# Conformal Portal to Dark Matter

Kunio Kanetaa<EMAIL_ADDRESS>Pyungwon Koa<EMAIL_ADDRESS>Wan-Il Parkb
<EMAIL_ADDRESS>aSchool of Physics, Korea Institute for Advanced Study,
Seoul 02455, Korea  bDivision of Science Education and Institute of Fusion
Science, Jeonbuk National University, Jeonju 54896, Korea

###### Abstract

We propose a new portal coupling to dark matter by taking advantage of the
nonminimally coupled portal sector to the Ricci scalar. Such a portal sector
conformally induces couplings to the trace of the energy-momentum tensor of
matters including highly secluded dark matter particles. The portal coupling
is so feeble that dark matter is produced by freeze-in processes of
scatterings and/or the decay of the mediator. We consider two concrete
realizations of the portal: conformally induced Higgs portal and conformally
induced mediator portal. The former case is compatible with the Higgs
inflation, while the latter case can be tested by dark matter direct detection
experiments.

††preprint: KIAS-P21021

##  I I. Introduction

Although the existence of the hypothetical matter, dark matter, is evident and
has extensively been studied so far, its particle nature remains unidentified
yet. The paradigm of the Weakly Interacting Massive Particle (WIMP) has led
investigations to uncover the dark matter for a long time Gunn  _et al._
(1978). The WIMP is an attractive Beyond-the-Standard-Model (BSM) particle in
that its nature, such as mass scale and interaction strength, is close to that
of the Standard Model particles, except that WIMP has to be neutral and stable
or long-lived enough to explain the observed dark matter relic density. In
modeling the WIMP, portal couplings to look into the dark matter sector from
the visible sector is considered as a useful concept, and a number of portals
have been considered so far, including Higgs portal Silveira and Zee (1985);
McDonald (1994); Burgess  _et al._ (2001); Davoudiasl  _et al._ (2005);
Mambrini (2011); Baek  _et al._ (2012a); Djouadi  _et al._ (2012); Lebedev
_et al._ (2012); Djouadi  _et al._ (2013); Baek  _et al._ (2013a, 2012b); Han
and Zheng (2015); Casas  _et al._ (2017), Z portal Arcadi  _et al._ (2015);
Escudero  _et al._ (2016); Kearney  _et al._ (2017); Ellis  _et al._ (2018),
and all other BSM related portals Bai  _et al._ (2009); Baek  _et al._
(2013b); Alves  _et al._ (2014); Arcadi  _et al._ (2014); Lebedev and Mambrini
(2014); Chowdhury  _et al._ (2019); Fuks  _et al._ (2020). The relic WIMP
density is solely determined by the annihilation rate of the WIMPs into
thermal particles by assuming that the WIMP is in thermal equilibrium at early
times, rendering ultraviolet (UV) conditions of the Universe irrelevant. The
WIMP interaction strength is on the other hand as large as being accessible by
using scatterings of the WIMP with nuclei, making direct detection possible.
However, direct dark matter detection experiments, such as LUX Akerib  _et
al._ (2017), PandaX Cui  _et al._ (2017), and XENON1T Aprile  _et al._ (2018),
have observed no signals so far, putting stringent constraints on the
parameter space of the WIMP scenario.

Feebly Interacting Massive Particle (FIMP) has been proposed as another type
of dark matter candidate Hall  _et al._ (2010), which sheds light on new
parameter spaces of theories and phenomenology of dark matter (see Bernal  _et
al._ (2017) for a review). Similar to the gravitino dark matter Nanopoulos
_et al._ (1983); Khlopov and Linde (1984), the FIMP is so feebly interacting
with thermal particles that it is never thermalized, which can be realized by
supposing that the FIMP sector is supposed to be highly secluded from the
Standard Model. For the relic FIMP density, instead of following the
equilibrium number density like the WIMP, the FIMP is produced, for instance,
from the annihilation of the thermal particles, which is one-way reactions as
the FIMP number density is too little to react inversely. Such highly secluded
dark matter sector is realized in many models that has super massive mediators
Mambrini  _et al._ (2013); Nagata  _et al._ (2015, 2017); Mambrini  _et al._
(2015); Berlin  _et al._ (2016a, b); Heikinheimo  _et al._ (2016); Benakli
_et al._ (2017); Dudas  _et al._ (2017a, b, 2018); Bernal  _et al._ (2018);
Bhattacharyya  _et al._ (2018); Banerjee  _et al._ (2019); Kang and Lee
(2020); Bernal  _et al._ (2020a); Heurtier and Huang (2019) or tiny couplings
Kaneta  _et al._ (2017a, b); Barman  _et al._ (2020); Ema  _et al._ (2016,
2018); Mambrini and Olive (2021); Hashiba and Yokoyama (2019); Ahmed  _et al._
(2020); Chianese  _et al._ (2021, 2021); Anastasopoulos  _et al._ (2020);
Barman and Bernal (2021). Depending on the type of relevant interactions, the
FIMP is dominantly produced in either infrared or UV regime. For instance, if
the production takes place through renormalizable couplings, the resulting
FIMP abundance is insensitive to the UV, whereas it may depend on the UV when
the production happens through higher dimensional operators.

In the FIMP models, although gravity is the weakest force in nature, the
gravity sector often has important effects on the FIMP productions. Besides
the direct contributions Ema  _et al._ (2016, 2018); Hashiba and Yokoyama
(2019); Ahmed  _et al._ (2020); Chianese  _et al._ (2020, 2021); Mambrini and
Olive (2021); Barman and Bernal (2021); Li  _et al._ (2021), dark matter
itself can arise from the space-time metric Anastasopoulos  _et al._ (2020);
Brax  _et al._ (2021a, b).

In this work, we propose a new portal to the FIMP dark matter, where the
portal coupling is induced via the conformal factor of the space-time metric
and takes advantage of having a particle which has a conformally induced
universal coupling to all the other particles via the trace of the energy-
momentum tensor. In particular, we show that the nonminimal coupling of the
Standard Model Higgs to the Ricci scalar automatically induces a feeble
coupling to dark matter and explains the observed dark matter density. We also
consider the case where a conformally induced scalar plays as a light/heavy
mediator to produce dark matter, in which some scenario can be tested by
direct detection experiments.

The paper is organized as follows. In section II, we define the model for both
the dark matter and conformally induced portal sectors, and discuss the dark
matter productions with generic reaction rates in section III. In sections IV
and V, we compute the dark matter number density produced via the conformally
induced Higgs portal and light/heavy mediator, respectively. The phenomenology
of dark matter and mediator is discussed in section VI, before concluding in
section VII.

##  II II.The Model

Our setup consists of three sectors: the Standard Model sector, the dark
matter sector, and the portal sector. To highlight the role of the portal
sector, we suppose that the dark matter sector does not have any direct
interactions to the Standard Model sector. In the following, first we consider
a generic framework for the portal sector, and then give a specific setup for
the dark matter sector. Throughout the paper, we use the mostly-minus
convention for the metric, $+---$.

###  II.1 II-A. The Portal Sector

The portal sector is introduced as a conformal factor of the metric,
$g_{\mu\nu}=C(\phi)\tilde{g}_{\mu\nu}$, where $g_{\mu\nu}$ defines the
Einstein frame in which the gravity sector takes the canonical form with the
Einstein-Hilbert action, and $\tilde{g}_{\mu\nu}$ defines the Jordan frame in
which the gravity sector is parametrized by

$\displaystyle S_{\rm
grav}=\frac{M_{P}^{2}}{2}\int\mathop{}\\!\mathrm{d}^{4}x\sqrt{-\tilde{g}}C(\phi)\tilde{R}.$
(1)

Here, $M_{P}$ is the reduced Planck scale ($M_{P}\simeq 2.4\times 10^{18}$
GeV), and $C(\phi)$ is an arbitrary function of a scalar field $\phi$. It
becomes clear shortly that a generic scalar field $\phi$ is taken as either
the Standard Model Higgs or an additional scalar which plays a role of a
mediator connecting between the Standard Model and the dark matter sector. The
conformal factor $C(\phi)$ is assumed to be expanded as

$\displaystyle C(\phi)\simeq 1+\delta C+\cdots.$ (2)

As discussed in detail in Appendix, the interactions among $\phi$, the
Standard Model, and the dark matter sectors are given in the form of

$\displaystyle{\cal L}_{\rm portal}=\frac{1}{2}\delta Cg^{\mu\nu}T_{\mu\nu},$
(3)

in the Einstein frame, where $T_{\mu\nu}$ contains the energy-momentum tensors
for all matter fields including $\phi$. The explicit definition and the detail
in deriving the theory in the Einstein frame are given in the Appendix, where
the energy-momentum tensor is introduced as a response to the variation of the
metric. 111 Given the existence of $C(\phi)\tilde{R}$ coupling at tree level,
there also be loop-level contributions that may enhance additional terms, such
as $\tilde{R}^{2}$ for instance Salvio and Mazumdar (2015); Calmet and Kuntz
(2016). Such radiative corrections involving gravity may give rise to
additional portal channels whose comprehensive study is however beyond the
scope of the paper.

###  II.2 II-B. The Dark Matter Sector

The dark matter sector is supposed to be secluded from the Standard Model
sector, namely, no direct interactions between those two. To be specific, we
consider three different scenarios where dark matter is a real scalar $X$, a
Dirac fermion $\chi$, or an Abelian vector $V_{\mu}$, whose Lagrangian is
respectively given by:

$\displaystyle{\cal
L}_{X}=\frac{1}{2}\partial_{\mu}X\partial^{\mu}X-\frac{1}{2}m_{X}^{2}X^{2},$
(4) $\displaystyle{\cal
L}_{\chi}=\overline{\chi}(i\cancel{\partial}-m_{\chi})\chi,$ (5)
$\displaystyle{\cal
L}_{V}=-\frac{1}{4}V^{\mu\nu}V_{\mu\nu}+\frac{1}{2}m_{V}^{2}V_{\mu}V^{\mu},$
(6)

where $V_{\mu\nu}\equiv\partial_{\mu}V_{\nu}-\partial_{\nu}V_{\mu}$.

For the vectorial dark matter case, the mass is put in by hand, or by
Stückelberg mechanism. In either case, perturbative unitarity in some
scattering channels will be violated at high energy (or high enough
temperature) and one has to consider UV completions. The simplest would be an
Abelian Higgs mechanism in the dark sector Baek  _et al._ (2013a), for which
Eq. (6) is replaced by

$\displaystyle{\cal L}_{V}$ $\displaystyle=$
$\displaystyle-\frac{1}{4}V_{\mu\nu}V^{\mu\nu}+\left(D_{\mu}\varphi\right)^{\dagger}D^{\mu}\varphi-\frac{\lambda_{\varphi}}{4\\!}\left(\varphi^{\dagger}\varphi-\frac{v_{\varphi}^{2}}{2}\right)^{2}$
(7) $\displaystyle+$ $\displaystyle\frac{\lambda_{\varphi
H}}{4}\left(\varphi^{\dagger}\varphi-\frac{v_{\varphi}^{2}}{2}\right)\left(H^{\dagger}H-\frac{v_{H}^{2}}{2}\right)$

where the dark Higgs $\varphi$ carries $U(1)_{D}$ charge equal to
$q_{\varphi}$.

Dark matter phenomenology in the UV completions can be different from the
effective field theory approach. Nice examples are the invisible Higgs decay
rate into a pair of vectorial dark matters, $\Gamma_{\rm inv}(H\rightarrow
VV)$, and its correlation with spin-independent dark matter-nuclear cross
section Baek  _et al._ (2014). In particular the behavior of $\Gamma_{\rm
inv}(H\rightarrow VV)$ is completely different in the limit $m_{V}\rightarrow
0$. Another example is the restoration of unitarity in the high energy
behavior of production cross sections for the vectorial dark matter at high
energy colliders such as ILC Ko and Yokoya (2016); Kamon  _et al._ (2017) and
LHC/100 TeV $pp$ collider Dutta  _et al._ (2018). Without including the dark
Higgs boson, the production rate will be too much overestimated not only at
high energy colliders but also at high temperature era in the early Universe.
Within this paper, however, we highlight only the conformal portal to the dark
sector, and thus we will take the dark Higgs mass above the
inflationary/cutoff scale so that it is never be produced after the end of
inflation. 222More precisely, there are two types of cutoff scales in our
framework for the vectorial dark matter. One is induced through $C(\phi)$,
which is the scale where the contributions from higher dimensional operators
in Eq. (3) blow up. The other is induced by the breaking of $U(1)_{D}$, which
we assume is above the other cutoff scale or the maximal temperature in the
Universe. After integrating out the dark Higgs (having a nonzero vacuum
expectation value), we obtain Eq. (6) as an effective theory where the only
$V_{\mu}$ is left as a light degree of freedom in the dark sector. In doing
so, we implicitly assume a hierarchy between the dark Higgs mass
($\sim\sqrt{\lambda_{\varphi}}v_{\varphi}$) and the vectorial dark matter mass
($\sim g_{D}v_{\varphi}$ with $g_{D}$ the gauge coupling of the $U(1)_{D}$),
namely, $\sqrt{\lambda_{\varphi}}\gg g_{D}$.

The energy-momentum tensor for each dark matter case is obtained as

$\displaystyle
T^{X}_{\mu\nu}=\frac{1}{2}(\partial_{\mu}X\partial_{\nu}X+\partial_{\nu}X\partial_{\mu}X)-g_{\mu\nu}{\cal
L}_{X},$ (8) $\displaystyle
T^{\chi}_{\mu\nu}=\frac{i}{4}\overline{\chi}(\gamma_{\mu}\overset{\leftrightarrow}{\partial}_{\nu}+\gamma_{\nu}\overset{\leftrightarrow}{\partial}_{\mu})\chi-
g_{\mu\nu}{\cal L}_{\chi},$ (9) $\displaystyle
T^{V}_{\mu\nu}=-g^{\alpha\beta}V_{\mu\alpha}V_{\nu\beta}-g_{\mu\nu}{\cal
L}_{V},$ (10)

where
$\overline{\chi}\overset{\leftrightarrow}{\partial}_{\mu}\chi\equiv\overline{\chi}(\partial_{\mu}\chi)-(\partial_{\mu}\overline{\chi})\chi$.
From Eq. (3) it is evident that $C(\phi)$ plays the role of the portal
connecting the Standard Model and the dark matter sector where the trace of
the energy-momentum tensor is given by

$\displaystyle
g^{\mu\nu}T^{X}_{\mu\nu}\simeq-(\partial^{\mu}X)(\partial_{\mu}X)$ (11)
$\displaystyle g^{\mu\nu}T^{\chi}_{\mu\nu}=m_{\chi}\overline{\chi}\chi,$ (12)
$\displaystyle g^{\mu\nu}T^{V}_{\mu\nu}=-2m_{V}^{2}V^{\mu}V_{\mu}.$ (13)

Here, for $X$ the mass term has been neglected, and for $\chi$ we have used
the equation of motion. In the following we consider two possible choices of
$C(\phi)$:

  1. (a)

$C(\phi=H)=1-\xi|H|^{2}/M_{P}^{2}$,

  2. (b)

$C(\phi)=e^{-\phi/f}$,

where $H$ is the Standard Model Higgs doublet, and thus (a) turns out to be a
conformally induced Higgs portal scenario. In the case of (b) we consider
$\phi$ as a real scalar mediator connecting between the Standard Model and the
dark matter sector.

##  III III. Dark Matter Production via scattering

The interactions between the Standard Model particles and the dark matter are
feeble, so the dark matter never thermalizes. In the following, we compute the
dark matter number density produced by the annihilation of the thermal
particles.

The reaction rate for the $12\to 34$ process whose amplitude is written as
${\cal M}$ is given by

$R(T)=\frac{S}{1024\pi^{6}}\int
f_{1}(E_{1})f_{2}(E_{2})E_{1}E_{2}\mathop{}\\!\mathrm{d}E_{1}\mathop{}\\!\mathrm{d}E_{2}\mathop{}\\!\mathrm{d}\cos\theta_{12}\int|{\cal
M}|^{2}\mathop{}\\!\mathrm{d}\Omega_{13},$ (14)

where $S$ represents the symmetry factor ($S=1/2$ when the particles 3 and 4
are identical). The particles 1 and 2 are assumed to be the Higgs or $\phi$ in
the most cases in the following, and therefore the distribution functions
$f_{1}$ and $f_{2}$ are taken as the Bose-Einstein distribution function. More
detail is explained in Appendix of Ref. Anastasopoulos  _et al._ (2020).

When the spin averaged squared amplitude is written in the form of

$\displaystyle|{\cal M}|^{2}=\frac{s^{k}}{\Lambda^{2k}}$ (15)

with an integer $k$ and a suppression scale $\Lambda$,333Notice that if the
reaction under consideration is not a 2 to 2 process, the squared amplitude
may not be written in the dimensionless form. In such cases, one should
directly compute the reaction rate by Eq. (14). the reaction rate is readily
computed as

$\displaystyle
R(T)=\frac{[2^{k}\Gamma(k+2)\zeta(k+2)T^{k+2}]^{2}}{128\pi^{5}(k+1)\Lambda^{2k}},$
(16)

where we assume that the annihilating particles in the thermal bath are
bosonic, while if they are fermion, a factor of $(1-2^{-(k+1)})^{2}$ should be
multiplied Brax  _et al._ (2021a).

Once the reaction rate is computed, the dark matter number density can be
obtained by solving the Boltzmann and Friedmann equations:

$\displaystyle\frac{\mathop{}\\!\mathrm{d}\rho_{\Phi}}{\mathop{}\\!\mathrm{d}t}+3H\rho_{\Phi}=-\Gamma_{\Phi}\rho_{\Phi},$
(17)
$\displaystyle\frac{\mathop{}\\!\mathrm{d}\rho_{R}}{\mathop{}\\!\mathrm{d}t}+4H\rho_{R}=+\Gamma_{\Phi}\rho_{\Phi},$
(18) $\displaystyle\frac{\mathop{}\\!\mathrm{d}n_{\rm
DM}}{\mathop{}\\!\mathrm{d}t}+3Hn_{\rm DM}=R(t),$ (19) $\displaystyle
3M_{P}^{2}H^{2}=\rho_{\Phi}+\rho_{R},$ (20)

where $\rho_{\Phi}$ and $\rho_{R}$ are the energy density of the inflaton
$\Phi$ and radiation, respectively. The total inflaton decay width is
parametrized by $\Gamma_{\Phi}$. The dark matter number density is represented
by $n_{\rm DM}$. Here, we assume the instantaneous thermalization of the
inflaton decay products.444Non-instantaneous reheating and/or thermalization
has been discussed in Refs. Barman  _et al._ (2021); Bernal  _et al._ (2019,
2020b); Di Marco  _et al._ (2018, 2019); Di Marco and Pradisi (2021); Elahi
_et al._ (2015); Garcia  _et al._ (2017); Garcia and Amin (2018); Harigaya
_et al._ (2014, 2019).

During the reheating epoch, the scale factor $a$ is proportional to $a\propto
T^{-8/3}$, and therefore, it is convenient to rewrite Eq. (19) as
follows555Note that we assume the absence of the radiation energy density
before the end of inflation.:

$\frac{\mathop{}\\!\mathrm{d}}{\mathop{}\\!\mathrm{d}T}\frac{n_{\rm
DM}}{T^{8}}=-\frac{8}{3}\frac{R(T)}{HT^{9}},$ (21)

where $H\simeq(2/5)\Gamma_{\Phi}(T/T_{\rm reh})^{4}$, the reheating
temperature is defined by $\rho_{\Phi}(T_{\rm reh})=\rho_{R}(T_{\rm reh})$
leading to $\Gamma_{\Phi}=(5/2)\alpha_{\rm reh}T_{\rm reh}^{2}/M_{P}$ with
$\alpha_{i}\equiv\sqrt{g_{*}(T_{i})\pi^{2}/90}$, and $g_{*}$ is the effective
degrees of freedom of $\rho_{R}$ Kaneta  _et al._ (2019)666Here, we have
assumed that the inflaton oscillation during the reheating is harmonic due to
a quadratic potential. For anharmonic inflaton oscillation cases, see Garcia
_et al._ (2020a, b)..

During the radiation domination epoch where $T<T_{\rm reh}$, Eq. (19) takes
the familiar form of

$\frac{\mathop{}\\!\mathrm{d}Y_{\rm
DM}}{\mathop{}\\!\mathrm{d}T}=-\frac{R(T)}{HT^{4}},$ (22)

where $Y_{\rm DM}\equiv n_{\rm DM}/T^{3}$, and $H=\alpha T^{2}/M_{P}$ with
$\alpha(T)\equiv\sqrt{g_{*}(T)\pi^{2}/90}$.

For a later convenience, we parametrize the reaction rate by

$R(T)\equiv\beta\frac{T^{n+6}}{\Lambda^{n+2}}$ (23)

where $\beta$ is a dimensionless coefficient. The resultant dark matter number
density depends on $n$, namely, the cases where $n<6$, $n=6$, or $n>6$. In our
case, $n$ always satisfies $n<6$, and thus we restrict ourselves to this case.
From Eq. (21), we obtain

$\frac{n_{\rm DM}(T_{\rm reh})}{T_{\rm reh}^{3}}=\frac{8}{3(6-n)}\frac{\beta
M_{P}T_{\rm reh}^{n+1}}{\alpha_{\rm reh}\Lambda^{n+2}}\equiv Y_{\rm DM}(T_{\rm
reh}),$ (24)

where we take $n_{\rm DM}(T_{\rm max})=0$ with $T_{\rm max}$ ($>T_{\rm reh}$)
being the maximal temperature. With $Y_{\rm DM}(T_{\rm reh})$, from Eq. (22),
$Y_{\rm DM}$ today is computed as

$Y_{\rm DM}=\frac{26+5n}{3(n+1)(6-n)}\frac{\beta M_{P}T_{\rm
reh}^{n+1}}{\alpha_{\rm reh}\Lambda^{n+2}},$ (25)

This expression is valid only for $n>-1$. Notice also that Eqs. (24) and (25)
rely on the fact that the reaction rate can be parametrized by a polynomial of
$T$, and thus cannot be applied to other cases, such as involving resonances.
By taking the dilution factor into account, the dark matter relic density can
be obtained by

$\Omega_{\rm DM}h^{2}\simeq 1.6\times 10^{8}Y_{\rm
DM}\left(\frac{g_{0}}{g_{*}}\right)\frac{m_{\rm DM}}{\rm GeV},$ (26)

where $g_{0}=g_{*S}(T_{0})=3.91$, and $g_{*}=g_{*S}(T_{*})$ will be taken at a
temperature $T_{*}$ where dark matter is dominantly produced.

##  IV IV. Conformally Induced Higgs Portal

From Eq. (3) and the choice of $\delta C=-\xi|H|^{2}/M_{P}^{2}$, the portal
coupling is given by

$\displaystyle{\cal L}_{\rm
portal}=-\frac{\xi}{2M_{P}^{2}}|H|^{2}g^{\mu\nu}T^{i}_{\mu\nu},$ (27)

where $i=X,\chi,V$. The relevant diagrams of the dark matter production for
each type of dark matter are shown in Fig. 1.

Figure 1: Feynman diagrams for the dark matter production in the conformal
Higgs portal scenario.

The scalar dark matter is produced via $HH^{\dagger}\to XX$, whose reaction
rate is given by

$R(T)=\beta_{0}\frac{T^{8}}{M_{P}^{4}},$ (28)

where $\beta_{0}=\xi^{2}\pi^{3}/{10800}\simeq 2.9\times 10^{-3}\xi^{2}$.

In the same way as the scalar dark matter, the reaction rate of
$HH^{\dagger}\to\chi\overline{\chi}$ is computed as

$R(T)=\beta_{1/2}\frac{m_{\chi}^{2}T^{6}}{M_{P}^{4}},$ (29)

where $\beta_{1/2}=\xi^{2}\zeta^{2}(3)/16\pi^{5}\simeq 3\times
10^{-4}\xi^{2}$. Notice that in contrast to the scalar dark matter case, the
reaction rate is suppressed by the dark matter mass because of the angular
momentum conservation, which requires the dark matter relatively massive to
explain the observed dark matter density.

For the massive vector dark matter, it is convenient to consider the
transverse and the longitudinal modes separately. The production rate of the
transverse polarization mode is given by

$R(T)=\beta_{1}^{T}\left(\frac{m_{V}}{M_{P}}\right)^{4}T^{4},$ (30)

where $\beta_{1}^{T}=\xi^{2}/{576}\pi\simeq 5.5\times 10^{-4}\xi^{2}$, leading
to the yield value

$Y_{V}^{T}\simeq\frac{\beta_{1}^{T}}{\alpha(m_{V})}\left(\frac{m_{V}}{M_{P}}\right)^{3},$
(31)

where we have assumed $m_{V}<T_{\rm reh}$.

On the other hand, the production of the longitudinal mode is readily computed
in the same manner as the scalar dark matter case, resulting in the same
expression given in Eq. (28), and thus the yield value becomes

$Y_{V}^{L}\simeq\frac{\beta_{1}^{L}}{\alpha_{\rm reh}}\left(\frac{T_{\rm
reh}}{M_{P}}\right)^{3},$ (32)

where $\beta_{1}^{L}=\beta_{0}$. Therefore, the number density of the
transverse mode becomes comparable to that of the longitudinal mode only when
$m_{V}\sim T_{\rm reh}$, otherwise the longitudinal mode is the dominant
component in the dark matter relic density, which coincides with the number
density of the scalar dark matter.

With the reaction rates computed for each type of dark matter, we obtain the
dark matter relic density from Eqs. (25) and (26) as

$\displaystyle\frac{\Omega_{\rm DM}^{\rm
scat}h^{2}}{0.1}\simeq\left(\frac{\xi}{10^{5}}\right)^{2}\left(\frac{106.75}{g_{\rm
reh}}\right)^{3/2}\left(\frac{T_{\rm reh}}{10^{11}~{}{\rm
GeV}}\right)^{3}\times\begin{cases}\left(\frac{m_{X,V}}{27~{}{\rm
PeV}}\right)&\text{ (scalar/vector dark matter)}\\\\[10.0pt]
\left(\frac{m_{\chi}}{12~{}{\rm EeV}}\right)^{3}&\text{ (fermionic dark
matter)}\end{cases},$ (33)

where $g_{i}\equiv g_{*}(T_{i})$.

###  IV.1 IV-D. Implication of Inflation

So far we have not specified the inflationary sector, whereas supposing a
concrete inflation model gives additional information on the dark matter
number density. For instance, if we consider the Higgs inflation scenario, the
nonminimal coupling is required to be $\xi\simeq 49000\sqrt{\lambda}$ and
$T_{\rm reh}\simeq(2\lambda/g_{\rm reh}\pi^{2})^{1/4}M_{P}/\sqrt{\xi}\simeq
2.3\times 10^{15}$ GeV Bezrukov and Shaposhnikov (2008), where $\lambda$ is
the Higgs quartic coupling and is taken as $\lambda\simeq 0.13$. For the
scalar (and vectorial) dark matter case, we thus obtain $m_{X}\simeq 71$ keV,
while the fermionic dark matter is predicted as $m_{\chi}\simeq 1.7$ PeV. Note
that, as we will discuss shortly, despite a large $\xi$ being required for
successful inflation, the scale where perturbative unitarity is violated is
above $T_{\rm reh}$, and dark matter is produced dominantly at $T\lesssim
T_{\rm reh}$ in our scenario. Thus the perturbative computation of the dark
matter production remains valid.

For a more generic inflation model, the dark matter production via the portal
coupling implies an upper bound on the maximal temperature ($T_{\rm max})$, so
that the cutoff scale involved should be above these temperatures. To be more
concrete, we consider a case that during the inflaton oscillation epoch the
inflaton potential can be approximated by
$V(\Phi)\simeq(1/2)m_{\Phi}^{2}\Phi^{2}$. As discussed in detail in Refs.
Garcia  _et al._ (2020a, b), by solving Eqs. (17), (18), and (20), we obtain

$\displaystyle T_{\rm max}=$ $\displaystyle 8.8\times 10^{14}~{}{\rm
GeV}\times y^{1/2}\left(\frac{106.75}{g_{*}(T_{\rm max})}\right)^{1/4}$ (34)
$\displaystyle\times\left(\frac{m_{\Phi}}{3\times 10^{13}~{}{\rm
GeV}}\right)^{1/4}\left(\frac{\rho_{\rm
end}}{0.175m_{\Phi}^{2}M_{P}^{2}}\right)^{1/8},$ $\displaystyle T_{\rm reh}=$
$\displaystyle 5.8\times 10^{14}~{}{\rm GeV}\times y\left(\frac{106.75}{g_{\rm
reh}}\right)^{1/4}$ (35) $\displaystyle\times\left(\frac{m_{\Phi}}{3\times
10^{13}~{}{\rm GeV}}\right)^{1/2},$

where we define $\Gamma_{\Phi}=(y^{2}/8\pi)m_{\Phi}$, and $\rho_{\rm end}$ is
the inflaton energy density at the end of the inflation. For instance, when
the inflaton potential is given by the Starobinsky-like potential Starobinsky
(1980), we obtain $\rho_{\rm end}=0.175m_{\Phi}^{2}M_{P}^{2}$ with
$m_{\Phi}\simeq 3\times 10^{13}~{}{\rm GeV}$ Ellis  _et al._ (2015) when the
end of the inflation is defined by
$\mathop{}\\!\mathrm{d}^{2}a/\mathop{}\\!\mathrm{d}t^{2}=0$ for the scale
factor $a$. By defining $\Lambda$ as a cutoff scale involved in the dark
matter production, we impose $T_{\rm max}\lesssim\Lambda$ to ensure the
unitarity, which results in

$\displaystyle\left(\frac{y}{1.3\times 10^{-2}}\right)^{1/2}$
$\displaystyle\lesssim$ $\displaystyle\left(\frac{\Lambda}{10^{14}~{}{\rm
GeV}}\right)\left(\frac{106.75}{g_{*}(T_{\rm
max})}\right)^{-1/4}\left(\frac{m_{\Phi}}{3\times 10^{13}~{}{\rm
GeV}}\right)^{-1/4}\left(\frac{\rho_{\rm
end}}{0.175m_{\Phi}^{2}M_{P}^{2}}\right)^{-1/8}.$ (36)

In other words, once $y$ is given within the limit of Eq. (36), the condition
$T_{\rm reh}<\Lambda$ is automatically satisfied. For instance, in the present
case, the cutoff scale is supposed to be $\Lambda\sim
M_{P}/\sqrt{\xi}=7.6\times 10^{15}(10^{5}/\xi)^{1/2}~{}{\rm GeV}$, and thus
the maximal and reheating temperatures needed to produce the right amount of
the dark matter number density can be well below the cutoff scale.

Indeed, for the scalar and vectorial dark matter, the scale in which
perturbative unitarity is violated is $\sim M_{P}/\sqrt{\xi}$, and for the
fermionic dark matter, the scale is $M_{P}^{2}/\xi m_{\chi}$. For a more
detailed discussion, see Appendix.

##  V V. Conformally induced mediator

Next, we consider the case where the conformally induced scalar $\phi$ plays a
role of a mediator: $C(\phi)=e^{-\phi/f}\simeq
1-\frac{\phi}{f}+\frac{\phi^{2}}{2f^{2}}-\cdots$. Notice that the coupling of
$\phi$ looks similar to that of the dilaton which appears as a Nambu-Goldstone
boson of spontaneously broken scale/conformal symmetry. If the scale
invariance is imposed to the theory, the trace of the energy-momentum tensor
is proportional to a mass scale related to the violation of the scale symmetry
Callan  _et al._ (1970); Coleman and Jackiw (1971). In our case, however, we
do not impose the scale invariance, and thus, $\phi$ couples to the trace of
the energy-momentum tensor which is not necessarily proportional to the
violation of the scale invariance, e.g., particle masses. For more detail,
see, for instance, Ref. Nakayama (2015). The relevant interactions are thus
given by

${\cal L}_{\rm
portal}=-\frac{1}{2}\left(\frac{\phi}{f}-\frac{\phi^{2}}{2f^{2}}+\cdots\right)g^{\mu\nu}(T^{H}_{\mu\nu}+T^{i}_{\mu\nu})$
(37)

with $i=X,\chi,V$, where the energy-momentum tensor for $H$ is given by

$\displaystyle T^{H}_{\mu\nu}=$ $\displaystyle
2(D_{\mu}H)^{\dagger}(D_{\nu}H)$ (38) $\displaystyle-
g_{\mu\nu}[(D^{\alpha}H)^{\dagger}(D_{\alpha}H)+{\cal L}_{\rm Yuk}],$
$\displaystyle{\cal L}_{\rm Yuk}\supset$ $\displaystyle-
y_{t}\overline{Q}_{L}\widetilde{H}t_{R}+{\rm h.c.},$ (39)

with omitting the Higgs potential, and $D_{\mu}$ being the covariant
derivative. Notice that the derivative term of the Higgs plays the essential
role since the relevant processes for the dark matter production take place at
high temperatures where the derivative coupling is allowed to acquire high
energies and hence enhances the reaction rates. For the derivative term in the
trace of $T^{H}_{\mu\nu}$, we write

$g^{\mu\nu}T^{H}_{\mu\nu}\supset-\sum_{i}^{4}(\partial^{\alpha}h_{i})(\partial_{\alpha}h_{i}),$
(40)

where $h_{i}$ represent the real degrees of freedom of $H$, and we have
dropped the gauge interaction terms in $D_{\mu}$.

Actual processes for the dark matter production depend on whether $\phi$ is in
the thermal bath and whether $\phi$ is lighter or heavier than the dark
matter. We summarize those cases in Tab. 1.

| thermal $\phi$ | nonthermal $\phi$
---|---|---
$m_{\phi}<2m_{\rm DM}$ | light-thermal | light-nonthermal
$m_{\phi}>2m_{\rm DM}$ | heavy-thermal | heavy-nonthermal
Table 1: Summary of four possible cases for the dark matter production.

Before discussing each case in detail, we make criteria to assess whether
$\phi$ is in the thermal bath or not. The reaction processes that bring $\phi$
into the thermal bath are the ones involving the top Yukawa coupling, shown in
Fig. 2. The reaction rate is computed as

$R_{\rm Yuk}(T)=\beta_{\rm Yuk}\frac{T^{6}}{f^{2}}$ (41)

with $\beta_{\rm Yuk}=567\zeta^{2}(3)y_{t}^{2}/256\pi^{5}\simeq 0.01$ Brax
_et al._ (2021b). In a radiation-dominated Universe, $H\propto T^{2}$ and
hence $R_{\rm Yuk}/n_{\gamma}H\propto T$ with
$n_{\gamma}=2\zeta(3)T^{3}/\pi^{2}$ being the number density of photon is
maximized at $T_{\rm reh}$. Then, evaluating the ratio at $T=T_{\rm reh}$ as

$\displaystyle\left.\frac{R_{\rm Yuk}}{n_{\gamma}H}\right|_{T=T_{\rm reh}}=$
$\displaystyle 0.3y_{t}^{2}\left(\frac{106.75}{g_{\rm reh}}\right)^{1/2}$ (42)
$\displaystyle\times\left(\frac{T_{\rm reh}}{10^{7}~{}{\rm
GeV}}\right)\left(\frac{10^{14}~{}{\rm GeV}}{f}\right)^{2},$

one can see whether $\phi$ had been thermalized (i.e., $\left.R_{\rm
Yuk}/n_{\gamma}H\right|_{T=T_{\rm reh}}>1$) or not for a given scale $f$. Note
that Eq. (42) is applicable for $T\gtrsim m_{h}$ with $m_{h}$ being the mass
of Higgs (but $y_{t}$ replaced to $y_{b}$ if $T\lesssim m_{t}$). It can also
be used to find the freeze-out temperature of $\phi$ ($T_{\rm fo}$) for a
given $f$ if $T_{\rm reh}>T_{\rm fo}$. 777The ratio should be evaluated at
$T_{\rm max}$ to be more precise, but we regard $T_{\rm reh}$ as a maximal
temperature in the following argument, since our results are not sensitive to
$T_{\rm max}$.

Figure 2: Single $\phi$ production channels. Note that in addition to these
diagrams, $t_{R}H(\overline{Q}_{L}H)\to Q_{L}\phi(\overline{t}_{R}\phi)$ also
exist.

Depending on its lifetime, the would-be abundance of $\phi$ is constrained by
BBN Kawasaki  _et al._ (2018) or CMB Slatyer and Wu (2017) even if it never
dominates the Universe. For example, if $m_{\phi}\gtrsim\mathcal{O}(10){\rm
MeV}$, dissociations of Deuterium is possible. Then, for $\tau_{\phi}\gtrsim
10^{8}{\rm sec}$, $m_{\phi}Y_{\phi}\lesssim 10^{-14}{\rm GeV}$ is required.
However, if $m_{\phi}\lesssim\mathcal{O}(1){\rm MeV}$, BBN bound is
irrelevant. Meanwhile, for $1<m_{\phi}/{\rm MeV}\lesssim 10$, the CMB
constraint on the lifetime of dark matter is $\tau_{\rm DM}\gtrsim 10^{25}{\rm
sec}$ Slatyer and Wu (2017). This bound can be interpreted as a constraint on
the abundance of $\phi$ before its decay:

$f_{\phi}\lesssim\frac{\tau_{\phi}}{\tau_{\rm DM}^{\rm bnd}}$ (43)

where $f_{\phi}$ is the would-be fractional abundance of $\phi$ relative to
the observed abundance of dark matter if the mass of $\phi$ were the same as
that of the decaying dark matter.

###  V.1 V-A. Light-Thermal $\phi$

When $\phi$ is lighter than the dark matter so that the decay into a pair of
the dark matter particles is kinematically forbidden, the main dark matter
production channel is the annihilation of either of $\phi$ or the Standard
Model Higgs (through $\phi$ exchange). In addition to the condition
$m_{\phi}<2m_{\rm DM}$, when $\phi$ is in the thermal bath, then a pair
annihilation of $\phi$ mainly produces dark matter, whose diagrams for the
scalar dark matter case are shown in Fig. 3. For the fermionic or vectorial
dark matter cases, the corresponding diagrams can be obtained by replacing $X$
with $\chi$ or $V_{\mu}$, respectively.

Figure 3: Feynman diagrams for the dark matter production processes when the
light $\phi$ is thermal.

The reaction rates for $\phi\phi\to XX/\chi\overline{\chi}$ are readily
obtained as

$\displaystyle R(T)=\begin{cases}\beta_{0}\frac{T^{8}}{f^{4}}&\text{(scalar
dark matter)}\\\ \beta_{1/2}\frac{m_{\chi}^{2}T^{6}}{f^{4}}&\text{(fermionic
dark matter)}\end{cases},$ (44)

where $\beta_{0}=\pi^{3}/19200\simeq 1.6\times 10^{-3}$ and
$\beta_{1/2}=\zeta^{2}(3)/32\pi^{5}\simeq 1.5\times 10^{-4}$. For the
vectorial dark matter case, as we see in the conformal Higgs portal scenario,
the longitudinal mode is the dominant component in the dark matter number
density, and thus the reaction rate becomes the same as the scalar dark matter
case. In the following, we thus consider only the scalar and fermionic dark
matter cases.

From Eqs. (25) and (26), the dark matter relic density can be computed as

$\displaystyle\frac{\Omega^{\rm scat}_{\rm
DM}h^{2}}{0.1}\simeq\left(\frac{106.75}{g_{\rm
reh}}\right)^{3/2}\left(\frac{10^{7}~{}{\rm
GeV}}{f}\right)^{4}\times\begin{cases}\left(\frac{T_{\rm reh}}{\rm
TeV}\right)^{3}\left(\frac{m_{X}}{{150}~{}{\rm keV}}\right)&\text{(scalar dark
matter)}\\\\[10.0pt] \left(\frac{T_{\rm reh}}{\rm
TeV}\right)\left(\frac{m_{\chi}}{10~{}{\rm GeV}}\right)^{3}&\text{(fermionic
dark matter)}\end{cases},$ (45)

where we have assumed $m_{\phi}\ll m_{X,\chi}$. For the scale of $f$ used in
Eq. (45), decoupling of $\phi$ from thermal bath takes place at $T\sim m_{h}$
thanks to bottom quark replacing top in Fig. 2. In this case, even if $\phi$
is lighter than ${\rm MeV}$ scale, its contribution to $\Delta N_{\rm eff}$
which is interpreted as extra neutrino contributions is negligible. However,
if $m_{\phi}<2m_{e}$ with $m_{e}$ being the mass of electron, $\phi$ can decay
dominantly to photons via 1-loop diagram. Such late decays of $\phi$ can cause
an era of matter-domination during the epoch of BBN, ruining it. In order to
avoid such a danger, we assume $m_{\phi}>2m_{e}$ in this case of light thermal
$\phi$.888Note that if $m_{\phi}\ll m_{\nu}$, such a light $\phi$ can not be
constrained by BBN and/or CMB. Instead, one may consider fifth-force searches
for such a long distance effect. (See, for instance, Refs. Adelberger  _et
al._ (2003); Kapner  _et al._ (2007).)

To make sure that the dark matter stays nonthermal, we compare the reaction
rate and the Hubble:

$\displaystyle\left.\frac{R}{n_{\gamma}H}\right|_{T=T_{\rm reh}}=$
$\displaystyle\left(\frac{106.75}{g_{\rm
reh}}\right)^{1/2}\left(\frac{10^{7}~{}{\rm
GeV}}{f}\right)^{4}\times\begin{cases}3.4\times 10^{-4}\left(\frac{T_{\rm
reh}}{\rm TeV}\right)^{3}\left(\frac{T_{\rm max}}{T_{\rm
reh}}\right)&\text{(scalar dark matter)}\\\\[10.0pt] 8.5\times
10^{-10}\left(\frac{T_{\rm reh}}{\rm
TeV}\right)\left(\frac{m_{\chi}}{10~{}{\rm GeV}}\right)^{2}&\text{(fermionic
dark matter)}\end{cases},$ (46)

from which it is evident that the dark matter abundance we obtained is
consistent with the nonthermal dark matter assumption. Notice that the scalar
dark matter case depends on $T_{\rm max}$ since $R\propto T^{8}$ while
$n_{\gamma}H\propto T^{7}$ for $T_{\rm reh}<T<T_{\rm max}$, and thus the ratio
is sensitive to $T_{\rm max}$. This is, however, not the case for the
fermionic dark matter case as $R$ is suppressed by $m_{\chi}^{2}$.
Nevertheless, the dark matter is dominantly produced at $T_{\rm reh}$.
Therefore, as long as $T_{\rm reh}\gtrsim m_{h}$, $\phi$ is in thermal
equilibrium, while the dark matter stays nonthermal. Note that in this case a
low scale inflation scenario may be preferred to avoid unitarity violation at
$T_{\rm max}$ since from Eqs. (34) and (35) the ratio $T_{\rm max}/T_{\rm
reh}$ increases for lower $T_{\rm reh}$. A more detail on the unitarity bound
in the conformal mediator scenario is discussed in Appendix.

###  V.2 V-B. Light-Nonthermal $\phi$

When supposing $\phi$ is light and nonthermal, the dark matter is produced via
the s-channel $HH^{\dagger}\to\phi\to XX/\chi\overline{\chi}$ processes whose
diagrams are shown in Fig. 4.

Figure 4: Feynman diagrams for the dark matter production processes for each
type of dark matter when $\phi$ is nonthermal and $m_{\phi}<2m_{\rm DM}$.

In the similar manner, we can readily compute the reaction rates whose
analytic expressions are the same as Eq. (44) but
$\beta_{0}=\pi^{3}/{43200}\simeq 7.2\times 10^{-4}$ and
$\beta_{1/2}=\zeta^{2}(3)/32\pi^{5}\simeq 1.5\times 10^{-4}$. Thus, from Eqs.
(25) and (26) with the reaction rate for each type of dark matter, we obtain

$\frac{\Omega_{\rm DM}^{\rm scat}h^{2}}{0.1}\simeq\left(\frac{106.75}{g_{\rm
reh}}\right)^{3/2}\left(\frac{M_{P}}{f}\right)^{4}\begin{cases}\left(\frac{T_{\rm
reh}}{10^{14}~{}{\rm GeV}}\right)^{3}\times\left(\frac{m_{X}}{{1.1}~{}{\rm
EeV}}\right)&\text{(scalar dark matter)}\\\\[10.0pt] \left(\frac{T_{\rm
reh}}{10^{14}~{}{\rm GeV}}\right)\times\left(\frac{m_{\chi}}{{3.3}\times
10^{12}~{}{\rm GeV}}\right)^{3}&\text{(fermionic dark matter)}\end{cases}.$
(47)

Conservatively speaking, even if the coherent mode is suppressed somehow, the
gravitational population of $\phi$ particles at the end of inflation is
expected to be $n_{\phi}^{\rm e}\sim\mathcal{C}H_{e}^{3}$ with
$\mathcal{C}=\mathcal{O}(10^{-3})$ as long as $m_{\phi}\lesssim H_{e}$ Ema
_et al._ (2018). Hence, the would-be abundance of $\phi$ before its decay well
after reheating is

$\displaystyle Y_{\phi}$ $\displaystyle\sim$
$\displaystyle\mathcal{C}\frac{T_{\rm reh}H_{\rm e}}{M_{\rm P}^{2}}$ (48)
$\displaystyle\sim$ $\displaystyle 10^{-20}\times\left(\frac{T_{\rm
reh}}{10^{7}{\rm GeV}}\right)\left(\frac{H_{\rm e}}{10^{13}{\rm GeV}}\right)$

where we used $\mathcal{C}=10^{-3}$ in the second line. It can be subject to
either BBN or CMB constraint.

###  V.3 V-C. Heavy-Thermal $\phi$

When $m_{\phi}>2m_{\rm DM}$, the dark matter is produced by the decay of
$\phi$, instead of the scattering of $\phi$ or the Standard Model Higgs. The
reaction rate can be written as

$\displaystyle R_{\rm D}(T)=\frac{m_{\phi}^{3}\Gamma_{\phi}{\rm Br}_{\rm
DM}}{2\pi^{2}}\int_{1}^{\infty}\frac{\sqrt{x^{2}-1}}{e^{zx}-1}dx,$ (49)

where $z\equiv m_{\phi}/T$, $\Gamma_{\phi}$ is the total decay width of
$\phi$, and the branching ratio to a pair of the dark matter is defined as

${\rm Br}_{\rm DM}\equiv\frac{2\Gamma_{\phi\to{\rm DM}}}{\Gamma_{\phi}}$ (50)

with a factor of 2 in the numerator being the number of dark matter particles
produced per decay. For simplicity, we suppose that $\phi$ is heavier than the
Standard Model Higgs, and thus the main decay channel is into a pair of the
Higgs bosons:

$\Gamma_{\phi\to HH^{\dagger}}\simeq\frac{m_{\phi}^{3}}{32\pi f^{2}}.$ (51)

Here, note that $\phi$ decays into 4 degrees of freedom of $H$ as long as the
decay happens before electroweak symmetry breaking. The decay width of $\phi$
into a pair of the scalar dark matters is estimated as $\Gamma_{\phi\to
XX}\simeq(1/4)\Gamma_{\phi\to HH^{\dagger}}$ where a factor of 1/4 arises from
the difference of the degrees of freedom of the final state. In the vectorial
dark matter case, the dominant component in the $\phi$ decay is the
longitudinal mode, and thus its decay width is the same as the scalar dark
matter case. For the fermionic dark matter, the decay width is given by
$\Gamma_{\phi\to\chi\overline{\chi}}\simeq m_{\chi}^{2}m_{\phi}/32\pi f^{2}$.

There are three scales characterizing the dark matter production: $m_{\phi}$,
$T_{\rm d}$, and $T_{\rm fo}$ with $T_{\rm d}$ and $T_{\rm fo}$ being the
temperature at which $\phi$ decays and the freeze-out temperature of $\phi$,
respectively. From Eq. (51) the decay temperature can be computed as

$H(T_{\rm d})=\Gamma_{\phi\to HH^{\dagger}}\Rightarrow T_{\rm
d}=\sqrt{\frac{M_{P}m_{\phi}^{3}}{32\pi\alpha_{\rm d}f^{2}}}.$ (52)

The freeze-out temperature of $\phi$ is readily obtained from Eq. (42) as

$\left.\frac{R_{\rm Yuk}}{n_{\gamma}H}\right|_{T=T_{\rm fo}}=1\Rightarrow
T_{\rm fo}=\frac{2\zeta(3)}{\pi^{2}}\frac{\alpha_{\rm fo}f^{2}}{\beta_{\rm
Yuk}M_{P}},$ (53)

leading to

$T_{\rm d}:T_{\rm fo}:m_{\phi}=(32\pi\alpha_{\rm
d})^{1/2}x:\frac{2\zeta(3)\alpha_{\rm fo}}{\pi^{2}\beta_{\rm Yuk}}x^{-2}:1$
(54)

with $x\equiv\sqrt{M_{P}m_{\phi}}/f$. It is important to notice that for
$x<(2\zeta(3)\alpha_{\rm fo}/\pi^{2}\beta_{\rm Yuk})^{1/2}(\simeq 9.1$ if
$g_{\rm fo}=106.75$), $\phi$ decouples before its number density $n_{\phi}$
receives exponential suppression, namely, $T_{\rm fo}>m_{\phi}$. In the same
manner, the region given by $x>(2\zeta(3)\alpha_{\rm fo}/\pi^{2}\beta_{\rm
Yuk})^{1/2}$ corresponds to $T_{\rm fo}<m_{\phi}$. These two cases are
distinctive for the dark matter production, in particular, for the production
dominated in the IR regime.

In either case, however, there also exists the dark matter production
dominated in the UV regime, namely, the scattering contribution from the
thermal $\phi$ at $T_{\rm reh}$. Such contribution is the same as given by Eq.
(45). As will be shown in the following, the scattering contribution becomes
in general non-negligible for larger $T_{\rm reh}$ and/or smaller $f$.

When $T_{\rm fo}>m_{\phi}$, the dominant contribution in the IR comes from the
decay of $\phi$ after the decoupling. In this case, the decay happens at
$T_{\rm d}<T_{\rm fo}$, and thus the number of produced $\phi$ is redshifted
by then. The number density of $\phi$ at $T_{\rm fo}$ is given by
$n_{\phi}(T_{\rm fo})=(\zeta(3)/\pi^{2})T_{\rm fo}^{3}$, and thus we obtain

$Y_{\rm DM}=\frac{n_{\rm DM}(T_{\rm d})}{T_{\rm d}^{3}}={\rm Br}_{\rm
DM}n_{\phi}(T_{\rm d})={\rm Br}_{\rm DM}\frac{g_{\rm d}}{g_{\rm
fo}}\frac{\zeta(3)}{\pi^{3}}.$ (55)

Therefore, including the scattering contribution, we obtain

$\displaystyle\frac{\Omega_{\rm DM}h^{2}}{0.1}\simeq$
$\displaystyle\left(\frac{106.75}{g_{\rm
fo}}\right)\begin{cases}\left(\frac{m_{X}}{14~{}{\rm MeV}}\right)\\\\[10.0pt]
\left(\frac{\rm TeV}{m_{\phi}}\right)^{2}\left(\frac{m_{\chi}}{19~{}{\rm
GeV}}\right)^{3}\end{cases}+\left(\frac{106.75}{g_{\rm
reh}}\right)^{3/2}\left(\frac{10^{10}~{}{\rm
GeV}}{f}\right)^{4}\times\begin{cases}\left(\frac{T_{\rm reh}}{10^{5}~{}{\rm
GeV}}\right)^{3}\left(\frac{m_{X}}{{330}~{}{\rm GeV}}\right)\\\\[10.0pt]
\left(\frac{T_{\rm reh}}{10^{5}~{}{\rm GeV}}\right)\left(\frac{m_{\chi}}{{\rm
TeV}}\right)^{3}\end{cases},$ (56)

where the second term corresponds to the scattering contribution, and the top
and bottom lines in each term correspond to the scalar and fermionic dark
matter cases, respectively.

When $T_{\rm fo}<m_{\phi}$, the out-of-equilibrium decay of $\phi$ does not
contribute as $n_{\phi}(T)$ is exponentially suppressed at $T=T_{\rm fo}$.
Instead, with $T_{\rm d}>T_{\rm fo}$, the freeze-in from the on-shell $\phi$
in the thermal bath may produce the dark matter, whose reaction rate is given
by Eq. (49). In particular, the dark matter is dominantly produced when $\phi$
is nonrelativistic, and thus we may use the Maxwell-Boltzmann distribution for
$\phi$, instead of the Bose-Einstein distribution, resulting in $R_{\rm
D}\simeq(m_{\phi}^{2}T\Gamma_{\phi}{\rm Br}_{\rm
DM}/2\pi^{2})K_{1}(m_{\phi}/T)$ with $K_{1}$ being the modified Bessel
function of the second kind. The abundance of dark matter can be obtained from
Eq. (22) with $R_{\rm D}$. Integrating the equation for $0<T<\infty$, we
obtain

$Y^{\rm dec}_{\rm DM}=\frac{3}{4\pi}\frac{M_{P}\Gamma_{\phi}{\rm Br}_{\rm
DM}}{\alpha(m_{\phi})m_{\phi}^{2}}.$ (57)

It is dominated by the contribution for $1\lesssim m_{\phi}/T\lesssim 10$.
Hence, as long as $T_{\rm fo}\ll m_{\phi}$, Eq. (57) provides a good enough
approximation for the abundance of dark matter. For the scalar dark matter, we
obtain

$\frac{\Omega_{X}^{\rm
dec}h^{2}}{0.1}\simeq\left(\frac{106.75}{g_{*}(m_{\phi})}\right)^{2}x^{4}\left(\frac{m_{X}}{3.3~{}{\rm
keV}}\right),$ (58)

where by noticing that $x>9.1$ (assuming $g_{*}(m_{\phi})=106.75$) and
$m_{X}\lesssim 10$ keV is excluded by the Lyman-$\alpha$ bound Narayanan  _et
al._ (2000); Viel  _et al._ (2005, 2013); Baur  _et al._ (2016); Iršič  _et
al._ (2017); Palanque-Delabrouille  _et al._ (2020); Garzilli  _et al._
(2019); Ballesteros  _et al._ (2021), we may conclude that the scalar dark
matter scenario is not viable in this parameter space due to the
overproduction of the dark matter. For the fermionic dark matter, we obtain

$\frac{\Omega^{\rm
dec}_{\chi}h^{2}}{0.1}\simeq\left(\frac{106.75}{g_{*}(m_{\phi})}\right)^{2}\left(\frac{x}{10}\right)^{4}\left(\frac{\rm
TeV}{m_{\phi}}\right)^{2}\left(\frac{m_{\chi}}{44~{}{\rm MeV}}\right)^{3}.$
(59)

Note that in addition to the decay contribution, there also exists the
scattering contribution as was the case when $T_{\rm fo}>m_{\phi}$.

###  V.4 V-D. Heavy-Nonthermal $\phi$

When $\phi$ is nonthermal, it is dominantly produced through the Yukawa
interactions, whose reaction rate is given by Eq. (41). Therefore, the yield
value of $\phi$ is readily computed from Eq. (25) as

$Y^{\rm Yuk}_{\phi}=\frac{\beta_{\rm Yuk}M_{P}T_{\rm reh}}{\alpha_{\rm
reh}f^{2}}.$ (60)

In addition to the Yukawa interaction contributions, the inverse decay
process, $HH^{\dagger}\to\phi$, may produce a single $\phi$ as well. Notice
that such a process is possible only when $\phi$ is heavier than the Higgs
boson. When $\phi$ is sufficiently heavier than the Standard Model Higgs, one
may take the massless limit for the Higgs. Under this setup, the reaction rate
is readily obtained from Eq. (14) as

$R_{\rm ID}(T)\simeq\frac{m_{\phi}^{6}}{32\pi^{3}f^{2}}I(z),$ (61)

where $I(z)$ is defined as

$\displaystyle I(z)=$
$\displaystyle\int_{1}^{\infty}\mathop{}\\!\mathrm{d}x_{+}\int_{\sqrt{x_{+}^{2}-1}}^{\sqrt{x_{+}^{2}-1}}\mathop{}\\!\mathrm{d}x_{-}$
(62)
$\displaystyle\times\left\\{\frac{1}{e^{\frac{z}{2}(x_{+}+x_{-})}-1}\frac{1}{e^{\frac{z}{2}(x_{+}-x_{-})}-1}\right\\}.$

To a good approximation, we may use

$\displaystyle I(z)\simeq
K_{5/2}(z)\simeq\begin{cases}3\sqrt{\frac{\pi}{2}}\frac{1}{z^{5/2}}\qquad(z\ll
1)\\\\[10.0pt] \sqrt{\frac{\pi}{2}}\frac{1}{z^{1/2}}e^{-z}\qquad(z\gtrsim
1)\end{cases},$ (63)

where $K_{5/2}(z)$ is the modified Bessel function of the second kind.
Therefore, for $T\gg m_{\phi}$, we obtain

$R_{\rm ID}(T)\simeq\frac{3m_{\phi}^{7/2}T^{5/2}}{32\sqrt{2}\pi^{5/2}f^{2}},$
(64)

resulting in

$Y^{\rm
ID}_{\phi}\simeq\frac{3M_{P}m_{\phi}}{80\sqrt{2}\alpha(m_{\phi})\pi^{5/2}f^{2}}.$
(65)

On the other hand, by comparing $Y^{\rm Yuk}_{\phi}$ and $Y^{\rm ID}_{\phi}$,
we find

$\frac{Y^{\rm ID}_{\phi}}{Y^{\rm Yuk}_{\phi}}\simeq 0.02\left(\frac{g_{\rm
reh}}{g_{*}(m_{\phi})}\right)^{3/2}\left(\frac{m_{\phi}}{T_{\rm reh}}\right),$
(66)

and thus, the inverse decay contribution turns out to be negligible in our
case. Therefore, the resultant dark matter relic abundance becomes

$\displaystyle\frac{\Omega^{\rm dec}_{\rm DM}h^{2}}{0.1}\simeq$
$\displaystyle\left(\frac{106.75}{g_{\rm
reh}}\right)^{3/2}\left(\frac{M_{P}}{f}\right)$ (67)
$\displaystyle\times\left(\frac{T_{\rm reh}}{10^{10}~{}{\rm
GeV}}\right)\left(\frac{{\rm Br}_{\rm DM}m_{\rm DM}}{140~{}{\rm GeV}}\right).$

##  VI VI. Phenomenology

When the mediator is lighter than the dark matter, direct detection
experiments may have sensitivity to the parameter spaces with relatively low
reheating temperature. The direct detection constraint on the dark matter-
nucleon scattering cross section usually assumes an effective contact
interaction. On the other hand, if the mediator mass is sufficiently light,
such as $m_{\phi}\lesssim 40$ MeV, the nuclear recoil through the light
mediator exchange can mimic the contact interaction event, which allows us to
look for the FIMP dark matter through the light mediator Fornengo  _et al._
(2011); Hambye  _et al._ (2018). Our analysis closely follows Ref. Hambye  _et
al._ (2018). The relevant formulas and parameters are summarized in Appendix.

The strategy to give a constraint by the XENON1T data Aprile  _et al._ (2018)
is the following. We use the experimental constraint on the dark matter-
nucleon cross section $\sigma_{\rm DM-n}$ to evaluate the differential recoil
rate given in Eq. (102). For the dark matter-nucleus differential cross
section in our case, we obtain

$\displaystyle b(q)=\frac{m_{N}m_{\rm DM}^{2}[Zm_{p}+(A-Z)m_{n}]^{2}}{32\pi
f^{4}(q^{2}+m_{\phi}^{2})^{2}}F^{2}(q),$ (68)

where $b(q)$ is defined in Eq. (101) with $q=\sqrt{2m_{N}E_{R}}$, and $m_{p}$
and $m_{n}$ are the proton and neutron masses, respectively. Note that for
both scalar and fermionic dark matter cases the expression for $b(q)$ becomes
the same. With this we can compute
$\mathop{}\\!\mathrm{d}R/\mathop{}\\!\mathrm{d}E_{R}$ for both experimental
input and theoretical prediction. Using the efficiency factor
$\epsilon(E_{R})$ taken from Ref. Aprile  _et al._ (2018), we minimize

$\displaystyle\Delta^{2}_{\rm DR}\equiv\frac{1}{R_{\rm
exp}^{2}}\int\mathop{}\\!\mathrm{d}E_{R}\epsilon^{2}(E_{R})\left[\frac{\mathop{}\\!\mathrm{d}R_{\rm
exp}}{\mathop{}\\!\mathrm{d}E_{R}}-\frac{\mathop{}\\!\mathrm{d}R_{\rm
th}}{\mathop{}\\!\mathrm{d}E_{R}}\right]^{2}$ (69)

by taking $m_{\rm DM}$ and $f$ as free parameters, where $R_{\rm exp}$ and
$R_{\rm th}$ are the recoil rates evaluated by XENON1T data and by Eqs. (68)
and (102), respectively.

Figure 5: Constraints from XENON1T. The black solid lines correspond to the
scalar dark matter with the light-thermal mediator, while the black dashed
lines show the fermionic dark matter case with the light-thermal mediator. The
red long-dashed line is the projection of the LZ experiment with 1000 days of
exposure.

Figure 5 shows the result of the direct detection constraint given by using
XENON1T data. The black solid lines depict the correct dark matter relic
density for the scenario with the scalar dark matter and the thermal $\phi$,
where we take $m_{\phi}=0$ to a good approximation up to $m_{\phi}\lesssim 40$
MeV. Note that our analysis does not apply for a heavier $\phi$, since the
recoil energy distribution is largely deformed by the effect of the nonzero
mass of $\phi$, and the direct detection experiments start losing the
sensitivity because of the absence of the enhancement in the cross section
from the light mediator. The black dashed lines are the case of the fermionic
dark matter with fixed $T_{\rm reh}$, while the black solid lines correspond
to the scalar dark matter case. Note that the vectorial dark matter is the
same as the scalar dark matter in the figure. The LZ projection with 1000 days
exposure is shown by the red dense-dashed line, where the data is taken from
Ref. Akerib  _et al._ (2020). Notice that the direct detection constraints on
$f$ scale as $m_{\rm DM}^{1/4}$ since the recoil rate is proportional to
$m_{\rm DM}/f^{4}$.

Before concluding the section, we consider possible cosmological limits on
such a light $\phi$. As mentioned above, our direct detection constraint
applies only when $m_{\phi}\lesssim 40$ MeV with which a dominant decay
channel is into a pair of electrons. By using the decay width $\Gamma_{\phi\to
ee}\simeq m_{e}^{2}m_{\phi}/(32\pi f^{2})$, the decay temperature is obtained
as

$T_{\rm d}=\sqrt{\frac{\Gamma_{\phi\to ee}M_{P}}{\alpha_{\rm d}}}.$ (70)

On the other hand, if $T_{\rm fo}\gtrsim m_{\phi}$, the energy density of the
produced $\phi$ dominates over the radiation at the temperature $T_{\rm
dom}=30\zeta(3)m_{\phi}/g_{\rm fo}\pi^{4}$. Requiring that $\phi$ should decay
before it dominates in the energy density, we obtain a limit given by

$\displaystyle T_{\rm dom}<T_{\rm d}\Rightarrow f$ $\displaystyle<$
$\displaystyle 3.8\times 10^{8}~{}{\rm GeV}\left(\frac{g_{\rm
fo}}{106.75}\right)$ (71) $\displaystyle\times\left(\frac{10.75}{g_{\rm
d}}\right)^{1/2}\left(\frac{10~{}{\rm MeV}}{m_{\phi}}\right)^{1/2}.$

Furthermore, $T_{\rm d}$ should also be higher than the BBN temperature
($T_{\rm BBN}\sim 1$ MeV), which leads to a constraint given by

$f\lesssim 10^{7}~{}{\rm GeV}\left(\frac{10.75}{g_{\rm
d}}\right)^{1/2}\left(\frac{m_{\phi}}{10~{}{\rm MeV}}\right)^{1/2}.$ (72)

Therefore, the region of $f\gtrsim 10^{7}$ GeV in Fig. 5 may conflict with
these constraints which can, however, be easily avoided if $m_{\phi}$ is
heavier. Nevertheless, in such a case, our direct detection constraint needs
to be replaced by the one based on a contact interaction approximation between
dark matter and the nucleus.

##  VII VII. Conclusion

In this work, a new portal coupling to dark matter is discussed, where the
portal sector is introduced through a conformal factor of the space-time
metric. The introduction of such conformal factor is equivalent to considering
a scalar field that couples to the Ricci scalar in the Jordan frame, and
consequently the scalar particle may couple to the trace of the energy-
momentum tensor of the matter sector, including the dark matter sector.
Supposing that the dark matter sector is secluded from the Standard Model
sector, we consider two examples of choosing the conformal factor: conformal
Higgs portal and conformally induced mediator portal.

In the conformal Higgs portal scenario, we show that the well-known Higgs
nonminimal coupling can produce the secluded dark matter via freeze-in. We
emphasize that this scenario is consistent with the Higgs inflation, and a
sharp prediction can be obtained on the dark matter mass: 71 keV for the
scalar (vector) dark matter and 1.7 PeV for the fermionic dark matter.

When the scalar field, $\phi$, induced through the conformal factor plays a
role of a mediator propagating between the dark matter sector and the Standard
Model, new dark matter production channels are allowed, depending on whether
or not $\phi$ is in the thermal bath, and whether or not the decay of $\phi$
into a pair of dark matter is kinematically allowed. In particular, we show
that when $\phi$ is light and thermalized, the dark matter direct detection
experiments can test the model. The CMB and BBN constraints on the late time
decay of $\phi$ are also discussed.

Acknowledgements: The work was supported by a KIAS Individual Grant No.
PG080301 (KK) and No. PG021403 (PK) at Korea Institute for Advanced Study, by
National Research Foundation of Korea (NRF) Grant No. NRF-2019R1A2C3005009
(PK), funded by the Korea government (MSIT), by Research Base Construction
Fund Support Program (WIP) funded by Jeonbuk National University in 2021, and
by Basic Science Research Program through the National Research Foundation of
Korea (NRF) funded by the Ministry of Education (No. 2017R1D1A1B06035959)
(WIP).

##  Appendix A APPENDIX

###  A.1 A. Conformally Introduced Scalar Sector

We here give an explicit correspondence between the conformally coupled scalar
sector in the Jordan frame and that in the Einstein frame. In the Jordan
frame, the action of the gravity and matter sectors is given by

$\displaystyle S=S_{\rm grav}+S_{\rm matt},$ (73) $\displaystyle S_{\rm
grav}=\frac{M_{P}^{2}}{2}\int\mathop{}\\!\mathrm{d}^{4}x\sqrt{-\tilde{g}}C(\phi)\tilde{R},$
(74) $\displaystyle S_{\rm
matt}=\int\mathop{}\\!\mathrm{d}^{4}x\sqrt{-\tilde{g}}{\cal L}_{\rm matt},$
(75)

where we defined $\tilde{g}\equiv{\rm det}(\tilde{g}_{\mu\nu})$ and
$\tilde{R}$ as the determinant of the metric and the Ricci scalar in the
Jordan frame, respectively. The metric defined in the Jordan frame,
$\tilde{g}_{\mu\nu}$, and that in the Einstein frame, $g_{\mu\nu}$, are
related with $\tilde{g}_{\mu\nu}=C^{-1}g_{\mu\nu}$, called the Weyl rescaling,
such that $S_{\rm grav}$ becomes the Einstein-Hilbert action. Indeed, under
the rescaling, the Ricci scalar transforms as

$\displaystyle C^{-1}\tilde{R}=$ $\displaystyle
R+\frac{(D-1)(D-2)}{4}g^{\mu\nu}(\nabla_{\mu}\ln C)(\nabla_{\nu}\ln C)$ (76)
$\displaystyle-(D-1)g^{\mu\nu}\nabla_{\mu}\nabla_{\nu}\ln C$

in $D$ dimensional space time999See, for instance, Appendix D of Wald (1984),
and thus we obtain

$\displaystyle S_{\rm
grav}=\frac{M_{P}^{2}}{2}\int\mathop{}\\!\mathrm{d}^{4}x\sqrt{-g}\left[R+\frac{3}{2}g^{\mu\nu}(\partial_{\mu}\ln
C)(\partial_{\nu}\ln C)\right],$

where the connection terms are omitted. Notice that the derivative term with
respect to $\ln C$ appears, which may play as a canonically normalized kinetic
term for $\phi$, when $C(\phi)=\exp\pm\sqrt{\frac{2}{3}}\frac{\phi}{M_{P}}$.
In this case therefore a kinetic term of $\phi$ in ${\cal L}_{\rm matt}$ is
not needed.

The relation between $\phi$ and the canonically normalized field $\chi$
depends on the choice of $C(\phi)$ and the kinetic term in ${\cal L}_{\rm
matt}$. For instance, we may define the kinetic term in ${\cal L}_{\rm matt}$
by

$\displaystyle{\cal L}_{\rm matt}\supset
M_{P}^{2}K\tilde{g}^{\mu\nu}(\partial_{\mu}C^{1/2})(\partial_{\nu}C^{1/2}),$
(78)

leading to the kinetic term in the Einstein frame given by

$\displaystyle S_{\phi,{\rm
kin}}=\frac{3+K}{4}M_{P}^{2}\int\mathop{}\\!\mathrm{d}^{4}x\sqrt{-g}g^{\mu\nu}(\partial_{\mu}\ln
C)(\partial_{\nu}\ln C),$

where $K$ is an arbitrary constant and chosen so that the kinetic term for
$\phi$ is canonically normalized. This choice gives $\chi=\phi$ when
$C(\phi)=\exp\pm\alpha\phi/M_{P}$ with $\alpha$ being an arbitrary constant,
and taking $K=2/\alpha^{2}-3$.

Another example is the case where the kinetic term is given by

${\cal L}_{\rm
matt}\supset\frac{1}{2}\tilde{g}^{\mu\nu}(\partial_{\mu}\phi)(\partial_{\nu}\phi)$
(80)

in the Jordan frame, leading to

$S_{\phi,{\rm
kin}}=\int\mathop{}\\!\mathrm{d}^{4}x\sqrt{-g}\frac{1}{2}\frac{1}{(1-\xi\phi^{2}/M_{P}^{2})^{2}}g^{\mu\nu}(\partial_{\mu}\phi)(\partial_{\nu}\phi),$
(81)

where we have taken $C(\phi)=1-\xi\phi^{2}/M_{P}^{2}$ and $\xi=1/6$, and thus
we obtain

$\displaystyle S_{\phi,{\rm
kin}}=\int\mathop{}\\!\mathrm{d}^{4}x\sqrt{-g}\frac{1}{2}g^{\mu\nu}(\partial_{\mu}\chi)(\partial_{\nu}\chi),$
(82)

where $\phi/M_{P}\equiv\sqrt{6}\tanh(\chi/\sqrt{6}M_{P})$.

In any cases of the choice of the kinetic term, the potential of $\phi$ in the
Einstein frame, $V$, is related to that in the Jordan frame, $\tilde{V}$, as
$V=\tilde{V}/C^{2}$ which we implicitly take $V=(1/2)m_{\phi}^{2}\phi^{2}$ in
the conformally induced mediator scenario in the text. Moreover, the field
displacement of $\phi$ from the origin is assumed to be zero during/at the end
of inflation, so that the coherent oscillation of $\phi$ does not come into
play in our discussion of dark matter production 101010There can be higher
order self-interaction terms of $\phi$, which lifts up $V$ so as to make
misalignment of $\phi$ small enough. .

To see the interactions among $\phi$ and the Standard Model particles, it is
convenient to consider the expansion $\tilde{g}_{\mu\nu}\simeq
g_{\mu\nu}+\delta g_{\mu\nu}$ with
$\tilde{g}_{\mu\nu}=C^{-1}g_{\mu\nu}\simeq(1-\delta
C)g_{\mu\nu}\Rightarrow\delta g_{\mu\nu}=-\delta C(\phi)g_{\mu\nu}$ where we
assume $C(\phi)$ can be expanded as $C(\phi)\simeq 1+\delta C(\phi)$. Thus,
$S_{\rm matt}$ can be expanded as

$\displaystyle S_{\rm matt}\simeq\int\mathop{}\\!\mathrm{d}^{4}x\sqrt{-g}{\cal
L}_{\rm matt}+\delta S_{\rm matt},$ (83) $\displaystyle\delta S_{\rm
matt}=\frac{1}{2}\int\mathop{}\\!\mathrm{d}^{4}x\sqrt{-g}T^{\rm
matt}_{\mu\nu}\delta g^{\mu\nu},$ (84)

where we defined

$T^{\rm matt}_{\mu\nu}=\frac{2}{\sqrt{-g}}\frac{\delta(\sqrt{-g}{\cal L}_{\rm
matt})}{\delta g^{\mu\nu}}.$ (85)

From $\tilde{g}^{\mu\nu}\simeq g^{\mu\nu}-\delta g^{\mu\nu}$ and $\delta
g^{\mu\nu}=\delta C(\phi)g^{\mu\nu}$, we obtain

$\delta S_{\rm matt}=\frac{1}{2}\int\mathop{}\\!\mathrm{d}^{4}x\sqrt{-g}\delta
C(\phi)g^{\mu\nu}T^{\rm matt}_{\mu\nu}.$ (86)

###  A.2 B. Perturbative Unitarity

Since our interest is in the reactions at high temperatures, the bound from
perturbative unitarity is one of the concerns. We here restrict ourselves to
consider the tree-level unitarity, which is given by

$|{\rm Re}~{}a_{J}(s)|<\frac{1}{2},$ (87)

where $a_{J}$ is the partial-wave amplitude for the total angular momentum
$J$, and a Mandelstam variable $s$ is the total energy2 in the CM frame.
$a_{J}(s)$ is related to the tree-level scattering amplitude ${\cal M}$
through

$\displaystyle a_{J}(s)=\frac{1}{32\pi}\int_{-1}^{1}{\cal
M}(s,\cos\theta)P_{J}(\cos\theta)\mathop{}\\!\mathrm{d}(\cos\theta),$ (88)

where $P_{J}(\cos\theta)$ is the Legendre polynomial of degree $J$, and
$\theta$ is the polar angle of the final particle reltive to the direction of
initial particles in the CM frame. We will consider ${\cal M}$ for $2\to 2$
processes with a scattering angle $\theta$ in each scenario of the conformal
Higgs portal and the conformal mediator portal.

It is convenient to give a generic formula for the tree-level unitarity bound
when the scattering amplitude can be parametrized by

${\cal M_{\rm S}}=\frac{s^{k/2}}{\Lambda^{k}}$ (89)

or

${\cal M_{\rm F}}=\frac{s^{k/2}}{\Lambda^{k/2+1}}\bar{v}(p_{2})u(p_{1}),$ (90)

where $k$ is an integer, $\Lambda$ is a cutoff scale, and the subscripts S and
F indicate scalars and fermions in the final state. For the scalar case, we
readily obtain the bound

$\sqrt{s}<(8\pi)^{1/k}\Lambda.$ (91)

For the fermion case, the only helicity amplitudes with the same helicity of
the two fermions can remain nonzero, and for each amplitude, by using
$\bar{v}(p_{2})u(p_{1})=\sqrt{s}$, we obtain

$\sqrt{s}<(8\pi)^{1/(k+1)}\Lambda.$ (92)

Since the reaction rates of our interest with the collision energy higher than
$T_{\rm max}$ is exponentially suppressed and becomes irrelevant due to the
thermal distribution function, we may take $\sqrt{s}\simeq T_{\rm max}$ when
discussing the unitarity limit. Unitarity limits in each case of dark matter
and portal scenarios are summarized in Tab. 2.

scenario | dark matter | relevant channel |  $k$ | $\Lambda$ | unitarity bound
---|---|---|---|---|---
Higgs portal | $X(V_{\mu})$ | $HH^{\dagger}\to XX$ | $1$ | $(2/\xi)^{1/2}M_{P}$ | $(16\pi/\xi)^{1/2}M_{P}$
| $\chi$ | $HH^{\dagger}\to\chi\bar{\chi}$ | $0$ | $2M_{P}^{2}/\xi m_{\chi}$ | $(16\pi/\xi)M_{P}^{2}/m_{\chi}$
light/heavy-thermal $\phi$ | $X(V_{\mu})$ | $\phi\phi\to XX$ | $1$ | $(2/\sqrt{3})f$ | $2(8\pi/3)^{1/2}f$
| $\chi$ | $\phi\phi\to\chi\bar{\chi}$ | $0$ | $2f^{2}/m_{\chi}$ | $16\pi f^{2}/m_{\chi}$
light-nonthermal $\phi$ | $X(V_{\mu})$ | $HH^{\dagger}\to XX$ | $1$ | $2f$ | $2(8\pi)^{1/2}f$
| $\chi$ | $HH^{\dagger}\to\chi\bar{\chi}$ | $0$ | $4f^{2}/m_{\chi}$ | $32\pi f^{2}/m_{\chi}$
Table 2: Summary of unitarity bounds. $k$ and $\Lambda$ are the parameters
appeared in Eqs. (89) and (90).

###  A.3 C. Basic Formulas for Direct Detection Bounds

We summarize formulas and relevant parameters used to give the constraints by
the direct detection experimental data. For a more comprehensive review, see,
for instance, Refs. Lewin and Smith (1996); Schumann (2019); Lin (2019).

Direct detection experiments look for the recoil events of dark matter
scattering off a target material, whose recoil rate $R$ is defined per unit
target mass and per unit time, and the differential recoil rate is defined by

$\frac{\mathop{}\\!\mathrm{d}R}{\mathop{}\\!\mathrm{d}E_{R}}=N_{T}\int\frac{\mathop{}\\!\mathrm{d}\sigma_{\rm
DM-N}}{\mathop{}\\!\mathrm{d}E_{R}}v\mathop{}\\!\mathrm{d}n_{\rm DM},$ (93)

where $E_{R}$ is the nucleus recoil energy, $N_{T}=N_{A}/A$ with the Avogadro
number $N_{A}=6.02\times 10^{23}$/g is the number of a nucleus of atom mass
$A$ in 1 g of substance, $\sigma_{\rm DM-N}$ is the dark matter-nucleus
scattering cross section, and $v$ is the dark matter velocity in the lab
frame.

For the local dark matter flux, we use

$\displaystyle v\mathop{}\\!\mathrm{d}n_{\rm DM}=\frac{\rho_{\rm
DM}^{0}}{m_{\rm DM}}vf(\vec{v})\mathop{}\\!\mathrm{d}^{3}\vec{v},$ (94)

where the local dark matter energy density $\rho_{\rm DM}^{0}=0.3$ GeV/cm3,
and $f(\vec{v})$ is assumed to be Maxwellian distribution:

$f(\vec{v})=N^{-1}e^{-|\vec{v}|^{2}/v_{0}^{2}}\theta(v_{\rm esc}-|\vec{v}|)$
(95)

with $v_{0}$ and $v_{\rm esc}$ being the dark matter velocity and the escape
velocity above which the dark matter fly away from the galaxy, respectively.
In our analysis, we take $v_{0}=220$ km/s and $v_{\rm esc}=550$ km/s. To
normalize as $\int f(\vec{v})\mathop{}\\!\mathrm{d}^{3}\vec{v}=1$, we may
choose the normalization factor

$\displaystyle N=\pi^{3/2}v_{0}^{3}\left[{\rm erf}\left(\frac{v_{\rm
esc}}{v_{0}}\right)-\frac{2}{\sqrt{v}_{0}}\frac{v_{\rm esc}}{v_{0}}e^{-v_{\rm
esc}^{2}/v_{0}^{2}}\right].$ (96)

Taking into account the velocity of the solar system,
$\vec{v}\to\vec{v}+\vec{v}_{S}$ with $|\vec{v}_{S}|=240$ km/s, we define

$\displaystyle g(v_{\rm min})$ $\displaystyle\equiv\int
v^{-1}f(\vec{v}+\vec{v}_{S})\mathop{}\\!\mathrm{d}^{3}\vec{v}$
$\displaystyle=\frac{\pi^{3/2}v_{0}^{3}}{2|\vec{v}_{S}|N}\left[{\rm
erf}\left(\frac{v_{\rm min}+|\vec{v}_{S}|}{v_{0}}\right)-{\rm
erf}\left(\frac{v_{\rm min}-|\vec{v}_{S}|}{v_{0}}\right)\right],$

where $v_{\rm min}\equiv(m_{N}E_{R}/2\mu_{N}^{2})^{1/2}$ with $m_{N}$ and
$\mu_{N}$ being the nucleus mass and the reduced mass of the dark matter and
nucleus, respectively.

The differential cross section for the dark matter-nucleus scattering in the
lab frame is given by

$\frac{\mathop{}\\!\mathrm{d}\sigma_{\rm
DM-N}}{\mathop{}\\!\mathrm{d}E_{R}}=\frac{\overline{|{\cal M}|^{2}}}{32\pi
m_{\rm DM}^{2}m_{N}v^{2}}|F(q)|^{2},$ (98)

where $\overline{|{\cal M}|^{2}}$ is the spin-averaged squared amplitude, and
$F(q)$ with $q=\sqrt{2m_{N}E_{R}}$ is the nuclear form factor which we take
Helm (1956)

$\displaystyle F(q)=3\frac{j_{1}(qr_{N})}{qr_{N}}e^{-(qs)^{2}/2}$ (99)

with $j_{1}$ being the spherical Bessel function of the first kind, and to a
good approximation $r_{N}^{2}=c^{2}+(7/3)\pi^{2}a^{2}-5s^{2}$, $c\simeq
1.23A^{1/3}-0.6$ fm, $a\simeq 0.52$ fm, and $s\simeq 0.9$ fm Lewin and Smith
(1996). The direct detection experiments, such as XENON1T Aprile  _et al._
(2018), put a bound on the dark matter-nucleon scattering cross section
$\sigma_{\rm DM-n}$ which is related to $\sigma_{\rm DM-N}$ by

$\frac{\mathop{}\\!\mathrm{d}\sigma_{\rm
DM-N}}{\mathop{}\\!\mathrm{d}E_{R}}=\frac{m_{N}}{2\mu_{n}^{2}}\frac{A^{2}}{v^{2}}\sigma_{\rm
DM-n}|F(q)|^{2},$ (100)

where $\mu_{n}$ is the reduced mass of the dark matter and nucleon. To
simplify the expression, we define

$\frac{\mathop{}\\!\mathrm{d}\sigma_{\rm
DM-N}}{\mathop{}\\!\mathrm{d}E_{R}}\equiv\frac{b(E_{R})}{v^{2}},$ (101)

which allows to write

$\frac{\mathop{}\\!\mathrm{d}R}{\mathop{}\\!\mathrm{d}E_{R}}=N_{T}\frac{\rho_{\rm
DM}^{0}}{m_{\rm DM}}b(E_{R})g(v_{\rm min}).$ (102)

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