# Scale Invariant FIMP Miracle

Basabendu Barman ID<EMAIL_ADDRESS>Centro de Investigaciones,
Universidad Antonio Nariño
Carrera 3 este # 47A-15, Bogotá, Colombia  Anish Ghoshal ID
<EMAIL_ADDRESS>Institute of Theoretical Physics, Faculty of
Physics, University of Warsaw,ul. Pasteura 5, 02-093 Warsaw, Poland  INFN -
Sezione Roma “Tor Vergata”, Via della Ricerca Scientifica 1, 00133, Rome,
Italy

###### Abstract

We study the freeze-in production of vector dark matter (DM) in a classically
scale invariant theory, where the Standard Model (SM) is augmented with an
abelian $U(1)_{X}$ gauge symmetry that is spontaneously broken due to the non-
zero vacuum expectation value (VEV) of a scalar charged under the $U(1)_{X}$.
Generating the SM Higgs mass at 1-loop level, it leaves only two parameters in
the dark sector, namely, the DM mass $m_{X}$ and the gauge coupling $g_{X}$ as
independent, and supplement with a naturally light dark scalar particle. We
show, for $g_{X}\sim\mathcal{O}\left(10^{-5}\right)$, it is possible to
produce the DM X out-of-equilibrium in the early Universe, satisfying the
observed relic abundance for $m_{X}\sim\mathcal{O}\left(\text{TeV}\right)$,
which in turn also determines the scalar mixing angle
$\sin\theta\sim\mathcal{O}\left(10^{-5}\right)$. The presence of such
naturally light scalar mediator with tiny mixing with the SM, opens up the
possibility for the model to be explored in direct search experiment, which
otherwise is insensitive to standard freeze-in scenarios. Moreover we show
that even with such feeble couplings, necessary for the DM freeze-in, the
scenario is testable in several light dark sector searches (e.g., in DUNE and
in FASER-II), satisfying constraints from the observed relic abundance as well
as big bang nucleosynthesis (BBN). Particularly, we find, regions of the
parameter space with $m_{X}$ $\gtrsim 1.8$ TeV are insensitive to direct
detection probes but still can become accessible in lifetime frontier
searches, courtesy to the underlying scale invariance of the theory.

PI/UAN-2021-698FT

###### Contents

  1. I Introduction
  2. II The Scale-Invariant Model
    1. II.0.1 Symmetries and particle content
    2. II.0.2 Minima of the scalar potential
    3. II.0.3 The scalar sector
    4. II.0.4 1-loop effective potential
    5. II.1 Perturbative constraints & RGE
  3. III Freeze-in production of the Dark Matter
    1. III.1 Freeze-in yield
    2. III.2 Viable dark matter parameter space
      1. III.2.1 Bound from the Big Bang nucleosynthesis (BBN)
      2. III.2.2 Dark matter direct detection
  4. IV Laboratory Phenomenology
  5. V Discussion and Conclusion
  6. VI Acknowledgement
  7. A Annihilation cross-sections for freeze-in production

##  I Introduction

The discovery of a Higgs-like boson of mass $\sim$ 125 GeV at the Large Hadron
Collider Aad et al. (2012a); Chatrchyan et al. (2012) has provided the last
missing piece of the Standard Model (SM) particle physics. Although the Higgs
mechanism Englert and Brout (1964); Higgs (1964); Guralnik et al. (1964)
satisfactorily provides masses to all the candidates within the SM in a gauge
invariant manner, the Higgs itself carries an ad-hoc negative mass term which
is provided by hand at the weak scale. As such, it is natural to assume that
the SM Higgs mass will be unstabilized against the Planck scale (or any cut-
off scale present in the UV) by quadratically diverging quantum corrections
unless large fine-tuned cancellation of the associated quadratic divergences
is imposed Susskind (1979). In fact, seeking resolutions to this so called
“naturalness problem” has been the major driving force behind numerous beyond
the SM (BSM) extensions, starting from the good old supersymmetry (SUSY),
compositeness, extra dimensions (for a review, see Ref. Giudice (2019)) 111In
context to non-local QFT with higher-derivatives, motivated from string field
theories, it has been found that the naturalness issue is somewhat reduced in
these theories due to the presence of UV-fixed point Ghoshal et al. (2018);
Buoninfante et al. (2019); Ghoshal (2019); Ghoshal et al. (2021).. In order to
stabilize the electroweak scale against the radiative corrections we can
promote scale invariance to be a symmetry of the action at the classical level
which eliminates the $\mu^{2}$ mass term or any other dimensionful parameters
from the theory222In a larger context, in BSM scenarios, assuming no scale is
fundamental in nature and all mass scales including the EW and Planck scales
are generated dynamically can be seen as direction of model-building for the
hierarchy problem in the SM Adler (1982); Coleman and Weinberg (1973a); Salvio
and Strumia (2014); Einhorn and Jones (2015); Einhorn and Jones (2016a, b);
Foot et al. (2008); Alexander-Nunneley and Pilaftsis (2010a); Englert et al.
(2013); Hambye and Strumia (2013a); Farzinnia et al. (2013); Altmannshofer et
al. (2015); Holthausen et al. (2013); Salvio and Strumia (2014); Einhorn and
Jones (2015); Kannike et al. (2015a); Farzinnia and Kouwn (2016); Kannike et
al. (2016). In context to cosmology scale invariance provides naturally flat
inflationary potentials Khoze (2013); Kannike et al. (2014); Rinaldi et al.
(2015); Salvio and Strumia (2014); Kannike et al. (2015a, b); Barrie et al.
(2016); Tambalo and Rinaldi (2017), and also leads to strong first-order phase
transitions in early universe and consequently high amplitude detectable
gravitational wave (GW) signals due to dominance of thermal corrections in
absence of tree-level mass terms Jaeckel et al. (2016); Marzola et al. (2017);
Iso et al. (2017); Baldes and Garcia-Cely (2019); Prokopec et al. (2019);
Brdar et al. (2019a); Marzo et al. (2019); Ghoshal and Salvio (2020). . In the
SM one can radiatively generate nonzero Higgs mass and spontaneous electroweak
symmetry breaking (EWSB) as pointed out by Coleman and Weinberg (CW) in their
seminal work in Ref. Coleman and Weinberg (1973b) but it making the Higgs
potential unbounded from below at one-loop given the experimentally observed
masses of top quark and electroweak gauge bosons. To resolve this problem we
thus need to look beyond the minimal SM, where additional bosonic
contributions333Recently the Neutrino Option idea considers threshold
corrections (as an alternative to bosonic corrections) to generate the Higgs
mass via fermionic loop Brivio and Trott (2017).. can lead to
phenomenologically successful models. A simple way to address this issue is to
generate the mass scale in a “hidden sector” and then transmit it to the SM
thus avoiding the direct CW relation between the SM gauge boson masses and the
mass of the SM Higgs. Such attempts have already been made with or without
introducing a suitable dark matter (DM) candidate in Refs. Foot et al. (2007);
Iso et al. (2009a, b); Okada and Orikasa (2012); Farzinnia et al. (2013);
Englert et al. (2013); Hambye and Strumia (2013b); Kang (2015a, b); Hambye and
Strumia (2013a); Kubo et al. (2014); Das et al. (2017); Humbert et al. (2015a,
b); Haba et al. (2016, 2015); Das et al. (2016); Hambye et al. (2018a); Kubo
et al. (2019); Brdar et al. (2019b); Yaser Ayazi and Mohamadnejad (2019);
Mohamadnejad (2020); Okada and Raut (2021); Mohapatra and Okada (2020a); Okada
et al. (2020); Mohapatra and Okada (2020b); Gialamas et al. (2021); Li and
Zhang (2021).

The SM also fails to provide a viable DM candidate, whose existence is already
been proven beyond any doubt from several astrophysical Zwicky (1933, 1937);
Rubin and Ford (1970); Clowe et al. (2006) and cosmological Hu and Dodelson
(2002); Aghanim et al. (2020a) evidences (for a review, see, e.g. Refs.
Jungman et al. (1996); Bertone et al. (2005); Feng (2010)). Therefore, one has
to look beyond the realms of the SM to explain the DM puzzle, if we assume it
is of particle origin. There already exists a very attractive scenario known
as the weakly interacting massive particle (WIMP), where the DM candidate
remains in thermal equilibrium in the early Universe down to a temperature at
which its number density becomes Boltzmann suppressed, and then the DM
chemically decouples from the thermal bath. This is the vanilla freeze-out
mechanism Kolb and Turner (1990), where the observed DM abundance is set
typically by the weak scale interaction strength which popularly goes by the
name “WIMP miracle.” Various experimental efforts like the direct detection
Aprile et al. (2018, 2020); Zhang et al. (2019) experiments typically search
for the recoil of nuclei from collisions with WIMPs from the local halo of DM
Goodman and Witten (1985); Ahlen et al. (1987), while attempts have also been
made to search for DM signal in collider (see, for example, Refs. Penning
(2018); Kahlhoefer (2017) and the references therein) experiments. However, no
significant excess have been found till date in either of these experiments
which can guarantee DM discovery. The observed DM abundance may have been
generated also out of equilibrium by the so-called freeze-in mechanism
McDonald (2002); Hall et al. (2010); Bernal et al. (2017). In this scenario,
the DM particle couples to the visible sector very feebly, so that chemical
equilibrium is never achieved. The DM particles are then produced by the decay
or annihilation of the bath particles, until the production ceased due to
cooling of the bath temperature below the relevant mass scale connecting the
DM to the visible sector. Typically, freeze-in can further be subdivided into
(i) Infra-red (IR), where the DM yield gradually builds up with time and
becomes important at lower temperature McDonald (2002); Hall et al. (2010);
Chu et al. (2012); Bernal et al. (2017); Duch et al. (2018); Biswas et al.
(2019); Barman et al. (2020a); Heeba et al. (2018), and (ii) Ultra-violate
(UV) Hall et al. (2010); Elahi et al. (2015); Chen and Kang (2018); Bernal et
al. (2019); Biswas et al. (2020); Barman et al. (2020b, c); Bernal et al.
(2020a, b); Barman et al. (2021) where the DM is dominantly produced at very
high temperature (as large as the reheating temperature of the Universe) and
the yield immediately saturates to a constant value. Because of the tiny
coupling strength, the DM particle produced via the freeze-in mechanism are
dubbed as the feebly interacting massive particle (FIMP), as opposed to the
WIMP. Even with such tiny portal couplings, in the presence of mediators which
connect the visible and the dark sectors, FIMPs may give rise to signatures
both in direct search Hambye et al. (2018b); Heeba and Kahlhoefer (2020) and
in collider Yaser Ayazi et al. (2016); Molinaro et al. (2014); Arcadi et al.
(2015, 2014); Hessler et al. (2017); Bélanger et al. (2019) experiments.
However, it is important to note here, none of these signatures directly
involve the coupling of the DM with the visible sector, that sets the DM relic
abundance, but too small to be probed at experiments.

In this work we show that a desirable UV-completion principle in particle
theory, like the scale invariance, can on one hand dynamically generate the
scales of electroweak and dark matter physics, thereby reducing the
naturalness problem of the SM Higgs mass, on the other hand, can also
incorporate a testable freeze-in DM scenario. To establish this, here we
consider the SM being extended by a $U(1)_{X}$ gauge symmetry, under which the
DM $(X_{\mu})$ transforms as an abelian gauge boson. The DM becomes massive
once this gauge symmetry is spontaneously broken by the vacuum expectation
value (VEV) of a new scalar $S$ that carries a non-zero $U(1)_{X}$ charge.
This gives rise to the typical Higgs portal model, where the DM communicates
with the visible sector via the portal coupling, with a strength proportional
to the scalar mixing. The DM renders stability as it carries an odd
$\mathbb{Z}_{2}$ charge, while all the SM fields are even under the
$\mathbb{Z}_{2}$. It is important to note that the conformal444We have used
the two terms “scale” invariance and “conformal” invariance interchangeably in
this paper since they are known to be classically equivalent in any four-
dimensional unitary and renormalizable field theory Gross and Wess (1970);
Callan et al. (1970); Coleman and Jackiw (1971). gauge extension to the SM
leads to extremely economical model building as one is left with only two
independent parameters in the model, one of which gets fixed entirely via
relic density requirements, thereby leaving very predictive light scalar
singlet hunt in several planned and upcoming intensity and lifetime frontier
experiments as a novel and direct probe of freeze-in scenarios. We find, in
the present scenario, correct DM abundance is achieved via freeze-in for dark
gauge coupling $g_{X}\sim 10^{-5}$ for dark matter mass $m_{X}\lesssim$ TeV,
ensuring the DM interaction rate always falls below the Hubble expansion rate.
The scale invariance also gives rise to a naturally light scalar with mass
$m_{2}\sim\mathcal{O}(\text{MeV})\ll m_{X}$ . This results in an enhanced
recoil rate in DM scattering experiments leading to a detectable DM signal
which is in sharp contrast to the usual FIMP paradigm, where DM direct
detection is simply impossible due to extreme smallness of the coupling of the
DM with the visible sector as mentioned earlier 555The exchange of such a
light mediator can also induce large self-interaction, a possibility that is
of much interest because it may alleviate the possible shortcomings of
collisionless dark matter on small scales (core/cusp Spergel and Steinhardt
(2000) and too-big-to-fail Kaplinghat et al. (2014) problems, see Tulin and Yu
(2018) for a review)..

The crucial outcome of our analysis is the experimental testability of the
FIMP model in low energy intensity frontier experiments, which typically
search for light dark sector particles, and are capable of probing extremely
small mixing angle. This is facilitated by the presence of the light scalar in
this model. We find, in order to satisfy bounds from relic density, spin-
independent direct detection and big bang nucleosynthesis (BBN), the mixing
within the scalar sector has to be $\sin\theta\sim
g_{X}\,\sim\mathcal{O}(10^{-5})$. Such a mixing angle is indeed within the
reach of experiments like DUNE Acciarri et al. (2015); Berryman et al. (2020),
FASER-II Feng et al. (2018); Ariga et al. (2019a, 2018, b), PS191 Bernardi et
al. (1986); Gorbunov et al. (2021), DarkQuest-Phase2 Batell et al. (2021),
MATHUSLA Curtin et al. (2019) and SHiP Anelli et al. (2015). Most importantly,
since the mixing $\theta$ is no more a free parameter (due to the scale
invariance of the theory), it rather can be parametrized in terms of $m_{X}$
and $g_{X}$, hence, the new gauge coupling that determines the freeze-in
abundance, is directly being probed at plethora of current and upcoming
experimental facilities. To the best of our knowledge, this has not been
realized before in the context of freeze-in DM. Particularly, we find out that
scale invariance uniquely dictates the mixing
$\sin\theta\sim\mathcal{O}\left(10^{-5}\right)$ that predicts potential
discovery at the laboratory experiments. This leads us to dub the scenario as
the “Scale Invariant FIMP Miracle.”

The paper is organized as follows. In Sec. II we have explained the underlying
model, elaborating its scale invariant features. The Renormalization Group
Equation (RGE) running of various couplings appearing in the theory are
discussed under subsection II.1. We then move on to the detailed discussion of
the DM parameter space in Sec. III, where the details of freeze-in yield are
mentioned in subsection III.1, together with the illustration of available
parameter space in subsection III.2. In Sec. IV we show the reach of several
experiments in probing the DM parameter space. Finally, in Sec. V we conclude
with an engaging discussion on our analysis.

##  II The Scale-Invariant Model

####  II.0.1 Symmetries and particle content

We consider a dark $U\left(1\right)_{X}$ gauge symmetry and a complex scalar
$S$ which is neutral under the SM gauge group but has unit charge under
$U\left(1\right)_{X}$666The phenomenology of abelian vector boson DM has been
widely studied in the literature both in the context of freeze-out and freeze-
in production without considering scale invariance (see, for example, Refs.
Servant and Tait (2003); Birkedal et al. (2006); Farzan and Akbarieh (2012);
Baek et al. (2013); Bian et al. (2014); Choi et al. (2013); Baek et al.
(2014a, b, c); Ko et al. (2014); Baek et al. (2014d); Ko and Tang (2015); Duch
et al. (2015); Beniwal et al. (2016); Kamon et al. (2017); Duch and
Grzadkowski (2017); Duch et al. (2018); Arcadi et al. (2017); Baek and Yu
(2018); Barman et al. (2020c)). In presence of scale invariance the
phenomenology of WIMP-like vector DM has also been studied Hambye and Strumia
(2013b); Carone and Ramos (2013); Khoze et al. (2014); Yaser Ayazi et al.
(2016); Karam and Tamvakis (2015, 2016); Khoze and Plascencia (2016); Baldes
and Garcia-Cely (2019); Yaser Ayazi and Mohamadnejad (2019), while FIMP case
has been discussed in Heikinheimo et al. (2017); Okada et al. (2020)..
Furthermore we impose a discrete symmetry $\mathbb{Z}_{2}$, under which all
the SM fields transform as even while the new fields transform as odd, such
that

$\begin{split}&X_{\mu}\to-X_{\mu};~{}~{}S\to S^{\star}.\end{split}$ (1)

. This forbids the gauge kinetic mixing with the SM $U(1)_{Y}$ gauge boson
$B_{\mu}$ ensuring the stability of $X_{\mu}$ vector which is our dark matter
(DM) candidate777In the absence of the $\mathbb{Z}_{2}$ symmetry, the dark
gauge boson can still account for all of the DM relic abundance if the kinetic
mixing is of the order $\epsilon\lesssim\mathcal{O}\left(10^{-8}\right)$ for
DM masses below twice the electron mass Arias et al. (2012); Caputo et al.
(2021), else the cosmological stability condition requires even smaller values
$(\lesssim 10^{-15})$ of the kinetic mixing parameter Bloch et al. (2017); Lin
(2019).. The relevant part of the total Lagrangian is given by

$\mathcal{L}\supset-\frac{1}{4}X_{\mu\nu}X^{\mu\nu}+\left|D_{\mu}S\right|^{2}-V\left(H,S\right)$
(2)

where $H$ is the $SU(2)_{L}$ scalar doublet and

$D_{\mu}=\partial_{\mu}+ig_{X}X_{\mu}$ (3)

is the covariant derivative of $S$ with $g_{X}$ as the new gauge coupling.
Since we assume classical scale invariance and renormalizibility, the only
terms allowed in the total effective scalar potential

$V\left(H,S\right)_{\text{cl}}=\lambda_{H}\left(H^{\dagger}H\right)^{2}+\lambda_{S}\left|S\right|^{4}-\lambda_{HS}\left(H^{\dagger}H\right)\left|S\right|^{2}.$
(4)

The dark and the visible sectors communicate via the renormalizable
$\lambda_{HS}$ coupling with the limit $\lambda_{HS}\to 0$ corresponds to the
case where the hidden sector completely decoupled from the visible sector.

####  II.0.2 Minima of the scalar potential

The stability of the potential can be verified by completing the square

$\begin{split}&V\left(H,S\right)=\lambda_{H}\Bigl{(}H^{\dagger}H-\frac{\lambda_{HS}}{2\lambda_{H}}\left|S\right|^{2}\Bigr{)}^{2}+\left|S\right|^{4}\Bigl{(}\lambda_{S}-\frac{\lambda_{HS}^{2}}{4\lambda_{H}}\Bigr{)}\end{split}$
(5)

which shows the potential is stable as long as

$4\lambda_{H}\lambda_{S}-\lambda_{HS}^{2}>0.$ (6)

.

Now, in the unitary gauge, the imaginary component of $S$ can be absorbed as
the longitudinal component of $X_{\mu}$. The tree level potential looks

$\displaystyle
V_{\text{tree}}=\frac{\lambda_{H}}{4}h^{4}-\frac{\lambda_{HS}}{4}h^{2}\,s^{2}+\frac{\lambda_{S}}{4}s^{4}$
(7)

where we write

$S=\frac{s}{\sqrt{2}}\,,H=\Bigl{\\{}0,\frac{h}{\sqrt{2}}\Bigr{\\}}^{T}.$ (8)

We now construct the Hessian matrix as:

$\mathcal{H}\equiv\frac{\partial^{2}V_{\text{cl}}}{\partial h_{i}\partial
h_{j}}.$ (9)

Then, the necessary and sufficient conditions for local minimum of
$V_{\text{cl}}$, corresponding to the vacuum: $\langle
S\rangle=v_{s}\,,\langle H\rangle=v_{h}$ are

$\displaystyle\frac{\partial V_{\text{cl}}}{\partial
h_{i}}\Big{|}_{v_{s},v_{h}}=0$ (10)
$\displaystyle\frac{\partial^{2}V_{\text{cl}}}{\partial^{2}h_{i}}\Big{|}_{v_{s},v_{h}}>0$
$\displaystyle\text{det}\Bigl{(}\mathcal{H}\left(v_{s},v_{h}\right)\Bigr{)}>0,$

where ‘det’ stands for the determinant of the Hessian matrix. These conditions
give rise to, for non-vanishing vacuum expectation values (VEVs)

$\lambda_{H}\,\lambda_{s}=\frac{1}{4}\lambda_{HS}^{2}$

and we obtain a relation between the VEVs888This relation establishes the
phenomenon of dimensional transmutation Foot et al. (2008).:

$\frac{v_{h}^{2}}{v_{s}^{2}}=\frac{\lambda_{HS}}{2\lambda_{H}}$ (11)

which defines a “flat direction”, along which $V_{\text{cl}}=0$ Gildener and
Weinberg (1976); Foot et al. (2007, 2008); Alexander-Nunneley and Pilaftsis
(2010b); Farzinnia et al. (2013); Chataignier et al. (2018); Mohamadnejad
(2020). Further including one-loop corrections will lift this flat direction
and generate the true physical vacuum (corresponding to the spontaneous
breaking of classical scale invariance). Since the classical potential is zero
along that direction, the one-loop correction necessarily dominates there.
Now, the first two relations in Eq. (10) require

$\lambda_{H}>0\,,\lambda_{s}>0\,,\lambda_{HS}>0,$ (12)

which also satisfies Eq. (6). However, the determinant of the Hessian turns
out to be zero, making the second derivative test inconclusive, and the point
$\\{v_{s},v_{h}\\}$ could be any of a minimum, maximum or saddle point. We
find, since at the minimum of the 1-loop effective potential
$V_{\text{cl}}\geq 0$ and $V_{\text{eff}}^{\text{1-loop}}<0$, the minimum of
$V_{\text{eff}}^{\text{1-loop}}$ along the flat direction (given by Eq. (11)),
where $V_{\text{tree}}=0$, is a global minimum of the full potential Yaser
Ayazi and Mohamadnejad (2019); Mohamadnejad (2020). Therefore, spontaneous
symmetry breaking (SSB) indeed occurs and we expand the fields around the
minima

$\begin{split}&S=\frac{1}{\sqrt{2}}\left(s+v_{s}\right),~{}~{}H=\frac{1}{\sqrt{2}}\Big{\\{}0,h+v_{h}\Big{\\}}^{T}.\end{split}$
(13)

As $S$ receives a non-vanishing VEV, the mass of the $U\left(1\right)_{X}$
gauge boson i.e., the DM can be expressed as

$\displaystyle m_{X}=g_{X}v_{s}\equiv
g_{X}\,\sqrt{2\lambda_{H}/\lambda_{HS}}\,v_{h}$ (14)

which implies, in order to have $m_{X}\sim\mathcal{O}\left(\text{TeV}\right)$
with $g_{X}\sim\mathcal{O}\left(10^{-5}\right)$, which will be required for
freeze-in, one needs $v_{s}\gg v_{h}\equiv v_{\text{EW}}\simeq 246$ GeV.

####  II.0.3 The scalar sector

To calculate the tree-level masses, we expand the potential in Eq. (7) around
the vacuum (Eq. (13)), and construct the mass matrix in the weak basis:

$\mathcal{M}^{2}=\left(\begin{array}[]{cc}2v_{h}^{2}\lambda_{H}&-v_{h}v_{s}\lambda_{HS}\\\
-v_{h}v_{s}\lambda_{HS}&2v_{s}^{2}\lambda_{s}\\\ \end{array}\right).$ (15)

We can then rotate it to the physical (mass) the basis via

$\begin{pmatrix}h_{1}\\\
h_{2}\end{pmatrix}=\begin{pmatrix}\cos\theta&-\sin\theta\\\
\sin\theta&\cos\theta\end{pmatrix}\,\begin{pmatrix}h\\\ s\end{pmatrix}$ (16)

where the mixing angle is given by

$\tan
2\theta=\frac{2v_{h}\,v_{s}}{v_{h}^{2}-v_{s}^{2}}\implies\sin\theta=\frac{v_{h}}{\sqrt{v_{h}^{2}+v_{s}^{2}}}\approx\frac{v_{h}}{v_{s}}$
(17)

for $v_{s}\gg v_{h}$. In tree-level, the field $h_{2}$ being along the flat
direction, is massless999This field with vanishing zeroth-order mass, is
dubbed as “scalon” Gildener and Weinberg (1976).. This can be readily seen by
obtaining the mass eigenvalues of the diagonalized mass matrix along the flat
direction using the mixing angle Eq. (17):

$m_{h_{1}}^{2}=v_{h}^{2}\,\left(2\lambda_{H}+\lambda_{HS}\right)\,,m_{h_{2}}^{2}=0$
(18)

where we note, in the limit $\lambda_{HS}\to 0$ we recover the standard
electroweak Higgs state. The massless scalar, on the other hand, will acquire
its radiative mass along the flat direction à la Coleman-Weinberg Coleman and
Weinberg (1973b), and thus becomes a pesudo-Nambu-Goldstone boson (pNGB) at
quantum level. The field $h_{1}$, on the other hand, is perpendicular to the
flat direction which we identify as the SM-like Higgs observed at the LHC with
a mass of $m_{h_{1}}=125$ GeV Mohamadnejad (2020). After spontaneous symmetry
breaking, we can express the couplings of the scalar potential in terms of
physical masses and mixings as

$\lambda_{H}=\frac{m_{h_{1}}^{2}}{2\,v_{h}^{2}}\cos^{2}\theta\,,\lambda_{HS}=-\frac{m_{h_{1}}^{2}}{2v_{h}\,v_{s}}\sin
2\theta\,,\lambda_{s}=\frac{m_{h_{1}}^{2}}{2v_{s}^{2}}\sin^{2}\theta$ (19)

while the VEV $v_{s}$ can be expressed as $v_{s}=m_{X}/g_{X}$ following Eq.
(14). Note that, the scale invariance helps to choose minimum number of
parameters, and we choose $\\{m_{X},g_{X}\\}$ as the free parameters for
further analysis. All other relevant parameters appearing in the theory can be
recasted in terms of $m_{X}$ and $g_{X}$:

$\displaystyle\lambda_{H}=\frac{m_{h_{1}}^{2}}{2v_{h}^{2}\left[1+\left(g_{X}/m_{X}\right)^{2}\,v_{h}^{2}\right]}\,,\lambda_{S}=\frac{\left(g_{X}/m_{X}\right)^{4}\,m_{h_{1}}^{2}v_{h}^{2}}{2\left[1+\left(g_{X}/m_{X}\right)^{2}\,v_{h}^{2}\right]}\,,\lambda_{HS}=\frac{g_{X}^{2}\left(m_{h_{1}}/m_{X}\right)^{2}}{1+\left(g_{X}/m_{X}\right)^{2}\,v_{h}^{2}}.$
(20)

We again remind the reader about the fact that all the renormalizable
couplings in the scalar potential are expressed in terms of the two free
parameters of the theory, thanks to the underlying classical scale invariance.

####  II.0.4 1-loop effective potential

The scale invariance of the theory gives rise to massless scalar field in the
classical level. One loop correction then breaks the scale invariance giving
mass to the massless eigenstate $h_{2}$. Following the approach of Gildener
and Weinberg Gildener and Weinberg (1976), the 1-loop effetive potential can
be approximately written as Foot et al. (2007, 2008); Farzinnia et al. (2013);
Yaser Ayazi and Mohamadnejad (2019); Mohamadnejad (2020); Kannike et al.
(2021)

$V_{\text{eff}}^{\text{1-loop}}=\alpha h_{2}^{4}+\beta
h_{2}^{4}\log\frac{h_{2}^{2}}{\mu^{2}}$ (21)

with $\alpha,\beta$ as dimensionless constants

$\alpha=\frac{1}{64\pi^{2}\,v^{4}}\sum_{j}g_{j}m_{j}^{4}\log\frac{m_{j}^{2}}{v^{2}}\,,\beta=\frac{1}{64\pi^{2}\,v^{4}}\sum_{j}\,g_{j}\,m_{j}^{4},$
(22)

where $g_{j}$ and $m_{j}$ are the tree-level mass and the internal degrees of
freedom of the $j^{\text{th}}$ particle, $v^{2}=v_{s}^{2}+v_{h}^{2}$ and $\mu$
is the renormalization scale. Minimizing Eq. (21) we find that the potential
has a non-trivial stationary point at

$\mu=v\,\exp\Biggl{(}\frac{\alpha}{2\beta}+\frac{1}{4}\Biggr{)}.$ (23)

Note that, this shows, the scale of the symmetry breaking is set by the
renormalization scale $\mu$. The stationary point is a minimum as long as
$\beta$ is positive definite. The 1-loop potential can now be re-written
utilizing Eq. (23) as

$V_{\text{eff}}^{\text{1-loop}}=\beta\,h_{2}^{4}\Biggl{[}\log\frac{h_{2}^{2}}{v^{2}}-\frac{1}{2}\Biggr{]}.$
(24)

With this we can now express the mass of $h_{2}$ as Coleman and Weinberg
(1973b); Gildener and Weinberg (1976)

$m_{h_{2}}^{2}=\frac{d^{2}V_{\text{eff}}^{\text{1-loop}}}{dh_{2}^{2}}\Big{|}_{v}=8\beta
v^{2}$ (25)

where again we see for $\beta>0$ this is positive definite. Considering
contributions from all standard and non-standard particles, we can write

$m_{h_{2}}^{2}=\frac{v_{h}^{4}}{8\pi^{2}v^{2}}\Bigl{(}\lambda_{H}^{4}+\frac{3}{8}g_{2}^{4}+\frac{3}{16}g_{2}^{4}\left(g_{2}^{2}+g_{1}^{2}\right)^{2}+3g_{X}^{4}\left(v_{s}/v_{h}\right)^{4}-3y_{t}^{4}\Bigr{)}$
(26)

where $g_{2}$ and $g_{1}$ are the gauge coupling corresponding to the SM
groups $SU(2)_{L}$ and $U(1)_{Y}$ gauge respectively, and $y_{t}$ is the SM
top Yukawa coupling101010Recent measurement Sirunyan et al. (2020) yields a
best fit value of $y_{t}=1.16^{+0.24}_{-0.35}$. Unless otherwise mentioned, we
have considered $y_{t}=1$.. Also note, the fermion contribution appears with a
relative sign. From Eq. (26) it is very important to note the role of the
$U(1)_{X}$ gauge boson, without which $m_{h_{2}}^{2}<0$ (or equivalently
$\beta<0$). Before concluding this section, let us summarize the present
model. Aside from the massive dark gauge boson, the scalar particle spectrum
consists of the CP-even state $h_{1}$, and an additional CP-even scalar
$h_{2}$, which is a pNGB of scale symmetry breaking, with radiatively induced
mass.

###  II.1 Perturbative constraints & RGE

There are three extra renormalizable couplings in this model on top of the SM
Higgs quartic coupling $\lambda_{H}$

$\\{\lambda_{S},\,\lambda_{HS},\,g_{X}\\}.$ (27)

The 1-loop $\beta$-functions extracted using PyR@TE Sartore and Schienbein
(2021), are given by

Figure 1:  RG evolution of different couplings appearing in the theory. The
Higgs self-coupling turns negative at $\mu\gtrsim 10^{9}$ GeV, signalling that
the SM Higgs potential becomes unstable below the Planck scale. We choose
$m_{t}=173.2$ GeV.

$\displaystyle\left(4\pi\right)^{2}\,\beta\left(\lambda_{H}\right)=24\lambda_{H}^{2}+\lambda_{HS}^{2}-3g_{1}^{2}\lambda_{H}-9g_{2}^{2}\lambda_{H}+\frac{3}{8}g_{1}^{4}+\frac{3}{4}g_{1}^{2}g_{2}^{2}+\frac{9}{8}g_{2}^{4}+12\lambda_{H}y_{t}^{2}-6y_{t}^{4}$
(28)
$\displaystyle\left(4\pi\right)^{2}\,\beta\left(\lambda_{S}\right)=20\lambda_{S}^{2}+2\lambda_{HS}^{2}-12g_{X}^{2}\lambda_{S}+6g_{X}^{4}$
$\displaystyle\left(4\pi\right)^{2}\,\beta\left(\lambda_{HS}\right)=12\lambda_{H}\lambda_{HS}+8\lambda_{HS}\lambda_{S}-4\lambda_{HS}^{2}-\frac{3}{2}g_{1}^{2}\lambda_{HS}-6g_{X}^{2}\lambda_{HS}-\frac{9}{2}g_{2}^{2}\lambda_{HS}+6y_{t}^{2}$
$\displaystyle\left(4\pi\right)^{2}\,\beta\left(g_{X}\right)=\frac{1}{3}g_{X}^{3}$

where we define $\beta\left(X\right)\equiv\mu\,\frac{dX}{d\mu}$. The
renormalization group (RG) equation for the top Yukawa coupling $y_{t}$ is

$\left(4\pi\right)^{2}\,\beta\left(y_{t}\right)=-8g_{3}^{2}y_{t}-\frac{9}{4}g_{2}^{2}y_{t}-\frac{17}{12}g_{1}^{2}y_{t}+\frac{9}{2}y_{t}^{3}.$
(29)

To solve the RG equations and determine the RG evolution of the couplings of
our models, we specify the initial conditions for the SM coupling constants at
$m_{t}$ (top Yukawa coupling $y_{t}$) and the SM gauge couplings initial
values are taken from Buttazzo et al. (2013). Having thus specified the
initial conditions for all couplings at the low scale $\mu=m_{t}$, we run them
up to the high scale $\mu=M_{\text{pl}}$. As we see from Eq. (28),
$\beta\left(\lambda_{S}\right)\sim+g_{X}^{4}$,
$\beta\left(\lambda_{HS}\right)\sim-g_{X}^{2}$ and
$\beta\left(g_{X}\right)\sim g_{X}^{3}$, hence for a very small $g_{X}$, which
is required for freeze-in production of the DM, the dark gauge coupling almost
remains fixed over the scale $\mu$, which also influences the running of the
coupling $\lambda_{HS}$. Thus, although $\lambda_{HS}$ contributes positively
to the beta function of $\lambda_{H}$, but that does not help to improve the
Higgs vacuum.

##  III Freeze-in production of the Dark Matter

The $\mathbb{Z}_{2}$-odd gauge boson of the dark $U(1)_{X}$ gauge group is the
DM candidate for the present model. In this section we aim to explore the
viable parameter space where the DM can be produced out of equilibrium from
the SM bath via freeze-in mechanism. As we have established in the model
framework, the minimum mass of the $U(1)_{X}$ gauge boson, as decided by the
scale invariance of the theory, is no less than 240 GeV. This clearly prevents
the DM production from decay of the scalar states and only 2-to-2 annihilation
of the bath particles are feasible way to account for the DM production.

Figure 2:  Feynman graphs for DM yield from 2-to-2 annihilation of the SM
particles. Here ‘SM’ stands for all the SM fermions, scalar and gauge bosons.

The Feynman diagrams that contribute to the freeze-in production of the DM
from the scattering of bath particles are shown in Fig. 2. We implemented this
model in LanHEP Semenov (2009) and utilized the model files to calculate
cross-sections via CalcHEP Belyaev et al. (2013). In the following sections we
will determine the DM yield and subsequently the present DM abundance by
numerically solving the Boltzmann equation, computing contributions due to
these processes.

###  III.1 Freeze-in yield

As it is known, the key for freeze-in DM production is to assume that DM was
not present in the early universe or the DM abundance was negligibly small
after reheating. In case where the DM is produced via generic 2-to-2
annihilation, the Boltzmann equation (BEQ) reads Duch et al. (2018); Barman et
al. (2020c)

$\begin{split}xHs\frac{dY_{X}^{\text{ann}}}{dx}&=\gamma_{\text{ann}},\end{split}$
(30)

where

$\displaystyle\gamma\left(a,b\to 1,2\right)$
$\displaystyle=\int\prod_{i=1}^{4}d\Pi_{i}\left(2\pi\right)^{4}\delta^{(4)}\biggl{(}p_{a}+p_{b}-p_{1}-p_{2}\biggr{)}f_{a}{{}^{\text{eq}}}f_{b}{{}^{\text{eq}}}\left|\mathcal{M}_{a,b\to
1,2}\right|^{2}$ (31)
$\displaystyle=\frac{T}{32\pi^{4}}g_{a}g_{b}\int_{s_{min}}^{\infty}ds~{}\frac{\biggl{[}\bigl{(}s-m_{a}^{2}-m_{b}^{2}\bigr{)}^{2}-4m_{a}^{2}m_{b}^{2}\biggr{]}}{\sqrt{s}}\sigma\left(s\right)_{a,b\to
1,2}K_{1}\left(\frac{\sqrt{s}}{T}\right),$

with $a,b(1,2)$ as the incoming (outgoing) states and $g_{a,b}$ are
corresponding degrees of freedom. Here
$f_{i}{{}^{\text{eq}}}\approx\exp^{-E_{i}/T}$ is the Maxwell-Boltzmann
distribution. The Lorentz invariant 2-body phase space is denoted by:
$d\Pi_{i}=\frac{d^{3}p_{i}}{\left(2\pi\right)^{3}2E_{i}}$. The amplitude
squared (summed over final and averaged over initial states) is denoted by
$\left|\mathcal{M}_{a,b\to 1,2}\right|^{2}$ for a particular $2\to 2$
scattering process. The lower limit of the integration over $s$ is
$s_{min}=\text{max}\biggl{[}\left(m_{a}+m_{b}\right)^{2},\left(m_{h_{1}}+m_{2}\right)^{2}\biggr{]}$.
Here we define the DM yield $Y_{X}=n_{X}/s$ as the ratio of DM number density
to the comoving entropy density in the visible sector since the DM is only
produced from the SM bath. The parameter $x=m_{X}/T$ describes the SM sector
temperature $T$, the Hubble parameter is denoted by $H$ and
$\gamma=\langle\sigma v\rangle n_{\text{eq}}^{2}$ is the so-called reaction
density Chu et al. (2012) for the SM particles annihilating into the DM. All
the relevant 2-to-2 annihilation cross-sections as a function of $s$ are
collected in Appendix A, and as we see, all of them are proportional to
$g_{X}^{4}$ in the lowest order of $g_{X}$. As mentioned before, the dark and
visible sector communicates only via the portal coupling $\lambda_{HS}$, which
in the scale invariant framework is not a free parameter (Eq. (20)). Thus, the
dependence of the annihilation cross-section on the dark gauge coupling
$g_{X}$ arises through the portal coupling.

Figure 3:  Top Left: Reaction densities as a function of the bath temperature,
where different colours indicate different SM initial states. The parameters
are chosen such that DM relic density is satisfied. Top Right: Evolution of DM
yield with $x=m_{X}/T$ where the asymptotic yield produces observed relic
abundance. Bottom Left: Comparison of DM freeze-in production rate with Hubble
parameter for different choices of $g_{X}$ (chosen arbitrarily) with fixed
$m_{X}=1$ TeV. Bottom Right: Same as bottom left but for a fixed
$g_{X}=10^{-4}$ with different choices of DM mass.

In the left panel of Fig. 3 we have shown the variation of the reaction
densities as a function of the temperature $T$. The red and green curves
correspond to SM leptons and quarks pair annihilating into the DM
respectively, while the blue and black curves correspond to annihilation of SM
gauge bosons and Higgs into a pair of DM. Note that, due to the presence of
$SU(2)_{L}$ gauge coupling the contribution from the SM gauge bosons is
maximum. On the other hand, the contact interaction (top right panel of Fig.
2) makes the scalar annihilation also important. Together, they imply, the
contribution to DM relic abundance from gauge boson and scalar annihilations
are dominant. In the right panel of Fig. 3 we illustrate the evolution of the
DM yield as a function of the dimensionless quantity $x=m_{X}/T$. We choose
two different benchmark points corresponding to two different DM masses such
that the yield at $x\to\infty$ gives rise to the Planck Aghanim et al. (2020b)
observed relic density. Here we see typical IR nature of the freeze-in, where
the yield freezes in at $x\sim 1$ and then becomes constant up to the present
temperature. Also note, for larger DM mass the yield is smaller which in turn
demands a larger $g_{X}$ to obtain the observed abundance. This can be
realized by considering an approximate analytical formulation of the IR yield
for the DM Hall et al. (2010); Duch et al. (2018); Barman et al. (2020c)

$Y_{X}\sim\sigma\left(T_{\text{FI}}\right)\,M_{\text{pl}}\,T_{\text{FI}}\sim\frac{g_{X}^{4}M_{\text{pl}}}{m_{X}^{5}}\,m_{h_{1}}^{4}$
(32)

where the first relation comes out from simple dimensional arguments, and
$\sigma\left(T_{\text{FI}}\right)$ is the production cross-section for the DM
at freeze-in temperature $T_{\text{FI}}$ which in the present scenario can be
approximated to be
$\sigma\left(T_{\text{FI}}\right)\sim\lambda_{HS}^{2}/T_{\text{FI}}^{2}\sim\frac{g_{X}^{4}m_{h_{1}}^{4}}{m_{X}^{6}}$.
We also consider $T_{\text{FI}}\sim m_{X}$ since for the SM plasma temperature
below the threshold $m_{X}$, the yield ceases to grow. From this approximate
relation it is clear that in order to produce a fixed $Y_{X}$ which can
provide the right abundance one has to choose a larger $g_{X}$ for a heavier
DM. However, we emphasize that this is just an approximate relation, and we
perform full numerical solution for solving the BEQ for yield. Before going
into the details of the DM parameter space, few comments in order

  * •

By demanding $m_{h_{2}}^{2}>0$ we put a lower bound on the vector DM mass:
$m_{X}\gtrsim 250$ GeV for $g_{X}\lesssim 10^{-5}$, which is required for
freeze-in production of the DM. This bound entirely arises from the scale-
invariance of the theory. For such choices of parameters we find $m_{h_{2}}\ll
m_{X}$. This ensures that it is not possible to produce the DM from the on-
shell decay of the new scalar. The bound on $m_{X}$ also tells that the SM-
like Higgs can not have on-shell decay into a pair of DM. Thus, the DM freeze-
in production in the present framework takes place entirely via 2-to-2
scattering of the bath particles.

  * •

One must ensure the DM remains out of equilibrium in early times such that the
freeze-in production condition $\Gamma_{\text{int}}<H$ is satisfied, where
$\Gamma_{\text{int}}=\langle\sigma v\rangle_{i}n_{\text{eq}}^{i}$ is the
reaction rate $(i\in\text{SM})$. We note, for
$g_{X}\sim\mathcal{O}\left(10^{-4}\right)$ the reaction rate always falls
below the Hubble rate $H$ ensuring the DM production takes place non-thermally
at high temperature. This is established from the bottom panel of Fig. 3.
Thus, for all benchmark values of $g_{X}$ and $m_{X}$ which give rise to right
DM abundance, the DM is safely produced out of thermal equilibrium.

  * •

As we will see, the observed DM abundance can be achieved for
$g_{X}\sim\mathcal{O}\left(10^{-5}\right)$ for DM mass up to 2 TeV. For such
$g_{X}$’s, we also find the scalar mixing
$\sin\theta\sim\mathcal{O}\left(10^{-5}\right)$, which implies the portal
coupling $\lambda_{HS}$ is extremely tiny, that in turn ensures that it is
impossible for the dark sector to equilibrate with the visible one. However,
the non-thermalization of the dark sector by itself is not guaranteed. If the
$XX\to h_{2}\,h_{2}$ process is very efficient then it will deplete the number
density of $X$ and form a dark plasma with a temperature, which is in general,
different from that of the SM temperature, within the dark sector. The non-
thermalization condition within the dark sector can be encoded via

$n_{X}\langle\sigma v_{XX\to h_{2}h_{2}}\rangle\leq H$ (33)

where $n_{X}$ is the number density of vector DM. Note that, the interaction
cross-section $\langle\sigma v_{XX\to h_{2}h_{2}}\rangle$ is proportional to
the dark gauge coupling $g_{X}^{4}$, and thus the condition can be trivially
satisfied for $g_{X}\sim\mathcal{O}\left(10^{-5}\right)$, which is of our
interest.

  * •

The DM, in principle, can be produced in two different regimes: (a) symmetric
phase: $T\gtrsim 160$ GeV, $\langle H\rangle=0$, where the SM gauge bosons are
massless and and (b) broken phase: $T<160$ GeV, $\langle H\rangle\neq 0$ with
massive SM gauge bosons. The total DM yield is then given by the sum of yields
in these two regimes. Since the scale invariance of the theory induces
symmetry breaking in both the sectors simultaneously, hence before the
electroweak symmetry is broken, the DM is massless and relativistic, as
opposed to the usual cases of freeze-in Duch et al. (2018); Barman et al.
(2020c). We aim to address this scenario in a future draft111111A study of
freeze-in production of scalar DM in the relativistic regime can be found in
Lebedev and Toma (2019)..

###  III.2 Viable dark matter parameter space

In order to compute the final DM yield, we have solved Eq. (30) numerically,
and obtain the relic abundance of the DM at the present epoch via

$\Omega_{X}h^{2}=\left(2.75\times
10^{8}\right)\left(\frac{m_{X}}{\text{GeV}}\right)Y_{X}(T_{0})$ (34)

where $T_{0}$ is the temperature at the present epoch, which corresponds to
$x\to\infty$. We must also remind that the Planck Aghanim et al. (2020b)
allowed relic density allows:

$\Omega_{\text{DM}}h^{2}=0.11933\pm 0.00091,$ (35)

which we will use to constrain the relic density allowed parameter space. The
presence of the (decaying) light scalar $h_{2}$ can give rise to two other
very important bounds in the present model along with the relic density, which
will constraint the model parameter space. We discuss them below.

####  III.2.1 Bound from the Big Bang nucleosynthesis (BBN)

Within the DM freeze-in regime, the mediator $h_{2}$ is much lighter than the
DM, and can be $\sim\mathcal{O}\left(1~{}\rm GeV\right)$ for DM mass
$m_{X}\sim\mathcal{O}\left(10~{}\rm TeV\right)$. Also, its coupling to the SM
fermions turns out to be $\theta\sim\mathcal{O}\left(10^{-5}\right)$. Since
$h_{2}$ decay to SM particles is suppressed by the small mixing angle, its
lifetime tends to be very long and it may cause problems in the early Universe
Zhang (2015). In particular there will be an upper bound on the
$h_{2}$-lifetime from nucleosynthesis. It is well known that quasi-stable
particles with decay times $\tau\gtrsim 0.1$ seconds may significantly perturb
the primordial light element nucleosynthesis occurring approximately between 1
and 1000 seconds after the birth of the universe. The decay lifetime of
$h_{2}$ into the SM states can thus potentially perturb the successful
predictions of light element yields accumulated in the early universe. If the
decay occurs after BBN with $\tau_{h_{2}}>1$ sec, entropy production has to be
less than $\sim 10\%$ (assuming no change to light element abundances due to
it) because the precision measurements of the baryon density through BBN and
CMB observations match very well Steigman (2012); Kaplinghat et al. (2014).
Therefore, the absence of significant entropy production after BBN will put a
strong bound on the lifetime $\tau_{h_{2}}$. If the mediator decays before
BBN, entropy production constraints are non-existent.

A fully quantitative analysis of these effects is beyond the scope of this
paper and can be found in, for example, Ref.Berger et al. (2016). Instead we
follow Kawasaki et al. (2000) and, to remain within the 2$\sigma$ limit of the
observed 4He abundance, require $\Gamma_{h_{2}}^{-1}\equiv\tau_{h_{2}}<1$ sec.
Note that, this is rather a conservative bound given the fact that the light
scalar never comes in equilibrium with the SM due to feeble portal coupling.
For $m_{h_{2}}\sim\mathcal{O}\left(1~{}\rm GeV\right)$, $h_{2}$ can decay on-
shell into a pair of light fermions, and also to photon and gluon final states
via loop. The partial decay widths to the SM final states of $h_{2}$ are given
by Djouadi (2008); Krnjaic (2016)

$\displaystyle\Gamma_{ff}=\frac{G_{F}\sin^{2}\theta\,N_{c}}{4\sqrt{2}}m_{f}^{2}m_{h_{2}}\Biggl{(}1-\frac{4m_{f}^{2}}{m_{h_{2}}^{2}}\Biggr{)}^{3/2}$
(36)
$\displaystyle\Gamma_{\gamma\gamma}=\frac{G_{F}\sin^{2}\theta}{128\sqrt{2}}\frac{\alpha^{2}m_{h_{2}}^{3}}{\pi^{3}}\Biggl{|}\sum_{f}N_{c}Q_{f}^{2}\mathcal{A}_{1/2}\left(x_{f}\right)+\mathcal{A}_{1}\left(x_{f}\right)\Biggr{|}^{2}$
$\displaystyle\Gamma_{gg}=\frac{G_{F}\sin^{2}\theta}{36\sqrt{2}}\frac{\alpha_{s}m_{h_{2}}^{3}}{\pi^{3}}\Biggl{|}\frac{3}{4}\sum_{q}\mathcal{A}_{1/2}\left(x_{q}\right)\Biggr{|}^{2}$

where $x_{i}=m_{h_{2}}^{2}/4m_{i}^{2}$, $N_{c}$ is the number of colors for a
given fermion species and $Q_{f}$ is its electromagnetic charge. Also,

$\displaystyle\mathcal{A}_{1/2}\left(x\right)=2\Bigl{[}x+\left(x-1\right)f(x)\Bigr{]}x^{-2}$
(37)
$\displaystyle\mathcal{A}_{1}(x)=-\Bigl{[}2x^{2}+3x+3\left(2x-1\right)f(x)\Bigr{]}x^{-2}$

with

$f(x)\equiv\begin{cases}\text{Arc}\sin^{2}\sqrt{x}&x\leq 1,\\\\[8.0pt]
-\frac{1}{4}\Bigl{[}\log\frac{1+\sqrt{1-x^{-1}}}{1-\sqrt{1-x^{-1}}}-i\pi\Bigr{]}^{2}&x>1.\end{cases}$
(38)

Now, since $\Gamma_{h_{2}}$ depends both on the mixing $\sin\theta$ and on
$m_{h_{2}}$, which is turn, are functions of $g_{X}$ and $m_{X}$, therefore it
is evident that the requirement $\tau_{h_{2}}<1$ sec will constraint the
resulting parameter space in the $g_{X}-m_{X}$ plane, especially ruling out
small $g_{X}$ values which can result in very long lifetime of the light
scalar.

####  III.2.2 Dark matter direct detection

Contrary to the usual notion that DM nuclear recoil cross sections are
suppressed by tiny portal couplings (which is required for DM freeze-in), and
therefore DM direct-detection experiments do not provide relevant constraints
in freeze-in framework, in the present set-up the DM-nucleon scattering cross-
section can be hugely amplified because it takes place via $t$-channel
mediation of the light scalar $h_{2}$ (along with the SM-like Higgs $h_{1}$).
The DM-nucleus scattering cross-section in this case has the form Duch et al.
(2018)

$\frac{d\sigma_{XN}}{dq^{2}}\left(q^{2}\right)=\frac{\sigma_{XN}}{4\mu_{XN}^{2}v^{2}}\,F\left(q^{2}\right)$
(39)

where $v_{\text{DM}}$ is the DM velocity in the lab frame and the DM-nucleus
reduced mass is given by $\mu_{XN}=m_{X}\,m_{N}/\left(m_{X}+m_{N}\right)$, and

$\sigma_{XN}=\frac{\lambda_{HS}^{2}\,f_{N}^{2}\,m_{X}^{2}\,m_{N}^{2}\,\mu_{XN}^{2}}{\pi\,m_{h_{1}}^{4}m_{h_{2}}^{2}\left(m_{h_{2}}^{2}+4\mu_{XN}^{2}\,v_{\text{DM}}^{2}\right)}$
(40)

is the total cross-section with the effective DM-nucleon coupling is
$f_{N}\approx 0.3$ Cline et al. (2013). Here the factor
$m_{h_{2}}^{2}+4\mu_{XN}^{2}v_{\text{DM}}^{2}$ in the denominator represents
the light mediator effects Duch et al. (2018, 2020). The function $F(q^{2})$
is defined as Geng et al. (2016)

$F(q^{2})=\frac{\Bigl{(}1+q_{\text{min}}^{2}/m_{h_{2}}^{2}\Bigr{)}\Bigl{(}1+q_{\text{ref}}^{2}/m_{h_{2}}^{2}\Bigr{)}}{\Bigl{(}1+q^{2}/m_{h_{2}}^{2}\Bigr{)}^{2}}$
(41)

which encodes the effects of the light mediator. Usually, $q_{\text{min}}^{2}$
is very small compared to the other scales appearing in the process, and thus
can be taken to be zero, while
$q_{\text{ref}}^{2}=4\mu_{XN}^{2}v_{\text{DM}}^{2}$ is related to the energy
thresholds of DM direct detection experiments. In the limit $m_{h_{2}}^{2}\gg
q^{2}\sim 4\mu_{XN}^{2}v_{\text{DM}}^{2}$, the form factor $F(q^{2})\approx 1$
and we recover the conventional DM-nucleon scattering cross-section for
contact interactions. But for $m_{h_{2}}^{2}\ll q^{2}$, Eq. (39) will have
extra $q^{2}$ dependence characterized by $F(q^{2})$. We again remind the
readers that the portal coupling $\lambda_{HS}$ is not a free parameter,
rather can be expressed in term of the dark gauge coupling $g_{X}$ following
the relations in Eq. (20). This makes the DM-nucleus scattering cross-section
$\sigma_{XN}\propto g_{X}^{4}$, which implies large $g_{X}$ will result in
large $\sigma_{XN}$, making the parameter space vulnerable from direct
detection bounds121212In deriving the direct search bounds we do not consider
RGE flow of $g_{X}$ from UV scale down to nuclear physics energy scale since
it remains almost fixed to a very small value (set by freeze-in), see Fig. 1..
We consider exclusion limits from XENON1T Aprile et al. (2018) (black solid
curve), and projected bounds from PandaX-4T Zhang et al. (2019) (black
dashed), LUX-ZEPLIN (LZ) (brown dashed) Akerib et al. (2020), XENONnT Aprile
et al. (2020) (black dotdashed) and DARWIN Aalbers et al. (2016) (black
dotted) experiments which provide upper limit on the DM-nucleon scattering
cross-section at 90% C.L.

Figure 4:  Left: The black straight line represents the contour satisfying
Planck observed relic density in the bi-dimensional plane of $g_{X}$-$m_{X}$.
Right: Relic density allowed parameter space (cyan region) projected in direct
search plane where exclusion limits from spin-independent direct search
experiments are portrayed (black solid, black thick dashed, black dot-dashed,
brown thick and black dotted curves) along with the expected discovery limit
corresponding to the so called “$\nu$-floor” from CE$\nu$NS of solar and
atmospheric neutrinos for a Ge target (orange dashed curve).

We can now proceed to explore the allowed parameter space for the DM
satisfying relic density and direct search limits. This is furnished in Fig.
4. The black thick straight line in the left panel of Fig. 4 shows the contour
satisfying observed relic abundance. Here we see, with increase in DM mass,
one needs a larger $g_{X}$ to satisfy the relic constraint
$\Omega_{X}h^{2}\simeq 0.12$. This can again be explained from the approximate
relation in Eq. (32), from which we find $\Omega_{X}h^{2}\propto m_{X}\times
Y_{X}\propto\left(g_{X}/m_{X}\right)^{4}$. Thus, it is evident, one needs to
tame down the gauge coupling in order to satisfy the observed abundance for a
lighter DM, and the dependence is linear. We see, up to a DM mass of
$m_{X}\sim 2$ TeV, right abundance is obtained in the vicinity of $g_{X}\sim
10^{-5}$. A part of the relic density allowed parameter space is also ruled
out from BBN constraint. This is shown by the gray region. As it is seen, for
a fixed $m_{X}$, increasing $g_{X}$ relaxes the BBN bound since it gives rise
to a larger mixing (and also a larger $m_{h_{2}}$ following Eq. (26)) that
results in a smaller lifetime (Eq. (36)). The same effect is also realized on
increasing $m_{X}$ for a fixed $g_{X}$. The BBN bound thus discards DM mass
below $m_{X}\lesssim 467$ GeV with $g_{X}\lesssim 2\times 10^{-5}$. In other
words, a lower bound on $g_{X}$ can be set solely from BBN, depending on the
corresponding DM mass.

In the right panel we show how much of the relic density allowed parameter
space survives spin-independent direct search constraints for DM mass up to 2
TeV. The right DM abundance is obtained within the cyan band where we have
$g_{X}\in\\{1\times 10^{-5}-8\times 10^{-5}\\}$, as we can see from the left
panel. The BBN bound can also be projected in $\sigma_{XN}-m_{X}$ plane, as
shown by the red points. Since with the increase in $g_{X}$ the DM-nucleon
scattering cross-section increases following Eq. (39), hence one expects
comparatively lower DM mass region to be safe from direct detection bound
where one requires lower $g_{X}$ to obtain the right abundance. However, a
lower $g_{X}$ also results in a lighter $m_{h_{2}}$ that enhances the direct
search cross-section in turn. Hence, we see, lower DM masses are tightly
constrained from several direct search experiments, while the bounds get
relaxed as we move on to heavier DM mass. Within this range of $g_{X}$ and
$m_{X}$ that produces right abundance, the mass of the new scalar turns out to
be $m_{h_{2}}\sim 30$ MeV (mentioned along the top horizontal axis), which is
of the order of typical momentum transfer scale $q\sim 10$ MeV in DM-nucleus
scattering. We see, the viable region of the parameter space lies below the
PandaX-4T exclusion limit for DM mass $m_{X}\gtrsim 642$ GeV with
$g_{X}\gtrsim 2.7\times 10^{-5}$. For $g_{X}\gtrsim 1.2\times 10^{-5}$
(corresponding $m_{X}\gtrsim 300$ GeV) the relic density allowed parameter
space lies below the exclusion limit from XENON1T, however, a major part of
the parameter space is ruled out by BBN (shown in red). One should note here,
it is possible to obtain the right relic abundance for a DM of mass as large
as $\sim\mathcal{O}(10~{}\rm TeV)$ with
$g_{X}\sim\mathcal{O}\left(10^{-4}\right)$, which, needless to mention, will
satisfy the direct search bound. In that case $m_{h_{2}}$ can be as large as
$\sim\mathcal{O}(1~{}\rm GeV)$. Hence, it is possible to have the scalar mass
around the GeV scale in the expense of making the DM heavy, satisfying both
relic abundance and direct detection constraints. As we shall see in the next
section, such light scalars can be probed in various ongoing and proposed
experiments. A large part of the parameter space, on the other hand, can be
potentially tested in the future XENONnT experiment. We also show limit from
the so called “$\nu$-floor” Billard et al. (2014) via the orange dashed lines,
below which the number of neutrino events due to coherent elastic neutrino-
nucleus scattering (CE$\nu$NS) is expected to be much larger than the number
of DM events, which prevents to identify DM signals with certainty. The
parameter space of our interest lies just below the $\nu$-floor for DM mass
$m_{X}\gtrsim 1.8$ TeV.

Before closing this section we would like to mention that in the present model
the $t$-channel scalar mediation can as well give rise to non-zero DM-electron
scattering cross-section. As it is known, experimental sensitivity to events
with a single or a few ionization electrons has been demonstrated with
XENON10Essig et al. (2017) and more recently DarkSide Agnes et al. (2018),
thus allowing for a novel way to probe light DM scattering off of electrons
with an existing experimental setup Essig et al. (2012). Now, an approximate
analytical expression for DM-$e$ scattering cross-section has the form:
$\sigma_{Xe}\propto\lambda_{HS}^{2}\,g_{X}^{2}\,\Bigl{(}\frac{m_{X}^{2}\,m_{e}}{m_{h_{1}}^{2}\,m_{h_{2}}^{2}}\Bigr{)}^{2}$,
which produces a cross-section
$\sim\mathcal{O}\left(10^{-50}\right)\text{cm}^{2}$ for a DM mass of 300 GeV
with $g_{X}=10^{-5}$ (where $\lambda_{HS}$ can be substituted using Eq. (20)).
Such a small cross-section is far below the present sensitivity of DM-electron
scattering experiments, hence we do not consider them here.

##  IV Laboratory Phenomenology

Figure 5:  The thick black straight line shows the DM parameter space
complying with the Planck observed relic abundance and satisfying spin-
independent direct search exclusion limit in $\sin\theta-m_{h_{2}}$ plane.
Experimental limits are shown from E949 Artamonov et al. (2008), CHARM Bergsma
et al. (1985), NA62 and (2017), FASER-I&II Feng et al. (2018); Ariga et al.
(2019a, 2018, b), FCC-hh Abada et al. (2019); Dev et al. (2017), ATLAS Aad et
al. (2012b, 2014); Chalons et al. (2016); Robens and Stefaniak (2015),
SeaQuest Berlin et al. (2018), LHCb Gligorov et al. (2018), KLEVER Moulson
(2019), DUNE Acciarri et al. (2015); Berryman et al. (2020), DarkQuest-Phase2
Batell et al. (2021), MATHUSLA Curtin et al. (2019), SHiP Anelli et al. (2015)
and PS191 Bernardi et al. (1986); Gorbunov et al. (2021). All proposed
experimental limits are denoted by dashed curves, while the existing limits
are in solid. Exclusion region from supernova cooling and BBN are also
depicted

To this end, we showed that the freeze-in vector DM in our model can give rise
to a viable parameter space satisfying bounds from relic density. We have also
realized that the scale invariance of the theory gives rise to a naturally
light scalar mediator which influences the nuclear recoil spectrum in direct
detection experiment, leading to the possibility of probing this model there-
in. In this section we investigate the prospects of searching for this light
scalar (and thus probing the DM parameter space) in intensity frontier and
lifetime frontier experiments which look for for light, weakly interacting,
electrically neutral long-lived particles. These experiments are capable of
probing extremely small mixing angles. Dark sectors with light degrees of
freedom can be probed with a variety of experiments at the luminosity
frontier, including proton Batell et al. (2009); deNiverville et al. (2011,
2012); Kahn et al. (2015); Adams et al. (2013); Soper et al. (2014); Dobrescu
and Frugiuele (2015); Coloma et al. (2016); deNiverville et al. (2017);
Aguilar-Arevalo et al. (2017, 2018); Alpigiani et al. (2018); Ariga et al.
(2018), electron Bjorken et al. (2009); Izaguirre et al. (2013); Diamond and
Schuster (2013); Izaguirre et al. (2014); Batell et al. (2014); Lees et al.
(2017); Berlin et al. (2019); Banerjee et al. (2019) and positron fixed target
facilities Accardi et al. (2021). Here we show, the allowed parameter space of
the DM lies well within the reach of several such experiments given the mass
and mixing of the light scalar131313Light dark sector scenarios have been
explored in great detail over the past decade Bramante et al. (2016);
Alexander et al. (2016); Battaglieri et al. (2017); Darmé et al. (2018);
Winkler (2019); Egana-Ugrinovic et al. (2020); Okada and Raut (2021);
Foroughi-Abari and Ritz (2020); Agrawal et al. (2021); Nardi et al. (2018)..
The noteworthy point here is that, the coupling responsible for the freeze-in
production is being directly probed at the experimental level, which, in
general, is not the case141414For probing freeze-in in collider experiments
the presence of a lightest odd sector particle (LOSP) is generally required
Hall et al. (2010).. However, given the underlying scale invariance, the
resulting parameter space is extremely constrained and predictive.

Figure 6:  Different colored bars represent the reach of the corresponding
experiments (mentioned along the horizontal axis) in probing the maximum mass
of the light scalar. The DM mass, satisfying relic density and spin-
independent direct detection bound, corresponding to different $m_{h_{2}}$
(following Eq. (26)) are mentioned along the vertical axis on the right side.

Let us now note that the requirement of $\Omega_{X}h^{2}\simeq 0.12$ typically
constraints the ratio $m_{X}/g_{X}$ since
$\Omega_{X}h^{2}\propto\left(g_{X}/m_{X}\right)^{4}$, as mentioned before
(c.f. Eq. (32)). This also implies, for fixed a $\Omega_{X}h^{2}$, the mixing
becomes constant since
$\sin\theta=v_{h}\Big{/}\sqrt{v_{h}^{2}+\left(m_{X}/g_{X}\right)^{2}}$ and
$g_{X}/m_{X}$ is determined from relic abundance. Thus, the scale-invariance
of the theory, together with the requirement of right relic abundance, fixes
$\sin\theta$ to a constant value. For all choices of $\\{m_{X},g_{X}\\}$ that
leaves the combination $m_{X}/g_{X}$ constant (proportional to
$\Omega_{X}h^{2}$), we find $\sin\theta\sim 10^{-5}$. This is what we are
referring to as a “miracle” in the present model, as the value of this mixing
angle is always fixed, as decided by the scale invariance of the theory. Thus,
in the $m_{h_{2}}-\sin\theta$ plane, the relic density allowed parameter space
is simply a straight line with a fixed $\sin\theta$ (independent of
$m_{h_{2}}$), determined from the observed relic density. This is shown by the
black horizontal straight line in the top panel of Fig. 5. For the given mass
range of $m_{h_{2}}$ the constraint from spin-independent direct search is
trivially satisfied since $m_{h_{2}}\geq 0.1$ GeV corresponds to DM mass
$m_{X}\gtrsim$ 3.5 TeV.

Figure 7:  Summary of all experimental bounds described in Sec. III.2 and Sec.
IV in the direct search plane, where the relic density allowed DM parameter
space shown via the thick cyan curve as in Fig. 4. The pink region is
disallowed from supernova bound on the mass of light scalar. The light blue
region in the background denotes the mass range that can only be probed in
lifetime frontier experiments. The colorful vertical dashed lines indicate the
ranges in which different intensity frontier experiments can search. Inset
shows the sensitivity of direct detection experiments up to DM mass of 2 TeV
(Fig. 4).

In Fig. 5 we summarize the result obtained from DM phenomenology, together
with the search reach of several planned/proposed experiments (contours with
the names of experiments indicated) in $m_{h_{2}}-\sin\theta$ plane. We see,
light scalar mediator in the present model is within the reach of CHARM
Bergsma et al. (1985), DUNE Acciarri et al. (2015); Berryman et al. (2020),
FASER-I&II Feng et al. (2018); Ariga et al. (2019a, 2018, b), PS191 Bernardi
et al. (1986); Gorbunov et al. (2021), DarkQuest-Phase2 Batell et al. (2021),
MATHUSLA Curtin et al. (2019) and SHiP Anelli et al. (2015) (see for example
Beacham et al. (2020) for a summary on these experiments) for the allowed
range of mass and mixing. Note that, the weakly coupled light scalar mediator
can be produced on shell during a supernova (SN) explosion and significantly
contribute to its energy loss, thereby shortening the duration of the
observable neutrino pulse emitted during core collapse. The most significant
such constraint arises from SN1987A Krnjaic (2016); Dev et al. (2020), that
excludes $m_{h_{2}}\sim 148$ MeV, as shown by the pink region labelled as
‘Supernova.’ Fig. 6 shows the sensitivity of the potential experiments (as
obtained from the left panel) in probing different ranges of $m_{h_{2}}$. For
each $m_{h_{2}}$ it is possible to choose suitable $\\{m_{X},g_{X}\\}$
(mentioned along the right vertical axis) such that right DM relic abundance
is obtained abiding all bounds discussed previously.

In Fig. 7, we have summarized the bounds from both DM direct search and
lifetime frontier experiments that we have discussed in Sec. III.2 and Sec. IV
respectively. The thick cyan curve denotes the relic density allowed parameter
space for the DM as we already found in Fig. 4. We have now extended this up
to $\sim 25$ TeV to accommodate the bounds on $m_{h_{2}}$ from lifetime
frontier experiments. We show the exclusion limits from spin-independent
direct detection experiments from XENON1T, and projected sensitivity limits
from PandaX-4T, LUX-ZEPLIN (LZ), XenonNT and DARWIN. As expected, for low DM
mass region (below 1 TeV) these bounds are severe but become rather weak for
heavier DM mass151515Direct search limit on superheavy DM have been explored
in Refs. Kavanagh (2018); Bramante et al. (2018); Clark et al. (2020).. A
light scalar with mass below $\sim 250$ MeV is ruled out from supernova
observations as indicated by the pink region. This corresponds to a DM mass
$\lesssim 6.4$ TeV. Above DM mass of $\sim 1.8$ TeV the relic density allowed
parameter space gets submerged into the $\nu$-floor, where separating DM
scattering from the background neutrino scattering is rather challenging.
However, it is interesting to note that in those regions several intensity
frontier experiments can provide excellent sensitivity. The vertical colored
dashed lines show the range of $h_{2}$ masses (equivalently DM mass) in
experiments like DUNE, CHARM, PS191, MATHUSLA, SHiP, FASER-II etc. Thus, even
though direct search experiments may lose sensitivity in exploring heavy DM
masses, but the presence of the light scalar can still provide potential
signals in the plethora of intensity and lifetime frontier experiments, and
thanks to the scale invariance of the theory, this automatically implies a
signature of the freeze-in DM, in this model.

##  V Discussion and Conclusion

Scale invariance at the classical level leads to alleviating the gauge
hierarchy problem as the observed scales (EW and DM scales) can be dynamically
generated at the quantum level, removing the perilous quadratic divergences
from the UV cut-off. In a conformal gauge extension of the SM we showed that
the observed cosmological DM abundance can also be satisfied. We investigated
the freeze-in production of dark matter (DM), which rests on the possibility
of DM particle being only feebly interacting with the visible SM sector; here
the DM abundance slowly builds up from, say, the collision of the bath of (SM)
particles. Although this mechanism is perfectly capable of explaining the
present observed relic abundance of DM, but due to its super weak coupling
with the visible sector, usually it is challenging to test such a framework in
laboratory experiments or astrophysical observations. In a minimal $U(1)_{X}$
gauge extended SM that incorporates a vector boson DM which receives mass due
to spontaneous symmetry breaking of $U(1)_{X}$ via radiative corrections known
as the Coleman-Weinberg mechanism, we showed a mass scale is transmuted from
the dark to the electroweak sector through the portal interactions. The
underlying scale invariance allows only two independent parameters, namely,
the DM mass $m_{X}\gtrsim 240$ GeV and the gauge coupling $g_{X}$. The new
scalar is also found to be naturally light with mass $m_{h_{2}}\ll m_{X}$. We
have shown that this set-up leads to DM production via freeze-in for suitable
choice of interaction strength, that is within the reach of several low energy
experiments. Considering $g_{X}\sim\mathcal{O}\left(10^{-5}\right)$ it is
possible to produce the entire observed DM relic abundance for
$m_{X}\sim\mathcal{O}\left(1~{}\rm TeV\right)$ via 2-to-2 scattering of the SM
particles in the thermal bath. Due to the presence of the light scalar
mediator (with mass $\sim$ MeV), the parameter space for freeze-in DM can be
constrained from spin-independent direct detection exclusion limits from
experiments like PandaX-4T , satisfying bounds from big bang nucleosynthesis
(BBN), ensuring that the light scalar decays into SM degrees of freedom with a
lifetime of less than a sec. For $m_{h_{2}}\sim\mathcal{O}\left(1~{}\rm
GeV\right)$, one has to choose a very heavy DM that automatically satisfies
direct detection bound.

The main findings of this paper are summarized in Figs. 5 & 7. In the former,
the requirement of obtaining the right abundance for the DM essentially fixes
the scalar mixing to a fixed value of
$\sin\theta\sim\mathcal{O}\left(10^{-5}\right)$, which is again attributed to
the the scale invariance of the theory. This gives rise to a simple straight
line contour in the $m_{h_{2}}-\sin\theta$ plane satisfying relic density and
direct detection bounds. But interestingly this particular mixing is within
the reach of several (proposed) lifetime and intensity frontier search
facilities for $m_{h_{2}}$ up to about 3.5 GeV. In Fig. 7, we show the
complementarity probes of such DM candidates, which tells us for massive DM
$(m_{X}\gtrsim 1.8~{}\text{TeV})$ the intensity frontier experiments can
provide more sensitivity than direct detection, even in probing parameter
space that goes beyond the $\nu$-floor. As a consequence, even though the
present model may produce null results in the scattering experiments but still
leaves a window to be probed in experimental frontiers. The scale invariant
$\text{SM}\times U(1)_{X}$ symmetry thus leads to a testable freeze-in
scenario satisfying all relevant constraints on the DM parameter space, where
the coupling responsible for freeze-in is being directly probed at
laboratories around the world.

We would finally like to conclude by pointing out that given such relevance of
scale invariance and the freeze-in mechanism of DM formation in early universe
along with the availability of planned and upcoming several light dark sector
(intensity frontier and lifetime frontier) experiments, we showed that due to
constrained relations between the model parameters ($\lambda_{HS}\propto
g_{X}^{4}$) it uniquely predicts the FIMP portal coupling at which the relic
density can be satisfied as well as be searched for in the laboratory
including DM direct detection experiments which otherwise is not motivated for
FIMP hunting. This ushers in a new era where UV-completion in BSM particle
physics model-buildings may lead to predictive FIMP dark matter candidates to
be tested in very near future.

##  VI Acknowledgement

The authors would like to thank Alexander Pukhov and Ahmad Mohamadnejad for
helping in implementing the model in LanHEP and in micrOmegas Belanger et al.
(2010), also Nicolás Bernal, Sudip Jana and Nobuchika Okada for useful
discussions. The authors would like to thank Dmitry Gorbunov for pointing out
a mistake in PS191 limit. BB received funding from the Patrimonio Autónomo -
Fondo Nacional de Financiamiento para la Ciencia, la Tecnología y la
Innovación Francisco José de Caldas (MinCiencias - Colombia) grant
80740-465-2020. This project has received funding /support from the European
Union’s Horizon 2020 research and innovation programme under the Marie
Skłodowska-Curie grant agreement No 860881-HIDDeN.

##  Appendix A Annihilation cross-sections for freeze-in production

Here we note down the analytical expressions for all the relevant 2-to-2
annihilation cross-section of the SM particles to DM final states. We denote
all SM leptons by $\ell$, quarks by $q$ and gauge boson masses by $m_{V}\in
m_{W},m_{Z}$. The total decay width of the SM-like Higgs is indicated by
$\Gamma_{h_{1}}\simeq 4$ MeV Khachatryan et al. (2017).

$\displaystyle\sigma\left(s\right)_{\ell\ell\to
XX}\simeq\frac{g_{X}^{4}\,m_{\ell}^{2}}{64\pi\,s}\,\frac{\sqrt{\left(s-4m_{X}^{2}\right)\,\left(s-4m_{\ell}^{2}\right)}}{\left(s-m_{h_{1}}^{2}\right)^{2}+\Gamma_{h_{1}}^{2}\,m_{h_{1}}^{2}}\Biggl{(}\frac{m_{h_{1}}^{2}-m_{h_{2}}^{2}}{s-m_{h_{2}}^{2}}\Biggr{)}^{2}\Biggl{(}\frac{s^{2}-4m_{X}^{2}s+12m_{X}^{4}}{\left(m_{X}^{2}+g_{X}^{2}v_{h}^{2}\right)^{2}}\Biggr{)}$
(42) $\displaystyle\sigma\left(s\right)_{qq\to
XX}\simeq\frac{g_{X}^{4}\,m_{q}^{2}}{192\pi\,s}\frac{\sqrt{\left(s-4m_{X}^{2}\right)\,\left(s-4m_{q}^{2}\right)}}{\left(s-m_{h_{1}}^{2}\right)^{2}+\Gamma_{h_{1}}^{2}\,m_{h_{1}}^{2}}\Biggl{(}\frac{m_{h_{1}}^{2}-m_{h_{2}}^{2}}{s-m_{h_{2}}^{2}}\Biggr{)}^{2}\Biggl{(}\frac{s^{2}-4m_{X}^{2}s+12m_{X}^{4}}{\left(m_{X}^{2}+g_{X}^{2}v_{h}^{2}\right)^{2}}\Biggr{)}$
$\displaystyle\sigma\left(s\right)_{VV\to
XX}\simeq\frac{g_{X}^{4}}{288\pi\,s}\,\sqrt{\frac{s-4m_{X}^{2}}{s-4m_{V}^{2}}}\,\Biggl{(}\frac{m_{h_{1}}^{2}-m_{h_{2}}^{2}}{s-m_{h_{2}}^{2}}\Biggr{)}^{2}\,\Biggl{(}\frac{1}{\left(s-m_{h_{1}}^{2}\right)^{2}+\Gamma_{h_{1}}^{2}\,m_{h_{1}}^{2}}\Biggr{)}$
$\displaystyle\Biggl{[}\frac{\left(s^{2}-4m_{X}^{2}s+12m_{X}^{4}\right)\left(s^{2}-4m_{V}^{2}s+12m_{V}^{4}\right)}{\left(m_{X}^{2}+g_{X}^{2}v_{h}^{2}\right)^{2}}\Biggr{]}$
$\displaystyle\sigma\left(s\right)_{h_{1}h_{1}\to
XX}\simeq\frac{g_{X}^{4}}{32\pi
s}\Biggl{(}\frac{m_{h_{1}}}{m_{X}}\Biggr{)}^{4}\frac{\left(s+2m_{h_{1}}^{2}\right)^{2}}{\left(s-m_{h_{1}}^{2}\right)^{2}+\Gamma_{h_{1}}^{2}\,m_{h_{1}}^{2}}\,\sqrt{\frac{s-4m_{X}^{2}}{s-4m_{h_{1}}^{2}}}\,\Biggl{(}1-\frac{4m_{X}^{2}}{s}+\frac{12m_{X}^{4}}{s^{2}}\Biggr{)}.$

The last expression is derived by keeping only the leading order in the double
expansion of $g_{X}$ and $m_{h_{2}}^{2}/m_{X}^{2}$.

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