aainstitutetext: Department of Physics, LEPP, Cornell University, Ithaca, NY
14853, USAbbinstitutetext: Department of Physics, The University of Chicago,
Chicago, IL 60637 , USA ccinstitutetext: Argonne National Laboratory, Lemont,
IL 60439, USAddinstitutetext: Rudolf Peierls Centre for Theoretical Physics,
University of Oxford, Parks Rd, Oxford OX1 3PJ, United Kingdom

# Dark Matter from a Conformal Dark Sector

Sungwoo Hong a,d Gowri Kurup a and Maxim Perelstein

(22 February, 2023)

###### Abstract

We consider theories in which a dark sector is described by a Conformal Field
Theory (CFT) over a broad range of energy scales. A coupling of the dark
sector to the Standard Model breaks conformal invariance. While weak at high
energies, the breaking grows in the infrared, and at a certain energy scale
the theory enters a confined (hadronic) phase. One of the hadronic excitations
can play the role of dark matter. We study a “Conformal Freeze-In”
cosmological scenario, in which the dark sector is populated through its
interactions with the SM at temperatures when it is conformal. In this
scenario, the dark matter relic density is determined by the CFT data, such as
the dimension of the CFT operator coupled to the Standard Model. We show that
this simple and highly predictive model of dark matter is phenomenologically
viable. The observed relic density is reproduced for a variety of SM operators
(“portals”) coupled to the CFT, and the resulting models are consistent with
observational constraints. The mass of the COFI dark matter candidate is
predicted to be in the keV-MeV range.

##  1 Introduction

The microscopic nature of dark matter (DM) is one of the most pressing issues
in fundamental physics, as no known elementary particle has the right
properties to make up DM. An interesting possibility is that DM particles are
part of a “dark sector”, a set of fields that are uncharged under the Standard
Model (SM) gauge group Dienes:2022zbh ; Essig:2013lka . A dark sector may
contain its own gauge interactions and matter fields, and may indeed have a
level of complexity and structure similar to or exceeding the SM. Such dark
sectors are very natural from a theoretical point of view, and in fact are
ubiquitous in string theory constructions incorporating the SM.

As there are very few theoretical constraints on the nature of the dark
sector, it is important to explore a wide range of possibilities that may lead
to viable DM candidates. In this paper, we will study the scenario where the
dark sector possesses conformal symmetry. Conformal field theories (CFT’s) are
generic in the landscape of quantum field theories, arising whenever
renormalization group evolution has a non-trivial attractive fixed point
CFT:BigBook ; Ginsparg:1988ui ; Rychkov:2016iqz . Moreover, while CFT’s are
generally strongly-coupled and cannot be studied via perturbative techniques,
the conformal symmetry is often sufficient to make non-trivial physical
predictions in these theories. In practice, this will allow us to construct
models of dark matter in which observables such as relic density are both
calculable and differ parametrically from the prediction of any perturbative
model of the dark sector. In fact, in many cases the only input needed from
the CFT side is the two-point function of the CFT operator coupled to the SM,
which is completely determined by the dimension of this operator and conformal
invariance.

Suppose that a conformally-invariant dark sector exists, and some energy is
injected into this sector in the early universe. Conformal symmetry implies
that in the expanding universe, the energy density of the dark sector will
scale as $\rho_{\rm dark}\propto a^{-4}$, where $a$ is the scale factor. This
scaling is that of radiation, not non-relativistic matter, leading to an
immediate objection to the idea of dark matter made out of a CFT. However,
very generically, we can expect the dark sector to interact, at some level,
with the non-conformal sector containing SM.111Here, we consider non-
gravitational coupling of the SM to the CFT. Models with gravitational
coupling of the two sectors were studied in Redi:2020ffc . These interactions
necessarily lead to breaking of the conformal symmetry in the dark sector.
While the SM-CFT coupling may be perturbatively small in the UV, it grows with
decreasing energy if the interaction involves a relevant operator
(dimension$<4$) in the CFT. Eventually, the conformal symmetry is completely
broken at an IR scale $M_{\rm gap}$. Below this scale the theory enters a
“hadronic” phase, with ordinary massive particle excitations in the spectrum.
These particles can play the role of Cold Dark Matter (CDM). While the DM
today consists of “normal” particles in this scenario, it is possible that the
processes that are responsible for populating the dark sector (thus fixing the
relic density of the DM) occurred when the dark sector was in the conformal
regime.

If the SM-CFT coupling is sufficiently strong for the two sectors to come to
thermodynamic equilibrium in the early universe, a rough estimate shows that
the observed relic density of DM requires $M_{\rm gap}\sim 10-100$ eV. This
scenario would lead to hot dark matter, ruled out by observations of large-
scale structure. (It is possible to avoid this conclusion if the DM can
effectively annihilate to the SM in the hadronic phase, but in that case, the
relic density would be completely determined by the ordinary particle physics
of the hadronic phase, not the CFT.) We will therefore focus on the case when
the CFT does not come into thermal equilibrium with the SM due to weakness of
the coupling between the two sectors. We assume that the CFT sector is not
populated by inflaton decays, since otherwise the DM relic density becomes
just an initial condition with no physical origin. (A model in which such
“asymmetric reheating” is realized naturally is discussed in Ref. Chiu:2022bni
.) The interactions with the SM then provide the main mechanism for populating
the CFT sector in the early universe. Such a non-thermal production mechanism
in the case of ordinary particles is known as “freeze-in”. The scenario
studied in this paper can then be described as “conformal freeze-in (COFI)”,
the term that was first introduced in Ref. Hong:2019nwd , where we considered
a specific realization of this scenario. In this paper, we present a
systematic study of the COFI mechanism, including several possible SM
operators, or “portals”, that can couple to the CFT dark sector, as well as
effects of operator mixing. We also include an updated analysis of
astrophysical constraints from stellar cooling and other sources. We find that
the COFI scenario is very generic and can occur for any of the portals we
consider, and in many cases the resulting DM candidate is phenomenologically
viable.

The rest of the paper is organized as follows. In Section 2, we describe the
model of the dark sector and its interactions with the SM underlying our
scenario. This includes the discussion of the CFT phase, the hadronic phase
that emerges at low energies after the conformal invariance is broken, and a
possible UV completion of the CFT by a gauge theory with a strongly-
interacting Banks-Zaks fixed point. In Section 3, we describe the cosmological
evolution of the dark sector in the COFI scenario, and calculate the dark
matter relic density. The figures in this section provide a snapshot of the
parameter space in various COFI models containing a viable dark matter
candidate, along with observational and theoretical constraints on these
models. The derivation of these constraints is presented in Section 4.
Finally, we summarize and conclude in Section 5. Technical details of
calculations of relic density and stellar cooling rates are contained in the
appendix.

##  2 Theoretical Framework

We consider a theory in which a Dark Sector (i.e. a set of fields with no
direct charges under SM gauge groups) is described by a Conformal Field Theory
(CFT) across a broad range of energy scales, between the “gap scale” $M_{\rm
gap}$ in the infrared (IR), and the ultraviolet (UV) cutoff $\Lambda_{\rm
UV}\gg M_{\rm gap}$. We discuss the theory in the CFT window and its
interactions with the Standard Model (SM) in Section 2.1. We describe the
mechanism that generates the gap scale and the physics at and below that scale
in Section 2.2. For completeness, we outline a possible UV completion above
$\Lambda_{\rm UV}$ in Section 2.3, although that theory is not directly
relevant for the discussion of dark matter.

###  2.1 Conformal Dark Sector

At energy scales between $M_{\rm gap}$ and $\Lambda_{\rm UV}$, the Dark Sector
is described by a CFT. We assume that the CFT contains an operator ${\cal
O}_{{}_{\rm CFT}}$ with a scaling dimension $d<4$, i.e. a relevant operator.
Generically the CFT is strongly coupled, and $d$ need not be integer. Further,
we assume that ${\cal O}_{{}_{\rm CFT}}$ is charged under a global symmetry
${\cal G}$ (for example a discrete $\mathbb{Z}_{2}$), which forbids a
Lagrangian term of the form $c\,{\cal O}_{{}_{\rm CFT}}$. Standard Model (SM)
fields are not charged under ${\cal G}$.

We consider a coupling between the SM and the dark CFT of the form

${\cal L}_{\rm int}=\frac{\lambda_{{}_{\rm CFT}}}{\Lambda_{{}_{\rm
CFT}}^{D-4}}\,{\cal O}_{{}_{\rm SM}}{\cal O}_{{}_{\rm CFT}}\,.$ (1)

where ${\cal O}_{{}_{\rm SM}}$ is an operator made out of SM fields. Here
$\lambda_{{}_{\rm CFT}}$ is a dimensionless constant, while $\Lambda_{{}_{\rm
CFT}}$ is a mass scale. Further,

$D=d+d_{{}_{\rm SM}}\,,$ (2)

where $d_{{}_{\rm SM}}$ is the scaling dimension of ${\cal O}_{{}_{\rm SM}}$.
The interaction term (1) explicitly breaks both conformal symmetry (since the
SM is not conformal), and the global symmetry ${\cal G}$. We consider the
regime where this interaction is small enough to consider this breaking
perturbatively, and work to leading order in the interaction strength.

Since the dark sector does not carry SM gauge charges, ${\cal O}_{{}_{\rm
SM}}$ must be gauge-invariant, but there are a priori no other restrictions on
this operator. For simplicity, we assume that at tree level, there is a single
SM operator interacting with the CFT via Eq. (1). (Of course, couplings
between ${\cal O}_{{}_{\rm CFT}}$ and other SM operators will generically be
induced by quantum corrections, as discussed below.) To illustrate the range
of possibilities, we consider several possible portal operators ${\cal
O}_{{}_{\rm SM}}$, which couple the CFT to quark, lepton, and gauge sectors of
the SM. We can classify these operators into two types: Type-I operators that
acquire a non-zero vacuum expectation value (VEV) in the IR, and Type-II
operators that do not. We consider three Standard Model operators in the class
of type-I operators:

  * •

Higgs portal, $\mathbf{H^{\dagger}H}$,

  * •

Quark portal, $\mathbf{HQ_{L}^{\dagger}q_{R}}$, and

  * •

Gluon portal, $\mathbf{G^{\mu\nu}G_{\mu\nu}}$.

The Higgs portal operator gets a VEV at the weak scale, while the quark and
gluon portals get VEVs at the QCD confinement scale. Further, we consider
three examples of type-II operators:

  * •

Lepton portal $\mathbf{HL^{\dagger}\ell_{R}}$,

  * •

Weak-gauge portal $\mathbf{W^{\mu\nu}W_{\mu\nu}}$, and

  * •

Hypercharge-gauge portal $\mathbf{B^{\mu\nu}B_{\mu\nu}}$.

All our examples involve relevant or marginal SM operators, which are expected
to be dominant at low energies. Also, all operators we consider are Lorentz
scalars. For an example of a non-scalar portal, namely ${\cal O}_{{}_{\rm
SM}}=B_{\mu\nu}$ which results in a composite dark photon, see Ref. COFI_DP.

In the case of quark and lepton portals, the flavor structure of the CFT
coupling to the SM needs to be specified. We will assume that the portal
operators are flavor-diagonal in the SM mass eigenbasis. The coupling to CFT
can then be written in this basis as

${\cal L}_{\rm int}=\frac{\lambda_{{}_{\rm CFT}}}{\Lambda_{{}_{\rm
CFT}}^{D-4}}\,{\cal O}_{{}_{\rm CFT}}\,\cdot\left(\sum_{i}\kappa_{i}{\cal
O}_{\rm Yuk}^{i}\right)\,,$ (3)

where the sum runs over the six flavors of SM quarks or three flavors of
charged leptons, and ${\cal O}_{\rm Yuk}^{i}$ is the SM Yukawa operator for
each flavor. The constants $\kappa_{i}$ encode the flavor dependence of the
CFT-SM interactions. Specifically, we will consider three cases:

  * •

Minimal Flavor Violation (MFV), with entries proportional to SM Yukawas:
$\kappa_{i}=y_{i}$, $i=1\ldots 6$ for quarks and $1\ldots 3$ for charged
leptons.

  * •

Democratic, with all entries the same: $\kappa_{i}=1$.

  * •

First-Generation Only: $\kappa_{i}=1$ for the first-generation quarks or
electrons, and 0 for the second and third generations.

###  2.2 CFT Breaking in the Infrared

Since the Dark Sector CFT contains a relevant operator ${\cal O}_{{}_{\rm
CFT}}$, the generic expectation is that the conformal symmetry is broken in
the infrared (IR). Specifically, if the Lagrangian contains a term

${\cal L}\,=\,c\,{\cal O}_{{}_{\rm CFT}}\,,$ (4)

where $c$ is a constant of mass dimension $4-d>0$, the conformal symmetry is
broken at the “gap” mass scale

$M_{\rm gap}\sim c^{1/(4-d)}\,.$ (5)

Here and below, we make use of Naive Dimensional Analysis (NDA) to estimate
various quantities of interest up to order-one factors. In most cases, more
precise analytic results are not available due to the strongly-coupled nature
of the underlying theory. NDA estimates will be sufficient to establish the
basic features of the dark matter model and establish its viability. At energy
scales below $M_{\rm gap}$, the theory is no longer conformal. In this
subsection, we will first estimate the gap scale for each of the six SM
portals, and then describe the physics at low energies below $M_{\rm gap}$.

####  2.2.1 Estimates of the Gap Scale

Global symmetry ${\cal G}$ forbids the deformation (4) within the CFT itself,
and the infrared breaking of the CFT is entirely due to its interaction with
the SM, Eq. (1). For each portal operator ${\cal O}_{{}_{\rm SM}}$, there are
several distinct contributions to $M_{\rm gap}$, with the NDA estimates for
each of them summarized in Table 1. Below, we will discuss each of these
contributions.

For type-I operators, a non-zero VEV directly leads to an effective Lagrangian
of the form (4), with a coefficient

$c=\frac{\lambda_{{}_{\rm CFT}}}{\Lambda_{{}_{\rm CFT}}^{D-4}}\langle{\cal
O}_{{}_{\rm SM}}\rangle.$ (6)

An estimate of the corresponding contribution to the gap scale $M_{\rm gap}$
for each of the three type-I portals is listed in the first column of Table 1.
We refer to this contribution as “tree-level”. Note that since these are NDA-
level estimates, all QCD condensates are simply taken to be $\Lambda_{\rm
QCD}$ to the appropriate power.

Figure 1: Contributions to conformal symmetry breaking via “radiative direct”
diagrams, in the Higgs, quark/lepton and gluon/weak boson portals
respectively. Blue circles indicate CFT operator insertions.

For both type-I and type-II operators, the deformation (4) is induced by
quantum corrections. For example, the leading contributions of this type for
Higgs, quark/lepton and gluon/weak boson/hypercharge boson portals are
illustrated in Fig. 1. We refer to these contributions as “radiative direct”.
The Feynman diagrams that contribute are generally UV-divergent, and the NDA
estimates of their contributions are proportional to powers of the scale
$\Lambda_{\rm SM}$ which serves as the UV cutoff of the SM loops. The LHC
constraints generally imply $\Lambda_{\rm SM}\mathrel{\>{\hbox{\raise
1.50696pt\hbox{$>$}}\kern-3.8889pt\kern-3.8889pt\hbox{\lower
2.79857pt\hbox{$\sim$}}}\>}1$ TeV. Note that if $\Lambda_{\rm SM}\gg 4\pi v$,
the observed weak scale requires strong fine-tuning. A similar fine-tuning may
or may not occur in the SM loop contributions to (4), and the gap scale in
this scenario is strongly model-dependent. For concreteness, we will use
$\Lambda_{\rm SM}=2\pi v\sim 1.5$ TeV in the estimates of this paper. The NDA
estimates of this contribution to $M_{\rm gap}$ for each portal are collected
in the second column of Table 1.

Figure 2: Diagrams that contribute to conformal symmetry breaking via mixing
with the Higgs, in the quark/lepton portal, the gluon portal and the
electroweak boson portal respectively. Blue circles indicate CFT operator
insertions.

Quantum corrections in the SM also introduce mixing among the SM operators. In
effect, for each choice of the portal operator in Eq. (1), interactions of
${\cal O}_{{}_{\rm CFT}}$ with all other gauge-invariant SM operators are
induced, with loop-suppressed coefficients. In particular, a coupling of the
CFT to the Higgs portal operator is always generated. The leading
contributions to this coupling for lepton, quark, gluon, weak and hypercharge
portals are illustrated in Fig. 2. Below the weak scale, this coupling induces
the deformation (4). We refer to this mechanism as “radiative mixing”. The NDA
estimates of the corresponding contribution to the gap scale for each portal
are summarized in the third column of Table 1. Mixing with the other two
type-I operators is also generically present, but their effect is subdominant
since $\Lambda_{\rm QCD}\ll v$.

Another potential source of radiative breaking of conformal symmetry is the
deformation

${\cal L}\,=\,c^{\prime}\,{\cal O}_{{}_{\rm CFT}}^{2}\,,$ (7)

which can also be generated through SM loops. For example, the relevant
diagrams for each portal are shown in Fig. 3. If ${\cal O}_{{}_{\rm CFT}}^{2}$
is a relevant operator (which in the large-$N$ limit corresponds to ${\cal
O}_{{}_{\rm CFT}}$ having $d\lesssim 2$), this leads to IR breaking of the
conformal symmetry and generation of the gap scale. The NDA estimates of the
resulting contribution to the gap scale are listed in the last column of Table
1.

Figure 3: Diagrams that contribute to conformal symmetry breaking via
generation of ${\cal O}_{{}_{\rm CFT}}^{2}$, in the Higgs portal, quark/lepton
portal, and the gluon/weak boson portal respectively. Blue circles indicate
CFT operator insertions.

${\cal O}_{{}_{\rm SM}}$ |  ${\cal O}_{{}_{\rm CFT}}$ from | Loop-induced | Loop-induced | ${\cal O}_{{}_{\rm CFT}}^{2}$
---|---|---|---|---
| $\left<{\cal O}_{{}_{\rm SM}}\right>$ | ${\cal O}_{{}_{\rm CFT}}$ |  ${\cal O}_{{}_{\rm CFT}}$ from $\left<H^{\dagger}H\right>$ |
$H^{\dagger}H$ | $\left(\frac{\lambda_{{}_{\rm CFT}}}{\Lambda_{{}_{\rm CFT}}^{d-2}}v^{2}\right)^{\frac{1}{4-d}}$ | $\left(\frac{\lambda_{{}_{\rm CFT}}}{\Lambda_{{}_{\rm CFT}}^{d-2}}\,\frac{\Lambda_{{}_{\rm SM}}^{2}}{16\pi^{2}}\right)^{\frac{1}{4-d}}$ | — | $\left(\frac{\lambda_{{}_{\rm CFT}}}{\Lambda_{{}_{\rm CFT}}^{d-2}}\frac{v}{m_{h}}\right)^{\frac{1}{2-d}}$
$HQ_{L}^{\dagger}q_{R}$ | $\left(\frac{\lambda_{{}_{\rm CFT}}}{\Lambda_{{}_{\rm CFT}}^{d}}v\Lambda_{{}_{\rm QCD}}^{3}\right)^{\frac{1}{4-d}}$ | $\left(\frac{\lambda_{{}_{\rm CFT}}}{\Lambda_{{}_{\rm CFT}}^{d}}\frac{(\sum_{i}\kappa_{i}y_{q_{i}})\,v^{2}\Lambda_{{}_{\rm SM}}^{2}}{(16\pi^{2})^{2}}\right)^{\frac{1}{4-d}}$ | $\left(\frac{\lambda_{{}_{\rm CFT}}}{\Lambda_{{}_{\rm CFT}}^{d}}\frac{(\sum_{i}\kappa_{i}y_{q_{i}})\,v^{2}\Lambda_{{}_{\rm SM}}^{2}}{16\pi^{2}}\right)^{\frac{1}{4-d}}$ | $\left(\frac{\lambda_{{}_{\rm CFT}}}{\Lambda_{{}_{\rm CFT}}^{d}}\sqrt{\frac{\Lambda_{{}_{\rm SM}}^{4}}{(16\pi^{2})^{2}}+\frac{v^{4}}{16\pi^{2}}}\right)^{\frac{1}{2-d}}$
$G^{\mu\nu}G_{\mu\nu}$ | $\left(\frac{\lambda_{{}_{\rm CFT}}}{\Lambda_{{}_{\rm CFT}}^{d}}\Lambda_{{}_{\rm QCD}}^{4}\right)^{\frac{1}{4-d}}$ | $\left(\frac{\lambda_{{}_{\rm CFT}}}{\Lambda_{{}_{\rm CFT}}^{d}}\,\frac{\Lambda_{{}_{\rm SM}}^{4}}{16\pi^{2}}\right)^{\frac{1}{4-d}}$ | $\left(\frac{\lambda_{{}_{\rm CFT}}}{\Lambda_{{}_{\rm CFT}}^{d}}\frac{(\sum_{i}y^{2}_{q_{i}})\,\alpha_{\rm\scriptstyle s}\,v^{2}\Lambda_{{}_{\rm SM}}^{2}}{64\pi^{3}}\right)^{\frac{1}{4-d}}$ | $\left(\frac{\lambda_{{}_{\rm CFT}}}{\Lambda_{{}_{\rm CFT}}^{d}}\frac{\Lambda_{{}_{\rm SM}}^{2}}{4\pi}\right)^{\frac{1}{2-d}}$
$HL^{\dagger}\ell_{R}$ | — | $\left(\frac{\lambda_{{}_{\rm CFT}}}{\Lambda_{{}_{\rm CFT}}^{d}}\frac{(\sum_{i}\kappa_{i}y_{\ell_{i}})\,v^{2}\Lambda_{{}_{\rm SM}}^{2}}{(16\pi^{2})^{2}}\right)^{\frac{1}{4-d}}$ | $\left(\frac{\lambda_{{}_{\rm CFT}}}{\Lambda_{{}_{\rm CFT}}^{d}}\frac{(\sum_{i}\kappa_{i}y_{\ell_{i}})\,v^{2}\Lambda_{{}_{\rm SM}}^{2}}{16\pi^{2}}\right)^{\frac{1}{4-d}}$ | $\left(\frac{\lambda_{{}_{\rm CFT}}}{\Lambda_{{}_{\rm CFT}}^{d}}\sqrt{\frac{\Lambda_{{}_{\rm SM}}^{4}}{(16\pi^{2})^{2}}+\frac{v^{4}}{16\pi^{2}}}\right)^{\frac{1}{2-d}}$
$W^{\mu\nu}W_{\mu\nu}$ | — | $\left(\frac{\lambda_{{}_{\rm CFT}}}{\Lambda_{{}_{\rm CFT}}^{d}}\,\frac{\Lambda_{{}_{\rm SM}}^{4}}{16\pi^{2}}\right)^{\frac{1}{4-d}}$ | $\left(\frac{\lambda_{{}_{\rm CFT}}}{\Lambda_{{}_{\rm CFT}}^{d}}\frac{g^{2}}{4}\frac{v^{2}\Lambda_{{}_{\rm SM}}^{2}}{16\pi^{2}}\right)^{\frac{1}{4-d}}$ | $\left(\frac{\lambda_{{}_{\rm CFT}}}{\Lambda_{{}_{\rm CFT}}^{d}}\frac{\Lambda_{{}_{\rm SM}}^{2}}{4\pi}\right)^{\frac{1}{2-d}}$
$B^{\mu\nu}B_{\mu\nu}$ | — | $\left(\frac{\lambda_{{}_{\rm CFT}}}{\Lambda_{{}_{\rm CFT}}^{d}}\,\frac{\Lambda_{{}_{\rm SM}}^{4}}{16\pi^{2}}\right)^{\frac{1}{4-d}}$ | $\left(\frac{\lambda_{{}_{\rm CFT}}}{\Lambda_{{}_{\rm CFT}}^{d}}\frac{g^{\prime\,2}}{4}\frac{v^{2}\Lambda_{{}_{\rm SM}}^{2}}{16\pi^{2}}\right)^{\frac{1}{4-d}}$ | $\left(\frac{\lambda_{{}_{\rm CFT}}}{\Lambda_{{}_{\rm CFT}}^{d}}\frac{\Lambda_{{}_{\rm SM}}^{2}}{4\pi}\right)^{\frac{1}{2-d}}$
Table 1: Standard Model operators and corresponding mass gaps generated
through different sources. $\Lambda_{{}_{\rm SM}}$ is the SM cutoff scale.

Depending on the parameters $\lambda_{{}_{\rm CFT}}$, $\Lambda_{{}_{\rm CFT}}$
and $d$, each of the conformal symmetry-breaking contributions listed in Table
1 may be dominant. We found that in the parameter space where the models
successfully reproduce the observed dark matter relic density via freeze-in,
${\cal O}_{{}_{\rm CFT}}^{2}$ deformations are sub-leading to ${\cal
O}_{{}_{\rm CFT}}$ deformations for all operators studied here. For the Higgs
portal, the tree-level contribution to the gap scale dominates. For quark and
lepton portals, the dominant source of conformal symmetry breaking is
radiative mixing. For gauge-boson portals (gluon, weak and hypercharge), the
radiative direct contribution is dominant. Note that for the quark and gluon
portals, radiative contributions dominate over the tree-level one; this is
primarily due to the hierarchy $v\gg\Lambda_{\rm QCD}$.

####  2.2.2 Physics Below the Gap Scale

Below the conformal symmetry breaking scale $M_{\rm gap}$, the dark sector is
populated by particle-like excitations which are hadronic composite states of
the original CFT degrees of freedom.222While a hadronic phase seems generic,
another possible IR phase suggested by certain five-dimensional CFT duals is a
“gapped continuum” Cabrer:2009we . For a recent example of viable dark matter
models with gapped continuum, see Csaki:2021gfm ; Csaki:2021xpy . Predicting
the spectrum of these excitations in a given CFT requires non-perturbative
analysis, which is outside the scope of this paper. Instead, we will make a
few simple, realistic assumptions about the properties of the low-energy
theory, which will be sufficient to estimate the dark matter density and other
quantities of interest up to order-one factors.

We assume that the lightest of the CFT composite states $\chi$ is stable on
cosmological time scales. This particle plays the role of dark matter.
Stability may be due to a conserved global (discrete or continuous) symmetry
under which $\chi$ (and possibly some other CFT composites) are charged, but
SM states are all neutral. Further, as in Hong:2019nwd , we posit that the DM
particle is a pseudo-Goldstone boson (PGB) of an approximate global symmetry
spontaneously broken at $M_{\rm gap}$. In this case, $m_{{}_{\rm DM}}\ll
M_{\rm gap}$ is natural, with the DM mass dictated by the amount of explicit
symmetry breaking.This is necessary to satisfy self-interaction constraints
Markevitch:2003at ; Lin:2019uvt , as will be discussed in Section 4. Notably,
both the PGB property and a $\mathbb{Z}_{2}$ global symmetry are in fact
realized for pions in QCD, although in that case the would-be stabilizing
symmetry is anomalous leading to $\pi^{0}\to 2\gamma$ decay. (For other
examples of models with dark pion playing the role of dark matter, see e.g.
Frigerio:2012uc ; Hochberg:2014kqa .)

Note that the ratio $r=m_{{}_{\rm DM}}/M_{\rm gap}$ is a free parameter of the
theory. Phenomenologically, the value of $r$ is bounded from above by the
self-interaction bound and from below by the warm dark matter constraint
(since very light DM states can disrupt structure formation). It turns out
that these considerations restrict $r$ to a parametrically narrow range, so
that the theory remains highly predictive with respect to the DM mass and
other relevant quantities. Fig. 14 illustrates this for one of the models
studied in this paper, while Section 4 explains these constraints in detail.

In addition to $\chi$, the low-energy theory generically contains a set of
bound states with masses $\sim M_{\rm gap}$. These states will couple to
$\chi$ and mediate both DM self-interactions and its interactions with the
Standard Model. We model these couplings as

$\mathcal{L}\sim
g_{\star}\rho^{\mu}\left(\chi^{\dagger}\partial_{\mu}\chi+{\rm
h.c.}\right)\,,$ (8)

for a vector mediator $\rho^{\mu}$, and

$\mathcal{L}\sim\frac{g_{\star}}{M_{\rm
gap}}\phi\left(\partial\chi\right)^{2},$ (9)

for a scalar mediator $\phi$. The characteristic coupling can be estimated in
the large-$N$ limit as

$g_{\star}\sim\frac{4\pi}{\sqrt{N}}.$ (10)

In a generic theory (such as QCD), both vector and scalar mesons will be
present with comparable masses.

The interactions of $\chi$ with the SM are obtained by matching the
interaction Lagrangian in the CFT phase, Eq. (1), to the low-energy effective
theory. Dimensional analysis and large-$N$ arguments suggest

${\cal O}_{{}_{\rm CFT}}\longrightarrow\,\frac{M_{\rm
gap}^{d-1}}{g_{\star}}\,\phi\,,$ (11)

while contributions from $\rho^{\mu}$ and $\chi$ are subdominant. This is seen
by first noting that ${\cal O}_{{}_{\rm CFT}}$ is a scalar operator with
scaling dimension $d$. Once the CFT confines, it is expected to “interpolate”
a scalar operator made up of canonically normalized field operators of
composite states. A single trace interpolation is given by the above equation
where $\phi$ is a gauge invariant operator for a composite scalar. The factor
$M_{\rm gap}^{d-1}$ is fixed by the dimensional analysis, while the factor
$1/g_{\star}$ is determined by the large-$N$ counting. Explicitly, in the
large-$N$ limit, $\langle{\cal O}_{{}_{\rm CFT}}{\cal O}_{{}_{\rm
CFT}}\rangle\sim\frac{N}{16\pi^{2}}=\frac{1}{g_{\star}^{2}}$, suggesting that
${\cal O}_{{}_{\rm CFT}}\propto\frac{1}{g_{\star}}$. For $\rho^{\mu}$ or
$\chi$, the interpolation relation is either that of a “descendant” or multi-
trace. This is simply because ${\cal O}_{{}_{\rm
CFT}}\sim\partial_{\mu}\rho^{\mu}$ by Lorentz invariance and ${\cal
O}_{{}_{\rm CFT}}\sim(\partial\chi)^{2}$ by the shift symmetry of $\chi$. This
amounts to raising the effective dimension with more suppression by inverse
powers of $M_{\rm gap}$, rendering them subdominant in the low-energy
effective theory.

###  2.3 Ultraviolet Completion

There exists a natural UV completion of a dark-sector CFT considered above:
SU(N) gauge theories with fixed points in the infrared a la Banks-Zaks Caswell
; BanksZaks .333The UV theory may be any gauge theory with an interacting IR
fixed point. The gauge group need not be $SU(N)$ and also we do not require
the fixed point to be weakly interacting. In the UV, an operator of this gauge
theory, for example, a fermion bilinear, is coupled to the SM. At some scale
$\Lambda_{{}_{\rm CFT}}$, there is a fixed point and the UV gauge theory has a
phase transition into the (generically strongly coupled) conformal phase.
${\cal O}_{{}_{\rm CFT}}$ is the operator in the conformal phase that
corresponds to the original operator of the gauge theory. The matching for the
example of a fermion bilinear operator is,

$\displaystyle{\cal L}_{\rm UV}=\frac{\lambda_{\rm BZ}}{M_{\rm BZ}^{d_{{}_{\rm
SM}}-1}}{\cal O}_{\rm SM}\bar{\Psi}\Psi\xrightarrow{\Lambda_{{}_{\rm
CFT}}}\frac{\lambda_{{}_{\rm CFT}}}{\Lambda_{{}_{\rm CFT}}^{D-4}}\,{\cal
O}_{{}_{\rm SM}}{\cal O}_{{}_{\rm CFT}}\,\Rightarrow\lambda_{{}_{\rm
CFT}}\approx\lambda_{\rm BZ}\left(\frac{\Lambda_{{}_{\rm CFT}}}{M_{\rm
BZ}}\right)^{d_{{}_{\rm SM}}-1}\,,$ (12)

where $M_{\rm BZ}$ is the UV cutoff scale of the gauge theory, $\lambda_{\rm
BZ}$ is the coupling and $\Psi$ is a fermion in the UV. We impose
$\lambda_{\rm BZ}\sim\cal{O}$(1) as a naturalness condition in all the models
we consider in the paper. Since $d_{{}_{\rm SM}}>1$ and $\Lambda_{{}_{\rm
CFT}}<M_{\rm BZ}$ , it is natural for $\lambda_{{}_{\rm CFT}}$ to be very
small. The dark sector is never in equilibrium with the Standard Model, and
dark sector energy density is produced through the freeze-in mechanism. In the
next section, we will show that this mechanism can provide dark matter with
the observed relic density.

##  3 Cosmology and Relic Density

In this section, we outline the cosmological history of the dark sector, and
estimate the resulting dark matter relic density for the six portal operators
in Table 1. We find that each portal operator can provide a phenomenologically
viable dark matter candidate. The key features of these candidates are
summarized in Table 2. Further, Figures 5 \- 10 and 12 below illustrate the
parameter space consistent with the observed dark matter density for each
portal. Phenomenological and theoretical constraints on the model will be
discussed in detail in Section 4.

###  3.1 Cosmological History of the Dark Sector

We consider the regime where the coupling between the SM and the dark sector
is sufficiently small that the two sectors are not in thermal equilibrium at
any time. At the end of inflation, the Standard Model sector is reheated to
temperature $T_{R}$. We assume that the inflaton does not couple to the dark
sector, so that the energy in the dark sector is zero at that time. (Without
this assumption, the dark matter density receives a contribution depending on
the details of the inflaton couplings and dynamics, and the model loses
predictivity.) After reheating, SM collisions and decays can populate dark
sector states via the interaction (1). We consider the “Conformal Freeze-In”
(COFI) scenario where

$M_{\rm gap}<T_{R}<\Lambda_{\rm CFT}\,,$ (13)

so that the dark sector is described by a CFT in this epoch. This allows us to
calculate energy transfer rates using the “unparticle” approach of Georgi
Georgi:2007ek ; Grinstein:2008qk . The energy transferred to CFT quickly
thermalizes due to strong coupling among the CFT states, but the CFT
temperature $T_{{}_{\rm D}}$ always remains below the SM plasma temperature
$T_{{}_{\rm SM}}$. The transfer of energy from the SM plasma to the conformal
dark sector continues until either the SM states coupled to the CFT become
non-relativistic and drop out of equilibrium, or the SM temperature drops
below the gap scale $M_{\rm gap}$. In either case, the dark sector eventually
undergoes a confining phase transition at $T_{{}_{\rm D}}\sim M_{\rm gap}$.
The energy stored in the CFT degrees of freedom is transferred to the
particle-like bound states of the dark sector, which then rapidly (compared to
Hubble timescale) decay down to stable dark matter states. Given the small
coupling of the dark sector to the SM, such decays would typically not involve
SM states, so that essentially all of the energy stored in the CFT at the time
of the phase transition ends up in dark matter.

Quantitative predictions of dark matter relic density in the COFI scenario are
obtained as follows. Energy transfer between the SM and CFT degrees of freedom
is described by a Boltzmann equation,

$\displaystyle\frac{d\rho_{{}_{\rm SM}}}{dt}+3H(\rho_{{}_{\rm SM}}+P_{{}_{\rm
SM}})=-\Gamma_{E}({\rm SM}\rightarrow{\rm CFT}),$ (14)

where $H$ is the Hubble expansion rate, $\rho_{{}_{\rm SM}}$ and $P_{{}_{\rm
SM}}$ are the energy density and pressure of the SM plasma, respectively, and
$\Gamma_{E}$ is the energy transfer rate per unit volume given by

$\displaystyle\Gamma_{E}({\rm SM}\rightarrow{\rm CFT})$ $\displaystyle=$
$\displaystyle\sum_{i,j}n_{i}n_{j}\langle\sigma(i+j\to{\rm CFT})v_{\rm
rel}E\rangle+\sum_{i}n_{i}\langle\Gamma(i\to{\rm CFT})E\rangle\,.$ (15)

Here the sums run over all SM degrees of freedom coupled to the CFT. The
cross-sections and decay rates can be evaluated using the “unparticle”
technique of Georgi Georgi:2007ek ; Grinstein:2008qk ; an explicit example of
such a calculation is given in Appendix A.1. In the COFI scenario, the dark
sector temperature $T_{D}$ remains well below the SM temperature, $T_{D}\ll
T_{{}_{\rm SM}}$, throughout the cosmological history. For this reason, we
have neglected the reverse energy transfer, from the CFT back to the SM
sector, in Eq. (14). Conformal symmetry of the dark sector guarantees that its
energy-momentum is traceless, $P_{{}_{\rm CFT}}=\frac{1}{3}\rho_{{}_{\rm
CFT}}$, and thus its energy density redshifts as radiation, $\rho_{{}_{\rm
CFT}}\propto a^{-4}$, as the universe expands. At the time when the CFT sector
is populated, the energy density in the SM sector is dominated by relativistic
matter, so that SM and CFT energy densities redshift in the same way. The
total energy of the two sectors can only change due to work done against the
expansion of the universe:

$\frac{d}{dt}\left(\rho_{{}_{\rm CFT}}+\rho_{{}_{\rm
SM}}\right)+4H\left(\rho_{{}_{\rm CFT}}+\rho_{{}_{\rm SM}}\right)=0.$ (16)

Subtracting Eq. (14), we find that the CFT energy density evolves according to

$\frac{d\rho_{{}_{\rm CFT}}}{dt}+4H\rho_{{}_{\rm CFT}}=\Gamma_{E}({\rm
SM}\rightarrow{\rm CFT})\,.$ (17)

Solving this equation, with the initial condition $\rho_{{}_{\rm CFT}}=0$ at
$T_{{}_{\rm SM}}=T_{R}$, yields the CFT energy density as a function of the SM
temperature $T$.

Figure 4: Dark sector energy density (normalized by SM energy density) vs.
temperature of the Standard Model plasma, for two different values of $D$. The
red curve ($D<9/2$) shows IR-dominant production, while the blue curve
($D>9/2$) shows UV-dominant production.

It is instructive to discuss an analytic solution of (17) for the simple case
when the energy transfer rate is given by

$\Gamma_{E}({\rm SM}\rightarrow{\rm CFT})\sim\frac{\lambda_{{}_{\rm
CFT}}^{2}}{\Lambda_{{}_{\rm CFT}}^{2(D-4)}}\,T_{{}_{\rm SM}}^{2D-3}.$ (18)

This scaling occurs when the SM temperature $T_{{}_{\rm SM}}$ is well above
all relevant SM energy scales (such as masses) and the mass gap of the dark
sector.444This regime, where the SM itself is approximately conformal, was
also considered in Ref. Redi:2021ipn . This can be easily shown via simple
dimensional analysis, keeping in mind that the SM temperature is the only
relevant dimensionful scale besides (the square of) the coupling to the dark
sector. Integrating (17), the energy density of the dark sector grows as

$\rho_{{}_{\rm CFT}}\sim\frac{M_{\rm pl}}{\Lambda_{{}_{\rm
CFT}}^{2D-8}}\;\left[\;T^{4}\;\left(\frac{T_{R}^{2D-9}-T^{2D-9}}{2D-9}\right)\right],$
(19)

where $M_{\rm pl}$ is the Planck mass.

For values of $D$ below the critical dimension $D=d_{{}_{\rm SM}}+d=4.5$, most
of the dark sector energy density is produced at low temperature (“in the
infrared”) and the dark matter relic density can be predicted without
knowledge of UV physics and the reheating temperature. (See Fig. 4.) This is
similar to the original freeze-in scenario of Hall et.al. Hall:2009bx . For
$D>4.5$, most of the dark sector energy density is produced soon after the
reheating. In this case, the predicted dark matter relic density does depend
on $T_{R}$. However, in practice this dependence is weak, due to the low
powers in the exponent for $T_{R}$ compared to the dependence on the mass gap,
as will be shown later in this section.

${\cal O}_{{}_{\rm SM}}$ | DM Mass | DM Mass | Dominant CFT | Dominant
---|---|---|---|---
| (Scalar Mediator) | (Vector Mediator) | Deformation | Production Mode
$H^{\dagger}H$ | 0.4 - 1.2 MeV | 40 - 400 keV | Tree-level | $h\rightarrow\mathrm{CFT}$
$HQ^{\dagger}q$ |  1st: ~~SN~~ All: 0.1 - 1 MeV MFV: 0.5 - 5 MeV |  1st: ~~SN~~ All: 50 - 200 keV MFV: 0.1 - 1 MeV | Radiative mixing | $q\bar{q}\rightarrow\mathrm{CFT}$
$HL^{\dagger}\ell_{R}$ |  1st: ~~WDM~~ All: 3 - 10 keV MFV: 10 - 100 keV |  1st: ~~WDM~~ All: ~~WDM~~ MFV: ~~WDM~~ | Radiative mixing | $\ell\bar{\ell}\rightarrow\mathrm{CFT}$
$G^{\mu\nu}G_{\mu\nu}$ | 0.2 - 2 MeV | 50 - 400 keV | Radiative direct | $gg\rightarrow\mathrm{CFT}$
$B^{\mu\nu}B_{\mu\nu}$ | 0.1 - 10 MeV | 0.05 - 1 MeV | Radiative direct | $\gamma\gamma\rightarrow\mathrm{CFT}$
Table 2: Summary table for each SM operator portal considered. In this table,
~~SN~~ stands for models that are ruled out by supernova cooling constraints,
and ~~WDM~~ stands for models that are ruled out by warm dark matter
constraints.

The Boltzmann equation (17), with energy transfer rates calculated within the
‘unparticle’ approach, is valid as long as $T_{{}_{\rm SM}}>M_{\rm gap}$
(required for the validity of the collision term) and $T_{D}>m_{\rm DM}$
(required for radiation-like Hubble term). As the universe expands and cools,
both conditions may become invalid, requiring modifications to the Boltzmann
equation. For $T_{D}<m_{\rm DM}$, we simply replace $4H\to 3H$ in the Hubble
term, since at these temperatures the dark sector is populated by non-
relativistic dark matter particles. For $M_{\rm gap}>T_{{}_{\rm SM}}>m_{\rm
DM}$, we consider dark matter production in the “hadronic phase”. The
corresponding collision term is calculated within the low-energy effective
theory discussed in Section 2.2. Note that production in the hadronic phase
only occurs if the SM particles interacting with the CFT are light (electrons
or photons); in all other cases, the relevant SM particles drop out of the
thermal bath at $T_{{}_{\rm SM}}>M_{\rm gap}$ and all production is in the CFT
regime. Moreover, we find that for all portal interactions considered here,
dark matter production in the hadronic phase is subdominant to production in
the CFT regime, with the exception of a small region in the parameter space of
the lepton-portal model.

We note that in the COFI scenario, it is possible that at some time in the
cosmological history $T_{{}_{\rm SM}}>M_{\rm gap}>T_{{}_{\rm D}}$. In this
regime, the thermal bath of the dark sector is described by particle-like
bound-state excitations. However, the energy transfer from the SM to the dark
sector can still be described within the unparticle approach, since the energy
transferred in a single collision is above $M_{\rm gap}$. This is analogous to
using the parton model to calculate (inclusive) rates of hadron production at
the LHC, even though no quark-gluon plasma is produced.

With the low-temperature modifications outlined above, Eq. (17) remains valid
to present day. Integrating this equation, with energy transfer rates
evaluated separately for each portal, provides predictions for current dark
matter relic density which can be compared with the observed value,
$\Omega\,h^{2}=0.1$. These predictions will be discussed in the rest of this
section.

###  3.2 Higgs Portal: ${\cal O}_{{}_{\rm SM}}=H^{\dagger}H$

There are multiple mechanisms of SM $\to$ dark sector energy transfer in the
$H^{\dagger}H$ portal model. For $T_{{}_{\rm SM}}$ between the reheating
temperature ($T_{R}$) and the weak scale, the leading mechanism is the
scattering process $HH\rightarrow$ CFT. After the electroweak phase
transition, one Higgs in the interaction term can be replaced with its VEV and
dark energy density will be produced through Higgs decay. Additionally, there
is production from quark and gluon fusion through a Higgs portal. Quark fusion
continues until the quarks fall out of the thermal bath. Other contributing
processes include heavy quark to light quark + CFT decay and pion annihilation
below $\Lambda_{{}_{\rm QCD}}$. These are subdominant due to phase space
factors and can be neglected. It can be shown that the Higgs decay process is
the dominant production mechanism, provided that production is IR dominated
with $D<4.5$ (or equivalently the CFT operator dimension $d<d_{*}=2.5$).

(a)  Figure 5: Dark matter relic density contours (red) and
observational/theoretical constraints, in the Higgs portal model, with a
scalar mediator (left) and a vector mediator (right). The solid red line
indicates parameters where the observed dark matter abundance is reproduced.

An analytic approximation for the relic density can be obtained by considering
only the dominant mode of production: Higgs decay. The collision term in the
Boltzmann equation is given by,

$\Gamma_{E}({\rm SM}\rightarrow{\rm CFT})=n_{h}\langle\Gamma(h\to{\rm
CFT})E\rangle\,=\,\frac{f_{d}\lambda_{{}_{\rm
CFT}}^{2}v^{2}m_{H}^{2(d-1)}T}{\Lambda_{{}_{\rm
CFT}}^{2d-4}}\,K_{2}(m_{H}/T),$ (20)

where $f_{d}=2^{-2d}\pi^{1/2-2d}\Gamma(d+1/2)/(\Gamma(d-1)\Gamma(2d))$, $v$ is
the Higgs VEV, $m_{H}$ is the Higgs boson mass and $K_{2}(x)$ is the modified
Bessel function of the second kind.

Using Eqs. (20) and (17), the current relic density of dark matter can be
calculated. This yields

$\displaystyle\frac{\Omega_{{}_{\rm DM}}h^{2}}{0.1}=\left[\frac{m_{{}_{\rm
DM}}}{1\mathrm{\
MeV}}\right]\left[\frac{\left(A\,f_{d}^{3}\,g_{*}^{-9/2}\right)^{1/4}}{10^{-5}}\right]\left[\frac{\left(\frac{M_{\rm
gap}}{m_{h}}\right)^{(6-\frac{3d}{2})}}{10^{-12}}\right].$ (21)

Here, $g_{*}\equiv g_{*}(m_{H})$ is the effective number of SM degrees of
freedom when $T_{{}_{\rm SM}}=m_{H}$ and $A$ is a model-dependent constant
that represents the number of degrees of freedom of the dark sector as
$\rho_{{}_{\rm CFT}}\equiv A\,m_{{}_{\rm DM}}^{4}$. We have used the mass gap
formula from Table 1 to convert the interaction coupling dependence to mass
gap dependence as,

$M_{\rm gap}=\left(\frac{\lambda_{{}_{\rm CFT}}}{\Lambda_{{}_{\rm
CFT}}^{d-2}}v^{2}\right)^{\frac{1}{4-d}}.$

The ratios in each bracket are $\mathcal{O}(1)$ for $1<d<2.5$ and
$A\sim\mathcal{O}(1)$. Thus, we expect a mass-gap for the Higgs portal model
at the MeV scale. For details of this calculation, see Appendix A.1. This
result is in good agreement with the numerical integration of the Boltzmann
equation.

The dark matter mass $m_{{}_{\rm DM}}$ and the dimension $d$ of the CFT
operator that produce the correct observed relic density are shown in Fig. 5.
Since the dark sector is mostly populated through Higgs decays which occur at
temperatures below the weak scale, the relic density is independent of the
reheating temperature or any other UV-scale parameters. Fig. 5 also shows
phenomenological and theoretical constraints on the model, which will be
discussed in detail in Section 4. We observe that the model produces a viable
DM candidate with masses $m_{{}_{\rm DM}}\sim 0.1-1$ MeV. In these figures, we
have fixed the value of $r=m_{{}_{\rm DM}}/M_{\rm gap}$ (see Section 2.2 for
the discussion of this parameter). The ratio $r$ is tightly constrained by the
combination of bounds from large-scale structure (warm dark matter) and dark
matter self-interactions. Given these bounds, $r$ can only be varied by a
factor of at most a few relative to the values shown. Such variation does not
have a strong effect on the predicted dark matter mass range.

###  3.3 Quark & Lepton Portals: ${\cal O}_{{}_{\rm SM}}=HQ^{\dagger}q,\
HL^{\dagger}\ell_{R}$

Above the weak scale, energy transfer from the SM to the dark sector occurs
via scattering processes $Hf\bar{f}\rightarrow\mathrm{CFT}$ and $Hf\rightarrow
f+\mathrm{CFT}$, where $f$ refers to quarks or leptons depending on the SM
operator used. The energy transfer rate in these channels peaks at high
temperatures, introducing dependence on the reheat temperature $T_{R}$. Below
the weak scale, ${\cal O}_{{}_{\rm SM}}$ is matched onto a dimension-3
bilinear fermion operator. The dominant process contributing to production of
CFT energy density is fermion annihilation $f\bar{f}\rightarrow\mathrm{CFT}$.
We find that for $T_{R}\mathrel{\>{\hbox{\raise
1.50696pt\hbox{$<$}}\kern-3.8889pt\kern-3.8889pt\hbox{\lower
2.79857pt\hbox{$\sim$}}}\>}$ few TeV, production below the weak scale is
dominant and the resulting DM relic density is independent of $T_{R}$. For
$D<4.5\Rightarrow d<1.5$, the energy transfer through fermion annihilation
peaks at low temperatures, while for $d>1.5$, temperatures of order the weak
scale dominate.

(a)  Figure 6: Dark matter relic density contours (red) and
observational/theoretical constraints, in the quark portal model with minimal
flavor violation couplings, with a scalar (vector) mediator on the left
(right).

(a)  Figure 7: Dark matter relic density contours (red) and
observational/theoretical constraints, in the quark portal model with
democratic couplings, with a scalar (vector) mediator on the left (right).

For the quark portal, conformal freeze-in continues until $T=\Lambda_{{}_{\rm
QCD}}$ or $T=M_{\rm gap}$, whichever happens first. For the lepton portal, it
continues until $T=m_{e}$ or $T=M_{\rm gap}$. Again, we assume that there are
dark pions that form the dark matter relic density we observe today, that are
a factor $r\sim 0.01$ (with scalar mediator) or $r\sim 0.001$ (with vector
mediator) lighter than the mass gap induced by the Standard Model deformation.
The dark sector energy density redshifts as radiation until $T_{{}_{\rm D}}$
hits $m_{{}_{\rm DM}}=m_{\chi}$, and redshifts as matter afterwards, until
today.

Notably, in the lepton portal, it is possible for the SM temperature at which
$T_{{}_{\rm D}}$ hits $m_{{}_{\rm DM}}$ to be higher than the stopping
temperature. In the short period when the universe cools from the former
temperature to the latter, the DM energy density is produced in the CFT phase,
but hadronizes quickly to matter and redshifts as matter. Additionally, in
parts of the parameter space of the lepton portal, production can also be
dominated by hadronic processes, where most of the energy density is produced
below $T=M_{\rm gap}$ through the processes involving the IR composite states.
This is the case for the grey shaded regions in Fig. 9. See Appendix A.2 for
details of thermally averaged hadronic cross-sections and production rates.

As discussed in Section 3, we consider three scenarios for flavor structure of
the quark/lepton portal couplings: Minimal Flavor Violation, Democratic, and
First-Generation Only. The three scenarios give different mass gap scales for
which the correct relic abundance is produced.

The energy density ($\rho_{{}_{\rm CFT}}$) produced through the dominant
process of fermionic scattering scales as follows for each structure:

  * •

First Generation Only: $\rho_{{}_{\rm CFT}}\sim M_{\rm
pl}\left(\frac{m^{4-d}}{\alpha^{2}v^{2}m_{1}}\right)^{2}\,T^{4}\,(v^{2d-3}-T^{2d-3})$
(22)

  * •

Democratic: $\rho_{{}_{\rm CFT}}\sim M_{\rm
pl}\left(\frac{m^{4-d}}{\alpha^{2}v^{2}\sum_{i}m_{i}}\right)^{2}\,T^{4}\,(v^{2d-3}-T^{2d-3})$
(23)

  * •

Minimal Flavor Violation: $\rho_{{}_{\rm CFT}}\sim M_{\rm
pl}\;m_{j}^{2}\left(\frac{m^{4-d}}{\alpha^{2}v^{2}\sum_{i}m_{i}^{2}}\right)^{2}\,T^{4}\,(v^{2d-3}-T^{2d-3})$
(24)

where $\Lambda_{\scriptstyle\rm SM}\equiv\alpha\,v$ and $m_{i}$ stands for the
relevant fermion masses. At the end of the freeze-in process for each
interacting fermion, $T={\rm Max}[m_{i},\,M_{\rm gap}]$ for the lepton portal
and $T={\rm Max}[m_{i},\,\Lambda_{\scriptstyle\rm QCD},\,M_{\rm gap}]$ for the
quark portal. Each of these contributions is summed and appropriately
redshifted to obtain the relic density. See Appendix A.1 for the relic density
equations for each flavor structure and portal.

Of the three scenarios, the MFV model is the least constrained, due to
suppressed couplings to the first generation of fermions. In the quark portal,
the First-Generation Only scheme is ruled out by supernova cooling constraints
from SN1987A data (for both scalar and vector mediators). The other four
models are viable and the plots are shown in Figs. 6 and 7.

In the lepton portal, the mass of the DM candidate with correct relic
abundance tends to be lower than in other models, and the bound on dark matter
free-streaming length from the Lyman-$\alpha$ forest data Irsic:2017ixq  plays
a major role in constraining the models. This is illustrated in Figs. 8 and 9.
The viability of COFI dark matter in this case depends on the details of the
model: for example, MFV and democratic models with a scalar mediator predict
$m_{\rm DM}\mathrel{\>{\hbox{\raise
1.50696pt\hbox{$>$}}\kern-3.8889pt\kern-3.8889pt\hbox{\lower
2.79857pt\hbox{$\sim$}}}\>}10$ keV and are consistent with observations, while
in other cases $m_{\rm DM}\sim 1$ keV and the models are ruled out.

(a)  Figure 8: Dark matter relic density contours (red) and
observational/theoretical constraints, in the lepton portal model with minimal
flavor violation couplings, with a scalar (vector) mediator on the left
(right).

(a)  Figure 9: Dark matter relic density contours (red) and
observational/theoretical constraints, in the lepton portal model with only
the first generation of leptons on the left and all generations of leptons on
the right, with a scalar mediator.

In summary, we find six models with allowed parameter space that reproduces
the relic density: quark portal with MFV or democratic coupling (both scalar
and vector mediators), and the scalar mediator lepton portal with MFV or
democratic couplings.

###  3.4 Gluon Portal: ${\cal O}_{{}_{\rm SM}}=G^{\mu\nu}G_{\mu\nu}$

(a)  Figure 10: Dark matter relic density contours (red) and
observational/theoretical constraints, in the gluon portal model, with a
scalar (vector) mediator on the left (right).

The dominant mode of populating the dark sector is through gluon annihilation,
$gg\rightarrow\mathrm{CFT}$. Additionally, there are subdominant processes of
production, through loop-induced quark annihilation. The dark sector energy
density produced via gluon annihilation scales as,

$\rho_{{}_{\rm CFT}}\sim M_{\rm
pl}\left(\frac{m^{4-d}}{16\pi^{2}\,\alpha^{4}v^{4}}\right)^{2}\,T^{4}\,(T_{R}^{2d-1}-T^{2d-1})$
(25)

As in the quark portal, production continues until $T=\Lambda_{{}_{\rm QCD}}$
or $T_{{}_{\rm D}}=M_{\rm gap}$, whichever happens first. The constraints on
the model parameter space are shown in Fig. 10. For analytic estimates of the
relic density, see Appendix A.1.

(a)  Figure 11: Dark matter mass that produces the observed relic density, as
a function of the reheating temperature, for various values of $d$, for the
gluon (left) and hypercharge (right) portals.

Since the operator $G^{\mu\nu}G_{\mu\nu}$ is of dimension $d_{{}_{\rm SM}}=4$,
and the CFT operator dimension $d\geq 1$ by unitarity, the dark sector energy
density is always dominated by the production at the highest available
temperature, i.e. the reheating temperature $T_{R}$ (see Section 3.1). The
predictions of these models thus depend on an additional parameter, $T_{R}$,
making it less predictive. However, in practice, the dependence of the
predicted dark matter candidate mass on $T_{R}$ is rather weak. As shown in
the derivation in Appendix A.1, the relic density of dark matter today scales
as

$\Omega\,h^{2}\propto\left(M_{\rm
gap}\right)^{7-\frac{3}{2}d}\,\left(T_{R}\right)^{\frac{3}{4}(2d-1)}.$ (26)

For relic density fixed to the observed value, the dependence of the inferred
mass gap on $T_{R}$ is given by

$\frac{\partial\log M_{\rm gap}}{\partial\log
T_{R}}=\left(\frac{3}{8}\left(\frac{2d-1}{3d-14}\right)\right).$ (27)

The logarithmic derivative is small throughout the range of $d$ considered
here. The relationship between the dark matter mass and the reheating
temperature is shown in Fig. 11, for various values of the CFT operator
dimension $d$.

###  3.5 Electroweak Boson Portal: ${\cal O}_{{}_{\rm
SM}}=W^{\mu\nu}W_{\mu\nu},\ B^{\mu\nu}B_{\mu\nu}$

(a)  Figure 12: Dark matter relic density contours (red) and
observational/theoretical constraints, in the hypercharge portal model, with a
scalar (vector) mediator on the left (right).

Since the phenomenology of both the weak SU(2)L (${\cal O}_{{}_{\rm
SM}}=W^{\mu\nu}W_{\mu\nu}$) and the hypercharge (${\cal O}_{{}_{\rm
SM}}=B^{\mu\nu}B_{\mu\nu}$) portals are similar, we consider only the case of
${\cal O}_{{}_{\rm SM}}=B^{\mu\nu}B_{\mu\nu}$ to illustrate the salient
features of the electroweak boson portal. The dominant production process is
that of vector boson annihilation, with the initial dynamical degrees of
freedom being hypercharge gauge bosons above the electroweak phase transition
(EWPT) and photons below EWPT. Subdominant processes include $Z$ boson decay
below the weak scale, and fermion annihilation through the electroweak portal.
Photon annihilation continues till $T_{{}_{\rm SM}}\sim M_{\rm gap}$, and the
dark matter redshifts as matter below $T_{{}_{\rm D}}\sim m_{{}_{\rm DM}}$.
Photon annihilation to CFT states produces dark sector energy density that
scales similarly to the gluon portal model;

$\rho_{{}_{\rm CFT}}\propto M_{\rm
pl}\,(1-\sin^{2}\theta_{w})\left(\frac{m^{4-d}}{16\pi^{2}\,\alpha^{4}v^{4}}\right)^{2}\,T^{4}\,(T_{R}^{2d-1}-T^{2d-1})$
(28)

where $\theta_{w}$ is the Weinberg angle. For analytic estimates of the relic
density, see Appendix A.1.

The viable parameter space and constraints on this model are shown in Fig. 12.
The value of $r$ is 0.1 and 0.01 respectively for scalar and vector mediators.
As in the gluon portal, the interaction term dimension $D$ is always $>5$ and
production is dominant at the reheating temperature $T_{R}$, making the relic
density dependent on an extra parameter. Due to the similarities with the
gluon portal, where vector boson annihilation in the UV determines the relic
density, equations (26) and (27) apply in this case as well. Fig. 11
demonstrates this scaling.

##  4 Dark Matter Phenomenology and Constraints

(a)  Figure 13: Left panel: Effective energy scale of the SM-CFT interaction
in the MFV lepton portal with a scalar mediator. Right panel: Effective
dimensionless strength of the SM-CFT coupling for the same portal, for SM
collision energies of order 100 GeV.

The interactions of the COFI dark matter candidate with the Standard Model
particles are extremely weak. The effective energy scale suppressing the SM-
CFT interaction is well above the weak scale

$\Lambda=(\lambda_{{}_{\rm CFT}})^{-\frac{1}{D-4}}\,\cdot\Lambda_{\rm
CFT}\,\sim 10^{10}-10^{15}~{}{\rm GeV},$ (29)

leading to tiny couplings of the DM particles to SM at energies of order the
weak scale and below. This is illustrated in Fig. 13, in the case of the
lepton portal in the MFV flavor scheme; other portals produce similar results.
As a result, no relevant constraints arise from direct, indirect, and collider
searches for DM. However, there are important phenomenological constraints on
the model from dark matter self-interaction and large-scale structure (which
are independent of the DM-SM coupling), as well as stellar cooling rates
(where the small coupling is compensated by large amount of SM particles in
the stellar bodies). These constraints will be considered in this section. We
will also outline theoretical constraints on the model parameter space related
to naturalness and CFT bootstrap bound.

###  4.1 Dark Matter Self-Interaction Bound

Observations of galactic clusters, such as the Bullet cluster, place an upper
bound on the cross-section of elastic scattering of non-relativistic DM
particles, $\sigma_{\rm SI}/m_{\chi}\lesssim 4500\;\mathrm{GeV}^{-3}$
Markevitch:2003at ; Lin:2019uvt . One generally expects that the hadronic
phase of our dark sector has characteristic coupling
$g_{\star}\sim\frac{4\pi}{\sqrt{N}}$. If the dark matter is a generic
composite state, the elastic scattering cross-section is of the order

$\sigma_{\rm SI}\sim\frac{g_{\star}^{4}}{8\pi M_{\rm gap}^{2}}.$ (30)

For the values of $M_{\rm gap}$ that produce the observed DM relic density,
$N\sim 10^{4}$ would be required for consistency with the observational bound.
Such large values of $N$ are possible, but theoretically unattractive. This
leads us to consider an alternative possibility that $g_{\star}\sim 1$ but the
DM state is not a (or a collection of) generic composite particle(s), but
rather is a derivatively-coupled PNGB. DM elastic scattering is mediated by
exchanges of a scalar or vector resonance with mass of order $M_{\rm gap}$.
Using the effective theory (9), the cross-section for the case of a scalar
mediator is estimated to be

$\sigma_{\rm SI}\sim\frac{r^{6}}{8\pi M_{\rm gap}^{2}},\;\;r=m_{\chi}/M_{\rm
gap}$ (31)

while for a vector mediator (using (8)),

$\sigma_{\rm SI}\sim\frac{r^{2}}{8\pi M_{\rm gap}^{2}}.$ (32)

Here $r=m_{{}_{\rm DM}}/M_{\rm gap}\ll 1$ is a model-dependent parameter. If
both vector and scalar mediators are present with similar masses, the vector
exchange will dominate. This is the case in QCD where $\rho$ exchange is the
main contribution to pion elastic scattering. However, for completeness, we
consider both scalar and vector mediator-dominated cases in our
phenomenological analysis. We find that in the scalar case, $r\sim
0.01\text{--}0.1$ is sufficient for consistency with observational bounds,
while in the vector case $r\mathrel{\>{\hbox{\raise
1.50696pt\hbox{$<$}}\kern-3.8889pt\kern-3.8889pt\hbox{\lower
2.79857pt\hbox{$\sim$}}}\>}10^{-2}$ is required. See Fig. 14 for an
illustration of allowed values of $r$ and its effect on the value of
$m_{{}_{\rm DM}}$ that produces the observed relic density, in one particular
model.

Figure 14: $r$ vs. $m_{{}_{\rm DM}}$ dependence for the lepton portal with the
MFV flavor structure, assuming a scalar mediator.

###  4.2 Warm Dark Matter Bound

Since dark matter in COFI models is light as well as relativistic in the early
universe, they can free-stream, leading to suppression of
structure/inhomogeneity below a certain length scale $\lambda_{\rm FS}$.
Observation of the existence of a DM halo of a certain size then puts an upper
bound on $\lambda_{\rm FS}$, and hence on the mass of DM. Typically,
observations of the Lyman-$\alpha$ flux spectra, which probes DM halos from
redshifts of $z\sim(3-5.5)$, are used to set such bounds. Depending on the
particular data set used and the systematics of the analysis, the current
bound is Irsic:2017ixq 555As discussed above, our DM generically has self-
interactions, while the analysis of Irsic:2017ixq  is based on an assumption
of collisionless DM. The bound for self-interacting dark matter is somewhat
weaker Egana-Ugrinovic:2021gnu , but the difference is not large enough to
affect the present discussion.,

$\displaystyle m_{{}_{\rm DM}}\gtrsim(3.5-5.3)\;{\rm keV}.$ (33)

This bound places a non-trivial constraint on certain COFI models with a
lepton portal, where the DM mass consistent with relic density is in the
$1-10$ keV range. For other portals, COFI DM candidates are much heavier and
this bound is irrelevant.

###  4.3 Stellar Cooling Bounds

In this section, we discuss constraints imposed on COFI dark matter models
from the evolution of stars. Dark matter candidates in the keV-MeV mass range
can be produced in collisions of SM states (nucleons and electrons) in stars.
In spite of weak DM-SM coupling, the production can be significant due to
large amount of matter in stars. Once a DM state is produced, the weak
coupling may allow it to escape the star without interacting, carrying away
energy. Systems supported by degenerate pressure, e.g. supernovae, have a
positive heat capacity and production of DM results in an extra cooling
mechanism. Systems supported mainly by thermal pressure, such as Main Sequence
(MS) stars, have a negative heat capacity. Energy carried by DM states
produced in the core does not necessarily lead to extra cooling, but still
affects the dynamics of the star, changing the time scale for each stage in
its evolution. In either case, existing observations provide constraints on
the rate of extra energy loss, which can be translated into bounds on new
physics.

In this work, we consider constraints from the following classes of stars:
Main Sequence (MS, e.g. Sun), Red Giant Branch (RGB), Horizontal Branch (HB),
and Supernova (SN).666It has been pointed out in Hardy:2016kme  that white
dwarfs (WD) and neutron stars (NS) give either comparable or weaker bounds and
we do not further consider them. For the purpose of computing bounds, each
system may be characterized by the core temperature $T$, mass density $\rho$
(which is dominated by nuclear mass density), electron number density $n_{e}$,
the degree of electron (and nucleon) degeneracy $E_{F}$ and $p_{F}$ (since the
Fermi energy and momentum are higher than the temperature only when
electrons/nucleons are degenerate), and composition of nuclear matter. From
these data and the form of interaction between new physics state and SM
states, one then computes energy loss rate per mass $\epsilon$, and compares
it to existing bounds. In our estimates, we will adopt the following benchmark
parameters for each class of stars Hardy:2016kme :

  * •

MS: $T\approx 1.3\;{\rm keV}$, $\rho\approx 156\;{\rm g\;cm^{-3}}$,
$n_{e}=6.3\times 10^{25}\;{\rm cm^{-3}}$, $\epsilon\lesssim 0.2\;{\rm
erg\;g^{-1}\;s^{-1}}$.

electrons not degenerate, nucleons not degenerate.

  * •

HB: $T\approx 10\;{\rm keV}$, $\rho\approx 10^{4}\;{\rm g\;cm^{-3}}$,
$n_{e}=3\times 10^{27}\;{\rm cm^{-3}}$, $\epsilon\lesssim 10\;{\rm
erg\;g^{-1}\;s^{-1}}$

electrons not degenerate, nucleons not degenerate.

  * •

RGB: $T\approx 10\;{\rm keV}$, $\rho\approx 10^{6}\;{\rm g\;cm^{-3}}$,
$n_{e}=3\times 10^{29}\;{\rm cm^{-3}}$, $\epsilon\lesssim 10\;{\rm
erg\;g^{-1}\;s^{-1}}$

electrons degenerate ($E_{F}\approx 144\;{\rm keV}$, $p_{F}\approx 409\;{\rm
keV}$), nucleons not degenerate.

  * •

SN: $T\approx 30\;{\rm MeV}$, $\rho\approx 3\times 10^{14}\;{\rm g\;cm^{-3}}$,
$n_{e}=1.8\times 10^{38}\;{\rm cm^{-3}}$, $\epsilon\lesssim 10^{19}\;{\rm
erg\;g^{-1}\;s^{-1}}$

electrons degenerate ($E_{F}\approx p_{F}\approx 344\;{\rm MeV}$), nucleons
nearly degenerate.

The production of DM in stars may occur in one of the two regimes. If energy
transferred from the SM into the dark sector in a single collision is above
$M_{\rm gap}$, the final state consists of CFT states and the cross-section
can be calculated using the unparticle approach. Following the collision, the
produced dark sector states quickly hadronize and decay, resulting in multiple
DM particles that share the transferred energy. If, on the other hand, energy
transferred from the SM into the dark sector in a single collision is between
$m_{{}_{\rm DM}}$ and $M_{\rm gap}$, the production occurs in the hadronic
phase and is estimated using the low-energy effective theory in the dark
sector, see Section 2.2. For each COFI portal and class of stars, we start by
determining which of the two regimes is appropriate, and proceed to estimate
the DM production cross-section and the resulting energy loss rate. In cases
where the energy loss argument imposes a relevant bound on the COFI scenario,
we also estimated the mean-free path $\ell_{\rm MFP}$ of the produced DM
particle in the star. If $\ell_{\rm MFP}$ is smaller than the star radius, the
DM will typically become trapped in the star, depositing its energy back into
the stellar material. In this case, the energy-loss bounds do not apply.

####  4.3.1 Quark and Gluon Portals

In this case, dark sector states are produced in stars primarily through
Bremsstrahlung in nucleon collisions. For $T<M_{\rm gap}$ (MS, HB, RGB), the
final state is the hadronic state of the confined CFT, while for $T>M_{\rm
gap}$ (SN), the process is energetic enough to produce CFT state directly.

####  $T<M_{\rm gap}$ (MS, HB, RGB)

We first consider the case with $T<M_{\rm gap}$. The matrix element for the
quark scalar operator in nucleons is given by (see for example Bishara:2017pfq
; Hisano:2017jmz )

$\langle N|\,\bar{q}q\,|N\rangle\equiv
f_{T_{q}}^{N}\,\frac{m_{N}}{m_{q}}=:C_{q}^{(N)}$ (34)

where $f_{T_{q}}^{N}$ is the mass fraction parameter of the quark $q$. Values
of $f_{T_{q}}^{N}$ can be found in Bishara:2017pfq ; Hisano:2017jmz . The
matrix element for the gluon scalar operator can be obtained from the trace
anomaly in QCD Bishara:2017pfq ; Hisano:2017jmz :

$\langle N|\,G_{\mu\nu}^{a}G^{\mu\nu
a}\,|N\rangle=-\frac{8}{9}\frac{\pi}{\alpha_{s}}m_{N}\left(1-\sum_{q}f_{T_{q}}^{N}\right)+\mathcal{O}(\alpha_{s})=:C_{G}^{(N)}.$
(35)

Together with the matching of CFT to its low-energy effective theory described
in Section 2.2, this provides the effective theories of nucleon-dark hadron
interactions. For quark portal, we have

$\mathcal{L}\sim\frac{\lambda_{{}_{\rm CFT}}}{\Lambda_{{}_{\rm
CFT}}^{d}}\,HQ^{\dagger}q\,{\cal O}_{{}_{\rm
CFT}}\to\mathcal{L}\sim\left(\frac{\lambda_{{}_{\rm CFT}}\,v\,M_{\rm
gap}^{d-1}}{\sqrt{2}g_{*}\,\Lambda_{{}_{\rm
CFT}}^{d}}\right)\left(f_{T_{q}}^{N}\,\frac{m_{N}}{m_{q}}\right)\bar{N}N\phi+\frac{g_{*}}{M_{\rm
gap}}\phi\left(\partial\chi\right)^{2}$ (36)

Below the gap scale, $\phi$ can be integrated out, yielding the nucleon-DM
coupling:

$\mathcal{L}\sim G_{N\chi}^{(q)}\,\,\bar{N}N\left(\partial\chi\right)^{2}\,,$
(37)

where

$G_{N\chi}^{(q)}\,=\,\frac{16\pi^{2}\sum_{q}\kappa_{q}C_{q}^{(N)}}{\left(\sum_{q}\kappa_{q}m_{q}\right)\Lambda_{\rm
SM}^{2}}\,.$ (38)

Here the sums run over quark flavors, and we have used the mass gap formula
for the quark portal.

For gluon portal, the effective theory of nucleon-dark hadron interactions

$\mathcal{L}\sim\frac{\lambda_{{}_{\rm CFT}}}{\Lambda_{{}_{\rm
CFT}}^{d}}\,G_{\mu\nu}^{2}\,{\cal O}_{{}_{\rm
CFT}}\to\mathcal{L}\sim\left(\frac{\lambda_{{}_{\rm CFT}}\,M_{\rm
gap}^{d-1}}{g_{*}\,\Lambda_{{}_{\rm
CFT}}^{d}}\right)\left(-\frac{\pi}{\alpha_{s}}\frac{8}{9}m_{N}\left(1-\sum_{q}f_{T_{q}}^{N}\right)\right)\bar{N}N\phi+\frac{g_{*}}{M_{\rm
gap}}\phi\left(\partial\chi\right)^{2}.$ (39)

Below $m_{\phi}\approx M_{\rm gap}$ we again integrate out the $\phi$. In
addition, since $f_{T_{q}}^{N}\sim 10^{-2}$, we neglect its contribution.
Using $\alpha_{s}(\Lambda_{\rm QCD})\approx 4\pi$ and thereby approximating
$C_{G}^{(N)}\approx-m_{N}/4$, we obtain the nucleon-DM coupling

$\mathcal{L}\sim G_{N\chi}^{(g)}\,\,\bar{N}N\left(\partial\chi\right)^{2}\,$
(40)

where

$G_{N\chi}^{(g)}=C_{G}^{(N)}\left(\frac{16\pi^{2}}{\Lambda_{\rm
SM}^{4}}\right).$ (41)

Using these effective couplings, energy loss rate due to DM emission can be
calculated in analogy with the well-known calculation for energy loss through
standard model neutrinos Raffelt:1996wa . The energy loss rate per unit volume
is given by

$\displaystyle\text{Quark-Portal
:}\;Q(\chi\chi)=Q(\bar{\nu}\nu)\,\left(\frac{T^{2}}{2G_{F}^{2}}\right)\,\left(\frac{G_{N\chi}^{(q)}}{\sum_{q}C_{q}^{(N)}}\right)^{2}\frac{C_{\chi\chi}}{C_{\bar{\nu}\nu}}$
$\displaystyle\text{Gluon-Portal
:}\;Q(\chi\chi)=Q(\bar{\nu}\nu)\,\left(\frac{T^{2}}{2G_{F}^{2}}\right)\,\left(\frac{G_{N\chi}^{(g)}}{\sum_{q}C_{q}^{(N)}}\right)^{2}\frac{C_{\chi\chi}}{C_{\bar{\nu}\nu}}$
(42)

Here $G_{F}$ is the Fermi constant. Explicit expressions for $Q(\bar{\nu}\nu)$
(the rate for neutrino-pair production in the standard model) and the
constants $C_{\bar{\nu}\nu}$ and $C_{\chi\chi}$ are given in the Appendix B.3,
along with the details of the calculation. The energy loss rate per unit mass
($\epsilon$) that we compare with the observed bounds is calculated as
$\epsilon=Q/\rho$.

####  $T>M_{\rm gap}$ (SN)

When $T>M_{\rm gap}$ such as in SN, the dark-sector states produced in nucleon
collisions are described by a CFT, and the production rate can be calculated
using the unparticle approach.777See Davoudiasl:2007jr  for a more heuristic
approach to calculating cosmological and astrophysical bounds with
unparticles. It is useful to normalize the energy loss rate using a simple
benchmark model of a light scalar particle $\phi$ coupled to nucleons through
a Yukawa interaction $\sim g\,\phi\,\bar{\psi}_{N}\psi_{N}$. In this case, the
energy loss per unit volume $Q(\phi)$ is well-known Raffelt:1996wa ; see
Appendix B for an analytic formula. The ratio $\epsilon(\text{\tiny
CFT})/\epsilon(\phi)$ can be reliably estimated by a procedure explained in
Appendix B. We obtain

$\epsilon(\text{\tiny CFT})\sim\frac{G_{\rm
eff}^{2}(m_{N}T)^{d-1}}{g^{2}}\frac{1}{(2\pi)^{2d-2}}\frac{\text{dof}_{\rm\scriptscriptstyle
CFT}}{\text{dof}_{\phi}}\frac{\langle\omega\rangle_{\rm\scriptscriptstyle
CFT}}{\langle\omega\rangle_{\phi}}\frac{Q(\phi)}{\rho}\,,$ (43)

where $d$ is the dimension of the CFT operator, while $\rho$ and $T$ are the
nucleon mass density and temperature in the SN core. Note that the dependence
on the coupling $g$ in the benchmark scalar model cancels out since
$Q(\phi)\propto g^{2}$. Here, $\frac{\text{dof}_{\rm\scriptscriptstyle
CFT}}{\text{dof}_{\phi}}$ denotes the ratio of the internal degrees of freedom
of the final state produced in the CFT and the benchmark scalar model, while
$\frac{\langle\omega\rangle_{\rm\scriptscriptstyle
CFT}}{\langle\omega\rangle_{\phi}}$ is the ratio of the average energy carried
by the corresponding final states. Explicit expressions for the effective
coupling $G_{\rm eff}$ in quark and gluon portals are given in the Appendix
B.3 (see Eqs. (103) and (104)). The factors
$\frac{\text{dof}_{\rm\scriptscriptstyle CFT}}{\text{dof}_{\phi}}$ and
$\frac{\langle\omega\rangle_{\rm\scriptscriptstyle
CFT}}{\langle\omega\rangle_{\phi}}$ can be determined only if CFT is fully
specified, but we expect that they will take values within the range $1\sim
d$. We use $1$ in the constraint plots of Section 3.

####  4.3.2 Higgs Portal

For Higgs portal, the COFI dark matter candidate has mass of order MeV, and
can only be produced in supernovae. Comparing $M_{\rm gap}$ in the Higgs
portal model to $T_{\rm SN}$, we learn that the production is in the CFT
regime. Again, the dominant production mechanism for dark states is
Bremsstrahlung in nucleon collisions. The relevant part of the Lagrangian is

$\mathcal{L}\sim\frac{\lambda_{{}_{\rm CFT}}\,v}{\sqrt{2}\Lambda_{{}_{\rm
CFT}}^{d-2}}\,h\,{\cal O}_{{}_{\rm CFT}}+\frac{\alpha_{s}}{12\pi
v}\,h\,G_{\mu\nu}^{a}G^{\mu\nu a}$ (44)

where the second term is the top-loop induced coupling between the Higgs and
gluon (see for example e.g. Gunion:1989we ; Harlander:2013oja ). Integrating
out the Higgs and using Eq. (35) yields the effective coupling

$\mathcal{L}\sim
C_{G}^{(N)}\left(\frac{\alpha_{s}}{6\sqrt{2}\pi}\right)\left(\frac{M_{\rm
gap}^{4-d}}{v^{2}\,m_{h}^{2}}\right)\bar{N}N\,{\cal O}_{{}_{\rm CFT}}.$ (45)

To get this form, we used the mass gap formula for the Higgs portal model. The
energy loss rate in the SN is calculated as in the gluon portal (see Appendix
B.3) and is given by

$\epsilon(\text{\tiny CFT})\sim\frac{G_{\rm
eff}^{2}(m_{N}T)^{d}}{g^{2}}\frac{1}{(2\pi)^{2d-2}}\frac{\text{dof}_{\rm\scriptscriptstyle
CFT}}{\text{dof}_{\phi}}\frac{\langle\omega\rangle_{\rm\scriptscriptstyle
CFT}}{\langle\omega\rangle_{\phi}}\frac{Q(\phi)}{\rho}$ (46)

where

$G_{\rm
eff}=C_{G}^{(N)}\left(\frac{\alpha_{s}}{6\sqrt{2}\pi}\right)\left(\frac{M_{\rm
gap}^{4-d}}{v^{2}m_{h}^{2}}\right).$ (47)

Numerically, emission from the SN core in the region of the model parameter
space relevant for COFI dark matter is well below the observational bound, so
that the Higgs portal scenario is unconstrained by stellar cooling
considerations.

####  4.3.3 Lepton Portal

Dark sector states are produced through their interactions with electrons in
the stellar medium. In all star systems other than the supernova, the electron
temperature is below $M_{\rm gap}$, so that the production is in the hadronic
phase of the dark sector. The effective theory of electron-dark hadron
interactions has the form

$\mathcal{L}\sim\frac{\lambda_{{}_{\rm CFT}}}{\Lambda_{{}_{\rm
CFT}}^{d}}\left(HL^{\dagger}\ell_{R}\right){\cal O}_{{}_{\rm
CFT}}\to\frac{\lambda_{{}_{\rm CFT}}\,v\,M_{\rm
gap}^{d-1}}{\sqrt{2}g_{*}\,\Lambda_{{}_{\rm
CFT}}^{d}}\,(\bar{e}e)\,\phi+\frac{g_{*}}{M_{\rm
gap}}\phi\left(\partial\chi\right)^{2}.\\\ $ (48)

Integrating out the scalar meson $\phi$ yields the electron-DM coupling:

$\mathcal{L}\sim\frac{16\pi^{2}\kappa_{e}}{\left(\sum\kappa_{\ell}m_{\ell}\right)\Lambda_{\rm
SM}^{2}}\left(\bar{e}e\right)\left(\partial\chi\right)^{2}$ (49)

where the sum runs over all charged lepton flavors.

The calculation of the energy loss rate due to DM emission is again similar to
the case of standard model neutrinos Raffelt:1996wa . The relevant process in
MS and HB stars is Compton scattering, $e^{-}\gamma\to e^{-}\chi\chi$. The
energy loss rate per unit mass is given by

$\epsilon=\frac{Q}{\rho}\sim\frac{9!\,2\alpha}{\left(\sum\kappa_{\ell}m_{\ell}\right)^{2}\Lambda_{\rm
SM}^{4}}\frac{Y_{e}}{m_{u}\,m_{e}^{2}}\,T^{10}.$ (50)

Here, $\alpha$ is the fine-structure constant, $Y_{e}$ is the electron number
fraction per baryon and $m_{u}=1.661\times 10^{-24}\;{\rm g}$ is the atomic
mass unit. The calculation of this rate is outlined in Appendix B.1.

In red giants, electrons are degenerate, and Compton scattering receives a
strong suppression by the final state Pauli-blocking effect (see Section 3.2
of Raffelt:1996wa  and footnote 8 in Appendix B.1). Instead, production by a
Bremsstrahlung process $e^{-}N\to e^{-}N\chi\chi$ is more efficient. The
energy loss rate per mass is given by

$\epsilon(\chi\chi)\sim\frac{\pi\,\alpha^{2}}{189}\left(\frac{Z^{2}}{Am_{u}}\right)\left(\frac{16\pi^{2}\,\kappa_{e}}{\left(\sum\kappa_{\ell}m_{\ell}\right)\Lambda_{\rm
SM}^{2}}\right)^{2}T^{8},$

where $Z$ is the charge of nuclei and $A$ is the atomic mass. The calculation
of this rate is outlined in Appendix B.2.

In the core of the supernova, temperature is sufficiently high for a thermal
population of positrons to exist. In this case, $e^{+}e^{-}$ annihilation
becomes the dominant production channel. Moreover, since $T>M_{\rm gap}$, the
produced dark-sector states are described by a CFT, and their production rate
is estimated using the unparticle approach.

The energy loss rate is given by

$Q(\text{\tiny CFT})\sim n_{e^{-}}n_{e^{+}}\langle\sigma vE\rangle\,,$ (51)

where the energy transfer rate $\langle\sigma vE\rangle$ is given by

$\langle\sigma vE\rangle\sim\left(\frac{\lambda_{{}_{\rm
CFT}}\,v}{\sqrt{2}\,\Lambda_{{}_{\rm
CFT}}^{d}}\right)^{2}\left(\frac{4\pi^{4}\,d(d^{2}-1)}{(2\pi)^{2d+1}}\right)E_{F}^{2d-3},$
(52)

where $E_{F}\approx 344{\rm MeV}$ is the electron Fermi energy. This is very
similar to the expression that was used in the calculation of relic density
produced during freeze-in, with the main difference being that the typical
collision energy is now of order $2E_{F}$ rather than $T$. The positron number
density is given by

$n_{e^{+}}=2\int\frac{d^{3}p}{(2\pi)^{3}}\frac{1}{e^{(E+\mu_{e^{-}})/T}+1}\approx
e^{-\beta\mu_{e^{-}}}\times n_{\rm th}$ (53)

with $n_{\rm th}$ being the equilibrium number density at $T=T_{\rm SN}$ with
Boltzmann distribution, and the chemical potential $\mu_{e^{-}}\approx E_{F}$.

Unsurprisingly, the MFV flavor scheme lepton portal models are not constrained
by supernovae due to the suppressed couplings to electrons and positrons. In
both the first generation and democratic flavor schemes however, the dark
matter particles end up trapped in the core of the supernova due to
significant interactions with the electrons in the plasma. Details of SN
trapping calculations can be found in Appendix B.6. As a result, there is no
relevant constraint from supernovae in any of the viable lepton portal models.

####  4.3.4 Hypercharge Portal

In this model, the dark matter candidate has a mass of order MeV, and only the
SN has a high enough temperature to produce dark-sector states. There are
three possible processes to consider: photon annihilation, $e^{+}e^{-}$
annihilation through a photon loop, and nucleon Bremsstrahlung.
Quantitatively, the photon annihilation turns out to be the most important
channel as explained in Appendix B.5. This is due to the loop- and
electromagnetic coupling-suppression for the $e^{+}e^{-}$ annihilation, and
phase space- and loop-suppression for the nucleon Bremsstrahlung. The energy
loss rate per volume from the photon annihilation is given by (see Appendix
B.5 for details)

$Q(\text{\tiny CFT})\sim n_{\gamma}^{2}\,\langle\sigma vE\rangle\sim
n_{\gamma}^{2}\;\left(\frac{\lambda_{{}_{\rm
CFT}}\cos^{2}\theta_{w}}{\Lambda_{{}_{\rm
CFT}}^{d}}\right)^{2}\left(\frac{16\,d^{2}(d^{2}-1)(d+2)}{(2d-1)(2\pi)^{2d+1}}\right)T_{\rm
SN}^{2d-1}.$ (54)

where $\cos\theta_{w}$ is the Weinberg angle.

###  4.4 Naturalness bound

In addition to the observational bounds discussed so far, we consider two
constraints on the model parameter space motivated by theoretical
considerations, the naturalness and “CFT bootstrap” bounds.

The effective coupling of the SM to the dark sector required to reproduce the
observed DM relic density is tiny, ${\cal O}(10^{-14}-10^{-11})$. The
naturalness bound is the requirement that such a coupling can be obtained in
the effective theory without invoking trans-Planckian mass scales or
unexplained small dimensionless parameters. As a concrete example, consider
the UV completion of the CFT in terms of a gauge theory with a BZ fixed point,
see Section 2.3. In addition to consistency requirements of the COFI scenario,
$T_{R}\mathrel{\>{\hbox{\raise
1.50696pt\hbox{$<$}}\kern-3.8889pt\kern-3.8889pt\hbox{\lower
2.79857pt\hbox{$\sim$}}}\>}\Lambda_{{}_{\rm CFT}}\mathrel{\>{\hbox{\raise
1.50696pt\hbox{$<$}}\kern-3.8889pt\kern-3.8889pt\hbox{\lower
2.79857pt\hbox{$\sim$}}}\>}M_{\rm BZ}$, naturalness requires

$M_{\rm Bz}\mathrel{\>{\hbox{\raise
1.50696pt\hbox{$<$}}\kern-3.8889pt\kern-3.8889pt\hbox{\lower
2.79857pt\hbox{$\sim$}}}\>}M_{\rm Pl},~{}~{}~{}~{}\lambda_{\rm BZ}\sim 1.$
(55)

In some of the COFI dark matter scenarios, there are parts of the parameter
space where these requirements cannot be satisfied; those regions are shaded
in green in the plots of Section 3. However, it is worth keeping in mind that
these bounds are model-dependent. Any amount of tuning, or alternative UV
completions, may lead to modifications of the naturalness bound.

###  4.5 Numerical CFT bootstrap bound

One of the attractive features of the COFI theories is that the small mass
scale in the CFT sector is generated dynamically. This occurs through a
combination of cosmological phase transitions in the SM sector followed by a
slow RG running of the CFT sector, and finally dimensional transmutation
within the CFT sector triggered by the $\mathcal{O}(1)$ CFT breaking effect.
Our analysis so far has been based on an assumption that the largest breaking
of the conformal invariance is from the interaction between the SM and the CFT
sector and associated operator mixing effects. In particular, we assumed that
the CFT scalar operator ${\cal O}_{{}_{\rm CFT}}$ appearing in the interaction
does not show up on its own in the UV Lagrangian. If it did, it would make the
CFT RG run from the onset and may result in a larger value of $M_{\rm gap}$
than what we have been taking.

As explained in Section 2.2, we may assume a $\mathbb{Z}_{2}$ discrete
symmetry in the CFT sector under which the particular ${\cal O}_{{}_{\rm
CFT}}$ is odd, hence can not be added to the UV Lagrangian. However, the CFT
may contain another $\mathbb{Z}_{2}$-even scalar operator of dimension $<4$,
which may not necessarily couple to the SM but would potentially generate a
large $M_{\rm gap}$ on its own. Such an operator would generically appear in
the OPE of two of ${\cal O}_{{}_{\rm CFT}}$ operators. A useful bound on this
indeed does exist in the numerical CFT bootstrap literature Poland:2018epd ;
Poland:2011ey . The idea is that given a scalar operator with scaling
dimension $d$, the numerical CFT bootstrap provides an upper bound on the
dimension of scalar operators that enter the OPE ${\cal O}_{{}_{\rm
CFT}}\times{\cal O}_{{}_{\rm CFT}}$. This latter dimension turns out to be
$\leq 4$ if $d\lesssim 1.6$. We indicate this bound by a dashed line on the
plots of the COFI parameter space in Section 3; the parameter space below the
line is potentially problematic. This bound is, however, somewhat model
dependent and can be evaded, for example, by assuming a larger global
symmetry, e.g. $\mathbb{Z}_{4}$, in the CFT sector.

##  5 Conclusions

In this paper, we have considered a dark sector that is invariant under
conformal symmetry, broken only by a weak coupling to the Standard Model. This
coupling leads to breaking of the conformal invariance in the infrared, at a
scale $M_{\rm gap}$. Below this scale, the dark sector is described by a
hadronic phase, with the lightest meson (dark pion) playing the role of dark
matter. Within a broad range of model and cosmological parameters, the dark
matter relic density is dominated by the energy transfer from the SM plasma to
the dark sector in the conformal regime. We have labeled this scenario
“Conformal Freeze-In” (COFI). We showed that the COFI scenario provides a
viable dark matter candidate, consistent with all phenomenological
constraints, for several choices of the SM portal operator primarily
interacting with the dark sector. We conclude that a conformal dark sector
minimally coupled to the SM can naturally produce the observed dark matter.

## Acknowledgements

We are grateful to Damon Binder, Jae Hyeok Chang, Thomas Hartman, Luca
Iliesiu, Hyung Do Kim, Eric Kuflik, Ofri Telem, Eliott Rosenberg, and Yiming
Zhong for useful discussions. SH thanks Wen Han Chiu and Liantao Wang for
useful discussions and collaboration on a related subject. SH also would like
to thank Nima Afkhami-Jeddi for helpful discussions about CFTs.

M.P. is supported by the U.S. National Science Foundation through grant
PHY-2014071. S.H. was supported by the DOE grants DE-SC-0013642 and DE-
AC02-06CH11357, as well as a Hans Bethe Post-doctoral fellowship at Cornell.
G.K. is supported by the Science and Technology Facilities Council with Grant
No. ST/T000864/1.

##  Appendix A Details of Calculations in Cosmology

###  A.1 Analytical Estimates of Relic Densities

Higgs Portal:

In this section, we show a brief derivation of Eq. (21), that relates observed
dark matter relic density to parameters in the theory in the Higgs portal. In
addition, the computation for Eq. (20) is shown in more detail. Using the same
procedure, analytical results for relic density can be computed for all
portals considered in this paper, and the results for other portals are
summarised at the end without going into technical details.

In the Higgs portal case, as mentioned before, below the critical dimension
$d_{*}\,=\,5/2$, dark matter production is dominated by the Higgs decay
process. At temperatures below the electroweak phase transition, the effective
interaction between the dark sector and the SM becomes,

${\cal L}_{\mathrm{int}}=\frac{\lambda_{{}_{\rm CFT}}}{\Lambda_{{}_{\rm
CFT}}^{D}}\frac{v}{\sqrt{2}}\ h\ {\cal O}_{{}_{\rm CFT}}.$ (56)

The energy transfer rate through this process is given by Eq. (20) and can be
computed as follows:

$\displaystyle
n_{h}\langle\;\Gamma(h\rightarrow\mathrm{CFT})\;E\rangle=\iint\,\mathrm{d}\Pi_{h}\,\mathrm{d}\Pi_{\mathrm{CFT}}\,f_{h}\,(2\pi)^{4}\,\delta^{4}(p_{h}-P)\,E_{h}\,|\mathcal{M}|^{2}.\
\ $ (57)

Here and below, $P=p_{\mathrm{CFT}}$ is the momentum carried by the dark
sector. The phase space for the CFT sector is chosen to be identical to that
of “unparticles” as prescribed by Georgi in Georgi:2007ek . Using Georgi’s
notation, we have,

$\displaystyle n_{h}\langle\;\Gamma(h\rightarrow\mathrm{CFT})\;E\rangle$
$\displaystyle=\iint\frac{\mathrm{d}^{3}\vec{p}_{h}}{(2\pi)^{3}2E_{h}}\
\frac{\mathrm{d}^{4}P}{(2\pi)^{4}}\mathrm{e}^{-\beta E_{h}}(2\pi)^{4}\
\delta^{4}(p_{h}-P)\ A_{d}\ (P^{2})^{d-2}\ E_{h}\ \frac{v^{2}}{4}\
\frac{\lambda_{{}_{\rm CFT}}^{2}}{\Lambda_{{}_{\rm CFT}}^{2d-4}}$
$\displaystyle=\frac{A_{d}\;v^{2}\;\lambda_{{}_{\rm
CFT}}^{2}}{4\Lambda_{{}_{\rm
CFT}}^{2d-4}}(m_{h}^{2})^{d-2}\int\frac{\mathrm{d}^{3}\vec{p}_{h}}{2(2\pi)^{3}}\,\exp(-\beta\sqrt{|\vec{p}_{h}|^{2}+m_{h}^{2}}),$
(58)

where,

$A_{d}=\frac{16\pi^{5/2}}{(2\pi)^{2d}}\frac{\Gamma(d+1/2)}{\Gamma(d-1)\Gamma(2d)}.$
(59)

Setting $p\equiv|\vec{p}_{h}|$ and simplifying gives

$\displaystyle n_{h}\langle\;\Gamma(h\rightarrow\mathrm{CFT})\;E\rangle$
$\displaystyle=$ $\displaystyle\frac{A_{d}\;v^{2}\;\lambda_{{}_{\rm
CFT}}^{2}\;(m_{h}^{2})^{d-2}}{4\Lambda_{{}_{\rm CFT}}^{2d-4}}\int 4\pi
p^{2}\frac{\mathrm{d}p}{2(2\pi)^{3}}\,\exp(-\beta\sqrt{p^{2}+m_{h}^{2}})$ (60)
$\displaystyle=$ $\displaystyle\frac{A_{d}\;v^{2}\;\lambda_{{}_{\rm
CFT}}^{2}\;(m_{h}^{2})^{d-2}}{32\pi^{2}\Lambda_{{}_{\rm CFT}}^{2d-4}}\int
p^{2}\,\mathrm{d}p\,\,\exp(-\beta\sqrt{p^{2}+m_{h}^{2}}).$

This integral represents a Bessel function of the second kind. Additionally,
in our notation, $f_{d}=A_{d}/16\pi^{2}$. Thus, on simplifying, we get,

$n_{h}\langle\Gamma(h\to{\rm CFT})E\rangle\,=\,\frac{f_{d}\,\lambda_{{}_{\rm
CFT}}^{2}v^{2}\,m_{h}^{2(d-1)}T}{\Lambda_{{}_{\rm
CFT}}^{2d-4}}\,K_{2}(m_{h}/T).$ (61)

The CFT energy density at any point in time (as a function of the Standard
Model bath temperature) can be obtained by integrating the Boltzmann equation
given in Eq. (17). To get a simple estimate, it suffices to do this
calculation in the relativistic approximation where the Higgs is assumed to be
massless and is described by a Maxwell-Boltzmann distribution. The process
roughly starts around the electroweak scale $\sim v$ and continues till the SM
temperature reaches the Higgs mass.

In the relativistic approximation (i.e., taking the limit $m_{h}\rightarrow 0$
in the thermal average calculation), the energy transfer rate in this process
is given by,

$n_{h}\langle\Gamma(h\to{\rm CFT})E\rangle\,=\,2f_{d}\,\lambda_{{}_{\rm
CFT}}^{2}\,v^{2}\,\frac{m_{h}^{2d-4}}{\Lambda_{{}_{\rm CFT}}^{2d-4}}\,T^{3}.$
(62)

We integrate the Boltzmann equation with this collision term, ignoring the
temperature dependence of $g_{*}$ for now, and enforcing the condition that
decays are inactive above the electroweak scale. Thus, we have,

$\rho_{{}_{\rm CFT}}(T)\,=\,\frac{2M_{*}f_{d}\lambda_{{}_{\rm
CFT}}^{2}}{3\sqrt{g_{*}(T)}v}\,\left(\frac{m_{h}}{\Lambda_{{}_{\rm
CFT}}}\right)^{2d-4}\,T^{4}\left(\frac{v^{3}}{T^{3}}-1\right),$ (63)

where $M_{*}=3\sqrt{5}/(2\pi^{3/2})\,M_{\rm pl}$, comes from the definition of
Hubble as $H=\sqrt{g_{*}}\;T^{2}/M_{*}$.

At $T\sim m_{h}$, as the Higgs falls out of the thermal bath, this process
becomes exponentially suppressed, and further production of dark sector energy
can be neglected for this analysis. The energy density present in the dark
sector then redshifts like radiation ($\rho\propto a^{-4}$) until its
temperature $T_{{}_{\rm D}}$ becomes comparable to the mass of the dark matter
candidate. After this point, it redshifts like matter ($\rho\propto a^{-3}$)
as required.

Thus,

$\rho_{{}_{\rm CFT}}(m_{h})\,=\,\frac{2M_{*}f_{d}\lambda_{{}_{\rm
CFT}}^{2}}{3\sqrt{g_{*}(m_{h})}v}\,\frac{m_{h}^{2d}}{\Lambda_{{}_{\rm
CFT}}^{2d-4}}\,\left(\frac{v^{3}}{m_{h}^{3}}-1\right),$ (64)

and

$\displaystyle\rho_{{}_{\rm CFT}}(T_{m})\,=\,\frac{2M_{*}f_{d}\lambda_{{}_{\rm
CFT}}^{2}\,g_{*}(T_{m})}{3(g_{*}(m_{h}))^{3/2}v}\,\left(\frac{m_{h}}{\Lambda_{{}_{\rm
CFT}}}\right)^{2d-4}\left(\frac{v^{3}}{m_{h}^{3}}-1\right)T_{m}^{4},$ (65)

where $T_{m}$ is the SM temperature at which the dark sector temperature
($T_{{}_{\rm D}}$) drops to the mass of the dark matter candidate. We also
define the CFT energy density at this temperature as $\rho_{{}_{\rm
CFT}}\equiv A\,m_{{}_{\rm DM}}^{4}$, where A represents a model-dependent
measure of the number of degrees of freedom of the CFT (times a constant =
$\pi^{2}/30$). Thus, the relic density is given by

$\rho_{\rm DM}(T_{0})\,=\,A\,m_{{}_{\rm
DM}}^{4}\frac{g_{*}(T_{0})T_{0}^{3}}{g_{*}(T_{m})T_{m}^{3}},$ (66)

where $T_{0}$ is the current CMB temperature. Additionally, from Eq. (65),
$T_{m}$ is given by,

$\displaystyle T_{m}^{4}=A\,m_{{}_{\rm
DM}}^{4}\left[\frac{2M_{*}f_{d}\lambda_{{}_{\rm
CFT}}^{2}\,g_{*}(T_{m})}{3(g_{*}(m_{h}))^{3/2}v}\,\left(\frac{m_{h}}{\Lambda_{{}_{\rm
CFT}}}\right)^{2d-4}\left(\frac{v^{3}}{m_{h}^{3}}-1\right)\right]^{-1}$ (67)

Using Eq. (67) in Eq. (66) gives the relic density of dark matter from the
Higgs portal in terms of other parameters in the theory.

Note that we use $g_{*}(T_{0})\sim g_{*}(T_{m})\sim{\cal O}(1)$. This is a
reasonable approximation, as both temperatures are below the QCD scale.
$g_{*}(m_{h})$, denoted as just $g_{*}$ below, is approximately
$\mathcal{O}(100)$. We also replace
$\left(\frac{v^{3}}{m_{h}^{3}}-1\right)\to{\cal O}(1)$ for this order-of-
magnitude estimate. Additionally, we substitute $M_{\rm gap}$ in the equation
instead of $\lambda_{{}_{\rm CFT}}$ and $\Lambda_{{}_{\rm CFT}}$ using the
mass gap equations. Taking the ratio of $\rho_{\rm DM}(T_{0})$ and the present
critical energy density gives Eq. (21):

$\displaystyle\frac{\Omega_{{}_{\rm DM}}h^{2}}{0.1}=\left[\frac{m_{{}_{\rm
DM}}}{1\mathrm{\
MeV}}\right]\left[\frac{\left(A\,f_{d}^{3}\,g_{*}^{-9/2}\right)^{1/4}}{10^{-5}}\right]\left[\frac{\left(\frac{M_{\rm
gap}}{m_{h}}\right)^{(6-\frac{3d}{2})}}{10^{-12}}\right].$ (68)

This simple estimate is in good agreement with the results of numerical
integration of Eq. (17).

Following the same procedure, the relic density can be calculated for each of
the other three portals. These equations are given below, neglecting
derivatives of $g_{*}$, but keeping all scales intact.

For the quark and lepton portals, the primary production process is that of
fermion annihilation below the weak scale, where the Higgs is replaced by its
VEV. The thermal averaging process can be repeated for $2\to$ CFT processes
as,

$n_{1}n_{2}\langle\sigma(f_{1}f_{2}\rightarrow\mathrm{CFT})v_{rel}E\rangle=g_{f}^{2}\,\frac{\lambda_{{}_{\rm
CFT}}^{2}}{\Lambda_{{}_{\rm
CFT}}^{2d}}\frac{4d(d^{2}-1)}{(2\pi)^{2d+1}}\;v^{2}\;T^{2d+3}$ (69)

where $g_{f}$ is the number of degrees of freedom of the fermion (considered
massless in this limit).

Integrating the Boltzmann equation, we get,

$\rho_{{}_{\rm CFT}}(T)=g_{f}^{2}\frac{\lambda_{{}_{\rm
CFT}}^{2}}{\Lambda_{{}_{\rm
CFT}}^{2d}}\frac{4d(d^{2}-1)}{(2d-3)(2\pi)^{2d+1}}\;T^{4}(T_{w}^{2d-3}-T^{2d-3})$
(70)

where $T_{w}$ is the weak scale temperature.

For the gluon and electroweak portals, the results are similar, since the
dominant process is that of vector boson scattering. However, the SM operator
is dimension 4, and $T_{w}$ is replaced by $T_{R}$ since production starts
right away after reheating, and these portals depend on the UV scale of
reheating. Thus, in the gluon and electroweak portals, we have,

$\rho_{{}_{\rm CFT}}(T)=g_{V}^{2}\frac{\lambda_{{}_{\rm
CFT}}^{2}}{\Lambda_{{}_{\rm
CFT}}^{2d}}\frac{d^{2}(d^{2}-1)(d+2)}{(2d-1)(2\pi)^{2d+1}}\;T^{4}(T_{R}^{2d-1}-T^{2d-1})$
(71)

Following the same procedure as described by Eqs. (64)-(68), we get the
following equations for relic densities in other portals.

Quark Portal:

  1. 1.

First Generation Only:

$\rho_{{}_{\rm DM}}(T_{0})=m_{{}_{\rm DM}}M_{\rm
gap}^{6-3d/2}\,A^{1/4}T_{0}^{3}\left[\frac{M_{*}}{\alpha^{4}v^{4}(m_{u}+m_{d})^{2}}\frac{16\,d(d^{2}-1)}{(2d-3)(2\pi)^{2d+1}}\,(v^{2d-3}-\Lambda_{\scriptstyle\rm
QCD}^{2d-3})\right]^{3/4}\\\ \hskip 45.00006pt$

  2. 2.

Democratic:

$\rho_{{}_{\rm DM}}(T_{0})=m_{{}_{\rm DM}}M_{\rm
gap}^{6-3d/2}\,A^{1/4}T_{0}^{3}\left[\frac{M_{*}}{\alpha^{4}v^{4}m_{\rm
top}^{2}}\frac{16\,d(d^{2}-1)}{(2d-3)(2\pi)^{2d+1}}\,(v^{2d-3}-\Lambda_{\scriptstyle\rm
QCD}^{2d-3})\right]^{3/4}\\\ \hskip 45.00006pt$

  3. 3.

Minimal Flavor Violation:

$\rho_{{}_{\rm DM}}(T_{0})=m_{{}_{\rm DM}}M_{\rm
gap}^{6-3d/2}\,A^{1/4}T_{0}^{3}\left[\frac{M_{*}\,m_{b}^{2}}{\alpha^{4}v^{4}m_{\rm
top}^{4}}\frac{16\,d(d^{2}-1)}{(2d-3)(2\pi)^{2d+1}}\,(v^{2d-3}-m_{b}^{2d-3})\right]^{3/4}\\\
\hskip 45.00006pt$

Note that in the MFV flavor structure, due to the dependence of the coupling
on the fermion mass, the heaviest fermion in the thermal bath below the weak
scale contributes more to production than the other flavors. This would be the
bottom quark in the quark portal.

Lepton Portal:

  1. 1.

First Generation Only:

$\rho_{{}_{\rm DM}}(T_{0})=m_{{}_{\rm DM}}M_{\rm
gap}^{6-3d/2}\,A^{1/4}T_{0}^{3}\left[\frac{M_{*}}{\alpha^{4}v^{4}m_{e}^{2}}\frac{16\,d(d^{2}-1)}{(2d-3)(2\pi)^{2d+1}}\,(v^{2d-3}-m_{e}^{2d-3})\right]^{3/4}\\\
\hskip 45.00006pt$

  2. 2.

Democratic:

$\rho_{{}_{\rm DM}}(T_{0})=m_{{}_{\rm DM}}M_{\rm
gap}^{6-3d/2}\,A^{1/4}T_{0}^{3}\left[\frac{M_{*}}{\alpha^{4}v^{4}m_{\tau}^{2}}\frac{16\,d(d^{2}-1)}{(2d-3)(2\pi)^{2d+1}}\,(v^{2d-3}-m_{e}^{2d-3})\right]^{3/4}\\\
\hskip 45.00006pt$

  3. 3.

Minimal Flavor Violation:

$\rho_{{}_{\rm DM}}(T_{0})=m_{{}_{\rm DM}}M_{\rm
gap}^{6-3d/2}\,A^{1/4}T_{0}^{3}\left[\frac{M_{*}}{\alpha^{4}v^{4}m_{\tau}^{2}}\frac{16\,d(d^{2}-1)}{(2d-3)(2\pi)^{2d+1}}\,(v^{2d-3}-m_{\tau}^{2d-3})\right]^{3/4}\\\
\hskip 45.00006pt$

Just as in the quark portal, in the lepton MFV case, the $\tau$-lepton
contributes most to the dark matter energy density.

Gluon Portal:

$\rho_{{}_{\rm DM}}(T_{0})=m_{{}_{\rm DM}}M_{\rm
gap}^{6-3d/2}\,A^{1/4}T_{0}^{3}\left[\frac{M_{*}}{256\,\pi^{4}\alpha^{8}v^{8}}\,\frac{36\,d^{2}(d^{2}-1)(d+2)}{(2d-1)(2\pi)^{2d+1}}\,T_{R}^{2d-1}\right]^{3/4}\\\
\hskip 45.00006pt$

Hypercharge Portal:

$\rho_{{}_{\rm DM}}(T_{0})=m_{{}_{\rm DM}}M_{\rm
gap}^{6-3d/2}\,A^{1/4}T_{0}^{3}\left[\frac{M_{*}\,\cos^{4}\theta_{w}}{256\,\pi^{4}\alpha^{8}v^{8}}\,\frac{16\,d^{2}(d^{2}-1)(d+2)}{(2d-1)(2\pi)^{2d+1}}\,T_{R}^{2d-1}\right]^{3/4}\\\
\hskip 45.00006pt$

As in the Higgs portal case examined previously in this appendix, these
analytical estimates are in good agreement (with DM mass within an order of
magnitude) with the numerically integrated results shown in Figs. 5, 6, 7, 8,
9, 10, and 12. Further, the DM mass dependence on reheating temperature as
shown in Figs. 11 and Eq. (26) can be shown using these relic density
estimates.

###  A.2 Estimates for Hadronic Production

While the exact nature of IR physics in COFI models depends on the details of
confinement and the hadronic spectrum, we can still calculate the thermally
averaged cross-sections in this regime up to $\mathcal{O}(1)$ factors,
assuming a simple model as described in Section 2.2.2. In this section, we
show an example calculation for the lepton portal in the democratic flavor
scheme, since hadronic production dominates in parts of the parameter space
that produces the observed DM relic density in this model. This is one of very
few COFI models with this property, and adding the hadronic contribution is
important in this case.

As explained earlier, we model the confined/hadronic regime as containing a
cosmologically stable pseudo-Nambu-Goldstone Boson, $\chi$, which acts as dark
matter and a mediator with mass $\sim M_{\rm gap}$. Note that ${\cal
O}_{{}_{\rm CFT}}$ is a scalar, and operator matching ensures that the scalar
mediator has the dominant coupling to the SM sector. Thus we ignore any
contributions from the vector mediator (if it exists). Using Eqns. (9) and
(11) the SM-DM interaction can be written as,

$\mathcal{L}\sim\frac{\lambda_{{}_{\rm CFT}}}{\Lambda_{{}_{\rm
CFT}}^{d+d_{{}_{\rm SM}}-4}}\,\frac{M_{\rm gap}^{d-1}}{g_{\star}}\,{\cal
O}_{{}_{\rm SM}}\,\phi+\frac{g_{\star}}{M_{\rm
gap}}\,\phi\,\left(\partial\chi\right)^{2},$ (72)

where $\phi$ is the scalar mediator, $\chi$ is the DM, and $g_{\star}$ is the
coupling ($=4\pi/\sqrt{N}$ in large-N theories).

Since the hadronic processes are only relevant below the confinement scale,
the mediator with mass $\sim M_{\rm gap}$ can be integrated out and we get,

$\mathcal{L}\sim\frac{\lambda_{{}_{\rm CFT}}}{\Lambda_{{}_{\rm
CFT}}^{d+d_{{}_{\rm SM}}-4}}\,M_{\rm gap}^{d-4}\,{\cal O}_{{}_{\rm
SM}}\left(\partial\chi\right)^{2}.$ (73)

In the lepton portal with democratic flavor scheme, using the relation between
the coupling and gap-scale, this simplifies to888Recall that the dominant
deformation that leads to confinement and generation of a mass gap in the
lepton portal is from radiative mixing with the Higgs operator.,

$\mathcal{L}\sim\frac{1}{\alpha^{2}v^{2}m_{\rm\scriptstyle
tot}}\,\bar{e}e\left(\partial\chi\right)^{2},$ (74)

where $m_{\rm\scriptstyle tot}=m_{e}+m_{\mu}+m_{\tau}$ is the sum of masses of
all the leptons running in the loop that generates the deformation of the CFT.
Since these hadronic processes occur at very low energies ($T<M_{\rm gap}$),
we only need to consider electrons as the other leptons are no longer in the
bath. The Higgs is also replaced by its VEV below the weak scale.

The energy transfer rate is then given by,

$\displaystyle n_{e}^{2}\;\langle$
$\displaystyle\sigma(e^{+}(p_{1})\,e^{-}(p_{2})\rightarrow\chi(p_{3})\,\chi(p_{4}))\;E\;\rangle$
$\displaystyle=$
$\displaystyle\iiiint\,\mathrm{d}\Pi_{e_{1}}\,\mathrm{d}\Pi_{e_{2}}\,\mathrm{d}\Pi_{\chi_{1}}\,\mathrm{d}\Pi_{\chi_{2}}\,f_{e_{1}}\,f_{e_{2}}\,(2\pi)^{4}\,\delta^{4}(\Sigma
p)\,(E_{1}+E_{2})\,|\mathcal{M}|^{2}$ $\displaystyle=$
$\displaystyle\prod_{i}\int\frac{d^{3}p_{i}}{(2\pi)^{3}E_{i}}\,e^{-\beta
E_{1}}\,e^{-\beta E_{2}}\,(2\pi)^{4}\delta^{4}(\Sigma
p)\,\left(\frac{1}{\alpha^{2}v^{2}m_{\rm\scriptstyle
tot}}\right)^{2}\,(E_{1}+E_{2})\,(p_{3}.p_{4})^{2}\,(p_{1}.p_{2})$
$\displaystyle=$
$\displaystyle\left(\frac{1}{\alpha^{2}v^{2}m_{\rm\scriptstyle
tot}}\right)^{2}\prod_{i}\int\frac{d^{3}p_{i}}{(2\pi)^{3}E_{i}}\,e^{-\beta(E_{1}+E_{2})}\,(2\pi)^{4}\delta^{4}(\Sigma
p)\,(E_{1}+E_{2})\,(p_{1}.p_{2})^{3},$ (75)

where $\Sigma p=p_{1}+p_{2}-p_{3}-p_{4}$, and in the last line, the particles
involved are approximated to be relativistic/massless.

Computing the phase space integrals and using the delta function as usual, one
gets,

$n_{e}^{2}\;\langle\sigma(e^{+}(p_{1})\,e^{-}(p_{2})\rightarrow\chi(p_{3})\,\chi(p_{4}))\;E\;\rangle=\left(\frac{1}{\alpha^{2}\,v^{2}\,m_{\rm\scriptstyle
tot}}\right)^{2}\,\frac{3240}{\pi^{8}}\,T^{13}.$ (76)

This ‘collision term’ can be plugged into the Boltzmann equation and
integrated to get the energy density of dark matter states ($\chi$) produced
in the hadronic phase:

$\rho_{{}_{\rm
DM}}(T)=\frac{M_{*}}{\sqrt{g(T)}}\left(\frac{1}{\alpha^{2}\,v^{2}\,m_{\rm\scriptstyle
tot}}\right)^{2}\frac{3240}{\pi^{8}}\,\frac{T^{4}}{5}\,(M_{\rm
gap}^{5}-T^{5}),$ (77)

where dark matter is assumed to redshift as radiation, and $M_{*}$ is as
defined in the previous subsection. The interaction term in the Lagrangian is
very irrelevant, and the power of temperature in this expression is high, as
one might expect. Thus, this process is dominant at the temperature it starts,
and we can use $(M_{\rm gap}^{5}-T^{5})\rightarrow M_{\rm gap}^{5}$ for
calculating the final contribution of the hadronic production process.
Additionally, we dropped terms of $\mathcal{O}(m_{e})$, since $m_{e}<M_{\rm
gap}$.

For most COFI models, hadronic production is very negligible. It is only
relevant in lepton portal models with significant couplings to electrons since
only electrons are light enough to persist in the thermal bath at such low
temperatures (unlike QCD states and Higgs bosons). The same argument could be
used in the case of the hypercharge portal, since photons are always present
in the SM plasma; however, the CFT-phase energy density production in this
case is proportional to positive powers of the reheating temperature which
easily overwhelms the hadronic production that is proportional to powers of
$M_{\rm gap}$. Thus, regions of parameter space with non-negligible hadronic
production exist only in IR-dominated regime ($d<1.5$) in the first-generation
and democratic flavor schemes in the lepton portal model.

##  Appendix B Derivation of Energy Loss Rates

###  B.1 Compton scattering in MS and HB

Figure 15: Feynman diagram for Compton scattering in the lepton portal.

In Raffelt:1996wa , the production of a neutrino ($\bar{\nu}\nu$) pair from
the Compton scattering of non-relativistic and non-degenerate electrons is
studied and the rate is999If electrons are degenerate, the rate is suppressed
by a factor  $F_{\rm deg}\sim\frac{3E_{F}T}{p_{F}^{2}}.$ However, Compton
scattering is important only in MS and HB starts in which electrons are not
degenerate. Also, throughout the calculation, we use the criterion that if the
photon plasma mass $\omega_{p}$ is less than $3T$ then we can neglect the
plasma mass effects. For more details, see Raffelt:1996wa

$\sigma(\bar{\nu}\nu)\sim\frac{\alpha}{8\pi^{2}}G_{F}^{2}m_{e}^{4}\left(\frac{\omega}{m_{e}}\right)^{4},$
(78)

where $\omega$ is the energy carried by the $\bar{\nu}\nu$ pair, which is
roughly $\omega\sim T$ (up to $\mathcal{O}(1)$). Using this result and the
effective theory Eq. (49), the cross-section for the production of
$\chi\chi$-pair is estimated to be

$\sigma(\chi\chi)\sim\frac{2\pi^{2}\alpha}{\left(\sum\kappa_{\ell}m_{\ell}\right)^{2}\Lambda_{\rm
SM}^{4}}m_{e}^{4}\left(\frac{\omega}{m_{e}}\right)^{6}.$ (79)

For non-relativistic and non-degenerate electrons, they are almost at rest,
and the energy loss rate per unit volume $Q$ can be approximated as

$Q\sim n_{e}\int
2\frac{d^{3}k_{\gamma}}{(2\pi)^{3}}\;\frac{\omega\,\sigma(\chi\chi)}{e^{\omega/T}-1},$
(80)

where $k_{\gamma}$ is the photon momentum and the factor 2 is for the two
photon polarization degrees of freedom. The computation proceeds by writing

$\sigma(\chi\chi)=\sigma_{*}\left(\frac{\omega}{m_{e}}\right)^{p},\;\sigma_{*}\equiv\frac{2\pi^{2}\alpha}{\left(\sum\kappa_{\ell}m_{\ell}\right)^{2}\Lambda_{\rm
SM}^{4}}m_{e}^{4}$ (81)

and $p=6$ for our case. Explicit computation given in Raffelt:1996wa  shows
that

$Q\sim\frac{(p+3)!\zeta(p+4)}{\pi^{2}}\frac{\sigma_{*}n_{e}T^{p+4}}{m_{e}^{p}}$
(82)

where $\zeta(n)$ is the Riemann zeta function. We can further use
$n_{e}=Y_{e}\frac{\rho}{m_{u}}$ (where $Y_{e}$ is the electron number fraction
per baryon and $m_{u}=1.661\times 10^{-24}\;{\rm g}$ is the atomic mass unit)
to finally get the energy loss rate per mass $\epsilon$:

$\epsilon=\frac{Q}{\rho}\sim\frac{9!2\alpha}{\left(\sum\kappa_{\ell}m_{\ell}\right)^{2}\Lambda_{\rm
SM}^{4}}\frac{Y_{e}}{m_{u}m_{e}^{2}}T^{10}.$ (83)

Here, we used the expression for $\sigma_{*}$ and $p=6$.

###  B.2 Bremsstrahlung from an electron in RGB

Figure 16: Feynman diagrams for relevant Bremsstrahlung processes.

The expression for $\epsilon$ for the production of a $\bar{\nu}\nu$ pair from
a degenerate electron line is given in Raffelt:1996wa  and is (with
$C_{V}\approx C_{A}=1$ and $F_{+}=1,F_{-}=0$ and assuming a single species of
nuclei of charge $Z$ and atomic mass $A$)

$\epsilon(\bar{\nu}\nu)\approx\frac{2\pi\alpha^{2}}{189}\left(\frac{Z^{2}}{Am_{u}}\right)G_{F}^{2}T^{6}.$
(84)

One reasonable estimation of the rate for the case of $\chi\chi$ production
can be made by comparing the effective coupling between the two cases. The
matching condition is

$\frac{G_{F}}{\sqrt{2}}\to\frac{16\pi^{2}\omega}{\left(\sum\kappa_{\ell}m_{\ell}\right)\Lambda_{\rm
SM}^{2}}$ (85)

where we have included one factor of $\omega$ to make up the right dimension.
That $\omega$, rather than $E_{F}$, is the right factor even in the degenerate
situation, is understood as follows. Unlike in $e^{-}e^{+}$ annihilation where
the final state energy is of the order $E_{F}$, in Bremsstrahlung, the final
states carry only $\omega\sim T$ because of final state Pauli-blocking for the
electron: essentially, while the electron has energy $\sim E_{F}$, the amount
of energy change by the momentum transfer, i.e. displacement in the Fermi
surface with radius $E_{F}$, is limited to $\sim T$. In the end, we get

$\epsilon(\chi\chi)\sim\frac{\pi\alpha^{2}}{189}\left(\frac{Z^{2}}{Am_{u}}\right)\left(\frac{16\pi^{2}}{\left(\sum\kappa_{\ell}m_{\ell}\right)\Lambda_{\rm
SM}^{2}}\right)^{2}T^{8}.$ (86)

We emphasize that our estimation is at the level of $\mathcal{O}(1)$ or even
an order of magnitude, due to non-trivial combinatoric factors and precise
values for $\omega/T$ and so on, that our computation does not take into
account.

###  B.3 Bremsstrahlung from a nucleon

This process is most relevant for quark-, gluon-, and Higgs-portal. For the
part of parameter space relevant for the relic density, the production is in
the form of hadrons of confined CFT in MS, HB, and RGB, while in SN, the final
state is CFT state.

#### Hadronic final state: MS, HB, RGB

We first derive rates for the case when the final states are hadrons of
confined CFT. The strategy is the same as before. We take the expressions
obtained for $\bar{\nu}\nu$ production and estimate for the $\chi\chi$
production by making necessary modifications. In Section 4 of Raffelt:1996wa ,
$Q(\bar{\nu}\nu)$ is shown to be

$Q(\bar{\nu}\nu)=\left(\frac{\sum_{q}C_{q}^{(N)}G_{F}}{\sqrt{2}}\right)^{2}\frac{n_{B}}{20\pi^{4}}\int_{0}^{\infty}d\omega\;\omega^{6}\;S_{\sigma}(-\omega)$
(87)

where in the non-degenerate (for nucleons) limit

$\displaystyle
S_{\sigma}(\omega)=\frac{\Gamma_{\sigma}}{\omega^{2}}s(\omega/T)\times\left\\{\begin{array}[]{ll}1\hskip
28.45274pt{\rm for}\;\omega>0\\\ e^{\omega/T}\;\;\;\;{\rm
for}\;\omega<0\end{array}\right.$ (90)
$\displaystyle\Gamma_{\sigma}=4\sqrt{\pi}\alpha_{\pi}^{2}n_{B}T^{1/2}m_{N}^{-5/2}$
(91) $\displaystyle s(x)\approx\sqrt{1+|x|\pi/4}$
$\displaystyle\alpha_{\pi}=\frac{m_{N}^{2}}{\pi m_{\pi}^{2}}\approx 15.$

The convention is that $\omega<0$ corresponds to the energy taken away from
the medium and $n_{B}=n_{p}+n_{n}$ is the nucleon number density. It may be
worth clarifying that the factor $C_{q}^{(N)}$ comes from the matching of
quark-neutrino four fermion interaction to nucleon-neutrino four Fermi
interaction. Hence, $C_{q}^{(N)}$ appearing here is literally the same as the
one in the effective theory Eq. (37). Performing the integration, Raffelt
showed that

$Q(\bar{\nu}\nu)\approx\frac{2048}{385\,\pi^{7/2}}\left(\sum_{q}C_{q}^{(N)}\right)^{2}G_{F}^{2}\,\alpha_{\pi}^{2}\,\frac{n_{B}^{2}}{m_{N}^{5/2}}T^{11/2}.$
(92)

The rate for $\chi\chi$ production can be obtained from this by (i) matching
the effective coupling and (ii) taking into account difference in the
integration. For quark- and gluon-portal, the matching of coupling becomes

$\displaystyle\text{Quark-Portal
:}\;\left(\frac{\sum_{q}C_{q}^{(N)}G_{F}}{\sqrt{2}}\right)^{2}\;\leftrightarrow\;\left(\frac{16\pi^{2}\sum_{q}\kappa_{q}C_{q}^{(N)}}{\left(\sum_{q}\kappa_{q}m_{q}\right)\Lambda_{\rm
SM}^{2}}\right)^{2}\left(\frac{\omega}{2}\right)^{2}$ (93)
$\displaystyle\text{Gluon-Portal
:}\;\left(\frac{\sum_{q}C_{q}^{(N)}G_{F}}{\sqrt{2}}\right)^{2}\;\leftrightarrow\;\left(C_{G}^{(N)}\frac{16\pi^{2}}{\Lambda_{\rm
SM}^{4}}\right)^{2}\left(\frac{\omega}{2}\right)^{2}$ (94)

where $\omega\sim T$ and the form factors $C_{q}^{(N)}$ and $C_{G}^{(N)}$ are
defined in Eq. (34) and (35), respectively.

Next, in the case of $\bar{\nu}\nu$ production, the integration in the
expression of $Q(\bar{\nu}\nu)$ is

$\int_{0}^{\infty}d\omega\;e^{-\omega/T}\;\omega^{4}\;s(-\omega/T)\equiv
C_{\bar{\nu}\nu}T^{5}$ (95)

In the case of $\chi\chi$ production, on the other hand, it is given by (other
than the $\omega$-independent part of effective coupling factors for which we
have already shown the matching)

$\frac{1}{4}\int_{0}^{\infty}d\omega\;e^{-\omega/T}\;\omega^{6}\;s(-\omega/T)\equiv
C_{\chi\chi}T^{5}\left(\frac{T}{2}\right)^{2}.$ (96)

Identifying $(T/2)^{2}=(\omega/2)^{2}$ in the matching of the coupling, we see
that a slightly more accurate ratio of the $\epsilon$’s requires a factor of
$C_{\chi\chi}/C_{\bar{\nu}\nu}$. More explicitly, we get

$\displaystyle\text{Quark-Portal
:}\;Q(\chi\chi)=Q(\bar{\nu}\nu)\frac{\left(\frac{16\pi^{2}\sum_{q}\kappa_{q}C_{q}^{(N)}}{\left(\sum_{q}\kappa_{q}m_{q}\right)\Lambda_{\rm
SM}^{2}}\right)^{2}\left(\frac{T}{2}\right)^{2}}{\left(\frac{\sum_{q}C_{q}^{(N)}G_{F}}{\sqrt{2}}\right)^{2}}\frac{C_{\chi\chi}}{C_{\bar{\nu}\nu}}$
(97) $\displaystyle\text{Gluon-Portal
:}\;Q(\chi\chi)=Q(\bar{\nu}\nu)\frac{\left(C_{G}^{(N)}\frac{16\pi^{2}}{\Lambda_{\rm
SM}^{4}}\right)^{2}\left(\frac{T}{2}\right)^{2}}{\left(\frac{\sum_{q}C_{q}^{(N)}G_{F}}{\sqrt{2}}\right)^{2}}\frac{C_{\chi\chi}}{C_{\bar{\nu}\nu}}$
(98)

where $Q(\bar{\nu}\nu)$ is given in Eq. (92). We make two comments. First,
since both $\bar{\nu}\nu$ and $\chi\chi$ productions involve two particles,
the phase space factor for the two cases are the same. The internal degrees of
freedom, however, can differ. In addition, while the neutrino production is
predominantly via axial coupling Raffelt:1996wa , the production of $\chi\chi$
pair is likely to be from a vector coupling. We do not think the latter two
factors will generate more than an order of magnitude effect. It would be
interesting to carry out improved stellar cooling computations for COFI, which
we leave for future investigations.

#### CFT final state: SN

Our strategy is based on the fact that the ratio $\epsilon(\text{\tiny
CFT})/\epsilon(\phi)$, the energy loss rate for COFI to that of real scalars,
can be estimated to a reasonable accuracy. To this end, (i) we need to compare
the effective couplings, (ii) use powers of $(m_{N}T)$ to make up the correct
dimensions, and (iii) powers of $(2\pi)$ to take into account difference in
phase space. In addition, one can improve the estimation by including (iv)
ratio of internal degrees of freedom of the energy carrying final states and
(v) ratio of thermal averaged energy $\langle\omega\rangle$ of the new physics
states. Regarding point (ii), we recall that nucleons are non-relativistic,
implying $p^{2}\approx 2m_{N}T$, and this in turn means that the
characteristic size of the energy transfer is $\omega\sim\sqrt{m_{N}T}$. Next,
the correct factors of $(2\pi)$ for the phase space: for each extra particle
we have $\frac{1}{(2\pi)^{3}}\times(2\pi)=\frac{1}{(2\pi)^{2}}$ where the
first factor is the naive one from the phase space integral measure and the
second factor of $(2\pi)$ is the result of extra angular integration. This
seemingly naive argument works even for the case of CFT final state. In this
case, we recall that the phase space associated with a dimension $d$ CFT
operator can be thought of as $d$ massless particles, and so we can estimate
the phase space factor as
$\sim\frac{1}{(2\pi)^{3d}}\times(2\pi)^{d-1}=\frac{1}{(2\pi)^{2d+1}}$, again
the first factor for $d$ number of naive phase space factor and the second
factor for $(d-1)$ extra angular integration. The final answer indeed agrees
with explicit computations once we adopt the Georgi’s choice for the
unparticle phase space density. To summarize, below we will estimate the ratio
$\epsilon(\text{\tiny CFT})/\epsilon(\phi)$ using

$\frac{\epsilon(\text{\tiny CFT})}{\epsilon(\phi)}=\frac{Q(\text{\tiny
CFT})}{Q(\phi)}\sim\frac{G_{\rm
eff}^{2}(m_{N}T)^{r}}{g^{2}}\frac{1}{(2\pi)^{2d-2}}\frac{\text{dof}_{\rm\scriptscriptstyle
CFT}}{\text{dof}_{\phi}}\frac{\langle\omega\rangle_{\rm\scriptscriptstyle
CFT}}{\langle\omega\rangle_{\phi}}$ (99)

where the exponent $r$ depends on the model and $g$ is the Yukawa coupling,
$\sim g\phi\bar{\psi}_{N}\psi_{N}$. Before we show our results, it is
instructive to present the known cases, confirming the validity of our
analysis scheme. To this end, let us compute the ratio
$\epsilon(\bar{\nu}\nu)/\epsilon_{a}$, i.e. the ratio of rate of neutrino-pair
production to that of axions. Without including the ratio of
$\langle\omega\rangle$’s it is given by

$\frac{\epsilon(\bar{\nu}\nu)}{\epsilon_{a}}=\frac{\left(\sum_{q}C_{q}^{(N)}G_{F}\right)^{2}(m_{N}T)^{2}}{g_{a}^{2}}\frac{1}{(2\pi)^{2}}\left(\frac{3\times
2\times
2}{1}\right)=\frac{3}{\pi^{2}}\frac{\left(\sum_{q}C_{q}^{(N)}G_{F}\right)^{2}(m_{N}T)^{2}}{g_{a}^{2}}$
(100)

where 3, 2, 2 are respectively family, spin, and $SU(2)_{L}$-doublet degrees
of freedom of neutrino. The same ratio can also be computed using explicit
expressions for $\epsilon(\bar{\nu}\nu)$ and $\epsilon_{a}$ (Eqns. (4.8),
(4.10), (4.23), and (4.24) in Raffelt:1996wa ) and the outcome is about twice
larger than our estimation. This factor of two, however, can also be explained
since $\langle\omega_{\bar{\nu}\nu}\rangle/\langle\omega_{a}\rangle$ is
numerically about 2.

We now present our results. First of all, $Q(\phi)$ for a degenerate nucleon
medium is given by (Eq. (4.13) in Raffelt:1996wa )

$Q(\phi)=\frac{g^{2}\alpha_{\pi}^{2}}{4\pi}\frac{44}{15^{3}}\left(\frac{T}{m_{N}}\right)^{4}p_{F}^{5}\,G_{\phi}(m_{\pi}/p_{F})$
(101)

where $\alpha_{\pi}$ is defined in Eq. (91) and numerically
$G_{\phi}(m_{\pi}/p_{f})\approx 0.8$. Also, since nucleons are borderline
degenerate in SN, we can use $p_{F}\approx m_{N}T$ (which in a sense we have
been using so far already). We can now use Eq. (99) with the effective
couplings

$\displaystyle\text{Higgs-Portal :}\;G_{\rm
eff}=C_{G}^{(N)}\left(\frac{\alpha_{s}}{6\sqrt{2}\pi}\right)\left(\frac{M_{\rm
gap}^{4-d}}{v^{2}m_{h}^{2}}\right)$ (102) $\displaystyle\text{Quark-Portal
:}\;G_{\rm eff}=\sum_{q}\kappa_{q}C_{q}^{(N)}\frac{16\pi^{2}M_{\rm
gap}^{4-d}}{\left(\sum_{q}\kappa_{q}m_{q}\right)\Lambda_{\rm SM}^{2}}$ (103)
$\displaystyle\text{Gluon-Portal :}\;G_{\rm
eff}=C_{G}^{(N)}\frac{16\pi^{2}M_{\rm gap}^{4-d}}{\Lambda_{\rm SM}^{4}}$ (104)

and $r=d-1$ for all cases since the mass dimension of $G_{\rm eff}$ is $1-d$
for all three cases. The factors $\frac{\text{dof}_{\rm\scriptscriptstyle
CFT}}{\text{dof}_{\phi}}$ and
$\frac{\langle\omega\rangle_{\rm\scriptscriptstyle
CFT}}{\langle\omega\rangle_{\phi}}$ depend on the details of the CFT and on a
general grounds they are expected to be within $1\sim d$.

###  B.4 Electron-positron annihilation

Figure 17: Feynman diagrams for relevant annihilation processes in the lepton
and hypercharge portals, in the CFT phase $(T>M_{\rm gap})$ .

The process of interest is the annihilation of $e^{-}e^{+}$ to a CFT final
state and this is relevant for the lepton-portal and hypercharge-portal models
in the SN. In lepton-portal it is via tree-level coupling, and for the
hypercharge-portal it is a one-loop generated coupling.

#### Lepton-portal

From the Lagrangian

$\mathcal{L}\sim\frac{\lambda_{{}_{\rm CFT}}v}{\sqrt{2}\Lambda_{{}_{\rm
CFT}}^{d}}\,\bar{e}e\,\mathcal{O}_{\rm CFT}$ (105)

we can estimate the energy transfer rate in the exactly the same way we do for
the freeze-in calculation

$\langle\sigma vE\rangle\sim\left(\frac{\lambda_{{}_{\rm
CFT}}v}{\sqrt{2}\Lambda_{{}_{\rm
CFT}}^{d}}\right)^{2}\left(\frac{4\pi^{4}d(d^{2}-1)}{(2\pi)^{2d+1}}\right)E_{F}^{2d-3},$
(106)

where $E_{F}\approx 344{\rm MeV}$ is the electron Fermi energy. From this we
get the energy loss rate per volume

$Q(\text{\tiny CFT})\sim n_{e^{-}}n_{e^{+}}\langle\sigma vE\rangle$ (107)

and hence $\epsilon$ by $\epsilon=Q/\rho$. For the number density of electrons
in SN, we use $n_{e^{-}}\approx 1.8\times 10^{38}\;{\rm cm^{-3}}$, while for
the positrons, we note that the process
$e^{-}e^{+}\leftrightarrow\gamma\gamma$ imposes the relationship among
chemical potentials, $\mu_{e^{+}}=-\mu_{e^{-}}$ and we know
$\mu_{e^{-}}\approx 344\;{\rm MeV}$. Using this, the number density for the
positron can be shown to be

$n_{e^{+}}=2\int\frac{d^{3}p}{(2\pi)^{3}}\frac{1}{e^{(E+\mu_{e^{-}})/T}+1}\approx
e^{-\beta\mu_{e^{-}}}\times n_{\rm th}$ (108)

with $n_{\rm th}$ being the equilibrium number density at $T=T_{\rm
SN}\,(=1/\beta)$ with Boltzmann distribution. We see that the positron number
density is suppressed compared to the thermal density by the factor
$e^{-\beta\mu_{e^{-}}}$.

#### Hypercharge-portal

The computation goes through the exact same steps as in the lepton-portal. To
this end, we first derive an effective action by computing the loop-diagram
shown in Fig. 17. The result is estimated to be

$\mathcal{L}\sim\left(\frac{\lambda_{{}_{\rm CFT}}}{\Lambda_{{}_{\rm
CFT}}^{d}}\right)\left(\frac{2e^{2}m_{e}}{\pi^{2}}\log\left(\frac{\Lambda_{\rm
SM}}{E_{F}}\right)\right)\bar{e}e{\cal O}_{{}_{\rm CFT}}$ (109)

The appearance of the log is due to the massless particle (i.e. photon)
running in the loop, and we used the external momentum to be $E_{F}$
appropriate for the degenerate electrons in the SN core. To get the energy
loss rate per mass, we now simply need to replace the effective coupling in
the lepton-portal computation:

$\frac{\lambda_{{}_{\rm CFT}}v}{\sqrt{2}\Lambda_{{}_{\rm
CFT}}^{d}}\to\left(\frac{\lambda_{{}_{\rm CFT}}}{\Lambda_{{}_{\rm
CFT}}^{d}}\right)\left(\frac{2e^{2}m_{e}}{\pi^{2}}\log\left(\frac{\Lambda_{\rm
SM}}{E_{F}}\right)\right).$ (110)

###  B.5 Photon annihilation

This process is relevant for the hypercharge portal at the core of SN. The
photon number density is that of a thermal Boltzmann distribution,

$n_{\gamma}\approx\frac{2\,\zeta(3)}{\pi^{2}}\,T_{\rm SN}^{3}$ (111)

and since the plasma frequency $\omega_{p}\sim 19\;{\rm MeV}$ is less than
$3\,T_{\rm SN}\sim(60\,\text{--}\,80)\;{\rm MeV}$, we ignore plasma mass
effects. The energy loss rate per volume is estimated to be

$Q(\text{\tiny CFT})\sim n_{\gamma}^{2}\langle\sigma vE\rangle\sim
n_{\gamma}^{2}\;\left(\frac{\lambda_{{}_{\rm
CFT}}\cos^{2}\theta_{w}}{\Lambda_{{}_{\rm
CFT}}^{d}}\right)^{2}\left(\frac{16\,d^{2}(d^{2}-1)(d+2)}{(2d-1)(2\pi)^{2d+1}}\right)T_{\rm
SN}^{2d-1}.$ (112)

This process is the dominant process for supernova cooling in hypercharge-
portal. To illustrate this, we compute the ratio

$\frac{Q(e^{-}e^{+}\to\text{\tiny CFT})}{Q(\gamma\gamma\to\text{\tiny
CFT})}\sim e^{-E_{F}/T_{\rm SN}}\left(\frac{E_{F}}{T_{\rm
SN}}\right)^{2d}\left(\frac{m_{e}}{T_{\rm
SN}}\right)^{2}\left(\frac{8\alpha\log\left(\frac{\Lambda_{\rm
SM}}{E_{F}}\right)}{\cos^{2}\theta_{w}}\right)\ll 1.$ (113)

Numerically, for $\Lambda_{\rm SM}=1$ TeV, the above ratio is $\lesssim
10^{-2}$ for $1\leq d\leq 3$.

###  B.6 Trapping in Supernovae

In this section, we describe how to estimate the cross-section required to
evaluate the trapping of hadronic states of the confined CFT at the core of
SN. For concreteness sake, we focus on the lepton-portal case, where trapping
is important. (See Dreiner:2013mua  for discussion of supernova constraints on
dark sectors.)

Figure 18: Production of CFT state via $e^{-}e^{+}$-annihilation in SN
followed by a hadronization into composite dark matter ($\chi$) plus
$\mathbb{Z}_{2}$-even final states (denoted as $X$) in lepton portal model.

Let us first discuss production of a pair of dark matter states $\chi$ near
the core of the supernova. We assume that the CFT is described by a large-$N$
gauge theory. When such a theory confines in the IR, we can use large-$N$
analysis which we follow. Schematically, when $M_{\rm
gap}<T_{\scriptstyle\mathrm{SN}}$, an annihilation of $e^{-}e^{+}$ produces
directly the state associated with ${\cal O}_{{}_{\rm CFT}}$ which can be
thought of as the “partonic” state of a confining CFT. Once these “partonic”
CFT states are produced, they will go through the hadronization process,
somewhat similar to the QCD jets. The situation is shown schematically in Fig.
18. Some of the hadronic states then can travel out of the supernova,
resulting in an extra mechanism for its cooling. Assuming $\mathcal{O}(1)$
fraction of energy is transferred to the dark matter state $\chi$ (which is
consistent with our freeze-in calculations), we can make a rough estimation as
follows. Here, we assume that the dark matter $\chi$ is a goldstone boson
created by a current operator $J_{\mu}$ of a broken global symmetry. We are
interested in the rate for ${\cal O}_{{}_{\rm CFT}}$ to result in a pair of
$J_{\mu}$’s which in turn “hadronizes” into the dark matter $\chi$ and other
hadronic states. This information will be used below to estimate the cross-
section responsible for trapping in SN. The rate for ${\cal O}_{{}_{\rm CFT}}$
to turn into a pair of $J_{\mu}$’s is encoded in the OPE (Operator Product
Expansion) coefficient

$J^{\mu}J_{\mu}\sim c\,{\cal O}_{{}_{\rm CFT}}+\cdots.$ (114)

where $c$ is the OPE coefficient which carries a scaling dimension of $6-d$
(recall that the conserved current has dimension 3 and acquires no anomalous
dimensions). In addition, the probability for a current $J_{\mu}$ to produce a
single $\chi$ can be summarized in an interpolation relation of the form (in
large-$N$ limit)

$J_{\mu}\sim\frac{1}{g_{\star}}M_{\rm gap}\partial_{\mu}\chi$ (115)

where the factor of $g_{\star}\sim\frac{4\pi}{\sqrt{N}}$ is inserted to be
consistent with large-$N$ behavior

$\langle JJ\rangle\sim\frac{N}{16\pi^{2}}\sim\frac{1}{g_{\star}^{2}}.$ (116)

The other factors are fixed by dimensional analysis and the fact that $\chi$
is a goldstone boson associated with a broken current $J_{\mu}$.

The matrix element for the pair production diagram in Fig. 18 is then
estimated to be

$\mathcal{M}\sim\left(\frac{v}{\lambda^{d}}\right)\cdot\frac{1}{c}\cdot\left(\frac{M_{\rm
gap}p_{\mu}}{g_{\star}}\right)^{2}\cdot\left(\bar{u}u\right)$ (117)

We wish to determine $g_{\star}$-dependence of the OPE coefficient $c$ which
is needed to figure out correct $g_{\star}$-counting for the rates. While
there is no fully rigorous and systematic means to answer this question,
“matching” between the above estimation and the fully-hadronic picture may be
used to get a reasonable assessment. To this end, we first note that at
leading order in $1/N$-expansion the hadronic cubic interaction vertex is
$\Gamma_{3}\sim g_{\star}$. This is understood by noting that

$\langle{\cal O}_{{}_{\rm
CFT}}JJ\rangle\sim\frac{N}{16\pi^{2}}\sim\frac{1}{g_{\star}^{2}},\;\;\;\;{\cal
O}_{{}_{\rm CFT}}\sim\frac{M_{\rm gap}^{d-1}}{g_{\star}}\phi$ (118)

where $\phi$ is a scalar meson interpolated by ${\cal O}_{{}_{\rm CFT}}$ with
mass of the order $M_{\rm gap}$. Using these together with Eq. (115), we get,

$\langle{\cal O}_{{}_{\rm
CFT}}JJ\rangle\sim\frac{1}{g_{\star}^{2}}\sim\left(\frac{M_{\rm
gap}^{d-1}}{g_{\star}}\right)\left(\frac{M_{\rm
gap}}{g_{\star}}\right)^{2}\cdot\Gamma_{3}\;\;\rightarrow\;\;\Gamma_{3}\sim
g_{\star}.$ (119)

With this information at hand, the matching to the fully-hadronic picture
gives

$\mathcal{M}\sim\left(\frac{v}{\Lambda^{d}}\right)\cdot\left(\frac{M_{\rm
gap}^{d-1}}{g_{\star}}\right)\cdot\frac{1}{M_{\rm gap}^{2}}\left(\frac{M_{\rm
gap}p_{\mu}}{g_{\star}}\right)^{2}\cdot\Gamma_{3}\cdot\left(\bar{u}u\right).$
(120)

Comparing this to Eq. (117) finally shows that

$c\sim M_{\rm gap}^{d-6}\;g_{\star}^{0}.$ (121)

Figure 19: Diagram relevant for trapping in SN, and the assumption of
factorization.

With these preparations, we now discuss trapping. Again, with a simplifying
assumption that most of the CFT energy is processed to the DM state $\chi$,
the relevant picture is: we have DM particles produced in the core of SN and
we are interested in the cross-section of $\chi+e^{-}\to\chi+e^{-}+X$, where
$X$ denotes any other collectively $\mathbb{Z}_{2}$-even states in the final
state. To the extent that the leading contribution comes from
$\chi+e^{-}\to\chi+e^{-}$,101010A simple argument based on phase space
suppression seems to support this assumption, although multiplicity of the
diagrams and any unknown non-perturbative physics could in principle
invalidate the claim. Here, we simply assume, which is certainly enough for
the stellar cooling bound, that at least $\mathcal{O}(1)$ contribution comes
from the simple $2\to 2$ process. a reasonable estimate is possible assuming
the factorization shown in Fig. 19. From the discussions given above about the
OPE, the lower part of the diagram is given by

$\langle\chi_{2}|{\cal O}_{{}_{\rm
CFT}}|\chi_{1}\rangle\sim\frac{1}{c}\langle\chi_{2}|J_{\mu}J^{\mu}|\chi_{1}\rangle\sim\frac{M_{\rm
gap}^{d-4}}{\hat{c}\,g_{\star}^{2}}\,p_{\chi_{1}}\cdot p_{\chi_{2}}$ (122)

where we have introduced a dimensionless quantity $\hat{c}$ defined by
$c=\hat{c}\,M_{\rm gap}^{6-d}$. The full matrix element is then computed to be

$\mathcal{M}\sim\left(\frac{\lambda
v}{\Lambda^{d}\sqrt{2}}\right)\left(\frac{M_{\rm
gap}^{d-4}}{\hat{c}\,g_{\star}^{2}}\,p_{\chi_{1}}\cdot
p_{\chi_{2}}\right)\bar{u}(k_{1})u(k_{2})$ (123)

with $k_{1,2}$ being the four-momentum of the incoming and outgoing electrons,
respectively. The cross-section can finally be estimated and one gets,

$\sigma\sim\frac{1}{8\pi\gamma\,\hat{c}\,g_{\star}^{4}}\frac{E_{F}^{4}}{\left(\sum_{\ell}\kappa_{\ell}m_{\ell}\;\alpha^{2}v^{2}\right)^{2}}$
(124)

where $E_{F}$ is the electron Fermi energy and $\gamma$, defined by
$E_{\chi}=\gamma\,E_{F}$, encodes the fraction of energy carried by the DM
$\chi$ upon CFT-hadronization. $v$ is the VEV of the Higgs and $\alpha$ is
defined by $\Lambda_{\rm\scriptscriptstyle SM}=\alpha\,4\pi v$. Also, we used
the formula for the $M_{\rm gap}$ given in Section 2. The mean free path is
obtained from

$\lambda_{\chi}=\frac{1}{n_{e}\sigma}$ (125)

and we use the optical depth criterion

$\int_{r_{0}}^{R_{c}}\frac{dr}{\lambda_{\chi}}\gtrsim\frac{2}{3}\;\;\;\Rightarrow\;\;\;\frac{3}{2}\frac{0.1R_{c}}{\lambda_{\chi}}\gtrsim
1\;\;\;\;\;\;\text{(trapped)}$ (126)

to assess the possibility of the trapping. To get the final expression, we
used $r_{0}\approx 0.9R_{c}$ and $R_{c}\approx 13\;{\rm km}$ is the radius of
the core Dreiner:2013mua .

In the lepton portal, models of both first generation and democratic flavor
schemes have trapping cross-sections many orders of magnitude above the
optical depth criterion, and the dark matter particles produced are completely
trapped. Thus, there is no relevant supernova constraint.

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