-R)+\alpha R^{2}+\beta
R_{\mu\nu}^{2}+\gamma R_{\mu\nu\rho\lambda}^{2}\Big{]},$ (2)

where $\kappa^{2}=16\pi G$ is the $d$-dimensional gravitational constant,
$\Lambda$ is the cosmological constant, and $\alpha$, $\beta$, $\gamma$ are
the higher-derivative couplings. Though we consider $d=4$ when we give the
results for the divergences, here we give the results for general dimension
$d$. It is sometimes more convenient to use a different basis for the higher-
derivative terms, namely $R^{2}$, the square of the Weyl tensor

$\displaystyle
C^{2}=R_{\mu\nu\alpha\beta}^{2}-\frac{4}{d-2}R_{\mu\nu}^{2}+\frac{2}{(d-1)(d-2)}R^{2},$
(3)

and the Gauss-Bonnet combination

$\displaystyle E=R_{\mu\nu\alpha\beta}^{2}-4R_{\mu\nu}^{2}+R^{2},$ (4)

which is topological for $d=4$ and vanishes identically for $d=3$. Conversely
we have

$\displaystyle
R_{\mu\nu\alpha\beta}^{2}=\frac{d-2}{d-3}C^{2}-\frac{1}{d-3}E+\frac{1}{d-1}R^{2},~{}~{}R_{\mu\nu}^{2}=\frac{d-2}{4(d-3)}(C^{2}-E)+\frac{d}{4(d-1)}R^{2}.$
(5)

Then the action has the alternative form

$\displaystyle S=\int
d^{d}x\sqrt{-g}\Big{[}\frac{1}{\kappa^{2}}(2\Lambda-R)+\frac{1}{2\lambda}C^{2}-\frac{1}{\rho}E+\frac{1}{\xi}R^{2}\Big{]},$
(6)

where

$\displaystyle\lambda=\frac{2(d-3)}{(d-2)(\beta+4\gamma)},~{}~{}\rho=\frac{4(d-3)}{(d-2)\beta+4\gamma},~{}~{}\xi=\frac{4(d-1)}{4(d-1)\alpha+d\beta+4\gamma}\
.$ (7)

or conversely

$\displaystyle\alpha=-\frac{1}{\rho}+\frac{1}{\xi}+\frac{1}{(d-1)(d-2)\lambda},~{}~{}\beta=\frac{4}{\rho}-\frac{2}{(d-2)\lambda},~{}~{}\gamma=-\frac{1}{\rho}+\frac{1}{2\lambda}.$
(8)

Note that in $d=3$, $C^{2}$ and $E$ both vanish identically. The couplings
$\lambda$, $\rho$ and $\xi$ have mass dimension $4-d$. In dimensions higher
than three, it is customary to define the dimensionless combinations

$\displaystyle\omega\equiv-\frac{(d-1)\lambda}{\xi},~{}~{}~{}\theta\equiv\frac{\lambda}{\rho}\
.$ (9)

We will apply the standard background field method, expanding the metric as

$\displaystyle g_{\mu\nu}=\bar{g}_{\mu\nu}+h_{\mu\nu}\ .$ (10)

In order to derive the effective action at the one-loop level, or to calculate
the one-loop beta functions, we need the expansion of the action to second
order in $h_{\mu\nu}$. For this purpose, it is useful to first make the
expansion of curvatures in the fluctuations, which are summarized in Appendix
A.1.

Using the formulae given in A.1, we find that the terms proportional to
$\alpha$ can be written in the form

$\displaystyle\alpha
h^{\mu\nu}\Big{[}\bar{\nabla}_{\mu}\bar{\nabla}_{\nu}\bar{\nabla}_{\alpha}\bar{\nabla}_{\beta}-\bar{g}_{\mu\nu}\Box\bar{\nabla}_{\alpha}\bar{\nabla}_{\beta}-\bar{g}_{\alpha\beta}\bar{\nabla}_{\mu}\bar{\nabla}_{\nu}\Box+\bar{g}_{\mu\nu}\bar{g}_{\alpha\beta}\Box^{2}-\bar{R}\bar{g}_{\nu\beta}\bar{\nabla}_{\alpha}\bar{\nabla}_{\mu}$
$\displaystyle-(2\bar{R}_{\mu\nu}-\bar{R}\bar{g}_{\mu\nu})\bar{\nabla}_{\alpha}\bar{\nabla}_{\beta}+2\bar{R}_{\mu\nu}\bar{g}_{\alpha\beta}\Box+\frac{1}{2}\bar{R}(\bar{g}_{\mu\alpha}\bar{g}_{\nu\beta}-\bar{g}_{\mu\nu}\bar{g}_{\alpha\beta})\Box-\bar{R}\bar{R}_{\alpha\beta}\bar{g}_{\mu\nu}$
$\displaystyle+2\bar{R}\bar{R}_{\mu\alpha}\bar{g}_{\beta\nu}+\bar{R}_{\mu\nu}\bar{R}_{\alpha\beta}-\frac{1}{4}J_{\mu\nu\alpha\beta}\bar{R}^{2}\Big{]}h^{\alpha\beta}\
.$ (11)

Here and in what follows, $\Box\equiv\bar{\nabla}_{\mu}\bar{\nabla}^{\mu}$ and
a bar indicates that the quantity is evaluated on the background; the indices
are raised, lowered and contracted by the background metric $\bar{g}$, and the
covariant derivative $\bar{\nabla}$ is constructed with the background metric.
The tensor $J$ is defined by

$\displaystyle
J_{\mu\nu\alpha\beta}=\delta_{\mu\nu,\alpha\beta}-\frac{1}{2}\bar{g}_{\mu\nu}\bar{g}_{\alpha\beta}$
(12)

where

$\displaystyle\delta_{\mu\nu,\alpha\beta}=\frac{1}{2}(\bar{g}_{\mu\alpha}\bar{g}_{\nu\beta}+\bar{g}_{\mu\beta}\bar{g}_{\nu\alpha})\equiv\hat{1},$
(13)

is the identity in the space of symmetric tensors.

The $\beta$ terms can be written in the form

$\displaystyle\beta h^{\mu\nu}$
$\displaystyle\Big{[}\frac{1}{2}\bar{\nabla}_{\mu}\bar{\nabla}_{\nu}\bar{\nabla}_{\alpha}\bar{\nabla}_{\beta}-\frac{1}{2}\bar{g}_{\mu\nu}\bar{\nabla}_{\alpha}\Box\bar{\nabla}_{\beta}-\frac{1}{2}\bar{g}_{\nu\beta}\bar{\nabla}_{\mu}\Box\bar{\nabla}_{\alpha}+\frac{1}{4}(\bar{g}_{\mu\alpha}\bar{g}_{\nu\beta}+\bar{g}_{\mu\nu}\bar{g}_{\alpha\beta})\Box^{2}$
(14)
$\displaystyle+\frac{1}{2}\bar{R}_{\nu\beta}\bar{\nabla}_{\alpha}\bar{\nabla}_{\mu}-2\bar{R}_{\mu}^{\rho}\bar{g}_{\nu\beta}\bar{\nabla}_{\rho}\bar{\nabla}_{\alpha}+\frac{3}{2}\bar{g}_{\alpha\beta}\bar{R}_{\rho\mu}\bar{\nabla}^{\rho}\bar{\nabla}_{\nu}+\bar{R}_{\mu\alpha\nu\beta}\Box$
$\displaystyle+\frac{1}{4}(2\bar{g}_{\mu\alpha}\bar{g}_{\nu\beta}-\bar{g}_{\mu\nu}\bar{g}_{\alpha\beta})\bar{R}^{\rho\lambda}\bar{\nabla}_{\rho}\bar{\nabla}_{\lambda}-\frac{3}{2}\bar{R}^{\rho}_{\beta}\bar{R}_{\rho\mu\nu\alpha}+\bar{R}_{\mu\rho\lambda\nu}\bar{R}_{\alpha}{}^{\rho\lambda}{}_{\beta}$
$\displaystyle+\frac{1}{2}\bar{g}_{\nu\beta}\bar{R}_{\mu\rho}\bar{R}^{\rho}_{\alpha}-\bar{g}_{\alpha\beta}\bar{R}_{\mu\rho}\bar{R}^{\rho}_{\nu}-\frac{1}{4}J_{\mu\nu\alpha\beta}\bar{R}_{\rho\sigma}^{2}\Big{]}h^{\alpha\beta}\
,$

and the terms proportional to $\gamma$ are

$\displaystyle\gamma h^{\mu\nu}$
$\displaystyle\Big{[}\bar{\nabla}_{\mu}\bar{\nabla}_{\nu}\bar{\nabla}_{\alpha}\bar{\nabla}_{\beta}+\bar{g}_{\mu\alpha}\bar{g}_{\nu\beta}\Box^{2}-2\bar{g}_{\nu\beta}\bar{\nabla}_{\mu}\Box\bar{\nabla}_{\alpha}-2\bar{g}_{\nu\beta}\bar{R}_{\mu\rho\lambda\alpha}\bar{\nabla}^{\rho}\bar{\nabla}^{\lambda}+3\bar{R}_{\mu\alpha\nu\beta}\Box$
(15)
$\displaystyle-4\bar{R}_{\mu\alpha\rho\nu}\bar{\nabla}^{\rho}\bar{\nabla}_{\beta}-4\bar{g}_{\nu\beta}\bar{R}_{\rho\mu}\bar{\nabla}^{\rho}\bar{\nabla}_{\alpha}-2\bar{g}_{\nu\beta}\bar{R}_{\mu\alpha}\Box+\bar{g}_{\mu\alpha}\bar{g}_{\nu\beta}\bar{R}_{\rho\lambda}\bar{\nabla}^{\rho}\bar{\nabla}^{\lambda}$
$\displaystyle-2\bar{g}_{\mu\nu}\bar{R}_{\rho\alpha\lambda\beta}\bar{\nabla}^{\lambda}\bar{\nabla}^{\rho}+4\bar{R}_{\mu\alpha}\bar{\nabla}_{\nu}\bar{\nabla}_{\beta}+2\bar{g}_{\nu\beta}\bar{R}_{\mu\lambda\rho\sigma}\bar{R}_{\alpha}{}^{\lambda\rho\sigma}-2\bar{g}_{\nu\beta}\bar{R}^{\rho\lambda}\bar{R}_{\mu\rho\alpha\lambda}$
$\displaystyle-\bar{R}_{\mu\lambda\rho\beta}\bar{R}_{\nu}{}^{\rho\lambda}{}_{\alpha}+3\bar{R}_{\mu\alpha}{}^{\rho\lambda}\bar{R}_{\nu\rho\beta\lambda}-3\bar{R}_{\mu\lambda\nu\rho}\bar{R}_{\alpha}{}^{\rho\lambda}{}_{\beta}-3\bar{R}^{\rho}_{\alpha}\bar{R}_{\mu\beta\nu\rho}$
$\displaystyle+2\bar{R}_{\mu\alpha}\bar{R}_{\nu\beta}-\bar{g}_{\mu\nu}\bar{R}_{\alpha}{}^{\rho\lambda\sigma}\bar{R}_{\beta\rho\lambda\sigma}-\frac{1}{4}J_{\mu\nu\alpha\beta}\bar{R}_{\rho\sigma\lambda\tau}^{2}\Big{]}h^{\alpha\beta}.$

It can be checked that when arranged in the Gauss-Bonnet combination
($\gamma=\alpha$, $\beta=-4\alpha$) and the background metric is maximally
symmetric, one obtains a total derivative. This gives a nontrivial check of
the results.

##  3 Gauge fixing and ghosts

The first attempt at calculating one-loop divergences was made in JT with
higher-order gauge fixing, but it was pointed out that it did not correctly
incorporate the FP ghosts. The problem is that an additional ghost
contribution was missing, which was first considered in FT1 ; FT2 . But it was
not clear why we need such additional ghosts, and later clarified in BC ; this
showed it must be there by carefully examining the one-loop amplitude in the
path integral formulation. Here we explain more elegant way of introducing the
correct FP ghosts including the additional ghosts for higher-derivative gauge
fixing Ohta:2020 , which is valid not only for one loop but also for all loops
because it is based on the exact BRST (Becchi-Rouet-Stora-Tyutin) symmetry of
the system.

The BRST transformation for the fields is found to be

$\displaystyle\delta_{B}g_{\mu\nu}$ $\displaystyle=$
$\displaystyle-\delta\lambda[g_{\rho\nu}\nabla_{\mu}c^{\rho}+g_{\rho\mu}\nabla_{\nu}c^{\rho}]\equiv-\delta\lambda{\cal
D}_{\mu\nu,\rho}c^{\rho},$ $\displaystyle\delta_{B}c^{\mu}$ $\displaystyle=$
$\displaystyle-\delta\lambda
c^{\rho}\nabla_{\rho}c^{\mu},~{}~{}~{}\delta_{B}\bar{c}_{\mu}=i\delta\lambda\,B_{\mu},~{}~{}~{}\delta_{B}B_{\mu}=0,$
(16)

which is nilpotent. Here $c^{\mu},\bar{c}_{\mu}$ and $B_{\mu}$ are the FP
ghost, anti-ghost and an auxiliary field, respectively, and $\delta\lambda$ is
an anticommuting parameter. The gauge fixing term and the FP ghost terms are
concisely written as

$\displaystyle{\cal L}_{GF+FP}/\sqrt{-\bar{g}}$ $\displaystyle=$
$\displaystyle-i\delta_{B}[\bar{c}_{\mu}Y^{\mu\nu}(\chi_{\nu}-\frac{a}{2}B_{\nu})]/\delta\lambda$
(17) $\displaystyle=$ $\displaystyle
B_{\mu}Y^{\mu\nu}\chi_{\nu}-i\bar{c}_{\mu}Y^{\mu\nu}(\nabla^{\lambda}{\cal
D}_{\lambda\nu,\rho}+b\nabla_{\nu}{\cal
D}_{\lambda,\rho}^{\lambda})c^{\rho}-\frac{a}{2}B_{\mu}Y^{\mu\nu}B_{\nu}$
$\displaystyle=$
$\displaystyle\frac{1}{2a}\chi_{\mu}Y^{\mu\nu}\chi_{\nu}-\frac{a}{2}{\tilde{B}}_{\mu}Y^{\mu\nu}{\tilde{B}}_{\nu}+i\bar{c}_{\mu}Y^{\mu\nu}\Delta_{gh,\nu\rho}c^{\rho},$

where

$\displaystyle\chi_{\mu}\equiv\nabla^{\lambda}h_{\lambda\mu}+b\nabla_{\mu}h,\hskip
14.22636pt\tilde{B}_{\mu}\equiv B_{\mu}-\frac{1}{a}\chi_{\mu},$ (18)

are the gauge fixing function and a field imposing the gauge condition,
respectively, $Y^{\mu\nu}$ is a derivative operator for higher-derivative
gauge fixing

$\displaystyle
Y_{\mu\nu}\equiv-\bar{g}_{\mu\nu}\Box-c\nabla_{\mu}\nabla_{\nu}+d\nabla_{\nu}\nabla_{\mu},\hskip
8.53581pt\Delta_{gh,\mu\nu}\equiv-\bar{g}_{\mu\nu}\Box-(1+2b)\nabla_{\mu}\nabla_{\nu}-R_{\mu\nu},~{}~{}$
(19)

$\Delta_{gh,\mu\nu}$ is the ghost kinetic term, and $a$, $b$, $c$ and $d$ are
gauge parameters. For one-loop calculation, we can replace $\nabla$ by
$\bar{\nabla}$. We see from (17) that we get the determinant factor $({\rm
Det}~{}Y^{\mu\nu})^{1/2}$ after we perform the path integral over
$\tilde{B}_{\mu}$ and FP ghosts $\bar{c}_{\mu}$ and $c^{\rho}$. (The factor
$Y^{\mu\nu}$ in the first term combines into the graviton contribution.) In
the standard way of Faddeev and Popov, the factor of $Y$ can be easily missed
JT , but in this formulation we see why there must be this factor. Note also
that the field $B_{\mu}$ is an auxiliary field for low-derivative gauge
fixing, but here it becomes dynamical due to the higher-derivative gauge
fixing with the factor $Y^{\mu\nu}$.

We choose the gauge parameters such that the nonminimal four derivative terms
$\bar{\nabla}_{\mu}\bar{\nabla}_{\nu}\bar{\nabla}_{\alpha}\bar{\nabla}_{\beta}$,
$\bar{g}_{\mu\nu}\Box\bar{\nabla}_{\alpha}\bar{\nabla}_{\beta}$ and
$\bar{g}_{\nu\beta}\bar{\nabla}_{\mu}\Box\bar{\nabla}_{\alpha}$ cancel. This
leads to the choice BS2 ; OP1

$\displaystyle
a=\frac{1}{\beta+4\gamma},~{}~{}~{}b=\frac{4\alpha+\beta}{4(\gamma-\alpha)},~{}~{}~{}c-d=\frac{2(\gamma-\alpha)}{\beta+4\gamma}-1.$
(20)

In order to simplify the gauge-fixing term, we will further choose $d=1$.
Then, the quadratic terms in the action can be written in the form
$h^{\mu\nu}{\cal K}_{\mu\nu,\alpha\beta}h^{\alpha\beta}$, where

$\displaystyle{\cal
K}=K\Box^{2}+D_{\rho\lambda}\bar{\nabla}^{\rho}\bar{\nabla}^{\lambda}+W.$ (21)

The explicit forms of the coefficients are

$\displaystyle(K)_{\mu\nu,\alpha\beta}=\frac{\beta+4\gamma}{4}\Big{(}\bar{g}_{\mu\alpha}\bar{g}_{\nu\beta}+\frac{4\alpha+\beta}{4(\gamma-\alpha)}\bar{g}_{\mu\nu}\bar{g}_{\alpha\beta}\Big{)},$
(22) $\displaystyle(D_{\rho\lambda})_{\mu\nu,\alpha\beta}$ (23)
$\displaystyle=$
$\displaystyle-2\gamma\bar{g}_{\nu\beta}\bar{R}_{\alpha\rho\lambda\mu}+4\gamma\bar{g}_{\rho\nu}\bar{R}_{\lambda\alpha\mu\beta}+(\beta+3\gamma)\bar{g}_{\rho\lambda}\bar{R}_{\mu\alpha\nu\beta}-(2\beta+4\gamma)\bar{g}_{\alpha\rho}\bar{g}_{\nu\beta}\bar{R}_{\mu\lambda}$
$\displaystyle-2\gamma\bar{g}_{\nu\beta}\bar{R}_{\mu\alpha}\bar{g}_{\rho\lambda}+\beta\bar{g}_{\mu\nu}\bar{g}_{\beta\rho}\bar{R}_{\alpha\lambda}-2\alpha\bar{g}_{\alpha\rho}\bar{g}_{\beta\lambda}\bar{R}_{\mu\nu}+2\alpha\bar{g}_{\mu\nu}\bar{g}_{\rho\lambda}\bar{R}_{\alpha\beta}+2\gamma\bar{g}_{\mu\nu}\bar{R}_{\alpha\rho\lambda\beta}$
$\displaystyle+\Big{(}\frac{\alpha}{2}\bar{R}-\frac{1}{4\kappa^{2}}\Big{)}(\bar{g}_{\mu\alpha}\bar{g}_{\nu\beta}\bar{g}_{\rho\lambda}-\bar{g}_{\mu\nu}\bar{g}_{\alpha\beta}\bar{g}_{\rho\lambda}-2\bar{g}_{\nu\beta}\bar{g}_{\mu\rho}\bar{g}_{\alpha\lambda}+2\bar{g}_{\mu\nu}\bar{g}_{\alpha\rho}\bar{g}_{\beta\lambda})$
$\displaystyle+2\gamma\bar{g}_{\nu\rho}\bar{g}_{\beta\lambda}\bar{R}_{\mu\alpha}+\Big{(}\frac{\beta}{2}+\gamma\Big{)}\bar{g}_{\mu\alpha}\bar{g}_{\nu\beta}\bar{R}_{\rho\lambda}-\frac{\beta}{4}\bar{g}_{\mu\nu}\bar{g}_{\alpha\beta}\bar{R}_{\rho\lambda},$
$\displaystyle(W)_{\mu\nu,\alpha\beta}$
$\displaystyle=\frac{3}{2}\gamma\bar{g}_{\nu\beta}\bar{R}_{\mu}{}^{\rho\lambda\sigma}\bar{R}_{\alpha\rho\lambda\sigma}+4\gamma\bar{R}_{\rho\alpha\mu\lambda}\bar{R}_{\nu\beta}{}^{\rho\lambda}-\gamma\bar{R}_{\rho\alpha\mu\lambda}\bar{R}^{\rho}{}_{\nu\beta}{}^{\lambda}+(\beta+5\gamma)\bar{R}_{\rho\mu\lambda\nu}\bar{R}^{\rho}{}_{\alpha}{}^{\lambda}{}_{\beta}$
$\displaystyle+6\gamma\bar{R}^{\rho}_{\mu}\bar{R}_{\rho\alpha\beta\nu}+\Big{(}\frac{\beta}{2}+\gamma\Big{)}\bar{R}_{\mu\alpha}\bar{R}_{\nu\beta}+\Big{(}\alpha\bar{R}-\frac{1}{2\kappa^{2}}\Big{)}\Big{(}\frac{1}{2}\bar{R}_{\mu\alpha\nu\beta}+\frac{3}{2}\bar{g}_{\beta\nu}\bar{R}_{\mu\alpha}-\bar{g}_{\mu\nu}\bar{R}_{\alpha\beta}\Big{)}$
$\displaystyle+\alpha\bar{R}_{\mu\nu}\bar{R}_{\alpha\beta}+\frac{1}{8}\Big{(}\alpha\bar{R}^{2}+\beta\bar{R}_{\rho\lambda}^{2}+\gamma\bar{R}_{\rho\lambda\sigma\tau}^{2}-\frac{1}{\kappa^{2}}(\bar{R}-2\Lambda)\Big{)}(\bar{g}_{\mu\nu}\bar{g}_{\alpha\beta}-2\bar{g}_{\mu\alpha}\bar{g}_{\nu\beta})$
$\displaystyle+\Big{(}\frac{5}{2}\beta+4\gamma\Big{)}\bar{g}_{\nu\beta}\bar{R}_{\mu\rho}\bar{R}^{\rho}_{\alpha}-\gamma\bar{g}_{\mu\nu}\bar{R}_{\alpha}{}^{\rho\lambda\sigma}\bar{R}_{\beta\rho\lambda\sigma}-\beta\bar{g}_{\alpha\beta}\bar{R}_{\mu\rho}\bar{R}^{\rho}_{\nu}-(\beta+4\gamma)\bar{g}_{\nu\beta}\bar{R}^{\rho\lambda}\bar{R}_{\mu\rho\alpha\lambda},$
(24)

where we have dropped terms with two derivatives acting on a background
curvature, and performed the symmetrizations $\mu\leftrightarrow\nu$,
$\alpha\leftrightarrow\beta$ and $(\mu,\nu)\leftrightarrow(\alpha,\beta)$.

In order to use the heat kernel formula, we have to put this operator into the
form

$\displaystyle{\cal H}=K^{-1}{\cal
K}=\Box^{2}+V_{\rho\lambda}\bar{\nabla}^{\rho}\bar{\nabla}^{\lambda}+U\ ,$
(25)

where

$\displaystyle(K^{-1})_{\mu\nu}{}^{\alpha\beta}=\frac{4}{\beta+4\gamma}(\delta_{\mu\nu}{}^{\alpha\beta}-\Omega\bar{g}_{\mu\nu}\bar{g}^{\alpha\beta}),$
(26)

with

$\displaystyle\Omega=\frac{4\alpha+\beta}{\Sigma},~{}~{}~{}\Sigma\equiv
4(\gamma-\alpha)+d(4\alpha+\beta).$ (27)

The form of the coefficients $V_{\rho\lambda}$ and $U$ is complicated. First
$V$ is given by

$\displaystyle V^{\rho\lambda}=\frac{4}{\beta+4\gamma}\sum_{i=1}^{20}b_{i}{\bf
k}_{i},$ (28)

where

$\displaystyle{\bf
k}_{1}=\bar{g}_{\nu\beta}\bar{g}^{\rho\lambda}\bar{R}_{\mu\alpha},~{}~{}~{}{\bf
k}_{2}=\delta_{\mu\nu,\alpha\beta}\bar{g}^{\rho\lambda},~{}~{}~{}{\bf
k}_{3}=\bar{g}^{\rho\lambda}\bar{R}_{\mu\alpha\nu\beta},~{}~{}~{}{\bf
k}_{4}=\delta_{\nu\beta}{}^{\rho\lambda}\bar{R}_{\mu\alpha},$
$\displaystyle{\bf
k}_{5}=\delta_{\nu\beta}{}^{\rho\lambda}\bar{g}_{\mu\alpha},~{}~{}~{}~{}{\bf
k}_{6}=\delta_{\mu\nu,\alpha\beta}\bar{R}^{\rho\lambda},~{}~{}~{}{\bf
k}_{7}=\frac{1}{2}(\delta_{\nu}^{(\rho}\bar{R}^{\lambda)}{}_{\alpha\beta\mu}+\delta_{\beta}^{(\rho}\bar{R}^{\lambda)}{}_{\mu\nu\alpha}),$
$\displaystyle{\bf
k}_{8}=\bar{g}_{\nu\beta}\delta^{(\rho}_{(\mu}\bar{R}^{\lambda)}_{\alpha)},~{}~{}~{}{\bf
k}_{9}=\bar{g}_{\nu\beta}\bar{R}_{(\alpha}{}^{\rho\lambda}{}_{\mu)},~{}~{}~{}{\bf
k}_{10}=\frac{1}{2}(\delta_{\alpha\beta}{}^{\rho\lambda}\bar{R}_{\mu\nu}+\delta_{\mu\nu}{}^{\rho\lambda}\bar{R}_{\alpha\beta}),$
$\displaystyle{\bf
k}_{11}=\bar{g}_{\mu\nu}\bar{R}_{\alpha}{}^{\rho\lambda}{}_{\beta},~{}~{}~{}{\bf
k}_{12}=\bar{g}_{\alpha\beta}\bar{R}_{\mu}{}^{\rho\lambda}{}_{\nu},~{}~{}~{}{\bf
k}_{13}=\bar{g}_{\mu\nu}\bar{g}^{\rho\lambda}\bar{R}_{\alpha\beta},~{}~{}~{}{\bf
k}_{14}=\bar{g}_{\alpha\beta}\bar{g}^{\rho\lambda}\bar{R}_{\mu\nu},$
$\displaystyle{\bf
k}_{15}=\bar{g}_{\mu\nu}\delta_{\alpha}^{\lambda}\bar{R}_{\beta}^{\rho},~{}~{}~{}~{}{\bf
k}_{16}=\bar{g}_{\alpha\beta}\delta_{\mu}^{\lambda}\bar{R}_{\nu}^{\rho},~{}~{}~{}{\bf
k}_{17}=\bar{g}_{\mu\nu}\delta_{\alpha\beta}{}^{\rho\lambda},~{}~{}~{}{\bf
k}_{18}=\bar{g}_{\alpha\beta}\delta_{\mu\nu}{}^{\rho\lambda},$
$\displaystyle{\bf
k}_{19}=\bar{g}_{\mu\nu}\bar{g}_{\alpha\beta}\bar{g}^{\rho\lambda},~{}~{}~{}{\bf
k}_{20}=\bar{g}_{\mu\nu}\bar{g}_{\alpha\beta}\bar{R}^{\rho\lambda},~{}~{}~{}$
(29)

and

$\displaystyle
b_{1}=-2\gamma,~{}~{}~{}b_{2}=\frac{\alpha}{2}\bar{R}-\frac{1}{4\kappa^{2}},~{}~{}~{}b_{3}=\beta+3\gamma,~{}~{}~{}b_{4}=2\gamma,~{}~{}~{}b_{5}=\frac{1}{2\kappa^{2}}-\alpha\bar{R},$
$\displaystyle
b_{6}=\frac{\beta}{2}+\gamma,~{}~{}~{}b_{7}=-4\gamma,~{}~{}~{}b_{8}=-2\beta-4\gamma,~{}~{}~{}b_{9}=-2\gamma,~{}~{}~{}b_{10}=-2\alpha,$
$\displaystyle
b_{11}=4\gamma\Omega_{3},~{}~{}~{}b_{12}=\gamma,~{}~{}~{}b_{13}=-\beta\Omega_{3},~{}~{}~{}b_{14}=\alpha,~{}~{}~{}b_{15}=2\beta\Omega_{3},~{}~{}~{}b_{16}=\frac{\beta}{2},$
$\displaystyle
b_{17}=2\alpha\Omega_{3}\bar{R}-\frac{\Omega_{1}-2\Omega}{2\kappa^{2}},~{}~{}~{}b_{18}=\frac{\alpha}{2}\bar{R}-\frac{1}{4\kappa^{2}},~{}~{}~{}b_{19}=-b_{17},~{}~{}b_{20}=-\beta\Omega_{3}.~{}~{}$
(30)

where

$\displaystyle\Omega_{1}=\frac{10\alpha+3\beta+2\gamma}{\Sigma},~{}~{}~{}\Omega_{3}=\frac{3\alpha+\beta+\gamma}{\Sigma},~{}~{}~{}$
(31)

with $\Sigma$ given in (27). Next,

$\displaystyle(U)_{\mu\nu,\alpha\beta}$
$\displaystyle=\frac{4}{\beta+4\gamma}\Bigg{[}\frac{3}{2}\gamma\bar{g}_{\nu\beta}\bar{R}_{\mu}{}^{\rho\lambda\sigma}\bar{R}_{\alpha\rho\lambda\sigma}-\gamma\bar{R}^{\lambda}{}_{\alpha\mu}{}^{\rho}\bar{R}_{\lambda\nu\beta\rho}+4\gamma\bar{R}_{\rho\alpha\mu\lambda}\bar{R}_{\nu\beta}{}^{\rho\lambda}$
$\displaystyle-3\gamma(\bar{R}_{\mu}^{\sigma}\bar{R}_{\sigma\alpha\nu\beta}+\bar{R}^{\sigma}_{\alpha}\bar{R}_{\sigma\mu\beta\nu})+\Big{(}\frac{\beta}{2}+\gamma\Big{)}\bar{R}_{\mu\alpha}\bar{R}_{\nu\beta}-\frac{\gamma}{2}\bar{g}_{\alpha\beta}\bar{R}_{\mu\rho\lambda\sigma}\bar{R}_{\nu}{}^{\rho\lambda\sigma}$
$\displaystyle+\frac{1}{4}S^{2}(\Omega_{1}\bar{g}_{\mu\nu}\bar{g}_{\alpha\beta}-\bar{g}_{\mu\alpha}\bar{g}_{\nu\beta})+\Big{(}\frac{\alpha}{2}\bar{R}-\frac{1}{4\kappa^{2}}\Big{)}(\bar{R}_{\mu\alpha\nu\beta}+3\bar{g}_{\nu\beta}\bar{R}_{\mu\alpha}-\bar{g}_{\alpha\beta}\bar{R}_{\mu\nu})$
$\displaystyle+\Big{(}\frac{5}{2}\beta+4\gamma\Big{)}\bar{g}_{\nu\beta}\bar{R}_{\mu\sigma}\bar{R}^{\sigma}_{\alpha}+(\beta+5\gamma)\bar{R}_{\rho\mu\lambda\nu}\bar{R}^{\rho}{}_{\alpha}{}^{\lambda}{}_{\beta}-\frac{\beta}{2}\bar{g}_{\alpha\beta}\bar{R}_{\mu\sigma}\bar{R}^{\sigma}_{\nu}$
$\displaystyle-\gamma\Omega_{1}\bar{g}_{\mu\nu}\bar{R}_{\alpha\rho\lambda\sigma}\bar{R}_{\beta}{}^{\rho\lambda\sigma}-\beta\Omega_{1}\bar{g}_{\mu\nu}\bar{R}_{\alpha\sigma}\bar{R}^{\sigma}_{\beta}+\alpha\bar{R}_{\mu\nu}\bar{R}_{\alpha\beta}+\Big{(}\frac{1}{\kappa^{2}}\Omega_{3}-\alpha\Omega_{1}\bar{R}\Big{)}\bar{g}_{\mu\nu}\bar{R}_{\alpha\beta}$
$\displaystyle+\frac{1}{4\kappa^{2}}(\bar{R}-4\Lambda)\Omega\bar{g}_{\mu\nu}\bar{g}_{\alpha\beta}-(\beta+4\gamma)\bar{g}_{\nu\beta}\bar{R}^{\rho\lambda}\bar{R}_{\mu\rho\alpha\lambda}\Bigg{]},$
(32)

where we have defined

$\displaystyle
S^{2}=\alpha\bar{R}^{2}+\beta\bar{R}_{\mu\nu}^{2}+\gamma\bar{R}_{\mu\nu\rho\lambda}^{2}-\frac{1}{\kappa^{2}}(\bar{R}-2\Lambda),$
(33)

These results are given in Refs. BS2 ; OP1 .

##  4 General formula for one-loop divergences

Correcting the errors in JT , the results for one-loop divergences were given
in FT1 ; FT2 , but they still contained errors, and this was corrected in AB .
Here we give the correct results using heat kernel expansion.

The partition function for the one-loop is obtained from the quadratic terms
of the action as

$\displaystyle Z={\rm Det}(\Delta)^{-1/2},$ (34)

for a real scalar field. and the effective action is given by

$\displaystyle\Gamma=-\log Z=\frac{1}{2}{\rm Tr}\log\Delta.$ (35)

If the fluctuations are anticommuting fields like the FP ghosts, the sign
should be opposite. If the fluctuations are complex fields (or independent two
hermitian fields), the front factor should be 1.

Suppose we know the eigenvalues and eigenfunctions of the operator $\Delta$:

$\displaystyle\Delta\phi_{n}=\lambda_{n}\phi_{n}.$ (36)

Then we can evaluate (35) as

$\displaystyle\frac{1}{2}{\rm
Tr}\log\Delta=\frac{1}{2}\sum_{n}\log\lambda_{n}.$ (37)

We define the zeta function for $\Delta$:

$\displaystyle\zeta_{\Delta}(s)=\sum_{n=1}^{\infty}\lambda_{n}^{-s}=\frac{1}{\Gamma(s)}\int_{0}^{\infty}dt\,t^{s-1}{\rm
Tr}(e^{-t\Delta}),$ (38)

and obtain

$\displaystyle\frac{1}{2}{\rm
Tr}\log\Delta=\left.-\frac{1}{2}\frac{d}{ds}\zeta_{\Delta}(s)\right|_{s=0}=-\frac{1}{2}\frac{d}{ds}\left[\frac{1}{\Gamma(s)}\int_{0}^{\infty}dt\,t^{s-1}{\rm
Tr}(e^{-t\Delta})\right]_{s=0}.$ (39)

For a differential operator of order $p$ in $d$ dimensions, the heat kernel
$e^{-t\Delta}$ has the expansion

$\displaystyle{\rm
Tr}(e^{-t\Delta})=\int\frac{d^{d}x}{(4\pi)^{d/2}}\sqrt{g}\sum_{n=0}^{\infty}b_{2n}(\Delta)\,t^{(2n-d)/p}.$
(40)

Typically we have the operators of $p=4$ and $p=2$. The formulae are given
separately. The divergent part of the effective action for $p=4$ operator is
evaluated as

$\displaystyle\Gamma^{(4)}$ $\displaystyle=$
$\displaystyle-\frac{1}{2}\int\frac{d^{d}x}{(4\pi)^{d/2}}\sqrt{g}\int_{1/\Lambda_{\rm
UV}^{4}}^{1/\mu^{4}}dt\left[t^{-\frac{d}{4}-1}b_{0}+t^{-\frac{d-2}{4}-1}b_{2}+\cdots+t^{-1}b_{d}+\cdots\right]$
(41) $\displaystyle=$
$\displaystyle-\frac{1}{2}\int\frac{d^{d}x}{(4\pi)^{d/2}}\sqrt{g}\left[\frac{\Lambda_{\rm
UV}^{d}}{d/4}b_{0}+\frac{\Lambda_{\rm
UV}^{d-2}}{(d-2)/4}b_{2}+\cdots+\log\frac{\Lambda_{\rm
UV}^{4}}{\mu^{4}}b_{d}+\mbox{ finite }\right],\qquad$

where the heat kernel coefficients are given in the subsection 4.1. Here
$\Lambda_{\rm UV}$ is an ultraviolet cutoff which should be distinguished from
the cosmological constant $\Lambda$.

The divergent part of the effective action for $p=2$ operator is given by

$\displaystyle\Gamma^{(2)}$ $\displaystyle=$
$\displaystyle-\frac{1}{2}\int\frac{d^{d}x}{(4\pi)^{d/2}}\sqrt{g}\int_{1/\Lambda_{\rm
UV}^{2}}^{1/\mu^{2}}dt\left[t^{-\frac{d}{2}-1}b_{0}+t^{-\frac{d}{2}}b_{2}+\cdots+t^{-1}b_{d}+\cdots\right]$
(42) $\displaystyle=$
$\displaystyle-\frac{1}{2}\int\frac{d^{d}x}{(4\pi)^{d/2}}\sqrt{g}\left[\frac{\Lambda_{\rm
UV}^{d}}{d/2}b_{0}+\frac{\Lambda_{\rm
UV}^{d-2}}{\frac{d}{2}-1}b_{2}+\cdots+\log\frac{\Lambda_{\rm
UV}^{2}}{\mu^{2}}b_{d}+\mbox{ finite }\right].\qquad$

These are relevant for the contributions from the ghost and the operator $Y$.

In our present case, the graviton contribution (25) is $p=4$ and the ghost
contributions are $p=2$. So the one-loop part of our effective action is

$\displaystyle\Gamma^{\rm 1-loop}$ $\displaystyle=$
$\displaystyle\frac{1}{2}{\rm Tr}\log{\cal H}-{\rm
Tr}\log{\Delta_{gh}}-\frac{1}{2}{\rm Tr}\log{Y}$ (43) $\displaystyle=$
$\displaystyle-\int\frac{d^{4}x}{2(4\pi)^{2}}\sqrt{g}\Big{[}\Lambda_{\rm
UV}^{4}\Big{(}b_{0}({\cal H})-b_{0}(\Delta_{gh})-\frac{1}{2}b_{0}(Y)\Big{)}$
$\displaystyle\hskip 28.45274pt+\Lambda_{\rm UV}^{2}\Big{(}2b_{2}({\cal
H})-2b_{2}(\Delta_{gh})-b_{2}(Y)\Big{)}$ $\displaystyle\hskip
28.45274pt+\log\frac{\Lambda_{\rm UV}^{2}}{\mu^{2}}\Big{(}2b_{4}({\cal
H})-2b_{4}(\Delta_{gh})-b_{4}(Y)\Big{)}\Big{]}.$

Now we start the evaluation of these contributions.

###  4.1 Contributions from $p=4$ minimal operator

First we have to evaluate the contribution from (25) which is $p=4$ minimal
operator. The coefficients in the heat kernel expansion for spin 2 are G :

$\displaystyle b_{0}({\cal H})$ $\displaystyle=$
$\displaystyle\frac{\Gamma(d/4)}{2\Gamma(d/2)}\frac{d(d+1)}{2}\;,$ (44)
$\displaystyle b_{2}({\cal H})$ $\displaystyle=$
$\displaystyle\frac{\Gamma((d-2)/4)}{2\Gamma((d-2)/2)}\,\mathrm{tr}\left[\frac{\bar{R}}{6}+\frac{1}{2d}V^{\mu}_{\mu}\right],$
(45) $\displaystyle b_{4}({\cal H})$ $\displaystyle=$
$\displaystyle\frac{\Gamma(d/4)}{2\Gamma((d-2)/2)}\mathrm{tr}\Bigl{[}\frac{\hat{1}}{90}\bar{R}_{\rho\lambda\sigma\tau}^{2}-\frac{\hat{1}}{90}\bar{R}_{\rho\lambda}^{2}+\frac{\hat{1}}{36}\bar{R}^{2}+\frac{1}{6}{\Omega}_{\rho\lambda}{\Omega}^{\rho\lambda}-\frac{2}{d-2}U$
(46)
$\displaystyle-\frac{1}{6(d-2)}(2\bar{R}_{\rho\lambda}V^{\rho\lambda}-\bar{R}V^{\rho}{}_{\rho})+\frac{1}{4(d^{2}-4)}(V^{\rho}{}_{\rho}V^{\lambda}{}_{\lambda}+2V_{\rho\lambda}V^{\rho\lambda})$
$\displaystyle+\frac{1}{15}\Box\bar{R}+\frac{d+4}{6(d^{2}-4)}\Box
V^{\rho}{}_{\rho}-\frac{2(d+1)}{3(d^{2}-4)}\bar{\nabla}^{\rho}\bar{\nabla}^{\lambda}V_{\rho\lambda}\Bigr{]},$

where $\hat{1}$ is the identity defined in (13) and ${\Omega}_{\rho\lambda}$
is the commutator of the covariant derivatives acting on the tensor
$h^{\alpha\beta}$:
${\Omega}_{\rho\lambda}=[\bar{\nabla}_{\rho},\bar{\nabla}_{\lambda}]$. The
traces should be taken over the space of symmetric tensors with the identity
(13). These give ${\rm tr}(\hat{1})=\frac{d(d+1)}{2},\,{\rm
tr}({\Omega}_{\rho\lambda}{\Omega}^{\rho\lambda})=-(d+2)\bar{R}_{\mu\nu\alpha\beta}^{2}$.

We need more explicit formulae for the traces. Because these are very
complicated for general dimensions, we give the results for $d=4$ and omit
total derivative terms. We find

$\displaystyle{\rm
tr}\,U=\delta^{\mu\nu,\alpha\beta}U_{\mu\nu,\alpha\beta}=A_{1}\bar{R}_{\mu\nu\rho\lambda}^{2}+A_{2}\bar{R}_{\mu\nu}^{2}+A_{3}\bar{R}^{2}-A_{4}\frac{\bar{R}}{\kappa^{2}}-A_{5}\frac{\Lambda}{\kappa^{2}},$
(47)

where

$\displaystyle
A_{1}=3,~{}~{}A_{2}=\frac{8}{3}+\frac{4\lambda}{\xi},~{}~{}A_{3}=\frac{1}{3}+\frac{2\lambda}{\xi},~{}~{}A_{4}=3\lambda,~{}~{}A_{5}=\frac{2}{9}(84\lambda-\xi).$
(48)

and

$\displaystyle{\rm
tr}\,(V^{\rho}_{\rho}\bar{R})=B_{1}\bar{R}^{2}-B_{2}\frac{\bar{R}}{\kappa^{2}},$
(49)

where

$\displaystyle
B_{1}=-\frac{68}{3}+\frac{32\lambda}{\xi},~{}~{}B_{2}=-20\lambda+\frac{2\xi}{3}.$
(50)

Next

$\displaystyle{\rm
tr}\,(V^{\rho\lambda}\bar{R}_{\rho\lambda})=C_{1}\bar{R}_{\mu\nu}^{2}+C_{2}\bar{R}^{2}-C_{3}\frac{\bar{R}}{\kappa^{2}},$
(51)

where

$\displaystyle
C_{1}=\frac{8}{3}-\frac{8\lambda}{\xi},~{}~{}C_{2}=-\frac{19}{3}+\frac{10\lambda}{\xi},~{}~{}C_{3}=-5\lambda+\frac{\xi}{6}.$
(52)

Finally

$\displaystyle\frac{1}{48}{\rm
tr}(V^{\rho}_{\rho}V^{\lambda}_{\lambda})+\frac{1}{24}{\rm
tr}(V_{\rho_{\lambda}}V^{\rho\lambda})=D_{1}\bar{R}_{\mu\nu\rho\lambda}^{2}+D_{2}\bar{R}_{\mu\nu}^{2}+D_{3}\bar{R}^{2}-D_{4}\frac{\bar{R}}{\kappa^{2}}+D_{5}\frac{1}{\kappa^{4}},~{}~{}~{}~{}$
(53)

where

$\displaystyle
D_{1}=6,~{}~{}D_{2}=\frac{2(18\lambda^{2}+6\lambda\xi+113\xi^{2})}{27\xi^{2}},~{}~{}D_{3}=\frac{576\lambda^{2}-240\lambda\xi-47\xi^{2}}{54\xi^{2}},$
$\displaystyle
D_{4}=\frac{-180\lambda^{2}+30\lambda\xi+\xi^{2}}{18\xi},~{}~{}D_{5}=\frac{180\lambda^{2}+\xi^{2}}{72}.$
(54)

###  4.2 Contribution from the $p=2$ nonminimal vector operator

To find the contribution from the ghost operator, we need the contribution
from $p=2$ nonminimal vector operator. The general form of the $p=2$
nonminimal operator is

$\displaystyle\Delta=-\bar{g}^{\mu\nu}\Box+a\bar{\nabla}^{\mu}\bar{\nabla}^{\nu}+X^{\mu\nu},$
(55)

The coefficients in the heat kernel expansion (40) have been calculated in GK
. For the general case, we have

$\displaystyle b_{0}=(1-a)^{-d/2}+d-1,\hskip 227.62204pt$ (56) $\displaystyle
b_{2}=\left(\frac{d^{2}-d-6}{6d}+(1-a)^{-d/2}\frac{6+(1-a)d}{6d}\right)\bar{R}+\frac{1-d-(1-a)^{-d/2}}{d}X,\hskip
48.36967pt$ (57) $\displaystyle b_{4}$ $\displaystyle=$
$\displaystyle\frac{d-16+(1-a)^{2-d/2}}{180}\bar{R}_{\mu\nu\rho\lambda}^{2}$
(58)
$\displaystyle+\frac{1}{180ad(d^{2}-4)}\Big{[}-360d-(d^{4}-d^{3}+116d^{2}-296d-360)a+(1-a)^{-d/2}[-360a$
$\displaystyle\hskip
28.45274pt+4(90-74a+28a^{2}+a^{3})d-60a(1-a)d^{2}-(1-a)^{2}ad^{3}]\Big{]}\bar{R}_{\mu\nu}^{2}$
$\displaystyle+\frac{1}{72ad(d^{2}-4)}\Big{[}72(2-a)+(16d-16d^{2}-d^{3}+d^{4})a$
$\displaystyle+(1-a)^{-d/2}[-72(2-a)-4a(a^{2}+4a-14)d+12a(1-a)d^{2}+a(1-a)^{2}d^{3}]\Big{]}\bar{R}^{2}$
$\displaystyle+\frac{1}{6ad(d^{2}-4)}\Big{[}\Big{\\{}-24+12a+ad^{2}-ad^{3}+(1-a)^{-d/2}[24-12a$
$\displaystyle+2a(a-4)d-a(1-a)d^{2}]\Big{\\}}\bar{R}X+2\Big{\\{}-12a+4(3-2a)d+5ad^{2}$
$\displaystyle+(1-a)^{-d/2}[12a-2(6-4a+a^{2})d+a(1-a)d^{2}]\Big{\\}}\bar{R}_{\mu\nu}X^{\mu\nu}$
$\displaystyle+3\Big{\\{}4-2a+ad+(1-a)^{-d/2}(-4+2a+ad)\Big{\\}}X^{2}+3\Big{\\{}4a-2(2+a)d-2ad^{2}$
$\displaystyle+ad^{3}+(1-a)^{-d/2}[-4a+2(2-a)d]\Big{\\}}X_{\mu\nu}X^{\mu\nu}\Big{]}+\mbox{
(total derivative terms)},$

where $X=X_{\mu}^{\mu}$.

Contribution from the the ghost operator $\Delta_{gh}$

For the ghost operator $\Delta_{gh}$ in (19), we see that we have
$a=-1-\frac{4\alpha+\beta}{2(\gamma-\alpha)}\equiv\sigma_{g}$ and
$X_{\mu\nu}=-\bar{R}_{\mu\nu}$ in (55). The above formulae give, for $d=4$,

$\displaystyle b_{0}(\Delta_{gh})$ $\displaystyle=$
$\displaystyle(1-\sigma_{g})^{-2}+3,$ (59) $\displaystyle b_{2}(\Delta_{gh})$
$\displaystyle=$
$\displaystyle\frac{6\sigma_{g}^{2}-13\sigma_{g}+10}{6(1-\sigma_{g})^{2}}\bar{R},$
(60) $\displaystyle b_{4}(\Delta_{gh})$ $\displaystyle=$
$\displaystyle\frac{-11}{180}\bar{R}_{\mu\nu\rho\lambda}^{2}-\frac{2\sigma_{g}^{2}+26\sigma_{g}-43}{90(1-\sigma_{g})^{2}}\bar{R}_{\mu\nu}^{2}+\frac{5\sigma_{g}^{2}-10\sigma_{g}+8}{36(1-\sigma_{g})^{2}}\bar{R}^{2}.$
(61)

Contribution from the the additional ghost operator $Y$

For the operator $Y$ in (19), we first note that
$[\bar{\nabla}_{\mu},\bar{\nabla}_{\nu}]\chi^{\nu}=-\bar{R}_{\mu\nu}\chi^{\nu}$.
Using this relation, we find that we have
$a=1+2\frac{\alpha-\gamma}{\beta+4\gamma}\equiv\sigma_{Y}$ and
$X_{\mu\nu}=\bar{R}_{\mu\nu}$ in (55), and the above formulae give, for $d=4$,

$\displaystyle b_{0}(Y)$ $\displaystyle=$
$\displaystyle(1-\sigma_{Y})^{-2}+3,$ (62) $\displaystyle b_{2}(Y)$
$\displaystyle=$ $\displaystyle\frac{3\sigma_{Y}-2}{6(1-\sigma_{Y})}\bar{R},$
(63) $\displaystyle b_{4}(Y)$ $\displaystyle=$
$\displaystyle\frac{-11}{180}\bar{R}_{\mu\nu\rho\lambda}^{2}+\frac{43}{90}\bar{R}_{\mu\nu}^{2}-\frac{1}{9}\bar{R}^{2}.$
(64)

##  5 One-loop divergences and asymptotic freedom

We are now ready to calculate the divergences in the quadratic curvature
theory on the general backgrounds. For this purpose, we need to know the heat
kernel coefficients. In this section, we restrict the dimension of our
spacetime to four.

Putting the results for ${\cal H}$ into eq. (46), we get

$\displaystyle b_{0}({\cal H})$ $\displaystyle=$ $\displaystyle 5,$ (65)
$\displaystyle b_{2}({\cal H})$ $\displaystyle=$
$\displaystyle\frac{\sqrt{\pi}}{2}\left[\frac{5\bar{R}}{3}+\frac{1}{8}\Big{(}B_{1}\bar{R}-B_{2}\frac{1}{\kappa^{2}}\Big{)}\right],$
(66) $\displaystyle b_{4}({\cal H})$ $\displaystyle=$
$\displaystyle\frac{1}{2}\Biggl{[}\bar{R}_{\mu\nu\rho\lambda}^{2}\left(-\frac{8}{9}-A_{1}+D_{1}\right)-\bar{R}_{\mu\nu}^{2}\left(\frac{1}{9}+A_{2}+\frac{1}{6}C_{1}-D_{2}\right)$
(67)
$\displaystyle+\bar{R}^{2}\left(\frac{5}{18}-A_{3}+\frac{1}{12}B_{1}-\frac{1}{6}C_{2}+D_{3}\right)+\frac{1}{\kappa^{2}}\bar{R}\left(A_{4}-\frac{1}{12}B_{2}+\frac{1}{6}C_{3}-D_{4}\right)$
$\displaystyle+\frac{1}{\kappa^{2}}\left(\Lambda\,A_{5}+\frac{1}{\kappa^{2}}D_{5}\right)\Biggr{]},$

where the constants $A_{i},B_{i},C_{i}$ and $D_{i}$ are given in (48) – (54).

Collecting other contributions from ghosts and $Y$, we finally get

$\displaystyle\Gamma^{\rm 1-loop}$ $\displaystyle=$
$\displaystyle-\int\frac{d^{4}x}{2(4\pi)^{2}}\Big{[}\Big{\\{}\frac{133}{20}C^{2}+\Big{(}10\frac{\lambda^{2}}{\xi^{2}}-5\frac{\lambda}{\xi}+\frac{5}{36}\Big{)}\bar{R}^{2}-\frac{196}{45}E$
(68) $\displaystyle\hskip
28.45274pt+\,\frac{(30\lambda-\xi)(4\lambda+\xi)}{12\xi\kappa^{2}}\bar{R}+2\frac{84\lambda-\xi}{9\kappa^{2}}\Lambda+\frac{180\lambda^{2}+\xi^{2}}{72\kappa^{4}}\Big{\\}}\log\frac{\Lambda_{\rm
UV}^{2}}{\mu^{2}}$ $\displaystyle\hskip
28.45274pt-\,\Big{\\{}\frac{144\lambda^{3}-24(7+6\sqrt{\pi})\lambda^{2}\xi+2(59+45\sqrt{\pi})\lambda\xi^{2}-(29+14\sqrt{\pi})\xi^{3}}{12(3\lambda-\xi)\xi^{2}}\bar{R}$
$\displaystyle\hskip
28.45274pt+\,\frac{\sqrt{\pi}(30\lambda-\xi)}{12\kappa^{2}}\Big{\\}}\Lambda_{\rm
UV}^{2}$ $\displaystyle\hskip
28.45274pt-\,\frac{2592\lambda^{4}-3456\lambda^{3}\xi+3024\lambda^{2}\xi^{2}-1248\lambda\xi^{3}+257\xi^{4}}{72\xi^{2}(3\lambda-\xi)^{2}}\Lambda_{\rm
UV}^{4}\Big{]}.$

Note that the coefficient of the Euler term $E$ is independent of any
coupling, in particular of its coupling $\rho$. This is to be expected,
because the Euler term is a topological term and is a total derivative itself,
and as such it does not contribute to the Hessian and therefore to quantum
effects FOP . Thus it is a universal result that it is independent of the
coupling $\rho$ whatever the approximation is (beyond one loop).

The correspondence between the cutoff and the dimensional regularization is
$\log\frac{\Lambda_{\rm UV}^{2}}{\mu^{2}}\leftrightarrow\frac{2}{4-d}$ and one
can try to compare the results with the existing literature (see for example,
AB ; BuchbinderS ).

Note also that there are strange terms with coefficients $\sqrt{\pi}$ in the
quadratic divergences. Indeed, if we consider the minimal operators
$F_{1}=\Box+P_{1}$ and $F_{2}=\Box+P_{2}$ and consider the quadratic
divergences of ${\rm Tr}\log(F_{1}F_{2})$, it appears that we get such factor
of $\sqrt{\pi}$ from the formula $b_{2}$ in (66), whereas if we write it as
${\rm Tr}\log F_{1}+{\rm Tr}\log F_{2}$, we do not get $\sqrt{\pi}$ as is
clear from (57) for the same quantity. This clearly indicates that the
coefficients of the quadratic divergences depend on how we calculate. Thus the
power divergences are not universal and do not lead to physical effects like
renormalization group scaling. The above results are just those obtained by
the naive application of the formulae but should not be taken seriously.

On the other hand, the logarithmic divergences are universal. From the
coefficients, we can determine the beta functions for the dimensionless
couplings: recall that together with the bare terms, the coefficients of
$C^{2}$ should give the renormalized coupling

$\displaystyle\frac{1}{2\lambda_{R}}=\frac{1}{2\lambda_{B}}-\frac{133}{(4\pi)^{2}40}\log\frac{\Lambda_{\rm
UV}^{2}}{\mu^{2}}.$ (69)

Since the bare coupling $\lambda_{B}$ does not depend on the renormalization
scale $\mu$, differentiation of the expression with respect to $\log\mu$ gives

$\displaystyle-\frac{1}{2\lambda_{R}^{2}}\mu\frac{d\lambda_{R}}{d\mu}=\frac{133}{(4\pi)^{2}20}.$
(70)

Omitting the subscript $R$, this gives the beta function of the coupling
$\lambda$:

$\displaystyle\mu\frac{d\lambda}{d\mu}=\beta_{\lambda}=-\frac{1}{(4\pi)^{2}}\frac{133}{10}\lambda^{2},$
(71)

Similarly we find the beta functions for other dimensionless couplings:

$\displaystyle\beta_{\xi}$ $\displaystyle=$
$\displaystyle-\frac{1}{(4\pi)^{2}}\left(10\lambda^{2}-5\lambda\xi+\frac{5}{36}\xi^{2}\right),$
$\displaystyle\beta_{\rho}$ $\displaystyle=$
$\displaystyle-\frac{1}{(4\pi)^{2}}\frac{196}{45}\rho^{2}.$ (72)

The first equation (71) tells us that the coupling $\lambda$ goes to zero from
positive $\lambda$. Similarly the other couplings also go to zero. This is
known as asymptotic freedom FT1 ; FT2 ; AB ; BS1 ; BS2 ; OP1 ; OP2 . However
it has recently been discovered that there are fixed point at finite values
for these couplings FOP in addition to these Gaussian fixed points. See,
however, KO .

##  6 Divergences for $f(R,R_{\mu\nu}^{2})$ gravity

As an interesting case of higher-derivative gravity, here we present the one-
loop divergences for $f(R,R_{\mu\nu})$ gravity on the Einstein space OPP :

$\displaystyle\bar{R}_{\mu\nu}=\frac{\bar{R}}{d}\bar{g}_{\mu\nu}.$ (73)

Moreover we consider a general parametrization of the fluctuations

$\displaystyle g_{\mu\nu}=\bar{g}_{\mu\nu}+\delta g_{\mu\nu}\ ,$ (74)

where the fluctuation is expanded:

$\displaystyle\delta g_{\mu\nu}=\delta g^{(1)}_{\mu\nu}+\delta
g^{(2)}_{\mu\nu}+\delta g^{(3)}_{\mu\nu}+\ldots\ ,$ (75)

where $\delta g^{(n)}_{\mu\nu}$ contains $n$ powers of $h_{\mu\nu}$. We will
parametrize the first two terms of the expansion as follows:

$\displaystyle\delta g^{(1)}_{\mu\nu}$ $\displaystyle=$ $\displaystyle
h_{\mu\nu}+m\bar{g}_{\mu\nu}h\ ,$ $\displaystyle\delta g^{(2)}_{\mu\nu}$
$\displaystyle=$ $\displaystyle\omega
h_{\mu\rho}h^{\rho}{}_{\nu}+mhh_{\mu\nu}+m\left(\omega-\frac{1}{2}\right)\bar{g}_{\mu\nu}h^{\alpha\beta}h_{\alpha\beta}+\frac{1}{2}m^{2}\bar{g}_{\mu\nu}h^{2}\
.$ (76)

It is convenient to use York decomposition

$\displaystyle
h_{\mu\nu}=h^{TT}_{\mu\nu}+\bar{\nabla}_{\mu}\xi_{\nu}+\bar{\nabla}_{\nu}\xi_{\mu}+\bar{\nabla}_{\mu}\bar{\nabla}_{\nu}\sigma-\frac{1}{d}\bar{g}_{\mu\nu}\bar{\nabla}^{2}\sigma+\frac{1}{d}\bar{g}_{\mu\nu}h,$
(77)

where

$\bar{\nabla}^{\mu}h^{TT}_{\mu\nu}=0\
;\qquad\bar{g}^{\mu\nu}h^{TT}_{\mu\nu}=0\
;\qquad\bar{\nabla}_{\mu}\xi^{\mu}=0\ .$

We then find that the Hessian is

$\displaystyle S^{(2)}=\int d^{d}x\sqrt{\bar{g}}$
$\displaystyle\left[h^{TT}_{\mu\nu}H^{TT}h^{TT\mu\nu}+\xi_{\mu}H^{\xi\xi}\xi^{\mu}+\sigma
H^{\sigma\sigma}\sigma+\sigma H^{\sigma h}h+hH^{h\sigma}\sigma\right.$ (78)
$\displaystyle\left.+hH^{hh}h\right],$

where

$\displaystyle H^{TT}$ $\displaystyle=$
$\displaystyle\frac{1}{4}\left[\left\\{\bar{f}_{X}\left({\Delta_{L}}_{2}-\frac{4\bar{R}}{d}\right)-\bar{f}_{R}\right\\}\left({\Delta_{L}}_{2}-\frac{2\bar{R}}{d}\right)-(1-2\omega)(1+md)\tilde{E}\right],$
(79) $\displaystyle H^{\xi\xi}$ $\displaystyle=$
$\displaystyle-\frac{(1-2\omega)(1+md)}{2}\left({\Delta_{L}}_{1}-\frac{2\bar{R}}{d}\right)\tilde{E},$
(80) $\displaystyle H^{\sigma\sigma}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left(\frac{d-1}{d}\right)^{2}\left[P{\Delta_{L}}_{0}\left({\Delta_{L}}_{0}-\frac{\bar{R}}{d-1}\right)+Q{\Delta_{L}}_{0}\right.$
(81) $\displaystyle\left.\
-\frac{d(1-2\omega)(1+md)}{2(d-1)}\tilde{E}\right]{\Delta_{L}}_{0}\left({\Delta_{L}}_{0}-\frac{\bar{R}}{d-1}\right),$
$\displaystyle H^{\sigma h}$ $\displaystyle=$
$\displaystyle\left(\frac{d-1}{d}\right)^{2}\frac{1+md}{2}\left[P\left({\Delta_{L}}_{0}-\frac{\bar{R}}{d-1}\right)+Q\right]{\Delta_{L}}_{0}\left({\Delta_{L}}_{0}-\frac{\bar{R}}{d-1}\right),$
(82) $\displaystyle H^{hh}$ $\displaystyle=$
$\displaystyle\left(\frac{d-1}{d}\right)^{2}\frac{(1+md)^{2}}{2}\left[P\left({\Delta_{L}}_{0}-\frac{\bar{R}}{d-1}\right)^{2}+\,Q\left({\Delta_{L}}_{0}-\frac{\bar{R}}{d-1}\right)\right.$
(83)
$\displaystyle\left.\qquad\qquad\qquad\qquad\qquad\qquad+\frac{d[(1+md)d-2(1-2\omega)]}{4(d-1)^{2}(1+md)}\tilde{E}\right],$

where ${\Delta_{L}}_{2}$, ${\Delta_{L}}_{1}$ and ${\Delta_{L}}_{0}$ are the
Lichnerowicz Laplacians defined as

$\displaystyle{\Delta_{L}}_{2}T_{\mu\nu}$ $\displaystyle=$
$\displaystyle-\bar{\nabla}^{2}T_{\mu\nu}+\bar{R}_{\mu}{}^{\rho}T_{\rho\nu}+\bar{R}_{\nu}{}^{\rho}T_{\mu\rho}-\bar{R}_{\mu\rho\nu\sigma}T^{\rho\sigma}-\bar{R}_{\mu\rho\nu\sigma}T^{\sigma\rho},$
$\displaystyle{\Delta_{L}}_{1}V_{\mu}$ $\displaystyle=$
$\displaystyle-\bar{\nabla}^{2}V_{\mu}+\bar{R}_{\mu}{}^{\rho}V_{\rho},$
$\displaystyle{\Delta_{L}}_{0}S$ $\displaystyle=$
$\displaystyle-\bar{\nabla}^{2}S.$ (84)

and the subscripts on $f$ denote derivatives with respect to its arguments:

$\displaystyle f_{R}=\frac{\partial f}{\partial\bar{R}}\ ,\quad
f_{X}=\frac{\partial f}{\partial X}\ ,\quad
f_{RR}=\frac{\partial^{2}f}{\partial\bar{R}^{2}}\ ,\quad
f_{RX}=\frac{\partial^{2}f}{\partial\bar{R}\partial X}\ ,\quad
f_{XX}=\frac{\partial^{2}f}{\partial X^{2}}\ ,$ (85)

with

$\displaystyle X\equiv\bar{R}_{\mu\nu}^{2},$ (86)

We have also used the shorthands

$\displaystyle P$ $\displaystyle=$
$\displaystyle\bar{f}_{RR}+\frac{4}{d^{2}}\bar{R}^{2}\bar{f}_{XX}+4\bar{R}\bar{f}_{RX}+\frac{d}{2(d-1)}\bar{f}_{X}$
(87) $\displaystyle Q$ $\displaystyle=$
$\displaystyle\frac{d-2}{2(d-1)}\bar{f}_{R}+\frac{3d^{2}-10d+8}{2d(d-1)^{2}}\bar{R}\bar{f}_{X}$
(88)

and

$\displaystyle\tilde{E}\equiv\bar{f}-\frac{2}{d}\bar{R}\bar{f}_{R}-\frac{4\bar{R}^{2}}{d^{2}}\bar{f}_{X}=0,$
(89)

is the field equation evaluated on the Einstein space (73).

Our gauge fixing is

$\displaystyle S_{GF}=\frac{1}{2a}\int
d^{d}x\sqrt{\bar{g}}\,\bar{g}^{\mu\nu}F_{\mu}F_{\nu},$ (90)

with

$\displaystyle
F_{\mu}=\bar{\bar{\nabla}}_{\alpha}{h^{\alpha}}_{\mu}-\frac{\bar{b}+1}{d}\bar{\bar{\nabla}}_{\mu}h\,,$
(91)

and $a$ and $\bar{b}$ are gauge parameters. This can be rewritten as

$\displaystyle S_{GF}=-\frac{1}{2a}\int
d^{d}x\sqrt{\bar{g}}\left[\xi_{\mu}\left({\Delta_{L}}_{1}-\frac{2\bar{R}}{d}\right)^{2}\xi^{\mu}+\frac{(d-1-b)^{2}}{d^{2}}\chi\;{\Delta_{L}}_{0}\left({\Delta_{L}}_{0}-\frac{\bar{R}}{d-1-b}\right)^{2}\chi\right],$

in terms of the new field

$\displaystyle\chi=\frac{(d-1){\Delta_{L}}_{0}-\bar{R}}{(d-1-b){\Delta_{L}}_{0}-\bar{R}}\sigma+\frac{b(1+dm)}{(d-1-b){\Delta_{L}}_{0}-\bar{R}}h\
,$ (93)

where $b=\bar{b}/(1+md)$.

The ghost action contains a nonminimal operator

$\displaystyle S_{gh}=i\int
d^{d}x\sqrt{\bar{g}}\,\bar{C}^{\mu}\left(\delta_{\mu}^{\nu}\bar{\nabla}^{2}+\left(1-2\frac{b+1}{d}\right)\bar{\nabla}_{\mu}\bar{\nabla}^{\nu}+\bar{R}_{\mu}{}^{\nu}\right)C_{\nu},$
(94)

We can rewrite this as

$\displaystyle S_{gh}=i\int
d^{d}x\sqrt{\bar{g}}\left[\bar{C}^{T\mu}\left({\Delta_{L}}_{1}-\frac{2\bar{R}}{d}\right)C^{T}_{\mu}+2\frac{d-1-b}{d}\bar{C}^{\prime
L}\left({\Delta_{L}}_{0}-\frac{\bar{R}}{d-1-b}\right)C^{\prime L}\right].$

in terms of the transverse and longitudinal parts of the ghost field:

$\displaystyle
C_{\nu}=C^{T}_{\nu}+\bar{\nabla}_{\nu}C^{L}=C^{T}_{\nu}+\bar{\nabla}_{\nu}\frac{1}{\sqrt{{\Delta_{L}}_{0}}}C^{\prime
L},$ (96)

and the same for $\bar{C}$. This change of variables has unit Jacobian.

$\displaystyle S_{gh}=i\int
d^{d}x\sqrt{\bar{g}}\left[\bar{C}^{T\mu}\left({\Delta_{L}}_{1}-\frac{2\bar{R}}{d}\right)C^{T}_{\mu}+2\frac{d-1-b}{d}\bar{C}^{\prime
L}\left({\Delta_{L}}_{0}-\frac{\bar{R}}{d-1-b}\right)C^{\prime L}\right].$

Unless we (1) set $\omega=\frac{1}{2}$, or (2) put $m=-\frac{1}{d}$ or (3) go
on shell, the effective action is gauge dependent. If we impose the on-shell
condition (89). the effective action is gauge independent and is independent
of $\omega$ and $m$. In this case, we find

$\displaystyle\Gamma$ $\displaystyle=$ $\displaystyle\frac{1}{2}{\rm
Tr}\log\left({\Delta_{L}}_{2}-\frac{4\bar{R}}{d}-\frac{\bar{f}_{R}}{\bar{f}_{X}}\right)+\frac{1}{2}{\rm
Tr}\log\left({\Delta_{L}}_{2}-\frac{2\bar{R}}{d}\right)$ (98)
$\displaystyle+\frac{1}{2}{\rm
Tr}\log\left({\Delta_{L}}_{0}-\frac{\bar{R}}{d-1}+\frac{Q}{P}\right)-\frac{1}{2}{\rm
Tr}\log\left({\Delta_{L}}_{1}-\frac{2\bar{R}}{d}\right).$

If $\bar{f}_{X}=0$, the first contribution is absent.

The divergent part of the effective action can be computed by the heat kernel
methods. On an Einstein background in four dimensions, with the help of the
heat kernel coefficients for the Lichnerowicz operator summerized in Appendix
A.2, the logarithmically divergent part is found to be OPP

$\displaystyle\Gamma_{\rm log}(\bar{g})$ $\displaystyle=$
$\displaystyle\frac{1}{720(4\pi)^{2}}\int
d^{4}x\,\sqrt{\bar{g}}\log\left(\frac{\Lambda_{\rm
UV}^{2}}{\mu^{2}}\right)\left[-826\bar{R}_{\mu\nu\rho\sigma}^{2}+509\bar{R}^{2}-\frac{300\bar{R}\bar{f}_{R}}{\bar{f}_{X}}-\frac{900\bar{f}_{R}^{2}}{\bar{f}_{X}^{2}}\right.$
$\displaystyle\left.+\frac{240\bar{R}(3\bar{f}_{R}+2\bar{R}\bar{f}_{X})}{8\bar{f}_{X}+12\bar{f}_{RR}+48\bar{R}\bar{f}_{RX}+3\bar{R}^{2}\bar{f}_{XX}}-\frac{320(3\bar{f}_{R}+2\bar{R}\bar{f}_{X})^{2}}{(8\bar{f}_{X}+12\bar{f}_{RR}+48\bar{R}\bar{f}_{RX}+3\bar{R}^{2}\bar{f}_{XX})^{2}}\right],$

where $\Lambda_{\rm UV}$ stands for a cutoff and we introduced a reference
mass scale $\mu$.

For the choice

$\displaystyle f(R,X)=\alpha\bar{R}^{2}+\beta X\ ,$ (100)

it reduces to

$\displaystyle\Gamma_{\rm log}(\bar{g})=\frac{1}{(4\pi)^{2}}\int
d^{4}x\,\sqrt{\bar{g}}\log\left(\frac{\Lambda_{\rm
UV}^{2}}{\mu^{2}}\right)\left[-\frac{413}{360}\bar{R}_{\mu\nu\rho\sigma}^{2}-\frac{1200\alpha^{2}+200\alpha\beta-183\beta^{2}}{240\beta^{2}}\bar{R}^{2}\right],$
(101)

which is the standard universal result in higher-derivative gravity.

On the other hand if we put

$\displaystyle f(R,X)=f(R)\ ,$ (102)

we obtain

$\displaystyle\Gamma_{\rm log}(\bar{g})=\frac{1}{(4\pi)^{2}}\int
d^{4}x\,\sqrt{\bar{g}}\log\left(\frac{\Lambda_{\rm
UV}^{2}}{\mu^{2}}\right)\left[-\frac{71}{120}\bar{R}_{\mu\nu\rho\sigma}^{2}+\frac{433}{1440}\bar{R}^{2}+\frac{\bar{f}_{R}\bar{R}}{12\bar{f}_{RR}}-\frac{\bar{f}_{R}^{2}}{36\bar{f}_{RR}^{2}}\right],$
(103)

which agrees with the results of RS ; Ohta .

## Acknowledgments

We would like to thank Andrei Barvinsky, Roberto Percacci and Ilya Shapiro for
valuable discussions. This work was supported in part by the Grant-in-Aid for
Scientific Research Fund of the JSPS (C) No. 16K05331, No. 20K03980, and by
the Ministry of Science and Technology, R. O. C. (Taiwan) under the grant MOST
111-2811-M-008-024.

## Appendix

###  A.1 Expansion of curvatures up to second order

Here we summarize our conventions and formulae necessary in the text. We give
these such that they are valid for any dimension $d$.

Our signature of the metric is $(-,+,\dots+)$ and the curvature tensors are
given as

$\displaystyle R^{\alpha}{}_{\beta\mu\nu}$ $\displaystyle=$
$\displaystyle\partial_{\mu}\Gamma^{\alpha}_{\beta\nu}-\partial_{\nu}\Gamma^{\alpha}_{\beta\mu}+\Gamma^{\alpha}_{\mu\lambda}\Gamma^{\lambda}_{\beta\nu}-\Gamma^{\alpha}_{\nu\lambda}\Gamma^{\lambda}_{\beta\mu},$
$\displaystyle R_{\mu\nu}$ $\displaystyle=$
$\displaystyle\bar{R}^{\alpha}{}_{\mu\alpha\nu}.$ (104)

The backgrounds are denoted with overbar. Expansion around the background
gives

$\displaystyle\Gamma^{\alpha}_{\mu\nu}$ $\displaystyle=$
$\displaystyle\bar{\Gamma}^{\alpha}_{\mu\nu}+\Gamma^{\alpha(1)}_{\mu\nu}+\Gamma^{\alpha(2)}_{\mu\nu},$
(105)

where

$\displaystyle\Gamma^{\alpha(1)}_{\mu\nu}$ $\displaystyle=$
$\displaystyle\frac{1}{2}(\bar{\nabla}_{\nu}h^{\alpha}{}_{\mu}+\bar{\nabla}_{\mu}h^{\alpha}{}_{\nu}-\bar{\nabla}^{\alpha}h_{\mu\nu}),$
(106) $\displaystyle\Gamma^{\alpha(2)}_{\mu\nu}$ $\displaystyle=$
$\displaystyle-\frac{1}{2}h^{\alpha\beta}(\bar{\nabla}_{\nu}h_{\mu\beta}+\bar{\nabla}_{\mu}h_{\nu\beta}-\bar{\nabla}_{\beta}h_{\mu\nu}).$
(107)

Note that

$\displaystyle\sqrt{-g}=\sqrt{-\bar{g}}\Big{[}1+\frac{1}{2}h+\frac{1}{8}(h^{2}-2h_{\mu\nu}^{2})+O(h^{3})\Big{]},$
(108)

where $h\equiv h_{\mu}{}^{\mu}$. We find, to the second order,

$\displaystyle R^{\mu}{}_{\nu\alpha\beta}$ $\displaystyle=$
$\displaystyle\bar{R}^{\mu}{}_{\nu\alpha\beta}+\bar{R}^{\mu}{}_{\nu\alpha\beta}^{(1)}+\bar{R}^{\mu}{}_{\nu\alpha\beta}^{(2)},$
$\displaystyle R^{\mu}{}_{\nu\alpha\beta}^{(1)}$ $\displaystyle=$
$\displaystyle\frac{1}{2}(\bar{\nabla}_{\alpha}\bar{\nabla}_{\nu}h^{\mu}_{\beta}-\bar{\nabla}_{\alpha}\bar{\nabla}^{\mu}h_{\nu\beta}-\bar{\nabla}_{\beta}\bar{\nabla}_{\nu}h^{\mu}_{\alpha}+\bar{\nabla}_{\beta}\bar{\nabla}^{\mu}h_{\nu\alpha})$
(109)
$\displaystyle+\frac{1}{2}\bar{R}_{\nu\gamma\alpha\beta}h^{\mu\gamma}+\frac{1}{2}\bar{R}^{\mu}{}_{\gamma\alpha\beta}h^{\gamma}_{\nu},~{}~{}~{}$
$\displaystyle R^{\mu}{}_{\nu\alpha\beta}^{(2)}$ $\displaystyle=$
$\displaystyle-\frac{1}{2}h^{\mu\gamma}\bar{\nabla}_{\alpha}(\bar{\nabla}_{\beta}h_{\nu\gamma}+\bar{\nabla}_{\nu}h_{\beta\gamma}-\bar{\nabla}_{\gamma}h_{\nu\beta})-\frac{1}{4}\bar{\nabla}_{\alpha}h^{\mu\gamma}(\bar{\nabla}_{\beta}h_{\nu\gamma}+\bar{\nabla}_{\nu}h_{\beta\gamma}-\bar{\nabla}_{\gamma}h_{\nu\beta})$
(110)
$\displaystyle+\;\frac{1}{4}\bar{\nabla}_{\gamma}h^{\mu}_{\alpha}(\bar{\nabla}_{\beta}h^{\gamma}_{\nu}+\bar{\nabla}_{\nu}h^{\gamma}_{\beta}-\bar{\nabla}^{\gamma}h_{\nu\beta})-\frac{1}{4}\bar{\nabla}^{\mu}h_{\alpha\gamma}(\bar{\nabla}_{\beta}h^{\gamma}_{\nu}+\bar{\nabla}_{\nu}h^{\gamma}_{\beta}-\bar{\nabla}^{\gamma}h_{\nu\beta})$
$\displaystyle-(\alpha\leftrightarrow\beta).$

Similarly

$\displaystyle R^{(1)}_{\mu\nu}$ $\displaystyle=$
$\displaystyle-\frac{1}{2}(\bar{\nabla}_{\mu}\bar{\nabla}_{\nu}h-\bar{\nabla}_{\mu}h_{\nu}-\bar{\nabla}_{\nu}h_{\mu}+\Box
h_{\mu\nu})-\bar{R}_{\alpha\mu\beta\nu}h^{\alpha\beta}+\frac{1}{2}\bar{R}_{\mu\alpha}h^{\alpha}_{\nu}+\frac{1}{2}\bar{R}_{\nu\alpha}h^{\alpha}_{\mu},$
$\displaystyle R^{(2)}_{\mu\nu}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\bar{\nabla}_{\mu}(h^{\alpha\beta}\bar{\nabla}_{\nu}h_{\alpha\beta})-\frac{1}{2}\bar{\nabla}_{\alpha}\\{h^{\alpha\beta}(\bar{\nabla}_{\mu}h_{\nu\beta}+\bar{\nabla}_{\nu}h_{\mu\beta}-\bar{\nabla}_{\beta}h_{\mu\nu})\\}$
$\displaystyle-\frac{1}{4}(\bar{\nabla}_{\mu}h^{\beta}_{\alpha}+\bar{\nabla}_{\alpha}h^{\beta}_{\mu}-\bar{\nabla}^{\beta}h_{\alpha\mu})(\bar{\nabla}_{\beta}h^{\alpha}_{\nu}+\bar{\nabla}_{\nu}h^{\alpha}_{\beta}-\bar{\nabla}^{\alpha}h_{\beta\nu})+\frac{1}{4}\bar{\nabla}_{\alpha}h(\bar{\nabla}_{\mu}h^{\alpha}_{\nu}+\bar{\nabla}_{\nu}h_{\mu}^{\alpha}-\bar{\nabla}^{\alpha}h_{\mu\nu}),$
$\displaystyle R^{(1)}$ $\displaystyle=$
$\displaystyle\bar{\nabla}_{\mu}h^{\mu}-\Box h-\bar{R}_{\mu\nu}h^{\mu\nu},$
$\displaystyle R^{(2)}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\bar{\nabla}_{\mu}(h^{\alpha\beta}\bar{\nabla}^{\mu}h_{\alpha\beta})-\frac{1}{2}\bar{\nabla}_{\alpha}\\{h^{\alpha\beta}(2h_{\beta}-\bar{\nabla}_{\beta}h)\\}-\frac{1}{4}(\bar{\nabla}_{\mu}h^{\beta}_{\alpha}+\bar{\nabla}_{\alpha}h^{\beta}_{\mu}-\bar{\nabla}^{\beta}h_{\alpha\mu})\bar{\nabla}_{\beta}h^{\alpha\mu}$
(111)
$\displaystyle+\;\frac{1}{4}(2h^{\alpha}-\bar{\nabla}^{\alpha}h)\bar{\nabla}_{\alpha}h+\frac{1}{2}h^{\alpha\beta}\bar{\nabla}_{\alpha}\bar{\nabla}_{\beta}h-\frac{1}{2}h_{\alpha}^{\mu}\bar{\nabla}_{\beta}(\bar{\nabla}^{\alpha}h_{\mu}^{\beta}+\bar{\nabla}_{\mu}h^{\alpha\beta}-\bar{\nabla}^{\beta}h^{\alpha}_{\mu})+\bar{R}_{\mu\nu}h^{\mu}_{\alpha}h^{\nu\alpha}$
$\displaystyle=$
$\displaystyle\frac{3}{4}\bar{\nabla}_{\alpha}h_{\mu\nu}\bar{\nabla}^{\alpha}h^{\mu\nu}+h_{\mu\nu}\Box
h^{\mu\nu}-h_{\mu}^{2}+h_{\mu}\bar{\nabla}^{\mu}h-2h_{\mu\nu}\bar{\nabla}^{\mu}h^{\nu}+h_{\mu\nu}\bar{\nabla}^{\mu}\bar{\nabla}^{\nu}h$
$\displaystyle-\;\frac{1}{2}\bar{\nabla}_{\mu}h_{\nu\alpha}\bar{\nabla}^{\alpha}h^{\mu\nu}-\frac{1}{4}\bar{\nabla}_{\mu}h\bar{\nabla}^{\mu}h+\bar{R}_{\alpha\beta\gamma\delta}h^{\alpha\gamma}h^{\beta\delta},$

where $h_{\mu}\equiv\bar{\nabla}^{\nu}h_{\mu\nu}$. Note that
$\bar{g}^{\mu\nu}R^{(1)}_{\mu\nu}\neq R^{(1)}$, because the latter has
additional contribution from $h^{\mu\nu}\bar{R}_{\mu\nu}$. When total
derivative terms are dropped, $R^{(2)}$ makes the contribution to the action

$\displaystyle R^{(2)}\simeq\frac{1}{4}(h_{\mu\nu}\Box h^{\mu\nu}+h\Box
h+2h_{\mu}^{2}+2\bar{R}_{\alpha\beta}h^{\alpha\gamma}h^{\beta}_{\gamma}+2\bar{R}_{\alpha\beta\gamma\delta}h^{\alpha\gamma}h^{\beta\delta}).$
(112)

We use the notation $\simeq$ to denote equality up to total derivatives.

###  A.2 Heat kernel coefficients for $p=2$ minimal operator

For minimal operator $\Delta=-\bar{\nabla}^{2}+\bm{E}$, the general formulae
for any spin are

$\displaystyle b_{0}={\rm tr}\,\hat{1},\qquad b_{2}=\frac{1}{6}\bar{R}\,{\rm
tr}\,\hat{1}-{\rm tr}\,\bm{E},$ $\displaystyle
b_{4}=\frac{1}{180}\Big{(}\bar{R}_{\mu\nu\alpha\beta}^{2}-\bar{R}_{\mu\nu}^{2}+\frac{5}{2}\bar{R}^{2}+6\bar{\nabla}^{2}\bar{R}\Big{)}{\rm
tr}\,\hat{1}+\frac{1}{2}{\rm tr}\,\bm{E}^{2}$ $\displaystyle\hskip
42.67912pt-\frac{1}{6}\bar{R}\,{\rm tr}\,\bm{E}+\frac{1}{12}{\rm
tr}\,\Omega_{\mu\nu}^{2}-\frac{1}{6}\bar{\nabla}^{2}{\rm tr}\,\bm{E},$ (113)

where $\Omega_{\mu\nu}=[\bar{\nabla}_{\mu},\bar{\nabla}_{\nu}]$ is the
curvature from the covariant derivatives for each spin, and the traces should
be taken using the identity $\eta_{\mu\nu}$ for the vector and (13) for the
symmetric tensor. The formulae in subsection 4.2 are for spin 1, but with an
additional nonminimal term.

Using these formulae (113) to the Lichnerowicz Laplacians (84), we find, for
spin 0

$\displaystyle b_{0}({\Delta_{L}}_{0})=1,\qquad
b_{2}({\Delta_{L}}_{0})=\frac{1}{6}\bar{R},$ $\displaystyle
b_{4}({\Delta_{L}}_{0})=\frac{1}{180}\Big{(}\bar{R}_{\mu\nu\alpha\beta}^{2}-\bar{R}_{\mu\nu}^{2}+\frac{5}{2}\bar{R}^{2}+6\bar{\nabla}^{2}\bar{R}\Big{)},$
(114)

and for spin 1

$\displaystyle b_{0}({\Delta_{L}}_{1})=d,\qquad
b_{2}({\Delta_{L}}_{1})=\frac{d-6}{6}\bar{R},$ $\displaystyle
b_{4}=\frac{d-15}{180}\bar{R}_{\mu\nu\alpha\beta}^{2}-\frac{d-90}{180}\bar{R}_{\mu\nu}^{2}+\frac{d-12}{72}\bar{R}^{2}+\frac{d-5}{30}\bar{\nabla}^{2}\bar{R},$
(115)

since
$\Omega_{\mu\nu}A_{\alpha}=\bar{R}_{\mu\nu\alpha\beta}A^{\beta}\equiv(\Omega_{\mu\nu})_{\alpha\beta}A^{\beta}$
and

$\displaystyle{\rm
tr}\,\Omega_{\mu\nu}^{2}=\bar{R}_{\mu\nu\alpha\rho}\bar{R}^{\mu\nu\rho\alpha}=-\bar{R}_{\mu\nu\alpha\beta}^{2}$
(116)

For spin 2, we find

$\displaystyle b_{0}({\Delta_{L}}_{2})$ $\displaystyle=$
$\displaystyle\frac{d(d+1)}{2},\qquad
b_{2}({\Delta_{L}}_{2})=\frac{d^{2}-11d-24}{12}\bar{R},$ $\displaystyle
b_{4}({\Delta_{L}}_{2})$ $\displaystyle=$
$\displaystyle\frac{d^{2}-29d+480}{360}\bar{R}_{\mu\nu\alpha\beta}^{2}-\frac{d^{2}-359d-1080}{360}\bar{R}_{\mu\nu}^{2}$
(117)
$\displaystyle+\frac{d^{2}-23d-48}{144}\bar{R}^{2}+\frac{d^{2}-9d-20}{60}\bar{\nabla}^{2}\bar{R},$

since
$(\Omega_{\mu\nu}^{2})_{\alpha\beta,\rho\sigma}=\bar{R}_{\mu\nu\alpha\gamma}\bar{R}^{\mu\nu\gamma}{}_{\rho}g_{\beta\sigma}+\bar{R}_{\mu\nu\alpha\rho}\bar{R}^{\mu\nu}{}_{\beta\sigma}+\bar{R}_{\mu\nu\beta\sigma}\bar{R}^{\mu\nu}{}_{\alpha\rho}+\bar{R}_{\mu\nu\beta\gamma}\bar{R}^{\mu\nu\gamma}{}_{\sigma}g_{\alpha\rho}$
and

$\displaystyle{\rm tr}(\Omega^{2}_{\mu\nu})$ $\displaystyle=$
$\displaystyle\frac{1}{2}\Big{(}\\{\mbox{sum over
}\alpha=\rho,\beta=\sigma\\}+\\{\mbox{sum over
}\alpha=\sigma,\beta=\rho\\}\Big{)}$ (118) $\displaystyle=$
$\displaystyle-(d+2)\bar{R}_{\mu\nu\alpha\gamma}^{2}.$

These are the results when the fields do not have any constraints. If the
fields have constraint such as transverse, suitable subtraction is needed for
spins 1 and 2.

Spin 0 does not have any constraint, so their heat kernel coefficients are the
same as above. For the transverse spin 1, we have
$b_{n}({\Delta_{L}}_{1}^{T})=b_{n}({\Delta_{L}}_{1})-b_{n}({\Delta_{L}}_{0})$.
Hence

$\displaystyle b_{0}({\Delta_{L}}_{1}^{T})=d-1,\quad
b_{2}({\Delta_{L}}_{1}^{T})=\frac{d-7}{6}\bar{R},\quad$ $\displaystyle
b_{4}({\Delta_{L}}_{1}^{T})=\frac{d-16}{180}\bar{R}_{\mu\nu\rho\sigma}^{2}-\frac{d-91}{180}\bar{R}_{\mu\nu}^{2}+\frac{d-13}{72}\bar{R}^{2}+\frac{d-6}{30}\bar{\nabla}^{2}\bar{R}.$
(119)

For the Einstein space, $b_{4}({\Delta_{L}}_{1}^{T})$ reduces to

$\displaystyle
b_{4}({\Delta_{L}}_{1}^{T})=\frac{d-16}{180}\bar{R}_{\mu\nu\rho\sigma}^{2}+\frac{5d^{2}-67d+182}{360d}\bar{R}^{2}+\frac{d-6}{30}\bar{\nabla}^{2}\bar{R}.$
(120)

Similarly those for transverse and traceless spin 2, we have
$b_{n}({\Delta_{L}}_{2}^{TT})=b_{n}({\Delta_{L}}_{2})-b_{n}({\Delta_{L}}_{1}^{T})-2b_{n}({\Delta_{L}}_{0})$,
so

$\displaystyle b_{0}({\Delta_{L}}_{2}^{TT})$ $\displaystyle=$
$\displaystyle\frac{(d+1)(d-2)}{2},\quad
b_{2}({\Delta_{L}}_{2}^{TT})=\frac{(d+1)(d-14)}{12}\bar{R},\quad$
$\displaystyle b_{4}({\Delta_{L}}_{2}^{TT})$ $\displaystyle=$
$\displaystyle\frac{d^{2}-31d+508}{360}\bar{R}_{\mu\nu\rho\sigma}^{2}-\frac{d^{2}-361d-902}{360}\bar{R}_{\mu\nu}^{2}$
(121)
$\displaystyle+\frac{(d+1)(d-26)}{144}\bar{R}^{2}+\frac{(d+1)(d-12)}{60}\bar{\nabla}^{2}\bar{R}.$

For the Einstein space, $b_{4}({\Delta_{L}}_{2}^{TT})$ reduces to

$\displaystyle b_{4}({\Delta_{L}}_{2}^{TT})$ $\displaystyle=$
$\displaystyle\frac{d^{2}-31d+508}{360}\bar{R}_{\mu\nu\rho\sigma}^{2}+\frac{5d^{3}-127d^{2}+592d+1804}{720d}\bar{R}^{2}$
(122) $\displaystyle+\frac{(d+1)(d-12)}{60}\bar{\nabla}^{2}\bar{R}.$

Finally we also need the following formulae which also follow from (113):

$\displaystyle b_{0}(\Delta+aR)$ $\displaystyle=$ $\displaystyle
b_{0}(\Delta),$ $\displaystyle b_{2}(\Delta+aR)$ $\displaystyle=$
$\displaystyle b_{2}(\Delta)-aRb_{0}(\Delta),$ (123) $\displaystyle
b_{4}(\Delta+aR)$ $\displaystyle=$ $\displaystyle
b_{4}(\Delta)-aRb_{2}(\Delta)+\frac{1}{2}a^{2}R^{2}b_{0}(\Delta).$

We can evaluate the one-loop divergent part (42) of the effective action (98)
by using eqs. (114), (119) – (123).

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