compat=1.0.0

TU-1140

KEK-TH-2379

UMN-TH-4112/22

FTPI-MINN-22-01

Scalar field couplings to quadratic curvature
and decay into gravitons

Yohei Emaa, Kyohei Mukaidab,c, Kazunori Nakayamad,e,f

a | _William I. Fine Theoretical Physics Institute, School of Physics and Astronomy,_
---|---
| _University of Minnesota, Minneapolis, MN 55455, USA_
b | _Theory Center, IPNS, KEK, 1-1 Oho, Tsukuba, Ibaraki 305-0801, Japan_
c | _Graduate University for Advanced Studies (Sokendai),_
| _1-1 Oho, Tsukuba, Ibaraki 305-0801, Japan_
d | _Department of Physics, Tohoku University, Sendai, Miyagi 980-8578, Japan_
e | _International Center for Quantum-field Measurement Systems for Studies of_
| _the Universe and Particles (QUP), KEK, 1-1 Oho, Tsukuba, Ibaraki 305-0801,
Japan_
f | _Kavli IPMU (WPI), University of Tokyo, Kashiwa 277-8583, Japan_

Any field, even if it lives in a completely hidden sector, interacts with the
visible sector at least via the gravitational interaction. In this paper, we
show that a scalar field in such a hidden sector generically couples to the
quadratic curvature via dimension five operators, i.e., $\phi R^{2}$, $\phi
R^{\mu\nu}R_{\mu\nu}$, $\phi R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}$, $\phi
R_{\mu\nu\rho\sigma}\tilde{R}^{\mu\nu\rho\sigma}$, because they can emerge if
there exist particles heavier than it. We derive these scalar couplings to the
quadratic curvature by integrating out heavy particles in a systematic manner.
Such couplings are of phenomenological interest since some of them induce the
scalar decay into the graviton pair. We point out that the decay of a scalar
field can produce a substantial amount of the stochastic cosmic graviton
background at high frequency since the suppression scale of these operators is
given by the mass of heavy particles not by the Planck scale. The resultant
graviton spectrum is computed in some concrete models.

###### Contents

  1. 1 Introduction
  2. 2 Scalar field couplings to quadratic curvature
    1. 2.1 Propagator and WKB expansion
    2. 2.2 Integrating out heavy fields
    3. 2.3 Relation to beta function as low energy theorem
  3. 3 Scalar decay into gravitons
    1. 3.1 Scalar decay rate into graviton pair
    2. 3.2 Cosmic graviton background spectrum from scalar decay
    3. 3.3 Some other models
  4. 4 Summary and discussion
  5. A Convention
  6. B Technical details of the WKB expansion
    1. B.1 Bitensor
    2. B.2 WKB expansion

##  1 Introduction

Among the four fundamental forces, the gravitational interaction is special
for its universality. Any ingredients, including the standard model (SM) and
possible dark sectors solving the puzzles in the SM, should at least interact
through gravity. Recently, there is a revival of attention to a dark sector
that interacts with the SM only through gravity [1, 2, 3, 4, 5, 6, 7, 8, 9,
10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24]. Most of the
studies focus on the gravitational interaction within the Einstein gravity or
the nonminimal coupling to gravity, namely up to the dimension four operators.
However, if the dark sector involves a scalar field, it can couple to the
quadratic curvature through dimension five operators, such as $\phi R^{2}$,
$\phi R^{\mu\nu}R_{\mu\nu}$, $\phi R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}$,
and $\phi R_{\mu\nu\rho\sigma}\tilde{R}^{\mu\nu\rho\sigma}$. These operators
lead to unique phenomena absent in the standard gravity [25, 26, 27, 28, 29,
30, 31, 32, 33, 34, 35, 36]. In particular, the last two operators,

$\displaystyle\mathcal{L}_{\phi
RR}=-\frac{\phi}{\Lambda}R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma},$ (1.1)
$\displaystyle\mathcal{L}_{\phi
R\widetilde{R}}=-\frac{\phi}{\Lambda}R_{\mu\nu\rho\sigma}\widetilde{R}^{\mu\nu\rho\sigma},$
(1.2)

induce a decay of $\phi$ into graviton pairs (see, _e.g._ , [37, 38]), which
is never realized in the Einstein gravity nor the nonminimal coupling to
gravity [1, 37, 38, 18].

Gravitational waves have been discovered by the LIGO/Virgo experiments [39]
and the Einstein gravity is confirmed at the classical level. The quantum
nature of the gravity, on the other hand, is not confirmed yet. The future
discovery of quantized field of the gravitational wave, the graviton, will
give us a clue to the quantum gravity. In the universe, there are several
phenomena that produce gravitons: scattering of Standard Model particles in
thermal plasma [40, 41, 42], bremsstrahlung production at the inflaton decay
[43, 44] and the amplification of the vacuum fluctuation during the reheating
era [1, 5, 4, 18]. The inflationary production of primordial gravitational
waves may also fall into this category [45]. There are several proposals to
detect high frequency gravitons [46, 47, 48, 49, 50, 51].

In this paper, we study a dark sector interacting with the SM only through
gravity whose lightest particle is a scalar field $\phi$. We show that
dimension five couplings of $\phi$ to the quadratic curvature is expected for
generic theories of this type unless prohibited by symmetry, because they are
generated when we integrate out particles heavier than $\phi$. We
systematically derive these couplings of $\phi$ to gravity based on the
Schwinger-DeWitt formalism. In particular, we clarify that the suppression
scale $\Lambda$ for these operators is associated with the mass of heavy
particles and hence can be much smaller than the Planck scale. As a result,
the amount of gravitational waves is far more enhanced, imprinting a high
frequency peak in the spectrum of gravitational waves. Also, in some
phenomenological applications, a relatively small suppression scale is
required to have interesting effects, which is, however, suffering from the
appearance of a ghost (see, _e.g._ , [31, 32]). Our understanding provides a
healthy UV completion to a small suppression scale, above which a heavy
particle just appears in the spectrum without having a ghost.

This paper is organized as follows. In Sec. 2 we systematically derive the
scalar coupling to the quadratic curvature: $R^{2}$, $R_{\mu\nu}R^{\mu\nu}$,
$R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}$ and
$R_{\mu\nu\rho\sigma}\widetilde{R}^{\mu\nu\rho\sigma}$ by integrating out
heavy particles, starting from the renormalizable couplings in the matter
sector. In Sec. 3 we show that the scalar decay into the graviton pair occurs
through the effective operator derived in Sec. 2. The graviton abundance from
the scalar decay can be significant in some concrete models and we derive
resultant cosmological graviton spectrum. In Sec. 4 we summarize our results.

##  2 Scalar field couplings to quadratic curvature

In this section, we derive the couplings of a scalar field to the quadratic
curvature terms induced after integrating out heavy intermediate fields.
Before going to actual computations, let us illustrate our basic strategy with
the following simple action:

$\displaystyle
S=\int\differential^{4}x\sqrt{-g}\left[\frac{M_{P}^{2}}{2}R+\frac{1}{2}g^{\mu\nu}\partial_{\mu}\phi\partial_{\nu}\phi+\frac{1}{2}g^{\mu\nu}\partial_{\mu}\chi\partial_{\nu}\chi-\frac{m^{2}+\lambda\phi}{2}\chi^{2}-V(\phi)\right],$
(2.1)

where $g_{\mu\nu}$ is the metric tensor with $g$ its determinant, $M_{P}$ is
the (reduced) Planck mass, $R$ is the Ricci scalar, $\chi$ is a heavy scalar
field to be integrated out and $\phi$ is a scalar field that eventually
couples to the quadratic curvature terms. By integrating out $\chi$, the one-
loop effective action is given by

$\displaystyle\Gamma[\phi]$
$\displaystyle=\frac{i}{2}\mathrm{Tr}\log\left[\Box+m^{2}+\lambda\phi\right],$
(2.2)

where $\Box=\nabla_{\mu}\nabla^{\mu}$ with $\nabla_{\mu}$ the covariant
derivative. Since we are interested in the term linear in $\phi$, we may
expand the effective action as

$\displaystyle\Gamma[\phi]$
$\displaystyle\simeq\frac{i\lambda}{2}\mathrm{Tr}\left[\frac{1}{\Box+m^{2}}\phi\right]=-\frac{\lambda}{2}\int\differential^{4}x\sqrt{-g}\left[D_{F}(x,x)\phi(x)\right],$
(2.3)

where the propagator and the state are defined as

$\displaystyle iD_{F}(x,x^{\prime})$ $\displaystyle=\langle
x|\frac{1}{\Box+m^{2}}|x^{\prime}\rangle,\quad\langle
x|x^{\prime}\rangle=\frac{1}{\sqrt{-g}}\delta^{(4)}(x-x^{\prime}).$ (2.4)

Since the $\Box$-operator contains the gravitons, the propagator depends on
the curvature tensors, which are then translated to the scalar field couplings
to the quadratic curvature terms through Eq. (2.3). This is schematically
illustrated in Fig. 1.

Figure 1: The schematic picture that the propagator induces the scalar field
couplings to the curvature tensors. The thick line on the left hand side
indicates the full propagator of the heavy intermediate particle, which
contains the curvature tensors as indicated on the right hand side. The vertex
insertion is indicated by the crossed dot, the scalar field by the dashed
lines, and the graviton by the wavy lines.

Thus our primary task is to compute the propagator. We consider the simple
example here, but the same strategy is applicable to a wider situation as we
shall see.

The rest of this section is organized as follows. In Sec. 2.1, we compute the
propagator of a field with a general spin. In particular, we use the
Schwinger-DeWitt formalism and derive the propagator to the lowest order in
the WKB expansion. The WKB expansion is equivalent to the derivative expansion
of the coupling, which is suitable for our purpose since we are interested in
the heavy mass limit of the fields that we integrate out. In Sec. 2.2, we
compute the scalar field couplings to the quadratic curvature terms after
integrating out heavy scalar/fermion/vector fields. Obviously the propagator
that we compute in Sec. 2.1 plays an essential role there. Finally we see that
our results are related to the beta functions in Sec. 2.3.

###  2.1 Propagator and WKB expansion

The propagator plays the central role in our derivation of the scalar field
couplings to the quadratic curvature terms. In this subsection, we review the
derivation of the propagator based on the Schwinger-DeWitt formalism (see
_e.g._ , [52, 53, 54] and references therein).

We define the propagator as

$\displaystyle\left(\Box+X\right)iD_{F}(x,x^{\prime})=\frac{\mathbbm{1}}{\sqrt{-g}}\delta^{(d)}(x-x^{\prime}),$
(2.5)

where we allow $X$ dependent not only on the mass term but on the curvature
tensors in general. Here $D_{F}$ is understood either scalar/bispinor/tensor
depending on the spin of a particle that we integrate out, and hence we use
the bold font for the unit operator in the right hand side. Furthermore, we
anticipate the UV divergences and hence we perform the dimensional
regularization. With the proper time $\tau$ the propagator is expressed as

$\displaystyle
D_{F}(x,x^{\prime})=\int_{0}^{\infty}\differential\tau\,K(\tau;x,x^{\prime}),$
(2.6)

where

$\displaystyle K(\tau;x,x^{\prime})$ $\displaystyle=\langle
x\rvert\exp\left[-i\tau\left(\Box+X\right)\right]\lvert
x^{\prime}\rangle\equiv\langle x,\tau|x^{\prime},0\rangle.$ (2.7)

The function $K$ satisfies the Schrödinger-type equation:

$\displaystyle i\frac{\partial}{\partial\tau}K(\tau;x,x^{\prime})$
$\displaystyle=\left(\Box+X\right)K(\tau;x,x^{\prime}),$ (2.8)

and this guarantees Eq. (2.6), where we use that $K(\tau;x,x^{\prime})\to 0$
as $\tau\to\infty$ due to the implicit $i\epsilon$-prescription. We now make
the WKB ansatz\natural1\natural1$\natural$1 This is the so-called $R$-summed
version of the WKB ansatz [55, 56].

$\displaystyle K(\tau;x,x^{\prime})=\frac{i}{\left(4\pi
i\tau\right)^{d/2}}\exp\left[-\frac{i\sigma(x,x^{\prime})}{2\tau}-i\tau\left(X-\frac{1}{6}R\right)\right]\Delta^{1/2}(x,x^{\prime})\Omega(\tau;x,x^{\prime}),$
(2.9)

as the solution of the Schrödinger equation, where $\sigma$ is Synge’s world
function and $\Delta$ the van Vleck-Morette determinant (see App. B). The
dimension of spacetime is promoted to be $d$ as we use the dimensional
regularization. We expand $\Omega$ as

$\displaystyle\Omega(\tau;x,x^{\prime})=\sum_{n=0}^{\infty}\left(i\tau\right)^{n}a_{n}(x,x^{\prime}),$
(2.10)

with the condition
$K(0;x,x^{\prime})=\mathbbm{1}\delta^{(d)}(x-x^{\prime})/\sqrt{-g}$ fixing

$\displaystyle a_{0}$ $\displaystyle=\mathbbm{1}.$ (2.11)

The propagator in the coincidence limit $x^{\prime}\to x$ is given in terms of
$a_{n}$ as

$\displaystyle\lim_{x^{\prime}\to x}D_{F}(x,x^{\prime})$
$\displaystyle=\frac{1}{(4\pi)^{d/2}}\sum_{n=0}^{\infty}[a_{n}]\left(\mathcal{M}^{2}\right)^{d/2-n-1}\Gamma\left(n-\frac{d}{2}+1\right),$
(2.12)

where we denote

$\displaystyle[a_{n}]$ $\displaystyle\equiv\lim_{x^{\prime}\to
x}a_{n}(x,x^{\prime}),\quad\mathcal{M}^{2}=X-\frac{1}{6}R.$ (2.13)

Remember that, for our purpose, we need the propagator only in the coincidence
limit since the effective action contains the trace of the spacetime (see Eq.
(2.3)). From this expression it is clear that the small $\tau$ expansion
corresponds to the derivative expansion since the higher order terms are
suppressed by the higher powers of $\mathcal{M}^{2}$. We can compute $[a_{n}]$
from the Schrödinger equation. We leave the derivation in App. B and show only
the final results here [57, 56, 58]:

$\displaystyle[a_{0}]$ $\displaystyle=\mathbbm{1},\quad[a_{1}]=0,$ (2.14)
$\displaystyle[a_{2}]$
$\displaystyle=\left[-\frac{1}{180}R_{\mu\nu}R^{\mu\nu}+\frac{1}{180}R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}+\frac{1}{12}W_{\mu\nu}W^{\mu\nu}+\frac{1}{6}\Box
X-\frac{1}{30}\Box R\right]\mathbbm{1},$ (2.15)

where the field $\Psi$ that is integrated out is assumed to satisfy

$\displaystyle\left[\nabla_{\mu},\nabla_{\nu}\right]\Psi=W_{\mu\nu}\Psi.$
(2.16)

###  2.2 Integrating out heavy fields

Equipped with the WKB expansion of the propagators, we now derive the scalar
field couplings to the quadratic curvature terms. We consider the cases that
the heavy intermediate particle is a scalar field, a fermion and a vector
field, in turn.

#### Scalar field.

We first integrate out a heavy scalar field. We may take the action as

$\displaystyle
S=\int\differential^{4}x\sqrt{-g}\left[\frac{M_{P}^{2}}{2}\left(1+\frac{\xi\chi^{2}}{M_{P}^{2}}\right)R+\frac{1}{2}g^{\mu\nu}\partial_{\mu}\phi\partial_{\nu}\phi+\frac{1}{2}g^{\mu\nu}\partial_{\mu}\chi\partial_{\nu}\chi-\frac{m^{2}+\lambda\phi}{2}\chi^{2}-V(\phi)\right],$
(2.17)

where we now include the nonminimal coupling between $\chi$ and the Ricci
scalar. As we explained at the beginning of this section, the effective
coupling is given by

$\displaystyle S_{\mathrm{eff}}$
$\displaystyle=-\frac{\mu^{4-d}}{2}\int\differential^{4}x\sqrt{-g}\left[\lambda\phi(x)\lim_{x^{\prime}\to
x}D_{F}(x,x^{\prime})\right],$ (2.18)

where $\mu$ is the scale that is introduced in the dimensional regularization,
and

$\displaystyle X=m^{2}-\xi
R,\quad\mathcal{M}^{2}=m^{2}-\left(\xi+\frac{1}{6}\right)R.$ (2.19)

With Eq. (2.12), we obtain the effective coupling to the quadratic curvature
terms as

$\displaystyle\mathcal{L}_{\mathrm{eff}}$
$\displaystyle=-\frac{1}{32\pi^{2}}\frac{\lambda\phi}{m^{2}}\left[\frac{1}{2}\left(\xi+\frac{1}{6}\right)^{2}R^{2}-\frac{1}{180}R_{\mu\nu}R^{\mu\nu}+\frac{1}{180}R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}\right],$
(2.20)

where we use that $W_{\mu\nu}=0$ for the scalar field. In particular, if we
assume that the mass of $\chi$ comes from the scalar field vacuum expectation
value (VEV), the original interaction may be written as

$\displaystyle\mathcal{L}_{\mathrm{mass}}=-\frac{\tilde{\lambda}}{2}\Phi^{2}\chi^{2},$
(2.21)

with $\Phi=v_{\phi}+\phi$. We then identify that (note that $\lambda$ is
dimensionful and $\tilde{\lambda}$ is dimensionless)

$\displaystyle
m^{2}=\tilde{\lambda}v_{\phi}^{2},\quad\lambda=2\tilde{\lambda}v_{\phi},~{}~{}\Rightarrow~{}~{}\frac{\lambda}{m^{2}}=\frac{2}{v_{\phi}}.$
(2.22)

We then obtain

$\displaystyle\mathcal{L}_{\mathrm{eff}}=-\frac{1}{16\pi^{2}}\frac{\phi}{v_{\phi}}\left[\frac{1}{2}\left(\xi+\frac{1}{6}\right)^{2}R^{2}-\frac{1}{180}R_{\mu\nu}R^{\mu\nu}+\frac{1}{180}R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}\right].$
(2.23)

#### Fermion.

We next consider the fermion. We may take the action as

$\displaystyle
S=\int\differential^{4}x\sqrt{-g}\,\bar{\psi}\left[i\not{\nabla}-m-\lambda\phi\right]\psi.$
(2.24)

The effective action is given by

$\displaystyle\Gamma[\phi]$
$\displaystyle=-i\mathrm{Tr}\log\left[i\not{\nabla}-m-\lambda\phi\right]=i\lambda\mathrm{Tr}\left[\frac{1}{i\not{\nabla}-m}\phi\right]+\cdots.$
(2.25)

We note that the spinor traces of odd $\gamma$s vanish, and hence

$\displaystyle\mathrm{Tr}\left[\frac{1}{i\not{\nabla}-m}\phi\right]$
$\displaystyle=\mathrm{Tr}\left[\frac{-i\not{\nabla}-m}{-i\not{\nabla}-m}\frac{1}{i\not{\nabla}-m}\phi\right]=-m\mathrm{Tr}\left[\frac{1}{\Box+X}\phi\right],\quad
X=m^{2}+\frac{R}{4}.$ (2.26)

We thus obtain the effective action as

$\displaystyle S_{\mathrm{eff}}$
$\displaystyle=\mu^{4-d}\int\differential^{4}x\sqrt{-g}\left[\lambda
m\phi(x)\,\mathrm{tr}\left[\lim_{x^{\prime}\to
x}D_{F}(x,x^{\prime})\right]\right],$ (2.27)

where the trace is over the spinor indices. In the fermion case we have (see
App. A)

$\displaystyle W_{\mu\nu}W^{\mu\nu}$
$\displaystyle=-\frac{i\gamma_{5}}{8}R_{\mu\nu\rho\sigma}\tilde{R}^{\mu\nu\rho\sigma}-\frac{1}{8}R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}.$
(2.28)

With Eq. (2.12), we then obtain the effective coupling as

$\displaystyle\mathcal{L}_{\mathrm{eff}}$
$\displaystyle=\frac{1}{16\pi^{2}}\frac{\lambda\phi}{m}\left[\frac{1}{72}R^{2}-\frac{1}{45}R_{\mu\nu}R^{\mu\nu}-\frac{7}{360}R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}\right].$
(2.29)

In particular, if we assume that the mass of the fermion comes from $\phi$, we
may take $m=\lambda v_{\phi}$ (here $\lambda$ is dimensionless) and we have

$\displaystyle\mathcal{L}_{\mathrm{eff}}$
$\displaystyle=\frac{1}{16\pi^{2}}\frac{\phi}{v_{\phi}}\left[\frac{1}{72}R^{2}-\frac{1}{45}R_{\mu\nu}R^{\mu\nu}-\frac{7}{360}R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}\right].$
(2.30)

Here it may be appropriate to comment on the gravitational chiral anomaly.
Indeed, the coupling induced by the gravitational anomaly can be computed
within the same framework. In order to see this point, we may consider the
following action:

$\displaystyle
S=\int\differential^{4}x\sqrt{-g}\,\bar{\psi}\left[i\not{\nabla}-me^{2i\gamma_{5}\phi/f}\right]\psi.$
(2.31)

We may expand the effective action to linear order in $\phi$ and obtain

$\displaystyle\Gamma[\phi]$
$\displaystyle=-i\mathrm{Tr}\left[\frac{1}{i\not{\nabla}-m}\left(-2i\gamma_{5}\frac{m\phi}{f}\right)\right].$
(2.32)

In the same way as above, this is rewritten as

$\displaystyle S_{\mathrm{eff}}$
$\displaystyle=\mu^{4-d}\int\differential^{4}x\sqrt{-g}\left[\frac{2im^{2}\phi}{f}\mathrm{tr}\left[\lim_{x^{\prime}\to
x}D_{F}(x,x^{\prime})\gamma_{5}\right]\right].$ (2.33)

Only the term proportional to $\gamma_{5}$ in $W_{\mu\nu}W^{\mu\nu}$
contributes and we thus get

$\displaystyle\mathcal{L}_{\mathrm{eff}}$
$\displaystyle=\frac{1}{192\pi^{2}}\frac{\phi}{f}R_{\mu\nu\rho\sigma}\tilde{R}^{\mu\nu\rho\sigma}.$
(2.34)

In order to see that this is consistent with the gravitational chiral anomaly,
we note that the axial phase in the mass can be rotated away by

$\displaystyle\psi\rightarrow e^{-i\gamma_{5}\phi/f}\psi.$ (2.35)

This rotation then induces [59]

$\displaystyle\mathcal{L}_{\mathrm{anom}}$
$\displaystyle=\frac{1}{192\pi^{2}}\frac{\phi}{f}R_{\mu\nu\rho\sigma}\tilde{R}^{\mu\nu\rho\sigma},$
(2.36)

through the gravitational chiral anomaly, in addition to the derivative
interaction between $\phi$ and $\psi$. Assuming that the derivative coupling
does not induce further threshold corrections (which is the case _e.g._ , in
the axion-gauge field system [60], and what we expect from the decoupling
theorem), this agrees with the effective coupling that we derived.

#### Vector field.

Finally we consider the vector boson. To be specific, we assume that the mass
originates from the Higgs mechanism and hence consider the abelian Higgs
model:

$\displaystyle S$
$\displaystyle=\int\differential^{4}x\sqrt{-g}\left[-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}+\absolutevalue{D_{\mu}\Phi}^{2}-\lambda\left(\absolutevalue{\Phi}^{2}-\frac{v_{\phi}^{2}}{2}\right)^{2}+\delta_{B}\left(\bar{c}F_{A}\right)\right],$
(2.37)

where $A_{\mu}$ is the gauge field with $F_{\mu\nu}$ its field strength,
$\delta_{B}$ is the BRST transformation, $F_{A}$ is the gauge fixing function
which we specify below, and

$\displaystyle D_{\mu}\Phi=\left(\partial_{\mu}+ieA_{\mu}\right)\Phi.$ (2.38)

We expand $\Phi$ as

$\displaystyle\Phi=\frac{v_{\phi}+\phi+i\chi}{\sqrt{2}}.$ (2.39)

The action is then given by

$\displaystyle\mathcal{L}$
$\displaystyle=-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}+\frac{\bar{m}_{A}^{2}}{2}A^{\mu}A_{\mu}+\frac{1}{2}\left(\partial\chi\right)^{2}-\frac{\lambda}{2}\left(2v_{\phi}\phi+\phi^{2}\right)\chi^{2}+\bar{m}_{A}A^{\mu}\partial_{\mu}\chi$
$\displaystyle+\frac{1}{2}\left(\partial\phi\right)^{2}-\frac{m_{\phi}^{2}}{2}\phi^{2}-g\chi
A^{\mu}\partial_{\mu}\phi+\delta_{B}\left(\bar{c}F_{A}\right)+\cdots,$ (2.40)

where

$\displaystyle\bar{m}_{A}^{2}=m_{A}^{2}\left(1+\frac{\phi}{v_{\phi}}\right)^{2},\quad
m_{A}^{2}=e^{2}v_{\phi}^{2},\quad m_{\phi}^{2}=2\lambda v_{\phi}^{2},$ (2.41)

and the dots indicate the higher order terms that are irrelevant at one-loop.

We now perform the derivative expansion. This is equivalent to taking
$m_{\phi}/m_{A}\to 0$ (or $e^{2}\gg\lambda$) in the current case. Although the
higher order terms in the small $m_{\phi}^{2}/m_{A}^{2}$ expansion can be
obtained with the higher order WKB expansions, here we keep only the leading
order terms. This in particular means that we should drop the $\phi$-dependent
mass term of $\chi$ that is proportional to $\lambda$ to be consistent. For
the same reason, we should also drop the $\chi$-$A$ mixing term that depends
on $\partial_{\mu}\phi$. With this in mind, the quadratic action of $\chi$ and
$A$ to the order of our interest is given by

$\displaystyle\mathcal{L}_{\chi A}$
$\displaystyle=-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}+\frac{\bar{m}_{A}^{2}}{2}A^{\mu}A_{\mu}+\frac{1}{2}\left(\partial\chi\right)^{2}+\bar{m}_{A}A^{\mu}\partial_{\mu}\chi+\delta_{B}\left(\bar{c}F_{A}\right).$
(2.42)

We take the gauge fixing function $F_{A}$ as

$\displaystyle
F_{A}=\frac{\zeta}{2}B-\nabla_{\mu}A^{\mu}+\zeta\bar{m}_{A}\chi,$ (2.43)

where $\zeta$ is the gauge fixing parameter. The BRST transformation is given
by

$\displaystyle\delta_{B}\bar{c}=B,\quad\delta_{B}B=0,\quad\delta_{B}A_{\mu}=-\partial_{\mu}c,\quad\delta_{B}\phi=-gc\chi,\quad\delta_{B}\chi=\bar{m}_{A}c.$
(2.44)

By integrating out the Nakanishi-Lautrup $B$-field, we then obtain

$\displaystyle\mathcal{L}$
$\displaystyle=-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}-\frac{1}{2\zeta}\left(\nabla_{\mu}A^{\mu}\right)^{2}+\frac{1}{2}\bar{m}_{A}^{2}A_{\mu}A^{\mu}+\frac{1}{2}\left(\partial\chi\right)^{2}-\frac{1}{2}\zeta\bar{m}_{A}^{2}\chi^{2}-\bar{c}\left(\Box+\zeta\bar{m}_{A}^{2}\right)c+\cdots.$
(2.45)

In the following we take the Feynman gauge $\zeta=1$ and then we get

$\displaystyle\mathcal{L}$
$\displaystyle=\frac{1}{2}A_{\mu}\left(g^{\mu\nu}\Box+g^{\mu\nu}\bar{m}_{A}^{2}+R^{\mu\nu}\right)A_{\nu}-\frac{1}{2}\chi\left(\Box+\bar{m}_{A}^{2}\right)\chi-\bar{c}\left(\Box+\bar{m}_{A}^{2}\right)c+\cdots.$
(2.46)

We may expand the effective action to the linear order in $\phi$ as

$\displaystyle S_{\mathrm{eff}}$
$\displaystyle=-\mu^{4-d}\int\differential^{4}x\sqrt{-g}\left[\mathrm{tr}\left[D_{F}^{(V)}\right]-D_{F}^{(S)}\right]\frac{m_{A}^{2}\phi}{v_{\phi}},$
(2.47)

where the superscript indicates whether a given propagator is that of a vector
or scalar field, and we use that the ghost is Grassmannian. We note that

$\displaystyle\left[\nabla_{\mu},\nabla_{\nu}\right]A_{\rho}$
$\displaystyle=-{R_{\mu\nu\rho}}^{\sigma}A_{\sigma}={\left(W_{\mu\nu}\right)_{\rho}}^{\sigma}A_{\sigma},$
(2.48)

and hence

$\displaystyle\mathrm{tr}\left[W_{\mu\nu}W^{\mu\nu}\right]$
$\displaystyle=-R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}.$ (2.49)

We thus obtain

$\displaystyle\mathcal{L}_{\mathrm{eff}}=-\frac{1}{16\pi^{2}}\frac{\phi}{v_{\phi}}\left[-\frac{1}{8}R^{2}+\frac{29}{60}R_{\mu\nu}R^{\mu\nu}-\frac{1}{15}R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}\right].$
(2.50)

###  2.3 Relation to beta function as low energy theorem

In the previous subsection we have derived the effective couplings of the
scalar field to the quadratic curvature terms. These coefficients are actually
controlled by the beta function. Indeed, if an intermediate particle is heavy
enough, its effect on the low energy theory is only through threshold
corrections. The interaction of a scalar field $\phi$ to the heavy particle of
the form $m(1+\phi/v_{\phi})$ modifies the energy scale at which the heavy
particle decouples in the constant $\phi$ limit, and hence the induced
effective couplings are given by the beta functions. This is analogous to the
Higgs couplings to photons in the massless Higgs limit [61].

In the case of our interest, the relevant beta functions are given by
[62]\natural2\natural2$\natural$2 The beta functions of the quadratic
curvature terms are computed _e.g._ , in [62, 63, 64, 65, 66, 67, 68, 69, 70],
although some references have opposite signs than others. Here we use the
result in [62] which agrees with our computation including the sign and hence
we believe is correct.

$\displaystyle\beta_{\alpha_{1}}$
$\displaystyle=\frac{\differential\alpha_{1}}{\differential\log\mu}=\frac{1}{16\pi^{2}}\left(-\frac{N_{S}}{2}\left(\xi+\frac{1}{6}\right)^{2}+\frac{N_{F}+20N_{V}}{144}\right),$
(2.51) $\displaystyle\beta_{\alpha_{2}}$
$\displaystyle=\frac{\differential\alpha_{2}}{\differential\log\mu}=\frac{1}{16\pi^{2}}\frac{N_{S}-2N_{F}-88N_{V}}{180},$
(2.52) $\displaystyle\beta_{\alpha_{3}}$
$\displaystyle=\frac{\differential\alpha_{3}}{\differential\log\mu}=-\frac{1}{16\pi^{2}}\frac{4N_{S}+7N_{F}-52N_{V}}{720},$
(2.53)

where $N_{S},N_{F}$ and $N_{V}$ are number of real scalars, Weyl fermions and
gauge bosons, respectively. We define the coefficients as

$\displaystyle\mathcal{L}=\alpha_{1}R^{2}+\alpha_{2}R_{\mu\nu}R^{\mu\nu}+\alpha_{3}R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma},$
(2.54)

and drop the contribution from the graviton loops. The RGE is different above
and below the scale at which the heavy particles decouple, and $\phi$ modifies
this scale as we explained above. Below the scale of the heavy particle, this
threshold correction is written as

$\displaystyle\Delta\alpha_{i}$
$\displaystyle=\beta_{\alpha_{i}}\log\left(1+\frac{\phi}{v_{\phi}}\right).$
(2.55)

By expanding this with respect to $\phi$, we obtain the effective Lagrangian
as

$\displaystyle\mathcal{L}_{\mathrm{eff}}$
$\displaystyle=\frac{\phi}{v_{\phi}}\left(\beta_{\alpha_{1}}R^{2}+\beta_{\alpha_{2}}R_{\mu\nu}R^{\mu\nu}+\beta_{\alpha_{3}}R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}\right).$
(2.56)

For the scalar field and fermion cases, we see that this effective Lagrangian
coincides with our computation in Sec. 2.2 with $N_{S}=1$ or $N_{F}=2$
(remember that $N_{F}$ is the number of Weyl fermions and we considered the
Dirac fermion in Sec. 2.2). For the vector field case, we take $N_{S}=N_{V}=1$
with $\xi=0$ and then the above coupling coincides with our result in Sec.
2.2. The additional scalar degree of freedom is understood as the contribution
from the longitudinal mode. Note that the same interpretation is applied to
the Higgs couplings to photons from the $W$-boson loop in [61]. We thus
establish the relation between the effective couplings and the beta functions,
which we may call the low energy theorem following [61].

Two comments are in order. First, there is actually no surprise that our
computation is related to the beta functions. This is rather built-in since
our method, if we do not expand with respect to $\phi$, is equivalent to
computing the full one-loop effective action, which after renormalization
encodes the information of the beta functions. See _e.g._ , [70] for the
effective potential and the beta functions of the SM in the curved spacetime
derived in this way. Second, although we focused on the term linear in $\phi$
and quadratic in the curvature tensors, the method is readily extended to
higher order terms. In particular, if we know a gravitational analogue of the
(full-order) Euler-Heiseinberg Lagrangian, we can easily derive couplings
between $n$ scalar fields and $m$ curvature tensors with $n$ and $m$
arbitrary. Unfortunately, to our knowledge, no such an effective Lagrangian is
known. Thus, although definitely doable order by order, it requires an
additional task to derive such an effective interaction. This is in contrast
to the Higgs couplings to photons that the Euler-Heisenberg Lagrangian is used
to derive such a generalized version of the low energy theorem [61].

##  3 Scalar decay into gravitons

In this section we study a scalar field decay into a graviton pair. First we
briefly explain the lowest order operators that induce the decay. Perhaps the
simplest interaction that one may think of is $\mathcal{L}=c\phi R$. This
coupling, however, does not induce the decay, contrary to the claims in [71,
72, 73]. The key point is that $R$ contains a linear term in the (scalar part
of) graviton and hence $c$ induces a kinetic mixing between the graviton and
$\phi$. Therefore one has to solve the kinetic mixing before discussing the
decay of $\phi$. Solving the kinetic mixing is equivalent to moving to the
frame without the nonminimal coupling, or the conformal frame, by the Weyl
transformation. One does not have any coupling between $\phi$ and two
gravitons in the conformal frame, meaning that there is no decay induced by
the operator $\phi R$ [1]. The next simplest possibility is the couplings
between $\phi$ and the quadratic curvature. Among them, $\phi R^{2}$ and $\phi
R_{\mu\nu}R^{\mu\nu}$ do not induce the decay in the flat spacetime [37, 38].
This is probably most easily seen by noting that the on-shell graviton
equation of motion gives $R=0$ and $R_{\mu\nu}=0$ in the flat spacetime. We
thus focus on the operators (1.1) and (1.2) in this section.

###  3.1 Scalar decay rate into graviton pair

In Sec. 2 we have shown that there appear scalar field couplings to the
quadratic curvature in the form of $\mathcal{L}_{\phi RR}$ (1.1) and/or
$\mathcal{L}_{\phi R\widetilde{R}}$ (1.2) after integrating out heavy
particles.\natural3\natural3$\natural$3 Note that we cannot remove these
couplings by the Weyl transformation.  These terms induce a scalar decay into
the graviton pair. Both operators give the same decay rate
as\natural4\natural4$\natural$4 This rate differs from the one in [38] by a
factor of 16. We believe that this difference originates from that [38] does
not properly normalize the graviton field.

$\displaystyle\Gamma(\phi\to
2h)=\frac{1}{4\pi}\frac{m_{\phi}^{7}}{\Lambda^{2}M_{\rm Pl}^{4}},$ (3.1)

Given a renormalizable interaction between $\phi$ and other scalars, fermions
or vector bosons, one can calculate the cutoff scale $\Lambda$, following the
procedure given in the previous section. For the scalar coupling
$\mathcal{L}_{\phi RR}$, from Eq. (2.56), we obtain

$\displaystyle\Lambda=v_{\phi}\frac{11520\pi^{2}}{4N_{S}+7N_{F}-52N_{V}},$
(3.2)

where for simplicity we assumed that the coupled scalar, fermion and vector
boson masses are given in the universal form by $m(1+\phi/v_{\phi})$ and
$N_{S},N_{F}$ and $N_{V}$ are the number of real scalar, Weyl fermion and
gauge boson that are decoupled above the scale $m_{\phi}$. Although the decay
rate into the graviton pair (3.1) might be small due to the $M_{\rm Pl}^{4}$
suppression, it is possible that the branching fraction of $\phi$ decay into
the graviton is sizable, or even dominant.

We emphasize that the scalar decay into the graviton pair is always allowed
unless it is prohibited by some symmetry. For example, let us consider the
action\natural5\natural5$\natural$5If the VEV of $\phi$ is zero, $v_{\phi}$ in
(3.2) should be regarded as $v_{\phi}=m/\lambda$. If, on the other hand, $m=0$
and $\phi$ has a finite VEV, $v_{\phi}$ in (3.2) is regarded as the VEV of
$\phi$.

$\displaystyle S=\int\differential^{4}x\sqrt{-g}\left[\frac{M_{\rm
Pl}^{2}}{2}R+\frac{1}{2}g^{\mu\nu}\partial_{\mu}\phi\partial_{\nu}\phi-V(\phi)+\bar{\psi}\left(i\not{\nabla}-m-\lambda\phi\right)\psi\right],$
(3.3)

where $\phi$ does not have interaction with the Standard Model particles,
_i.e._ , it lives in the completely hidden sector. If the fermion $\psi$ is
heavier than $\phi$, the scalar particle $\phi$ cannot decay into the fermion
pair. However, one can integrate out the heavy fermion and obtain the
effective operators of the form (1.1) and (1.2) as we explicitly computed in
Sec. 2.2, which causes the scalar decay into the graviton pair. Thus the
scalar $\phi$ is not stable even in such a case. Not only heavy fermion, but
also scalar boson or vector boson loops also contribute to the scalar decay
into the graviton pair. Below we show a concrete example that exhibits
significant graviton production through the scalar decay.

###  3.2 Cosmic graviton background spectrum from scalar decay

An example is the dark Higgs field $\Phi$ which spontaneously breaks dark
gauged U(1) symmetry:

$\displaystyle\mathcal{L}=\left|(\partial_{\mu}+ieA_{\mu})\Phi\right|^{2}-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}-\lambda\left(|\Phi|^{2}-\frac{v_{\phi}^{2}}{2}\right)^{2}.$
(3.4)

We assume that there are no direct couplings between the dark sector and the
Standard Model sector. If $e^{2}>\lambda$, the radial mode of the Higgs (which
is denoted by $\phi$) cannot perturbatively decay into the gauge boson pair.
In such a case, $\phi$ can only decay into the graviton through the effective
operator $\mathcal{L}_{\phi RR}$ (1.1). The decay rate is estimated by
inserting $N_{F}=0$ and $N_{S}=N_{V}=1$ in (3.2) as

$\displaystyle\Gamma(\phi\to 2h)=8\times 10^{-17}\,{\rm
GeV}~{}\lambda\left(\frac{m_{\phi}}{10^{13}\,{\rm GeV}}\right)^{5},$ (3.5)

where the dark Higgs mass is given by $m_{\phi}=\sqrt{2\lambda}v_{\phi}$. In
terms of the cosmic temperature $T_{\rm dec}$ at which the Hubble parameter
$H$ becomes equal to $\frac{1}{3}\Gamma(\phi\to 2h)$, we obtain

$\displaystyle T_{\rm dec}\simeq 4\,{\rm
GeV}~{}\lambda^{1/2}\left(\frac{m_{\phi}}{10^{13}\,{\rm
GeV}}\right)^{5/2}\left(\frac{106.75}{g_{*}(T_{\rm dec})}\right)^{1/4},$ (3.6)

assuming the radiation-dominated universe when $\phi$ decays, where $g_{*}(T)$
denotes the relativistic degrees of freedom at the temperature $T$. Let us
suppose that $H_{\rm inf}<m_{\phi}<m_{\rm inf}$, where $H_{\rm inf}$ and
$m_{\rm inf}$ are the inflationary Hubble scale and the inflaton mass,
respectively. In such a case, the dark Higgs already sits at the potential
minimum and the U(1) symmetry is already broken during inflation. After
inflation ends, $\phi$ particles are produced during the inflaton oscillation
through the gravitational effect [1]. The abundance, in terms of the energy
density-to-entropy density ratio, is given by\natural6\natural6$\natural$6
Here we present a contribution from the inflaton coherent oscillation, which
is also interpreted as the gravitational inflaton annihilation [1, 4, 9, 11,
23]. The gravitational annihilation of Standard Model particles in thermal
bath also produces scalar particles [2, 6, 7, 8], but it is subdominant in the
most parameters we are interested in.

$\displaystyle\frac{\rho_{\phi}}{s}\simeq 2\times 10^{-3}\,{\rm
GeV}\left(\frac{m_{\phi}}{10^{13}\,{\rm GeV}}\right)\left(\frac{H_{\rm
inf}}{10^{12}\,{\rm GeV}}\right)\left(\frac{T_{\rm R}}{10^{12}\,{\rm
GeV}}\right),$ (3.7)

where $T_{\rm R}$ denotes the reheating temperature of the
universe.\natural7\natural7$\natural$7 We assume that the gauge boson mass
$ev_{\phi}$ is larger than the inflaton mass $m_{\rm inf}$ in order to
suppress the gravitational gauge boson production by the inflaton oscillation
[13].  Thus the dark Higgs particle abundance can be substantial when it
decays and the energetic gravitons are efficiently produced. Depending on the
parameter choice, the dark Higgs can be dominant component of the universe
before it decays and the graviton energy density becomes so large that it
conflicts with the upper bound on the dark radiation energy density at the
Big-bang Nucleosynthesis or the recombination epoch. The energy density of the
graviton, in terms of the density parameter
$\Omega_{h}\equiv\rho_{h}/\rho_{\rm crit}$ where $\rho_{h}$ and $\rho_{\rm
crit}$ denote the present graviton and critical energy density of the
universe, is given by

$\displaystyle\Omega_{h}$ $\displaystyle=\Omega_{\rm
rad}\left(\frac{g_{*s}(T_{\rm
dec})}{g_{*0}}\right)\left(\frac{g_{*s0}}{g_{*s}(T_{\rm
dec})}\right)^{4/3}\frac{4}{3T_{\rm dec}}\frac{\rho_{\phi}}{s}$ (3.8)
$\displaystyle\simeq 2\times 10^{-8}\,\lambda^{-1/2}\left(\frac{10^{13}\,{\rm
GeV}}{m_{\phi}}\right)^{3/2}\left(\frac{H_{\rm inf}}{10^{12}\,{\rm
GeV}}\right)\left(\frac{T_{\rm R}}{10^{12}\,{\rm GeV}}\right),$ (3.9)

where $\Omega_{\rm rad}$ denotes the present density parameter of the
radiation, $g_{*}$ $(g_{*s})$ is the effective degrees of freedom for the
energy (entropy) density, and $g_{*0}$ ($g_{*s0}$) is its present value.

Let us calculate the present-day energy spectrum of the graviton produced by
the scalar decay. The present day graviton energy spectrum is given by

$\displaystyle\frac{\differential\rho_{h}}{\differential\ln
E}=E^{2}\int\frac{\differential z}{H(z)}\Gamma(\phi\to
2h)n_{\phi}(z)a^{3}(z)\frac{\differential N_{h}}{\differential E^{\prime}},$
(3.10) $\displaystyle\frac{\differential N_{h}}{\differential
E^{\prime}}=2\delta\left(E^{\prime}-\frac{m_{\phi}}{2}\right),~{}~{}~{}~{}~{}~{}~{}~{}E^{\prime}\equiv(1+z)E,$
(3.11)

where $H(z)$ is the Hubble parameter at the redshift $z$. The integral is
explicitly performed as

$\displaystyle\frac{\differential\rho_{h}}{\differential\ln
E}=\frac{16E^{4}}{m_{\phi}^{4}}\frac{\Gamma(\phi\to
2h)\rho_{\phi}(z_{d})}{H(z_{d})},$ (3.12)

where $1+z_{d}\equiv m_{\phi}/(2E)$. The energy density $\rho_{\phi}$ is given
by

$\displaystyle\rho_{\phi}(z)=\left(\frac{\rho_{\phi}}{s}\right)s(z)\exp\left(-\Gamma_{\rm
tot}t(z)\right),$ (3.13)

where $\Gamma_{\rm tot}$ denotes the total decay width of $\phi$ and
$\rho_{\phi}/s$ on the right-hand-side evaluated before $\phi$ starts to
decay. If the two graviton mode is the only possible decay channel,
$\Gamma_{\rm tot}=\Gamma(\phi\to 2h)$. It is convenient to plot it in terms of
the density parameter

$\displaystyle\frac{\differential\Omega_{h}}{\differential\ln
E}\equiv\frac{1}{\rho_{\rm crit}}\frac{\differential\rho_{h}}{\differential\ln
E},$ (3.14)

where $\rho_{\rm crit}$ denotes the present critical energy density of the
universe. The peak frequency is given by

$\displaystyle f_{\rm peak}=\frac{m_{\phi}}{4\pi}\frac{a(T_{\rm
dec})}{a_{0}}\simeq 2\times 10^{22}\,{\rm
Hz}~{}\lambda^{-1/2}\left(\frac{10^{13}\,{\rm GeV}}{m_{\phi}}\right)^{3/2}.$
(3.15)

Fig. 2 shows the present graviton spectrum from the dark Higgs decay for (a):
$(m_{\phi},H_{\rm inf},T_{\rm R})=(10^{13},10^{12},10^{12})$, (b):
$(10^{13},10^{11},10^{11})$ and (c): $(10^{12},10^{11},10^{11})$ in GeV unit.
We have taken $v_{\phi}=10m_{\phi}$ (or $\lambda=5\times 10^{-3}$). We used
the fitting formula in [74] for the effective degrees of freedom for the
entropy density $g_{*s}$ and energy density $g_{*}$ at the redshift $z_{d}$. A
small modulation is seen in the spectrum due to the $z_{d}$ dependence of
$g_{*}$ and $g_{*s}$. As seen from the figure, the typical frequency is so
high that the current gravitational wave detectors are not sensitive. The
reason is that the scalar decays at relatively late epoch and the effect of
redshift is relatively small.

---
Figure 2: The present graviton spectrum from the dark Higgs decay for (a):
($m_{\phi}$,$H_{\rm inf}$,$T_{\rm R}$) = ($10^{13}$,$10^{12}$,$10^{12}$), (b):
$(10^{13}$,$10^{11}$,$10^{11}$) and (c): $(10^{12}$,$10^{11}$,$10^{11}$) in
GeV unit.

###  3.3 Some other models

Below we list several models that may predict significant amount of graviton
through the scalar decay.

#### Axion.

In the previous subsection we considered the radial mode of the dark Higgs
decaying into the graviton pair. Not only the radial mode, but also the
axionic component has the same decay mode. Let us consider a simple model with
global chiral U(1) symmetry like the Peccei-Quinn symmetry:

$\displaystyle
S=\int\differential^{4}x\sqrt{-g}\,\left[g^{\mu\nu}\partial_{\mu}\Phi^{\dagger}\partial_{\nu}\Phi-\lambda\left(|\Phi|^{2}-\frac{v_{\phi}^{2}}{2}\right)^{2}+\bar{\psi}i\not{\nabla}\psi-(\lambda\Phi\bar{\psi}_{R}\psi_{L}+{\rm
h.c.)}\right].$ (3.16)

The axion-like particle $a$ appears as an angular component of $\Phi$ after
the symmetry breaking: $\Phi=(v_{\phi}/{\sqrt{2}})e^{ia/v_{\phi}}$. The axion
obtains a mass due to either non-perturbative effects if the fermion $\psi$
are charged under some hidden gauge symmetry with strong coupling or explicit
breaking terms of the global U(1). If $\psi$ (and hidden sector gauge fields
if any) is heavier than the axion, the only possible decay mode of the axion
is that into the graviton pair. The relevant effective axion-graviton coupling
is given in (2.34) and the axion decay rate into the graviton pair is given by
Eq. (3.1) with $\Lambda=192\pi^{2}v_{\phi}$. The decay temperature of the
axion is estimated as

$\displaystyle T_{\rm dec}\simeq 2\times 10^{-3}\,{\rm
GeV}~{}\left(\frac{m_{a}}{10^{12}\,{\rm
GeV}}\right)^{7/2}\left(\frac{10^{13}\,{\rm
GeV}}{v_{\phi}}\right)\left(\frac{10}{g_{*}(T_{\rm dec})}\right)^{1/4}.$
(3.17)

The axion abundance is estimated by assuming the standard misalignment
production mechanism:

$\displaystyle\frac{\rho_{a}}{s}=\frac{1}{8}T_{\rm R}\left(\frac{a_{i}}{M_{\rm
Pl}}\right)^{2},$ (3.18)

where $a_{i}$ denotes the initial axion amplitude. Here we assumed that the
axion starts to oscillate before the completion of the reheating:
$m_{a}\gtrsim T_{\rm R}^{2}/M_{\rm Pl}$. Otherwise, $T_{\rm R}$ in (3.18)
should be replaced with $\sim\sqrt{m_{a}M_{\rm Pl}}$. Since the axion
dominantly decays into the graviton in our setup, the graviton abundance is
calculated in the same way as (3.8) and given by

$\displaystyle\Omega_{h}\simeq 9\times 10^{-6}\left(\frac{T_{\rm
R}}{10^{8}\,{\rm GeV}}\right)\left(\frac{v_{\phi}}{10^{13}\,{\rm
GeV}}\right)^{3}\left(\frac{10^{12}\,{\rm
GeV}}{m_{a}}\right)^{7/2}\left(\frac{a_{i}}{v_{\phi}}\right)^{2}.$ (3.19)

Thus one can avoid the graviton overproduction for appropriate parameter
choices, while it is also possible that a significant amount of cosmic
graviton background exists in the present universe. If one would neglect the
axion decay mode into the graviton, it is completely stable and it would
dominate the universe leading to inconsistent cosmology. Taking account of the
gravitational decay mode can make such a model viable.

#### Inflaton.

Next we consider the possibility of the inflaton decay into the graviton
pair.\natural8\natural8$\natural$8 The inflaton “annihilation” into the
graviton pair in the Einstein gravity and its extensions has been considered
in Refs. [1, 4, 18].  In this case, the relic graviton abundance is roughly
given by $\Omega_{h}\sim\Omega_{\rm rad}\times{\rm Br}_{\phi\to 2h}$, where
${\rm Br}_{\phi\to 2h}$ denotes the branching ratio of the inflaton decay into
the graviton pair. It is estimated as

$\displaystyle{\rm Br}_{\phi\to 2h}=\frac{\Gamma(\phi\to
2h)}{\sqrt{\frac{\pi^{2}g_{*}}{10}}\frac{T_{\rm R}^{2}}{M_{\rm Pl}}}\simeq
5\times 10^{-21}\left(\frac{m_{\phi}}{10^{13}\,{\rm
GeV}}\right)^{7}\left(\frac{10^{17}\,{\rm
GeV}}{\Lambda}\right)^{2}\left(\frac{10^{10}\,{\rm GeV}}{T_{\rm
R}}\right)^{2}.$ (3.20)

In order to avoid the constraint on the abundance of dark radiation from the
cosmic microwave background observation, we need ${\rm Br}_{\phi\to
2h}\lesssim 0.1$. Note that, typical inflation models such as hilltop
inflation models require nontrivial relation between the inflaton mass
$m_{\phi}$ and its VEV $v_{\phi}$ (and hence $\Lambda$) once the condition for
reproducing the observed large scale density perturbation is imposed. Under
this constraint, we find that the branching ratio is typically negligibly
small unless the reheating temperature is very low.

One of the possible inflation models that may predict a substantial branching
ratio into the graviton pair is the so-called $\alpha$-attractor inflation
model [75, 76, 77, 78]. Let us modify the kinetic term of the dark Higgs in
the Lagrangian (3.4) in order to identify the dark Higgs as the inflaton:

$\displaystyle\mathcal{L}=\frac{\left|(\partial_{\mu}+ieA_{\mu})\Phi\right|^{2}}{\left(1-\frac{|\Phi|^{2}}{M^{2}}\right)^{2}}-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}-\lambda\left(|\Phi|^{2}-v_{\phi}^{2}\right)^{2}.$
(3.21)

The potential becomes effectively flat after the canonical rescaling of the
Higgs field at large field value and inflation happens there. If we take
$v_{\phi}<M<\sqrt{2}v_{\phi}$, the potential energy of the inflaton during
inflation is lower than that of the origin $\Phi=0$ and hence the symmetry is
never restored after inflation. A similar inflation model was considered in
Ref. [79] in the context of global U(1), instead of the gauged U(1). A crucial
difference is that the inflaton decays into the Goldstone boson (axion) pair
in the former case while the inflaton decay to the gauge boson pair is
kinematically forbidden for $e^{2}>\lambda$ in the latter case. In this model,
we have a constraint on the parameter in order to reproduce the observed
density perturbation of the universe [79]:

$\displaystyle\lambda\simeq 0.03\left(\frac{10^{14}\,{\rm
GeV}}{v_{\phi}}\right)^{2}\left(\frac{x}{1-x^{2}}\right)^{2},$ (3.22)

where $x\equiv v_{\phi}/M$ and its range is $1/\sqrt{2}<x<1$. The inflaton
mass is calculated as

$\displaystyle m_{\phi}=2\sqrt{\lambda}v_{\phi}(1-x^{2})\simeq 2\times
10^{13}\,{\rm GeV}\times x.$ (3.23)

In this scenario, the inflaton should dominantly decay into the visible sector
for successful reheating. It can be realized by introducing a small portal
coupling between the Standard Model Higgs and the dark Higgs or a small
kinetic mixing between the Standard Model photon and the dark photon. The
decay rate into the Standard Model sector is parametrized by the reheating
temperature $T_{\rm R}$. By substituting these values into (3.20) using
$\Lambda$ estimated by (3.2), we have ${\rm Br}_{\phi\to
2h}\sim\mathcal{O}(1)$ for $T_{\rm R}\sim 10\,{\rm GeV}$.

#### Decaying dark matter.

Let us suppose that a scalar in the hidden sector is dominant component of the
dark matter. Without a direct coupling to the Standard Model sector, it can
decay into the graviton pair through the term (1.1) unless prohibited by
symmetry.\natural9\natural9$\natural$9 Here we do not consider violation of
global symmetries due to quantum gravity.  From the decay rate (3.1), the
lifetime is

$\displaystyle\tau\simeq 2\times 10^{17}\,{\rm sec}\,\left(\frac{10^{10}\,{\rm
GeV}}{m_{\phi}}\right)^{7}\left(\frac{\Lambda}{M_{\rm Pl}}\right)^{2}.$ (3.24)

Thus the lifetime can be comparable to the present age of the universe while
the right amount of dark matter is obtained through the gravitational
production (3.7) (see also Refs. [9, 80, 81, 2, 6]). In such a case it is a
simple candidate of the decaying dark matter.\natural10\natural10$\natural$10
A worth-mentioning possibility is $\phi\to hh^{*}\to h+\mathrm{SM}$ with
$h^{*}$ denoting an off-shell graviton and $\mathrm{SM}$ denoting SM
particles. This process involves SM particles in the final state and thus is
in principle more suitable for detection. However, the decay rate is
suppressed further by $m_{\phi}^{2}/M_{P}^{2}$ (and the three-body phase space
factor), and our estimation indicates that the flux from this channel is not
sizable enough to be observed by the current cosmic ray experiments.

##  4 Summary and discussion

In this paper, we demonstrated that a scalar field $\phi$ can generically
couple to the quadratic curvature (unless prohibited by symmetry) because such
couplings are generated if there exist intermediate particles heavier than
$\phi$. We systematically derived higher dimensional couplings of $\phi$ to
gravity by integrating out heavy scalar, fermion, and vector fields, based on
the Schwinger-DeWitt formalism at the lowest order in the WKB expansion. We
confirmed that our results are consistent with the beta functions of the
quadratic curvature [62] since the effects of heavy fields should be imprinted
in threshold corrections.

One of the most interesting consequences is a decay of $\phi$ into graviton
pairs, which is induced by the quadratic curvature in the form of
$\mathcal{L}_{\phi RR}$ (1.1) and/or $\mathcal{L}_{\phi R\widetilde{R}}$
(1.2). Let us note a typical frequency range of the present-day graviton
spectrum from various sources. The dark Higgs decay as studied in the main
text produces the peak graviton frequency of $f_{\rm peak}\sim 10^{22}\,{\rm
Hz}$ as shown in Eq. (3.15) and Fig. 2. In the case of decaying dark matter,
even higher frequency is expected: $f_{\rm peak}\sim 10^{31}\,{\rm Hz}$. On
the other hand, the graviton spectrum from the Standard Model thermal plasma
predicts $f_{\rm peak}\sim 10^{11}\,{\rm Hz}$ [40, 42]. The inflaton
annihilation into the graviton pair also predicts the spectrum that extends to
$f\sim 10^{12}\,{\rm Hz}$ [18]. It shows that there are quite rich sources of
high-frequency gravitons that can be probes of the early universe and high
energy physics.

As mentioned in the introduction, there are several phenomenological
implications of the couplings between $\phi$ and the quadratic curvature other
than the graviton production. For instance, the Chern-Simons coupling, $\phi
R_{\mu\nu\rho\sigma}\tilde{R}^{\mu\nu\rho\sigma}$, can lead to gravitational
leptogenesis if we identify $\phi$ as inflaton [30]. This coupling induces the
production of chiral gravitational waves during inflation, which results in
the $B-L$ asymmetry through the gravitational anomaly. However, it is known
that $\phi R_{\mu\nu\rho\sigma}\tilde{R}^{\mu\nu\rho\sigma}$ leads to the
ghost instability, which severely constrains the efficiency of leptogenesis
[31, 32]. In connection with this, the prediction is highly sensitive to the
UV physics and hence depends on other higher dimensional operators involving
$\phi$ and curvature [36]. Our results suggest that such ghost instability can
be absent in a healthy UV completion, where a new fermion appears above the
instability scale without having a ghost. Also, the coefficients of relevant
higher dimensional operators are fixed when we integrate out a heavy fermion.
It would be interesting to study the production of chiral gravitational waves
[82] in a concrete healthy UV completion. The same coupling is also discussed
in the context of topological superconductor [83, 84, 85].

Our results are readily extended to higher order terms both in $\phi$ and
curvature. In the context of Schwinger effect, it is known that the Schwinger
current has a contribution which is only suppressed by a power of mass
although the particle production should exhibit an exponential suppression for
heavy masses. This contribution can be elegantly understood as higher
dimensional operators obtained from integrating out heavy particles, _i.e._ ,
the Euler-Heisenberg Lagrangian [86, 87]. Similar power-suppressed behavior is
also observed in the calculation of two point functions for heavy particles,
$m^{2}>R$, in the curved spacetime [54]. We expect that they are not related
to the production of particles, rather should be identified with higher
dimensional operators, gravitational analogue of the Euler-Heisenberg
Lagrangian. We will come back to this issue elsewhere.

## Acknowledgments

YE is supported in part by U.S. Department of Energy Grant No. desc0011842. KM
was supported by MEXT Leading Initiative for Excellent Young Researchers Grant
No. JPMXS0320200430. This work was supported by JSPS KAKENHI Grant Nos.
18K03609 (KN) and 17H06359 (KN). The Feynman diagrams are drawn with TikZ-
Feynman [88].

##  Appendix A Convention

In this appendix, we summarize our conventions. We work with the mostly-minus
convention for the spacetime metric. In particular, the Minkowski metric is
given by

$\displaystyle\eta_{ab}=\mathrm{diag}\left(+1,-1,-1,-1\right).$ (A.1)

We define the geometrical quantities as

$\displaystyle{\Gamma^{\mu}}_{\nu\rho}$
$\displaystyle=\frac{1}{2}g^{\mu\alpha}\left(\partial_{\nu}g_{\rho\alpha}+\partial_{\rho}g_{\nu\alpha}-\partial_{\alpha}g_{\nu\rho}\right),$
(A.2) $\displaystyle{R_{\mu\nu\rho}}^{\sigma}$
$\displaystyle=\partial_{\mu}{\Gamma^{\sigma}}_{\nu\rho}-\partial_{\nu}{\Gamma^{\sigma}}_{\mu\rho}+{\Gamma^{\alpha}}_{\nu\rho}{\Gamma^{\sigma}}_{\alpha\mu}-{\Gamma^{\alpha}}_{\mu\rho}{\Gamma^{\sigma}}_{\alpha\nu},$
(A.3) $\displaystyle R_{\mu\nu}$
$\displaystyle={R_{\mu\alpha\nu}}^{\alpha}=\partial_{\mu}{\Gamma^{\alpha}}_{\alpha\nu}-\partial_{\alpha}{\Gamma^{\alpha}}_{\mu\nu}+{\Gamma^{\alpha}}_{\beta\nu}{\Gamma^{\beta}}_{\alpha\mu}-{\Gamma^{\alpha}}_{\mu\nu}{\Gamma^{\beta}}_{\alpha\beta},$
(A.4) $\displaystyle R$ $\displaystyle=g^{\mu\nu}R_{\mu\nu}.$ (A.5)

This fixes the sign convention for the Ricci scalar. In particular, the
Einstein-Hilbert action comes with plus, and the conformal coupling
corresponds to $\xi=-1/6$ with this convention. The covariant derivative acts
on a vector field as

$\displaystyle\nabla_{\mu}v_{\nu}=\partial_{\mu}v_{\nu}-\Gamma_{\mu\nu}^{\rho}v_{\rho}.$
(A.6)

The commutator is then given by

$\displaystyle\left[\nabla_{\mu},\nabla_{\nu}\right]v_{\rho}=-{R_{\mu\nu\rho}}^{\sigma}v_{\sigma}.$
(A.7)

We thus obtain

$\displaystyle\left(W_{\mu\nu}W^{\mu\nu}\right)_{\alpha\beta}$
$\displaystyle=R_{\mu\nu\alpha\rho}{R^{\mu\nu\rho}}_{\beta},$ (A.8)

for a vector field. The covariant derivative acting on a spinor is given by

$\displaystyle\nabla_{\mu}\psi$
$\displaystyle=\left(\partial_{\mu}+\frac{1}{4}\omega_{\mu}^{ab}\gamma_{ab}\right)\psi,$
(A.9)

where our convention of the gamma matrix is

$\displaystyle\left\\{\gamma^{a},\gamma^{b}\right\\}$
$\displaystyle=2\eta^{ab},\quad\gamma_{ab}=\frac{1}{2}\left(\gamma_{a}\gamma_{b}-\gamma_{b}\gamma_{a}\right),\quad\gamma_{5}=i\gamma^{0}\gamma^{1}\gamma^{2}\gamma^{3}=-\frac{i}{4!}\epsilon^{abcd}\gamma_{a}\gamma_{b}\gamma_{c}\gamma_{d},$
(A.10)

with $\epsilon^{0123}=+1$, and we use $a,b,\cdots$ for the local Lorentz
indices and $\mu,\nu,\cdots$ for the spacetime indices. The spin connection is
defined as

$\displaystyle\omega_{\mu}^{ab}$
$\displaystyle=e^{a}_{\nu}\left[\partial_{\mu}e^{\nu
b}+{\Gamma^{\nu}}_{\sigma\mu}e^{\sigma b}\right],$ (A.11)

with the vierbein defined as

$\displaystyle g_{\mu\nu}=e_{\mu}^{a}\eta_{ab}e_{\nu}^{b}.$ (A.12)

Note that this is equivalent to the compatibility equation:

$\displaystyle\nabla_{\mu}e_{\nu}^{a}=\partial_{\mu}e_{\nu}^{a}+{\omega_{\mu}}^{ab}e_{\nu
b}-{\Gamma^{\rho}}_{\mu\nu}e_{\rho}^{a}=0.$ (A.13)

The Riemann tensor is equally expressed by the spin connection as

$\displaystyle{R_{\mu\nu}}^{ab}=-\partial_{\mu}{\omega_{\nu}}^{ab}+\partial_{\nu}{\omega_{\mu}}^{ab}-{\omega_{\mu}}^{ac}{\omega_{\nu
c}}^{b}+{\omega_{\nu}}^{ac}{\omega_{\mu c}}^{b}.$ (A.14)

It indeed satisfies

$\displaystyle R_{\mu\nu\rho\sigma}=e_{\rho a}e_{\sigma b}{R_{\mu\nu}}^{ab},$
(A.15)

which one can show from
$\left[\nabla_{\mu},\nabla_{\nu}\right]e^{a}_{\rho}=0$. It follows that the
spinor satisfies

$\displaystyle\left[\nabla_{\mu},\nabla_{\nu}\right]\psi=-\frac{1}{4}{R_{\mu\nu}}^{ab}\gamma_{ab}\psi.$
(A.16)

We thus have

$\displaystyle W_{\mu\nu}W^{\mu\nu}$
$\displaystyle=-\frac{i\gamma_{5}}{8}R_{\mu\nu\rho\sigma}\tilde{R}^{\mu\nu\rho\sigma}-\frac{1}{8}R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma},$
(A.17)

where the dual Riemann tensor is defined as

$\displaystyle\tilde{R}_{\mu\nu\rho\sigma}$
$\displaystyle=\frac{1}{2}{\epsilon_{\mu\nu}}^{\alpha\beta}R_{\alpha\beta\rho\sigma}.$
(A.18)

We can also easily show that

$\displaystyle\not{\nabla}^{2}=\Box+\frac{R}{4}.$ (A.19)

##  Appendix B Technical details of the WKB expansion

In this appendix we outline the computation of the coefficients of the WKB
expansion $a_{n}$ in the coincidence limit. Our goal is to derive Eqs. (2.14)
and (2.15).

###  B.1 Bitensor

We first summarize basic properties of Synge’s world function and the related
bitensors that are needed for the WKB approximation. We mainly follow [89].

#### Synge’s world function.

The world function $\sigma(x,x^{\prime})$ is defined as

$\displaystyle\sigma(x,x^{\prime})=\frac{1}{2}\left(\lambda_{1}-\lambda_{0}\right)\int_{\lambda_{0}}^{\lambda_{1}}d\lambda\,g_{\mu\nu}(z(\lambda))\frac{dz^{\mu}}{d\lambda}\frac{dz^{\nu}}{d\lambda},$
(B.1)

where $z(\lambda)$ is the geodesic that connects $x$ and $x^{\prime}$ and
$\lambda$ the affine parameter with $z(\lambda_{1})=x$ and
$z(\lambda_{0})=x^{\prime}$. We assume that there is only one such geodesic
that is true in the coincidence limit $x^{\prime}\rightarrow x$. One can show
with the geodesic equation that

$\displaystyle\epsilon\equiv
g_{\mu\nu}\frac{dz^{\mu}}{d\lambda}\frac{dz^{\nu}}{d\lambda},$ (B.2)

is constant along the geodesic, and hence

$\displaystyle\sigma(x,x^{\prime})=\frac{\epsilon}{2}(\lambda_{1}-\lambda_{0})^{2}=\frac{s^{2}}{2},$
(B.3)

where $s$ is the proper distance that is well-defined if $x$ and $x^{\prime}$
are time like.

We now consider the derivative of $\sigma$ with respect to $x$ and
$x^{\prime}$. We may use the notation that

$\displaystyle\sigma_{\alpha}=\frac{\partial\sigma}{\partial
x^{\alpha}},\quad\sigma_{\alpha^{\prime}}=\frac{\partial\sigma}{\partial
x^{\alpha^{\prime}}},$ (B.4)

and so on. We first take $x^{\mu}$ to $x^{\mu}+\delta x^{\mu}$ with $\delta
x^{\mu}$ infinitesimal. In this case the geodesic is modified as
$z^{\mu}(\lambda)\rightarrow z^{\mu}(\lambda)+\delta z^{\mu}(\lambda)$ with
$\delta z^{\mu}(\lambda_{0})=0$ and $\delta z^{\mu}(\lambda_{1})=\delta
x^{\mu}$. Note that the affine parameter is rescaled such that it again runs
from $\lambda_{0}$ to $\lambda_{1}$. One can show with the geodesic equation
that the variance of $\sigma$ comes only from the boundary, and is explicitly
given as

$\displaystyle\delta\sigma=\left(\lambda_{1}-\lambda_{0}\right)\left[g_{\mu\nu}\frac{dz^{\mu}}{d\lambda}\delta
z^{\nu}\right]_{\lambda_{0}}^{\lambda_{1}}.$ (B.5)

This indicates that

$\displaystyle\sigma_{\alpha}=\left(\lambda_{1}-\lambda_{0}\right)g_{\alpha\mu}t^{\mu},\quad\sigma_{\alpha^{\prime}}=-\left(\lambda_{1}-\lambda_{0}\right)g_{\alpha^{\prime}\mu^{\prime}}{t^{\prime}}^{\mu},$
(B.6)

where $g_{\alpha\mu}$ is the metric tensor at $x$ and
$g_{\alpha^{\prime}\mu^{\prime}}$ at $x^{\prime}$, and

$\displaystyle
t^{\mu}=\left.\frac{dz^{\mu}}{d\lambda}\right\rvert_{\lambda=\lambda_{1}},\quad{t^{\prime}}^{\mu}=\left.\frac{dz^{\mu}}{d\lambda}\right\rvert_{\lambda=\lambda_{0}}.$
(B.7)

We then obtain

$\displaystyle\sigma^{\alpha}\sigma_{\alpha}$
$\displaystyle=\sigma^{\alpha^{\prime}}\sigma_{\alpha^{\prime}}=2\sigma.$
(B.8)

#### van Vleck-Morette determinant.

Next we introduce the van Vleck-Morette determinant. For this purpose we first
introduce the metric bitensor $g_{\mu\nu^{\prime}}$. In general, the metric
tensor is expressed by the vierbein as

$\displaystyle g_{\mu\nu}=\eta_{ab}e^{a}_{\mu}e^{b}_{\nu}.$ (B.9)

We introduce a function $g_{\mu\nu^{\prime}}$ that depends both on $x$ and
$x^{\prime}$ as

$\displaystyle
g_{\mu\nu^{\prime}}(x,x^{\prime})=\eta_{ab}e^{a}_{\mu}(x)e^{b}_{\nu^{\prime}}(x^{\prime}).$
(B.10)

We then define

$\displaystyle{\Delta^{\alpha^{\prime}}}_{\beta^{\prime}}=-{g^{\alpha^{\prime}}}_{\alpha}{\sigma^{\alpha}}_{\beta^{\prime}},\quad\Delta=\mathrm{det}{\Delta^{\alpha^{\prime}}}_{\beta^{\prime}}.$
(B.11)

By using that $\mathrm{det}e^{a}_{\mu}=\sqrt{-g}$, this is equally expressed
as

$\displaystyle\Delta=-\frac{\mathrm{det}\left(-\sigma_{\alpha\beta^{\prime}}\right)}{\sqrt{-g}\sqrt{-g^{\prime}}}.$
(B.12)

We now show $(\Delta\sigma^{\alpha})_{;\alpha}=d\Delta$. We take the
derivative of $\sigma=\sigma^{\mu}\sigma_{\mu}/2$ twice and get

$\displaystyle\sigma_{\alpha\beta^{\prime}}={\sigma^{\mu}}_{\alpha}\sigma_{\mu\beta^{\prime}}+\sigma^{\mu}\sigma_{\mu\alpha\beta^{\prime}}.$
(B.13)

The primed and unprimed derivatives commute with each other. Moreover, since
$\sigma$ is a scalar function, we have $\sigma_{\mu\nu}=\sigma_{\nu\mu}$. As a
result we obtain

$\displaystyle{\Delta^{\alpha^{\prime}}}_{\beta^{\prime}}={g^{\alpha^{\prime}}}_{\alpha}g_{\mu\nu^{\prime}}\sigma^{\mu\alpha}{\Delta^{\nu^{\prime}}}_{\beta^{\prime}}+\sigma^{\mu}\left({\Delta^{\alpha^{\prime}}}_{\beta^{\prime}}\right)_{;\mu}.$
(B.14)

By multiplying ${(\Delta^{-1})^{\beta^{\prime}}}_{\gamma^{\prime}}$ and taking
the trace of $\alpha^{\prime}$ and $\gamma^{\prime}$, we obtain

$\displaystyle d$
$\displaystyle={\sigma^{\alpha}}_{\alpha}+\sigma^{\alpha}\left(\ln\Delta\right)_{;\alpha},$
(B.15)

which is equivalent to $(\Delta\sigma^{\alpha})_{;\alpha}=d\Delta$, or

$\displaystyle
d\Delta^{1/2}=\Delta^{1/2}{\sigma^{\alpha}}_{\alpha}+2\sigma^{\alpha}(\Delta^{1/2})_{\alpha}.$
(B.16)

#### Coincidence limit.

Our primary interest is $\sigma$ and $\Delta$ in the coincidence limit
$x^{\prime}\rightarrow x$. We assume that this limit exits independent of the
direction that $x^{\prime}$ approaches to $x$. For our purpose its enough to
consider the unprimed derivatives and hence we focus on the unprimed indexed
quantities. We first note that

$\displaystyle[\sigma]=[\sigma_{\alpha}]=0.$ (B.17)

Eq. (B.8) is our basic tool to study the higher derivative terms. We take the
derivative of Eq. (B.8) and obtain

$\displaystyle\sigma^{\alpha}(\sigma_{\alpha\beta}-g_{\alpha\beta})=0.$ (B.18)

Since $\sigma^{\alpha}\propto t^{\alpha}$ and we assume that the limit is
independent of $t^{\alpha}$, we obtain

$\displaystyle[\sigma_{\alpha\beta}]=g_{\alpha\beta}.$ (B.19)

We next take the derivative of Eq. (B.8) three times and obtain

$\displaystyle[\sigma_{\gamma\alpha\beta}]+[\sigma_{\beta\alpha\gamma}]=0.$
(B.20)

The first two indices of $\sigma_{\alpha\beta\gamma}$ commute with each other
and hence this is equivalent to

$\displaystyle
2[\sigma_{\alpha\beta\gamma}]+[\left[\nabla_{\beta},\nabla_{\gamma}\right]\sigma_{\alpha}]=2[\sigma_{\alpha\beta\gamma}]-[{R_{\beta\gamma\alpha}}^{\delta}\sigma_{\delta}]=0.$
(B.21)

The curvature is well-behaved in the coincidence limit, and hence we obtain

$\displaystyle[\sigma_{\alpha\beta\gamma}]=0.$ (B.22)

In the same way, we can show that

$\displaystyle[\sigma_{\alpha\beta\gamma\delta}]$
$\displaystyle=S_{\alpha\beta\gamma\delta}\equiv\frac{1}{3}\left(R_{\alpha\gamma\beta\delta}+R_{\alpha\delta\beta\gamma}\right),$
(B.23) $\displaystyle[\sigma_{\alpha\beta\gamma\delta\mu}]$
$\displaystyle=\frac{3}{4}\left[\nabla_{\mu}S_{\alpha\beta\gamma\delta}+\nabla_{\delta}S_{\alpha\beta\gamma\mu}+\nabla_{\gamma}S_{\alpha\beta\delta\mu}\right],$
(B.24) $\displaystyle[\Box^{3}\sigma]$
$\displaystyle=\frac{4}{15}\left(R_{\mu\nu}R^{\mu\nu}-R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}\right)+\frac{8}{5}\Box
R.$ (B.25)

We now move to the coincidence limit of the determinant. It is trivial to see
that\natural11\natural11$\natural$11 Here we actually use
$[\sigma_{\alpha\beta^{\prime}}]=-g_{\alpha\beta}$ where the minus sign
originates from the opposite sign of $[\sigma_{\alpha}]$ and
$[\sigma_{\alpha^{\prime}}]$.

$\displaystyle[\Delta^{1/2}]=1.$ (B.26)

In order to compute the coincidence limit with derivatives, we use Eq. (B.16).
By taking the derivative of this equation once, we obtain

$\displaystyle[(\Delta^{1/2})_{\alpha}]=0.$ (B.27)

By taking the derivative twice we obtain\natural12\natural12$\natural$12 It
seems both the Riemann tensor and the Ricci tensor have the opposite sign as
[57]. In our case it correctly reproduces the factor $\xi+1/6$ so should be
consistent.

$\displaystyle[(\Delta^{1/2})_{\alpha\beta}]=-\frac{1}{6}R_{\alpha\beta}.$
(B.28)

In the same way, we obtain

$\displaystyle[(\Delta^{1/2})_{\alpha\beta\gamma}]$
$\displaystyle=-\frac{1}{12}\left[\nabla_{\gamma}R_{\alpha\beta}+\nabla_{\beta}R_{\gamma\alpha}+\nabla_{\alpha}R_{\beta\gamma}\right],$
(B.29) $\displaystyle[\Box^{2}(\Delta^{1/2})]$
$\displaystyle=\frac{1}{30}\left(-R_{\mu\nu}R^{\mu\nu}+R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}\right)-\frac{1}{5}\Box
R+\frac{1}{36}R^{2}.$ (B.30)

###  B.2 WKB expansion

We now compute the coefficients of the WKB expansion $a_{n}$ in the
coincidence limit. By substituting Eq. (2.9) to the Schrödinger equation, we
obtain

$\displaystyle 0$
$\displaystyle=i\frac{\partial\Omega}{\partial\tau}+\frac{i}{\tau}\sigma^{\alpha}\Omega_{\alpha}-\Delta^{-1/2}\Box\left(\Delta^{1/2}\Omega\right)$
$\displaystyle+2i\tau(\mathcal{M}^{2})^{\alpha}\Delta^{-1/2}\nabla_{\alpha}\left(\Delta^{1/2}\Omega\right)+\left[\sigma^{\alpha}(\mathcal{M}^{2})_{\alpha}-\frac{R}{6}+i\tau\Box\tilde{X}-(i\tau)^{2}(\mathcal{M}^{2})^{\alpha}(\mathcal{M}^{2})_{\alpha}\right]\Omega.$
(B.31)

By expanding $\Omega$ as

$\displaystyle\Omega(\tau;x,x^{\prime})=\sum_{n=0}^{\infty}\left(i\tau\right)^{n}a_{n}(x,x^{\prime}),$
(B.32)

we obtain the recursion relations

$\displaystyle\mathbbm{1}$ $\displaystyle=a_{0},$ (B.33) $\displaystyle 0$
$\displaystyle=\sigma^{\alpha}a_{0;\alpha},$ (B.34) $\displaystyle 0$
$\displaystyle=a_{1}+\sigma^{\alpha}a_{1;\alpha}+\Delta^{-1/2}\Box\left(\Delta^{1/2}a_{0}\right)-\left(\sigma^{\alpha}(\mathcal{M}^{2})_{\alpha}-\frac{R}{6}\right)a_{0},$
(B.35) $\displaystyle 0$
$\displaystyle=2a_{2}+\sigma^{\alpha}a_{2;\alpha}+\Delta^{-1/2}\Box\left(\Delta^{1/2}a_{1}\right)$
$\displaystyle-\left(\sigma^{\alpha}(\mathcal{M}^{2})_{\alpha}-\frac{R}{6}\right)a_{1}-2(\mathcal{M}^{2})^{\alpha}\Delta^{-1/2}\nabla_{\alpha}\left(\Delta^{1/2}a_{0}\right)-\left(\Box\mathcal{M}^{2}\right)a_{0},$
(B.36) $\displaystyle 0$
$\displaystyle=\left(n+3\right)a_{n+3}+\sigma^{\alpha}a_{n+3;\alpha}+\Delta^{-1/2}\Box\left(\Delta^{1/2}a_{n+2}\right)-\left(\sigma^{\alpha}(\mathcal{M}^{2})_{\alpha}-\frac{R}{6}\right)a_{n+2}$
$\displaystyle-2(\mathcal{M}^{2})^{\alpha}\Delta^{-1/2}\nabla_{\alpha}\left(\Delta^{1/2}a_{n+1}\right)-\left(\Box\mathcal{M}^{2}\right)a_{n+1}+(\mathcal{M}^{2})^{\alpha}(\mathcal{M}^{2})_{\alpha}a_{n},~{}~{}n\geq
0.$ (B.37)

Notice that we used the bold font for $a_{0}$ since it lives in the Lorentz
tensor space in general.

We now take the coincidence limit. We need up to four derivatives of $a_{0}$
to compute the terms of our interest. We first take the derivative of Eq.
(B.34) once and obtain

$\displaystyle[a_{0;\alpha}]$ $\displaystyle=0.$ (B.38)

Next, we take the derivative of Eq. (B.34) twice and obtain

$\displaystyle[a_{0;\alpha\beta}]+[a_{0;\beta\alpha}]=0.$ (B.39)

We also note that, if a particle of one’s interest has a finite spin, its
covariant derivative acting on $a_{0}$ satisfies

$\displaystyle\left[\nabla_{\alpha},\nabla_{\beta}\right]a_{0}=W_{\alpha\beta}a_{0},$
(B.40)

where $W_{\alpha\beta}$ depends on the spin and is a combination of the
curvature tensors. This together with Eq. (B.39) tell us that

$\displaystyle[a_{0;\alpha\beta}]$
$\displaystyle=-\frac{1}{2}W_{\alpha\beta}\mathbbm{1}.$ (B.41)

We then take the derivative of Eq. (B.34) three times and obtain

$\displaystyle[a_{0;\alpha\beta\gamma}]+[a_{0;\beta\alpha\gamma}]+[a_{0;\gamma\alpha\beta}]=0.$
(B.42)

We have

$\displaystyle a_{0;\beta\alpha\gamma}$
$\displaystyle=a_{0;\alpha\beta\gamma}+\nabla_{\gamma}\left(W_{\alpha\beta}a_{0}\right),$
(B.43) $\displaystyle a_{0;\alpha\gamma\beta}$
$\displaystyle=a_{0;\alpha\beta\gamma}-{R_{\beta\gamma\alpha}}^{\rho}a_{0;\rho}+W_{\beta\gamma}a_{0;\alpha},$
(B.44)

and hence we obtain

$\displaystyle[a_{0;\alpha\beta\gamma}]$
$\displaystyle=-\frac{1}{3}\left(W_{\alpha\beta;\gamma}+W_{\alpha\gamma;\beta}\right)\mathbbm{1}.$
(B.45)

We finally take the derivative of Eq. (B.34) four times. After a similar (but
longer) manipulation as above we obtain

$\displaystyle\left[\Box^{2}a_{0}\right]=\frac{1}{2}W_{\mu\nu}W^{\mu\nu}.$
(B.46)

By substituting these results in the recursion relations, we obtain

$\displaystyle[a_{0}]$ $\displaystyle=\mathbbm{1},\quad[a_{1}]=0,$ (B.47)
$\displaystyle[a_{2}]$
$\displaystyle=\left[-\frac{1}{180}R_{\mu\nu}R^{\mu\nu}+\frac{1}{180}R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}+\frac{1}{12}W_{\mu\nu}W^{\mu\nu}+\frac{1}{6}\Box
X-\frac{1}{30}\Box R\right]\mathbbm{1},$ (B.48)

in agreement with [57, 56, 58].

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