# Freeze-in Dark Matter Through Forbidden Channel in $U(1)_{B-L}$

Partha Konar  Rishav Roshan  Sudipta Show

###### Abstract

We examine a scenario for freeze-in production of dark matter, which occurs
due to the large thermal correction to the mass of a decaying mediator
particle present in the thermal bath of the early Universe. We show that the
decays, which are kinematically forbidden otherwise, can open up at very high
temperatures and dominate the dark matter production. We explore such
forbidden production of dark matter in the minimal $U(1)_{B-L}$ model,
comparing dark matter phenomenology in the context of forbidden frozen-in with
the standard picture.

##  1 Introduction

The existence of a non-luminous and non-baryonic form of matter popularly
known as dark matter (DM) is already well established by the observations like
galaxy rotation curve [1], Bullet cluster [2], etc. Besides, the amount of
dark matter at present has been measured quite precisely in the experiments
like Planck [3], WMAP [4] by using the Standard Model of cosmology, i.e., the
$\Lambda$CDM model. However, the exact nature of dark matter and its
production mechanism remain a mystery and an open question. Since the Standard
Model(SM) fails to provide a viable DM candidate, one needs to go beyond the
Standard Model(BSM) of particle physics in order to accommodate a feasible
dark matter.

Among the existing dark matter models, the most popular ones are those that
accommodate dark matter in the form of the weakly interacting massive particle
(WIMP) [5, 6, 7, 8, 9, 10, 11, 12, 13, 14]. Here, the DM interactions with the
SM lies in the weak scale. In the early Universe, such a kind of DM remained
in thermal equilibrium with the hot and dense SM plasma. When the temperature
of the Universe cooled and fell below its mass, its number density started to
dilute, which continued till the time its annihilation rate became smaller
than the expansion rate of the Universe. Subsequently, its number density
freezes out, and the resulting yield gets saturated. Interestingly, the
dynamics of the freeze-out mechanism can also be affected once the thermal
corrections are incorporated, as shown in [15]. An exciting aspect of the
freeze-out is that it is dominated by physics at low energies, as it occurs
near a temperature that is 20 - 25 times smaller than the DM mass. Hence the
formation mechanism does not carry any information about the history of the
early Universe.

Despite such interesting features, the WIMP paradigm still remains in tension
due to its null detection at various direct [16, 17, 18], indirect [19] and
collider [20, 21] frontiers. These null results motivate us to explore various
other possibilities of DM production. Among them, the most popular one is the
freeze-in production of the DM from the thermal bath. Here, one assumes: (i) a
negligible initial abundance of DM in the early Universe and (ii) a very
feeble interaction of the DM with the thermal plasma. Such feeble interactions
are responsible for the gradual production of DM from the decay or scattering
of the bath particles. Furthermore, the smallness of the same interactions
also ensures that the DM never thermalizes with the bath. Once the number
density of the particles responsible for the DM production becomes Boltzmann
suppressed, the comoving number density of the DM ceases, and its abundance
freezes in. This type of DM is popularly known as _feebly interacting massive
particle_ or FIMP [22, 23]. Depending on the nature of DM interactions with
thermal bath, the freeze-in scenario can be categorized as (i) the Ultra-
Violet (UV) freeze-in [22, 24, 25, 26, 27] and (ii) the Infrared (IR) freeze-
in [22, 23, 28, 29, 30, 31]. In a UV freeze-in scenario, the DM interacts with
the bath particles only via non-renormalizable ($i.e.$, higher dimensional)
operators. In this picture, dark matter is produced solely through the
scattering of the bath particles in the early Universe, and the dark matter
abundance is highly sensitive to physics at high temperatures, particularly to
the reheating temperature of the Universe. Contrary to the UV freeze-in, in IR
freeze-in, the dark matter is connected with the visible sector only via
renormalizable interaction. Here, dark matter production is dominated
primarily at the temperature around the mass of the heavier particle involved
in the interactions.

In this work, we focus on DM production, where it has renormalizable
interactions with the thermal bath (IR freeze-in). From here onwards, we will
refer to the freeze-in scenario where the DM production dominantly takes place
at a temperature near about the mass of the decaying111Assuming the production
of the DM through the scattering of the bath particles remains sub-dominant in
comparison the production via decay. bath particle as a _standard freeze-in_
or SFI. Deviating from this scenario, in the present analysis, we consider the
production of DM in a parameter regime where its production remains
kinematically forbidden in the SFI framework. Darmé et al. studied such a
mechanism recently for DM production in Ref. [32]. The interesting feature of
this particular production mechanism is the involvement of thermally corrected
masses [33, 34, 35, 36, 32, 37, 38] of the particles participating in the DM
production. Here, one considers that the mediator is not only a part of the
hot thermal plasma, but it may also acquire a sizable thermal mass. In the
early Universe, when the temperature was extremely high, the thermal mass of
the mediator can have differed substantially from its mass at vacuum, i.e.,
the thermal effects must have dominated the mediator’s mass. Analogous to the
SFI, here, the initial population of dark matter is assumed to be zero or
negligibly small, and it is produced gradually from the mediator’s decay. At a
sufficiently high temperature, the mediator can acquire large thermal mass,
and the condition: $M_{\text{mediator}}(T)>2M_{\text{DM}}$ can easily be
achieved. The dark matter then can be copiously produced from this decay, even
if such a process remains kinematically forbidden at low temperatures. This
alternative approach of DM production can be indexed as _forbidden freeze-in_
(FFI). This new FFI scenario can open up an exciting and new paradigm of dark
matter phenomenology.

This article aims to explore the FFI scenario in a minimal $U(1)_{B-L}$
extension [39, 40, 41, 42, 43, 28, 44, 45, 46, 47] of the SM. As is well
known, the $B-L$ extension necessitates the introduction of three right-handed
neutrinos (RHN) to make the model free from the triangular anomaly. Unlike the
Type-I seesaw [48, 49, 50, 51, 52, 53, 54], here, the bare mass term for the
RHNs are not allowed at tree level. Hence, in order to make the RHNs massive,
they are required to couple to an SM gauge singlet (complex) scalar
appropriately charged under the $U(1)_{B-L}$ symmetry. These RHNs become
massive once the $B-L$ scalar acquires a non-zero vacuum expectation value
(vev) and spontaneously breaks the $U(1)_{B-L}$ symmetry. In addition, the
$B-L$ gauge boson also becomes massive after the breaking of $B-L$ symmetry.
It is interesting to point out that the $B-L$ setup can provide a common
solution to three of the most important issues of present-day particle physics
and cosmology, $,i.e.,$ the non-zero neutrino mass [55, 56, 57, 58, 54, 9],
baryogenesis via leptogenesis [59, 60, 61, 62, 63, 9, 64] from the decay of
heavier RHNs and dark matter (WIMP/FIMP). The WIMP type DM in the context of
the $B-L$ extension has been thoroughly studied [65, 66, 6]. Here, the
lightest RHN (non-trivially charged under a $Z_{2}$ symmetry) plays the role
of a DM [67]. Even though such extension can explain all three outstanding
issues under the same umbrella, the DM phenomenology still remains highly
constrained. The RHN dark matter relic density can only satisfy the Planck
limit [3] near the resonance regimes [65, 66]. An interesting alternative is
to consider the lightest RHN as FIMP type DM (SFI). This possibility is also
vastly explored in the literature [68, 43, 28, 44, 6, 69] and unlike the WIMP
scenario, here a sizable mass range is allowed222We would like to point out
that in a recent study [70], the authors have shown that Lyman$-\alpha$ bound
can also exclude DM mass $\lesssim\mathcal{O}(15~{}\text{keV})$ if produced
through a freeze-in mechanism. for DM. Contrary to this, in the present setup,
we follow the FFI approach to study the freeze-in production of dark matter
from the kinematically disallowed decay of the scalar that gets a significant
thermal mass correction while maintaining equilibrium with the hot thermal
plasma in the early Universe.

This paper is organized as follows. In Section 2, we introduce the model part
while Section 3 describes in detail the thermal mass correction of the
mediator. Different theoretical and experimental constraints deemed relevant
here are described in Section 4. Next, we present the forbidden freeze-in
production of dark matter and the estimation of numerical results in Section 5
and finally, we summarize our findings in Section 6.

##  2 The scenario

The present scenario explores the possibility of a $U(1)_{B-L}$ extension of
the SM gauge symmetry. Here, the particle content is extended by adding three
right-handed neutrinos $N_{i}$ ($i=1,2,3$) together with a complex scalar $S$,
all of them charged under the $U(1)_{B-L}$ symmetry. In addition, the SM
leptons and quarks also carry $U(1)_{B-L}$ charges of $-1$ and $+\frac{1}{3}$,
respectively. Further, invoking an additional unbroken discrete $Z_{2}$
symmetry and making one of the RHN (say, $N_{1}$) non-trivially charged under
it ensures its stability by forbidding its interactions with the SM leptons
and Higgs. Being stable, $N_{1}$ contributes as a suitable DM candidate in the
present setup. On the other hand, the remaining BSM particles and SM particles
carry a positive charge under this $Z_{2}$. In Table 1, we present the charges
of all the BSM fields under the different symmetry groups.

Field |  $SU(2)_{L}\times U(1)_{Y}$ |  $Y_{BL}$ |  $\mathbb{Z}_{2}$
---|---|---|---
$N_{1}$ | (1, 0) | -1 | $-$
$N_{2},N_{3}$ | (1, 0) | -1 | +
$S$ | (1, 0) | 2 | +
Table 1: The additional fields and their quantum numbers under different
symmetry groups. Here, $Y_{BL}$ refers to the $U(1)_{B-L}$ charge.

These $B-L$ charge assignments also eliminate the possibility of triangular
$B-L$ gauge anomalies in our model [71]. With the given particle spectrum and
the gauge symmetries, the most general renormalizable and gauge invariant
Lagrangian for the present setup can be written as,

$\displaystyle\mathcal{L}$ $\displaystyle=$
$\displaystyle\mathcal{L}_{KE}+\mathcal{L}_{y}-V(\phi,S)$ (2.1)

where kinetic terms $\mathcal{L}_{KE}$ for the BSM fields are given as,

$\displaystyle\mathcal{L}_{KE}$ $\displaystyle=$
$\displaystyle|D_{\mu}S|^{2}+\sum_{i=1,2,3}\bar{N_{i}}i\gamma^{\mu}D_{\mu}N_{i}-\frac{1}{4}Z_{\mu\nu}Z^{\mu\nu},$
(2.2)

with $Z^{\mu\nu}=\partial^{\mu}Z_{BL}^{\nu}-\partial^{\nu}Z_{BL}^{\mu}$, and
$D_{\mu}=\partial_{\mu}+i\,[Yg^{\prime}+Y_{BL}\,g_{BL}]\,(Z_{BL})_{\mu}$.
Here, we work in the pure $U(1)_{B-L}$ model, where $g^{\prime}$ is considered
to be zero. This choice of $g^{\prime}=0$ forbids $Z$-$Z_{BL}$ mixing at the
tree level333The gauge kinetic mixing is highly constrained by electroweak
precision measurements demands it to be $\lesssim 10^{-4}$ [72].. Finally,
$g_{BL}$ denotes the $U(1)_{B-L}$ gauge coupling.

Moving on to the scalar part of the Lagrangian, the most general
renormalizable scalar potential for this setup is given by

$\displaystyle V(\phi,S)$ $\displaystyle=$
$\displaystyle-\mu_{\phi}^{2}\phi^{\dagger}\phi-\mu_{S}^{2}|S|^{2}+\frac{\lambda_{\phi}}{2}(\phi^{\dagger}\phi)^{2}+\lambda_{\phi
S}(\phi^{\dagger}\phi)|S|^{2}+\lambda_{S}|S|^{4}.$ (2.3)

For $\mu_{S}^{2}>0$, the CP even component of $B-L$ scalar
$S=\frac{1}{\sqrt{2}}(v_{BL}+\phi_{S})$ develops a non-zero vacuum expectation
value $v_{BL}$ and breaks the $U(1)_{B-L}$ symmetry. This breaking ensures
Majorana masses for the RHNs (discussed latter) together with an additional
massive $B-L$ gauge boson $Z_{BL}$. The masses of the $B-L$ scalar
($\phi_{S}$) and gauge boson after the $B-L$ symmetry breaking is expressed as
444After the breaking of $B-L$ symmetry, $\phi$ also obtains mass due to the
presence of $\lambda_{\phi S}$ interaction. We do not write that mass term
explicitly as its presence does not alter the present analysis.,

$\displaystyle m_{S}^{2}$ $\displaystyle=$ $\displaystyle
2\,\lambda_{S}\,v_{BL}^{2},$ (2.4a) $\displaystyle M_{Z_{BL}}$
$\displaystyle=$ $\displaystyle 2\,g_{BL}\,v_{BL}.$ (2.4b)

On the other hand, Electroweak Symmetry Breaking (EWSB) is triggered for
$\mu_{\phi}^{2}$ when the $CP$-even components of $\phi$ receive a vev $v$.
The minimization conditions for the potential in Eq. 2.3 are given below:

$\displaystyle\mu_{\phi}^{2}$ $\displaystyle=$
$\displaystyle\frac{\lambda_{\phi}}{2}v^{2}+\frac{\lambda_{\phi
S}}{2}v_{BL}^{2},$ (2.5a) $\displaystyle\mu_{S}^{2}$ $\displaystyle=$
$\displaystyle\frac{\lambda_{\phi S}}{2}v^{2}+\lambda_{S}\,v_{BL}^{2}.$ (2.5b)

After the EWSB, scalar doublet in the present setup can be parametrized as

$\displaystyle\phi$ $\displaystyle=$ $\displaystyle\begin{pmatrix}0\\\
\frac{1}{\sqrt{2}}(v+\phi_{h})\end{pmatrix}.$ (2.6)

Subsequent to the EWSB, a non-zero $\phi_{h}-\phi_{S}$ mixing leads to the
following mass terms

$\displaystyle
V\supset\frac{1}{2}\begin{pmatrix}\phi_{h}&\phi_{S}\end{pmatrix}\begin{pmatrix}\lambda_{\phi}\,v^{2}&\lambda_{\phi
S}\,v\,v_{BL}\\\ \lambda_{\phi
S}\,v\,v_{BL}&2\lambda_{S}\,v_{BL}^{2}\end{pmatrix}\begin{pmatrix}\phi_{h}\\\
\phi_{S}\end{pmatrix}.$ (2.7)

The mass matrix is diagonalised using

$\displaystyle\begin{pmatrix}\phi_{h}\\\
\phi_{S}\end{pmatrix}=\begin{pmatrix}c_{\theta}&s_{\theta}\\\
-s_{\theta}&c_{\theta}\end{pmatrix}\begin{pmatrix}h\\\ s\end{pmatrix}$ (2.8)

with

$\displaystyle\tan{2\theta}$ $\displaystyle=$
$\displaystyle\frac{-2\,\lambda_{\phi
S}\,v\,v_{BL}}{\lambda_{\phi}\,v^{2}-2\,\lambda_{S}\,v^{2}_{BL}}.$ (2.9)

The mass eigenstates ($h,s$) then have masses

$\displaystyle m^{2}_{h,s}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\Big{[}\big{(}\lambda_{\phi}v^{2}+2\lambda_{S}v^{2}_{BL}\big{)}\pm\sqrt{(\lambda_{\phi}v^{2}-2\lambda_{S}v^{2}_{BL}\big{)}^{2}+4\lambda_{\phi
S}^{2}v^{2}v_{BL}^{2}}\Big{]}.$ (2.10a)

Here we consider physical scalar $h$ as the SM like Higgs boson with mass
$m_{h}=125.09$ GeV [73]. The various model parameters are expressible in terms
of the physical quantities as follows:

$\displaystyle\lambda_{\phi}$ $\displaystyle=$
$\displaystyle\frac{(m^{2}_{h}c^{2}_{\theta}+m^{2}_{s}s^{2}_{\theta})}{v^{2}},$
(2.11a) $\displaystyle\lambda_{\phi S}$ $\displaystyle=$
$\displaystyle\frac{(m^{2}_{s}-m^{2}_{h})s_{\theta}c_{\theta}}{v\,v_{BL}},$
(2.11b) $\displaystyle\lambda_{S}$ $\displaystyle=$
$\displaystyle\frac{(m^{2}_{h}s^{2}_{\theta}+m^{2}_{s}c^{2}_{\theta})}{2v_{BL}^{2}}.$
(2.11c)

The $\phi_{S}-\phi_{h}$ mixing angle is highly constrained, and the current
experiments demand it to be small (see Section 4). As this mixing angle does
not play any significant role in the present context, we have kept
$s_{\theta}$ fixed at $10^{-3}$ throughout this work, such that it satisfies
the experimental constraints. In the limit of sufficiently small
$\phi_{S}-\phi_{h}$ mixing, one obtains $\phi_{S}\simeq s$, and $m_{S}\simeq
m_{s}$.

Next, the Yukawa interactions for the present scenario is expressed as,

$\displaystyle-\mathcal{L}_{y}\supset
y_{11}\bar{N_{1}^{c}}N_{1}S+y_{\alpha\beta}\bar{N_{\alpha}^{c}}N_{\beta}S+h_{i\alpha}\overline{l_{L}}\tilde{\phi}N_{\alpha}+h.c.,$
(2.12)

with $\alpha,~{}\beta=2,~{}3$ and $i=e,\mu,\tau$. As discussed earlier,
$N_{1}$ being $Z_{2}$ odd remains stable, unlike the other two RHNs $N_{2}$
and $N_{3}$, which can decay into the scalar and the SM leptons ($l$) through
the third term of Eq. 2.12 if kinematically allowed. The existence of $N_{2}$
and $N_{3}$ in the present setup can also explain the origin of non-zero
neutrino masses together with baryogenesis via leptogenesis. In addition, EWSB
gives rise to the following mass matrix for $N_{1,2,3}$.

$\displaystyle M_{N}=\sqrt{2}~{}v_{BL}\begin{pmatrix}y_{11}&0&0\\\
0&y_{22}&y_{23}\\\ 0&y_{23}&y_{33}\end{pmatrix}.$ (2.13)

To demonstrate our point without losing the generality, we consider $y_{23}=0$
for simplicity in the rest of the analysis, in which case $M_{N}$ is diagonal
with masses

$\displaystyle M_{i}=\sqrt{2}~{}y_{ii}\,v_{BL}.$ (2.14)

For simplicity, we assume the other two RHNs to be nearly mass degenerate for
the rest of the analysis and consider $y_{22}\simeq y_{33}=y$. Finally, for
our analysis purpose, we choose the following sets of independent parameters:

$\displaystyle\\{m_{s},M_{1},y,v_{BL},g_{BL},s_{\theta}\\}.$

##  3 Thermal corrections

This section briefly comments on the thermal corrections to the masses of
relevant particles. These corrections play a non-trivial role in understanding
the DM phenomenology of the present setup. In the early Universe, when the
temperature of the thermal soup was very high, the thermal corrections [74,
75, 37] to the masses of the particles in the bath must have been very large.
In general, any particle that couples in the thermal bath with the primordial
plasma is expected to obtain a mass proportional to the temperature of the
Universe provided the condition $T>m_{i}$ is satisfied, here $m_{i}$ denotes
various mass scales involved in the theory [76].

SM particles are expected to be in equilibrium with the thermal plasma at high
temperatures. In the present set up we also assume that the particles like the
scalar $S$ and the heavier RHNs $N_{2,3}$ remained in equilibrium with the
thermal plasma due to their sizable interaction strengths in the early
Universe. Hence, their masses are expected to obtain thermal corrections at
high temperature. On the other hand, DM candidate $N_{1}$ in this model
interacts very feebly with the thermal bath and never enters thermal
equilibrium. Due to this reason, the thermal correction to its mass remains
negligible even at high temperatures. For example, considering $U(1)_{B-L}$
breaking scale $v_{BL}\sim\mathcal{O}(10^{10}~{}\text{GeV})$ with a fixed DM
mass $M_{1}\sim 500~{}\text{GeV}$, one obtains $y_{11}\sim 3\times
10^{-8}~{}\text{GeV}$, following Eq. 2.14. With such a feeble interaction, the
thermal corrections to $N_{1}$ mass at a temperature $T>>M_{1}$ remains
negligible $i.e.~{}M_{1}(T)=\sqrt{M_{1}^{2}+(y_{11}^{2}/16)T^{2}}\simeq
M_{1}$. Finally, the setup also demands a very feeble
$g_{BL}\sim\mathcal{O}(10^{-8})$. Such a small $g_{BL}$ also prevents $Z_{BL}$
from entering into the equlibrium and hence its thermal mass can also be
negelected. These choices of couplings will be further clarified in Section 5.

Figure 1: One-loop diagram contributing dominantly towards thermally corrected
mass of scalar $s$.

Now we discuss the thermal corrections to the mass of $s$ as it plays a
crucial role in the DM phenomenology of the present construct. Note that
several processes can provide thermal contributions to the mass of $s$. For
example, one can have self-energy corrections with $s$, $H$, $Z_{BL}$ and
$N_{i}$ coming in the loop, which can contribute to the thermally corrected
mass of $s$. These contributions can be denoted as $\Pi^{2}_{s}(T)$,
$\Pi^{2}_{H}(T)$, $\Pi^{2}_{Z_{Bl}}(T)$ and $\Pi^{2}_{N_{i}}(T)$ respectively.
The present work demands a very large $v_{BL}$ to ensure the feeble
interaction of the DM with $s$. This, in turn, also makes the couplings like
$\lambda_{S}$ and $\lambda_{\phi S}$ negligibly small (see Eq. 2.11, with
$v_{BL}\sim\mathcal{O}(10^{10}~{}\text{GeV})$, $\lambda_{S}\sim 10^{-13}$ and
$\lambda_{\phi S}\sim 10^{-10}$). The smallness of these couplings guarantees
that $\Pi^{2}_{s}$, $\Pi^{2}_{H}$ remains significantly small in comparison to
$\Pi_{N_{i}}^{2}(T)$ (Note that the $N_{i}-N_{i}-S$ coupling ($y$) can be
quite large) and can be ignored. Next, the set up also demands a very small
$g_{BL}$ and hence the contributions of $\Pi^{2}_{Z_{BL}}(T)$ can also be
safely ignored. The thermal contribution to the mass of $s$ from the diagram
shown in Fig. 1 is given as [74, 37]:

$\displaystyle\Pi_{N_{i}}^{2}(T)$ $\displaystyle=$
$\displaystyle\frac{y_{ii}^{2}}{6}T^{2}.$ (3.1)

Finally, the effective mass of the scalar can be expressed as,

$\displaystyle M_{s}(T)$ $\displaystyle=$
$\displaystyle\sqrt{m_{s}^{2}+\Pi_{s}^{2}(T)+\Pi_{H}^{2}(T)+\Pi_{Z_{BL}}^{2}(T)+\Pi_{N_{i}}^{2}(T)}.$
(3.2)

One can similarly calculate the masses of $N_{2,3}$ in terms of temperature by
incorporating all relevant contributions. In Section 5, we describe the
importance of the thermal corrections in the context of the DM phenomenology.

##  4 Theoretical and experimental constraints

###  4.1 Theoretical constraints

The scalar potential discussed in Eq. 2.3 must remain bounded from below in
various directions in the field space. Stability of vacuum can be ensured if
the quartic couplings satisfy the following conditions:

$\displaystyle\lambda_{\phi}>0,~{}\lambda_{S}>0,~{}\lambda_{\phi
S}+\sqrt{2\lambda_{\phi}\lambda_{S}}>0.$ (4.1)

On the other hand, to keep the model parameters perturbative, the parameters
must obey:

$\displaystyle|\lambda_{i}|<4\pi,~{}|g_{i}|<\sqrt{4\pi},~{}|y_{i}|<\sqrt{4\pi},$
(4.2)

Where $g_{i}$ and $y_{i}$ denote the gauge, and the Yukawa couplings and
$\lambda_{i}$ represent the scalar quartic couplings involved in the
calculation.

###  4.2 Experimental Constraints

  * I.

Relic density and direct detection: Due to the presence of a DM, the model is
subjected to the constraints coming from the Planck experiment [3]:

$\displaystyle\Omega_{\text{DM}}h^{2}$ $\displaystyle=$ $\displaystyle
0.120\pm 0.001.$ (4.3)

Additionally, the model is also exposed to the constraints imposed by the
direct detection experiments like LUX [16], PandaX-II [17] and Xenon-1T [18].
Elaborated discussions on the dark matter phenomenology are presented in
Section 5.

  * II.

LHC diphoton searches: In presence of the mixing between $h$ and $s$, the tree
level interactions of the SM Higgs with the SM fermion and gauge bosons get
modified. In such a scenario, the signal strength in the di-photon channel
then takes a form:

$\displaystyle\mu_{\gamma\gamma}=c^{2}_{\theta}\frac{BR_{h\to\gamma\gamma}}{BR_{h\to\gamma\gamma}^{\text{SM}}}\simeq
c^{2}_{\theta}\frac{\Gamma_{h\to\gamma\gamma}}{\Gamma_{h\to\gamma\gamma}^{\text{SM}}}.$
(4.4)

LHC sets a limit on this new mixing angle as $|\sin{\theta}|\leq 0.36$ [77].

  * III.

LEP bound and opposite sign di-lepton search at LHC: Since the SM fermions are
charged under $U(1)_{B-L}$ symmetry and interact directly with the
$U(1)_{B-L}$ gauge boson $Z_{BL}$, the footprints of $Z_{BL}$ can be obtained
in the collider searches. The null detection of such signature severely
constrains the ratio $M_{Z_{BL}}/g_{BL}$. The exclusion limit from LEP-II [78,
79] on this ratio is:

$\displaystyle\frac{M_{Z_{BL}}}{g_{BL}}\geq 7~{}\text{TeV}.$ (4.5)

On the other hand, one should also observe the constraints coming from
opposite-sign di-lepton searches at LHC, which primarily excludes the model
for $150~{}\text{GeV}<M_{Z_{BL}}<3$ TeV [65, 80], depending on the size of
$g_{BL}$. In this work, the $B-L$ gauge boson is treated as a FIMP which in
turn demands $g_{BL}$ to be very small, and hence the stringent constraints,
as discussed above, can easily be evaded.

  * IV.

Invisible Higgs decay: In this model, SM Higgs can also decay to the RHNs,
$Z_{BL}$ and also to the BSM scalar, if kinematically allowed. These extra
decay modes can contribute towards invisible Higgs decay. In such a situation,
we need to employ the bound on the invisible Higgs decay width as [81]:

$\displaystyle Br(h\rightarrow\rm{Invisible})<0.11,$ (4.6a)
$\displaystyle\frac{\Gamma(h\rightarrow\rm{Invisible})}{\Gamma(h\rightarrow
SM)+\Gamma(h\rightarrow\rm{Invisible})}<0.11.$ (4.6b)

where $\Gamma(h\rightarrow\rm{Invisible})=\Gamma(h\rightarrow BSM)$ when
$m_{i}<~{}\frac{m_{h}}{2}$ with $i=N_{1},N_{2},N_{3},Z_{BL},s$ and
$\Gamma(h\rightarrow SM)=4.2$ MeV. However, in our present analysis, we
primarily focus on the parameter space where $m_{i}>~{}\frac{m_{h}}{2}$. So
the above constraint is not applicable.

##  5 Dark Matter Phenomenology

Null detection of WIMP dark matter in the direct [16, 17, 18] and indirect
search experiments [19] has motivated the community to explore the various
exotic realization of DM. Among such possibilities, the popular one is the
FIMP-type DM, where the DM never comes in equilibrium with the thermal soup.
Here, the initial abundance of the DM is assumed to be zero (or negligible).
As the Universe cools down, its feeble interaction with the bath helps in its
gradual production from decays or scatterings of the bath particles. Such a
weaker strength of coupling ensures that the DM interaction rate invariably
remains smaller than the Hubble expansion rate ($H$), $i.e.~{}\Gamma_{int}<H$.
Studies of such a FIMP type DM establish a condition where the maximum DM
production takes place when the temperature of the thermal bath is of the
order or below the mass of the mother particle responsible for the production
of the DM. Unlike the standard freeze-in scenario, in the present up, DM
production can be enhanced at early times if thermal corrections to the mass
of the mother particle are included. This mechanism of DM production can be
dubbed as the _forbidden freeze-in_. Here, the DM production channel, which
was otherwise forbidden or kinematically disallowed in the standard freeze-in
(SFI), now becomes allowed once the thermal correction to the mass of the
mother particle is incorporated.

The present setup explores the $U(1)_{B-L}$ extension of the SM where the
lightest RHN ($N_{1}$), which is odd under a $Z_{2}$ symmetry, plays the role
of FIMP dark matter. Here, the production of $N_{1}$ can take place from the
decay of $s$, $h$ (physical scalars obtained after the mixing between
$\phi_{S}$ and $\phi_{h}$ after the EWSB) and $Z_{BL}$. All such relevant
production channels of $Z_{BL}$ and $N_{1}$ are depicted in Fig. 2. The feeble
interaction of $N_{1}$ is assured by choosing a relatively large $v_{BL}$
($\Gamma_{s\to N_{1}N_{1}}\propto~{}y_{11}^{2}c_{\theta}^{2}\propto
M_{1}^{2}c_{\theta}^{2}/v_{BL}^{2}$) and a relatively smaller $g_{BL}$
($\Gamma_{Z_{BL}\to N_{1}N_{1}}\propto~{}g_{BL}^{2}$). Note that, due to the
smallness of $g_{BL}$, the $B-L$ gauge boson $Z_{BL}$ also never thermalizes
with bath and is produced feebly from the decay of $s$ and $h$. Hence, in
order to study the evolution of dark matter with the expansion of the
Universe, one needs to solve a set of coupled Boltzmann equations while taking
into account the evolution of $Z_{BL}$ as well. The coupled Boltzmann
equations are expressed as,

Figure 2: Possible production channels of $Z_{BL}$ and the DM candidate
$N_{1}$.

$\displaystyle\frac{dY_{Z_{BL}}}{dx}$ $\displaystyle=$
$\displaystyle\frac{1}{Hx}\Bigg{[}\theta(M_{s}(m_{s}/x)-2M_{Z_{BL}})~{}\langle\Gamma_{s\to
Z_{BL}Z_{BL}\rangle}Y_{s}^{EQ}-\langle\Gamma_{Z_{BL}\to~{}all}\rangle
Y_{Z_{BL}}\Bigg{]},$ (5.1a) $\displaystyle\frac{dY_{N_{1}}}{dx}$
$\displaystyle=$ $\displaystyle\frac{1}{Hx}\Bigg{[}\langle\Gamma_{Z_{BL}\to
N_{1}N_{1}}\rangle
Y_{Z_{BL}}+\theta(M_{s}(m_{s}/x)-2M_{1})~{}\langle\Gamma_{s\to
N_{1}N_{1}}\rangle Y_{s}^{EQ}\Bigg{]},$ (5.1b)

Here $x=m_{s}/T$, where $T$ and $H=1.67\sqrt{g_{*}}\frac{T^{2}}{M_{Pl}}$
denotes the temperature and expansion rate of the Universe respectively.
Whereas $Y_{j}=n_{j}/\mathfrak{s}$ denotes the comoving number density of the
different species ($j=s,Z_{BL},N_{1}$) involved with $\mathfrak{s}$ being the
entropy density. $Y_{s}^{EQ}$ signifies the equlibrium density of $s$. Next,
$\langle\Gamma_{i}\rangle$ with $i=s,Z_{BL}$ represents the thermally
averaged555Since $Z_{BL}$ never thermalizes with the plasma, one should
properly consider the non-thermal distribution function $(f_{Z_{BL}})$ for
$Z_{BL}$ in order to calculate its thermally averaged decay width [28]. In
such a scenario
$\langle\Gamma_{Z_{BL}}\rangle=\frac{\int(\frac{M_{Z_{BL}}}{E_{Z_{BL}}})\Gamma_{Z_{BL}\rightarrow
AA}f_{Z_{BL}}(p,T)d^{3}p}{\int f_{Z_{BL}}(p,T)d^{3}p}$. decay widths [28]
where

$\displaystyle\Gamma_{s\longrightarrow Z_{BL}Z_{BL}}$ $\displaystyle=$
$\displaystyle\frac{g_{BL}^{2}c_{\theta}^{2}}{8\pi}\frac{M_{s}^{3}(T)}{M_{Z_{BL}}^{2}}~{}\Big{(}1-\frac{4M^{2}_{Z_{BL}}}{M^{2}_{s}(T)}\Big{)}^{1/2}~{}\Big{(}1-\frac{4M^{2}_{Z_{BL}}}{M^{2}_{s}(T)}+\frac{12M^{4}_{Z_{BL}}}{M^{4}_{s}(T)}\Big{)},$
(5.2a) $\displaystyle\Gamma_{s\longrightarrow N_{1}N_{1}}$ $\displaystyle=$
$\displaystyle\frac{M_{s}(T)}{32\pi}~{}y_{11}^{2}c_{\theta}^{2}~{}\Big{(}1-\frac{4M^{2}_{1}}{M^{2}_{s}(T)}\Big{)}^{3/2},$
(5.2b) $\displaystyle\Gamma_{Z_{BL}\longrightarrow N_{1}N_{1}}$
$\displaystyle=$
$\displaystyle\frac{M_{Z_{BL}}}{24\pi}~{}g^{2}_{BL}~{}\Big{(}1-\frac{4M^{2}_{1}}{M^{2}_{Z_{BL}}}\Big{)}^{3/2},$
(5.2c) $\displaystyle\Gamma_{Z_{BL}\longrightarrow f\bar{f}}$ $\displaystyle=$
$\displaystyle\frac{M_{Z_{BL}}}{12\pi}~{}g^{2}_{BL}~{}\Big{(}1+\frac{2M^{2}_{f}}{M^{2}_{Z_{BL}}}\Big{)}\Big{(}1-\frac{4M^{2}_{f}}{M^{2}_{Z_{BL}}}\Big{)}^{1/2}.$
(5.2d)

Note that, due to the large $B-L$ breaking scale, BSM particles gain their
masses in the early Universe, and hence the $Z_{BL}$ is mainly produced
through the decay of $s$. At this stage, we would also like to mention that
due to the feeble interaction ($y_{11}$) of the DM with $s$ and large value of
both $s$ and $N_{i}$ masses at high temperature the production of $N_{1}$ is
dominated by the decay of $s$, while its production from scattering processes
like $N_{i}N_{i}\to N_{1}N_{1}$ or $hh(ss)\to N_{1}N_{1}$ remains subdominant
and can be neglected. Finally, we have also ensured that rate of the
scattering processes like $N_{1}N_{i}\to N_{1}N_{i}$ and $N_{1}h(s)\to
N_{1}h(s)$ remains several orders of magnitude smaller than the Hubble
expansion rate. For example we found that $\Gamma_{N_{1}N_{i}\to
N_{1}N_{i}}/H(T)\sim 10^{-20}$ at $T\simeq 10^{8}$ GeV which shows that the
$N_{1}$ never enters thermal equilibrium even at high temperatures.

In the present setup, we are interested in exploring the production of both
$Z_{BL}$ and $N_{1}$ through the forbidden channels. These channels become
effective once thermal corrections to the mass of $s$ are incorporated and
remain active only till the point these decays are kinematically allowed, this
is ensured by the use of $\theta-$function in Eq. 5.1. Once the asymptotic
yield of the DM $Y_{N_{1}}(x_{\infty})$ is obtained after solving the
Boltzmann equation, we can use it to calculate the relic density of the DM as,

$\displaystyle\Omega_{N_{1}}h^{2}$ $\displaystyle=$ $\displaystyle 2.75\times
10^{8}\bigg{(}\frac{M_{1}}{\rm{GeV}}\bigg{)}Y_{N_{1}}(x_{\infty}),$ (5.3)

where $x_{\infty}$ indicates the asymptotic value of $x$ after the DM freeze-
in.

Figure 3: Variation of thermally corrected mass $M_{s}(x)$ of second scalar
with a dimensionless quantity $x=\frac{m_{s}}{T}$ for three different values
Yukawa coupling $y$. In the left panel, we demonstrate a scenario where the
$B-L$ gauge boson $Z_{BL}$ and the DM candidate $N_{1}$ can only be produced
via the FFI mechanism. In contrast, the right panel depicts a picture where
the $Z_{BL}$ can only be produced via the FFI mechanism, but $N_{1}$ can be
produced through both FFI and SFI.

To understand the DM phenomenology more evidently, we categorize our study
into two cases in terms of possible mass hierarchies: (A)
$M_{Z_{BL}}>M_{1}>m_{s}$, and (B) $M_{Z_{BL}}>m_{s}>M_{1}$ so that the effect
of FFI and its benefits over SFI becomes visible. We demonstrate the
importance of these two cases in Fig. 3. Here, we show the variation of
thermally corrected scalar mass $M_{s}(x)$ in terms of dimensionless parameter
$x=m_{s}/T$ for three different choices of Yukawa couplings $y$. Note that,
while generating Fig. 3 we followed a conservative limit where it is assumed
that the thermal correction to the mass of $s$ remains significant till the
temperature $T\sim M_{2,3}\sim yv_{BL}$. Below this temperature, the thermally
corrected mass of $s$ coincides with the bare mass value [76]. We follow the
same principle in presenting the rest of our analysis. In Fig. 3 the dashed
horizontal line represents the fixed values of different mass parameters,
$2M_{Z_{BL}}$ (in purple), $2M_{1}$ (in magenta), and $m_{s}$ (in orange)
which helps to understand the mass hierarchy. The pink shaded region shows the
parameter space where the FIMP type particles can also be produced if allowed
in the SFI scenario. It is evident from the left panel of Fig. 3 that the
production of the FIMP type particles ($Z_{BL}$ and $N_{1}$) can only take
place through the mechanism of FFI if one considers the mass hierarchy
$M_{Z_{BL}}>M_{1}>m_{s}$. On contrary to this, in the right panel of Fig. 3,
we consider a situation where the mass of the dark matter $i.e.~{}M_{1}$ lies
below $M_{s}(T)=m_{s}$. Primary condition on scalar mass $M_{s}(T)=2M_{1}$
ensures the production of the dark matter both from the decay of $s$ and
$Z_{BL}$ in the forbidden freeze-in scenario and only through $s$ in a
standard freeze-in scenario. $s\to Z_{BL}Z_{BL}$ remains forbidden in this
case due to the choice of mass hierarchy considered. This case also provides a
clear distinction between the FFI and SFI scenarios. Next, we solve the set of
coupled Boltzmann equations (Eq. 5.1) numerically to study the evolution of
$Z_{BL}$ and $N_{1}$ with the expansion of the Universe for these two cases.

###  5.1 Case A: Complete FFI region when $M_{Z_{BL}}>M_{1}>m_{s}$

In this mass hierarchy, $s$ being the lightest BSM particle, it neither decays
to $Z_{BL}$ nor to $N_{1}$ in a typical SFI scenario. Once the thermally
corrected mass of $s$ is taken into account, the left panel of Fig. 3
demonstrates that $s$ can be heavy enough to produce both $Z_{BL}$ and $N_{1}$
through the FFI mechanism. We also provide Table 2 for a better understanding
of this picture.

Decay-Channels | SFI |  FFI
---|---|---
$s\to Z_{BL}Z_{BL}$ | X | ✓
$s\to N_{1}N_{1}$ | X | ✓
$Z_{BL}\to N_{1}N_{1}$ | X | ✓
Table 2: List of processes contributing to dark matter and $Z_{BL}$ production
in a standard freeze-in (SFI) and forbidden freeze-in (FFI) scenario for a
mass hierarchy $M_{Z_{BL}}>M_{1}>m_{s}$. $s\to Z_{BL}Z_{BL}$ remains forbidden
within this mass hierarchy for the SFI scenario, which in turn suggests that
$Z_{BL}\to N_{1}N_{1}$ is also forbidden even though this decay remains
kinematically allowed.

(a)

(b)

(c)

Figure 4: Evolution of generated yield of ${Z_{BL}}$ (dotted lines) and dark
matter ${N_{1}}$ (solid lines) with respect to a dimensionless parameter
$x=\frac{m_{s}}{T}$. The values of different parameters controlling the DM
phenomenology are mentioned at the top of each plot in a case study for
complete FFI region, observed for mass hierarchy $M_{Z_{BL}}>M_{1}>m_{s}$.
Thick black dashed line represents the yield of the DM corresponding to the
observed relic density.

To facilitate our discussion, in Fig. 4a, we show the variation of
$Y_{Z_{BL}}$ and $Y_{N_{1}}$ with a dimensionless quantity
$x=\frac{m_{s}}{T}$. The values of different parameters controlling the DM
phenomenology are mentioned at the top of each plot. One notices that the
production of the $Z_{BL}$ which can occur through the decay of the second
scalar $s$ is kinematically forbidden with the given choices of $m_{s}$ and
$M_{Z_{BL}}$ if the thermal corrections are not incorporated. Looking at the
Eq. 2.11, one finds that for a large $v_{BL}$ as required in this setup, the
couplings $\lambda_{\phi}$ and $\lambda_{\phi S}$ remains significantly small,
on the other hand, the setup also demands a very small $g_{BL}$; hence the
contribution of $\Pi_{s}^{2}(T),\Pi_{H}^{2}(T)$ and $\Pi_{Z_{BL}}^{2}(T)$ in
Eq. 3.2 remains almost negligible. On the other hand, the scalar field can
acquire a sizeable thermal mass depending on the choices of BSM Yukawa
couplings ($y_{22}~{}\propto~{}M_{2}/v_{BL}$ and
$y_{33}~{}\propto~{}M_{3}/v_{BL}$). This is also consistent with our
expectation that the masses of the other two RHNs must be quite heavy to
explain the non-zero neutrino masses and leptogenesis through Type-I seesaw.
It is expected that, with an adequate choice of the $y$ (with $y_{22}\simeq
y_{33}=y$), one can easily obtain a scenario: $M_{s}(T)>2M_{Z_{BL}},2M_{1}$
and thereafter, the decay of $s$ can produce $Z_{BL}$ and $N_{1}$. This can
also be seen in Fig. 4a. Next, with the choices of parameters considered, the
production of $N_{1}$ can also proceed via the decay of $Z_{BL}$. In Fig. 4a,
the evolution of $Z_{BL}$ (dotted) and $N_{1}$ (solid) are shown for three
different choices of $y$, $i.e.~{}y=10^{-3}~{}(\text{red}),y=7\times
10^{-4}~{}(\text{blue}),y=10^{-4}~{}(\text{green})$. With $y=10^{-3}$, the
thermally corrected mass of $s$ is expected to be large. The larger mass leads
to a relatively larger decay widths for the processes $s\to Z_{BL}Z_{BL}$ and
$s\to N_{1}N_{1}$ (in comparison to smaller $y$ values) which in turn
generates relatively larger yields of $Z_{BL}$ and $N_{1}$ in Fig. 4a. One
notices that the abundance of $Z_{BL}$ gradually increases due to its
production from the decay of $s$, then saturates (plateau) once its production
rate becomes comparable to its decay rate. Finally, it falls as its decay to
the SM fermions, and the DM overtakes its production. However, the abundance
of $N_{1}$ increases slowly till the time (first bend) when the temperature of
the Universe becomes of the order of the heavier RHN masses $i.e.~{}T\simeq
M_{i}$ ($i.e.~{}x\simeq 7\times 10^{-6}$ for $y=0.001$), after which its
production from the decay of $s$ becomes kinematically forbidden, and its
yield saturates. This is because, at this point, the dominant contribution to
the thermally corrected mass of $s$ becomes insignificant (as also discussed
in Section 3), and $M_{s}(T)$ falls back to the bare mass value $m_{s}$.
Subsequently, a relatively sharper rise is observed in its yield due to its
production from $Z_{BL}$, and finally, its abundance saturates (at around
$T\simeq M_{Z_{BL}}~{}i.e.~{}x\simeq 10^{-1}$ ) once the decay of $Z_{BL}$ is
completed. It is interesting to point out that the production of $Z_{BL}$ from
the decay of $s$ starts much earlier in comparison to the production of
$N_{1}$. This happen because the $s$ decays dominantly to $Z_{BL}$ and sub-
dominantly to $N_{1}$ (see Eq. 5.2). Similar behavior is observed in the
evolution of $Z_{BL}$ for smaller $y$, but with a relatively smaller yield.
With a small $y$, the thermal correction to the mass of $s$ also remains
small. This, in turn, reduces the decay width of $s$. Unlike the scenario with
a relatively larger $y$, now $N_{1}$ production ceases when the decay $s\to
N_{1}N_{1}$ becomes kinematically disallowed at a relatively later time ( as a
smaller $y$ corresponds to a smaller value of $M_{2,3}$, hence a larger $x$).
It again starts getting produced as the $Z_{BL}\rightarrow N_{1}N_{1}$ becomes
operational. Finally, DM abundance freezes in once the decay of $Z_{BL}$ is
complete. The thick dashed horizontal black line (in each plot) indicates the
abundance of dark matter for which the relic density satisfies the Planck
experimental limit.

Next, in Fig. 4b we show the evolution of the dark matter for three different
combinations of $v_{BL}$ and $g_{BL}$ while keeping $M_{Z_{BL}}$ fixed at
$1~{}\text{TeV}$. Here, one finds that for a choice of smaller $v_{BL}$ (and a
larger $g_{BL}$), both the FFI production channels get enhanced, leading to an
overabundant $N_{1}$ (as $\Gamma(s\to N_{1}N_{1})\propto\frac{1}{v_{BL}^{2}}$
and $\Gamma(Z_{BL}\to N_{1}N_{1})\propto g_{BL}^{2}$). Hence, one can
accommodate the correct yield of the DM by tuning these two parameters
appropriately, as seen from the blue curve. Lastly, Fig. 4c shows the effect
of different DM masses on its evolution. For a choice with $M_{1}=500$ GeV,
the only source of its production is the decay of $s$. The moment this decay
stops, the DM yield becomes constant. In such a case, it is difficult for the
DM to satisfy the measured relic at the Planck experiment.

###  5.2 Case B: Partial FFI region when $M_{Z_{BL}}>m_{s}>M_{1}$

Decay-Channels | SFI |  FFI
---|---|---
$s\to Z_{BL}Z_{BL}$ | X | ✓
$s\to N_{1}N_{1}$ | ✓ | ✓
$Z_{BL}\to N_{1}N_{1}$ | X | ✓
Table 3: List of processes contributing to dark matter production in a
standard freeze-in (SFI) and forbidden freeze-in (FFI) scenario for a mass
hierarchy $M_{Z_{BL}}>m_{s}>M_{1}$. $s\to Z_{BL}Z_{BL}$ remains forbidden
within this mass hierarchy for the SFI scenario, which in turn suggests that
$Z_{BL}\to N_{1}N_{1}$ is also forbidden even though this decay remains
kinematically allowed.

We now aim to study the DM phenomenology with the above mass hierarchy where
FFI decay modes are open partially, as also shown in the right panel of Fig 3.
Hence evolution process of the DM indicates a distinct direction in the FFI
scenario compared to SFI. Unlike the previous case, DM can now be produced
directly from the decay of $s$, even if $M_{s}(T)\simeq m_{s}$ is satisfied.
However, the production of the $Z_{BL}$ can only be possible through the
forbidden freeze-in mechanism from the decay of $s$666Although the production
of $Z_{BL}$ can proceed through $2\rightarrow 2$ scatterings, its abundance
remains almost negligible as the production cross-section depends on
$g_{BL}^{4}$. . For a better understanding, in Table 3 we provide all the
relevant decay channels required for the production of $Z_{BL}$ and $N_{1}$
for FFI and SFI. Next, we demonstrate the importance of forbidden freeze-in
(FFI) over the standard freeze-in (SSI) in Fig. 5.

Figure 5: Evolution of generated yield of ${N_{1}}$ with respect to a
dimensionless parameter $x=\frac{m_{s}}{T}$. The solid blue line depicts the
production of $N_{1}$ in an FFI scenario which can satisfy the Planck
experimental limit on the relic density for the given choice of parameters.
With the same choice of parameters, the DM remains under-abundant for an SFI
scenario, as shown by the dashed blue line. The values of different parameters
controlling the DM phenomenology are mentioned at the top of each plot in a
case study for partial FFI region, observed for mass hierarchy
$M_{Z_{BL}}>m_{s}>M_{1}$. The thick black dashed line represents the abundance
of the DM corresponding to the observed relic density.

This figure shows a comparison between the production of the DM in the SFI
scenario (dashed blue line) and the FFI scenario (solid blue line). Here, with
the given choice of parameters, the $Z_{BL}$ can never be produced from the
decay of $s$ in the SFI scenario. Hence its abundance remains almost
negligible (as it can also be produced through scatterings). With such an
insignificant yield, $Z_{BL}$ contribution in producing the DM will always
remain sub-dominant in comparison to the DM production coming from the $s$
decay. Hence, the DM yield saturates as soon as its production from the $s$
decay stops. In this situation, it may become difficult for the DM to satisfy
the correct order of relic density. On the other hand, with the incorporation
of FFI, DM can be further produced from the decay of both $s$ and $Z_{BL}$ in
an adequate amount to satisfy the correct relic density with the given choice
of parameters.

Finally, we also like to comment on the detection prospect of the model under
consideration. The spontaneous breaking of the $U(1)_{B-L}$ symmetry at a high
energy scale leads to the formation of Nambu-Goto _cosmic strings_ [82]. Once
formed, the collisions and self-interactions of strings produce non-self-
interacting string loops, which further oscillates and radiates their energy
in the form of gravitational wave (GW). The incoherent superposition of such
continuous emission results in stochastic GW signals. This GW signal can be
detected at the present and future GW detectors like pulsar timing arrays
(PTAs), NANOGrav [83], PPTA [84], EPTA [85], IPTA [86], LISA [87], LIGO [88]
etc. The searches of GWs can increase the predictability of the present setup.
The detailed study of GWs is beyond the scope of the present work, and we plan
to take it as a future endeavor.

##  6 Summary and conclusions

In this paper, we study the phenomenology of feebly interacting massive
particles as dark matter in a minimal $U(1)_{B-L}$ extension of the SM. The
role of DM is played by the lightest of the three right-handed neutrinos,
which in turn are introduced to make the model free from the triangular
anomaly. The other two heavier RHNs can generate non-zero neutrino masses and
matter-antimatter asymmetry of the Universe through Type-I seesaw. Here, an
unbroken $Z_{2}$ symmetry ensures the stability of the DM. The setup also
requires a complex $SU(2)_{L}$ singlet scalar charged under the $B-L$
symmetry. After obtaining a non-zero vev, the scalar breaks the $U(1)_{B-L}$
spontaneously and simultaneously makes the RHNs together with a $B-L$ gauge
boson massive.

Due to their feeble interactions with the bath particles, both the DM
candidate ($N_{1}$) and $B-L$ gauge boson ($Z_{BL}$) never comes in
equilibrium with the thermal bath. Contrary to this, the complex scalar
mediator remains in the thermal equilibrium with the bath due to its not-so-
small interactions with the bath particles and contributes to the gradual
freeze-in production of $N_{1}$ and $Z_{BL}$. Moreover, if kinematically
allowed, the DM production is further dominated by the $Z_{BL}$ decay.

Although FIMP-type DM is studied in the $B-L$ framework, the thermal
corrections to the mediator masses were never taken into account.
Incorporating such corrections to the mediator mass at high temperature opens
up a new paradigm for a FIMP-type DM phenomenology. Simultaneously, it also
opens up an attractive possibility of producing the DM in a kinematically
forbidden region of the standard freeze-in (SFI) picture. In this work, we
explore this exciting possibility. With this in mind, we categorized our study
into two cases depending on the mass hierarchy of these particles. All other
likely mass hierarchies can be summed up within these two possibilities.

The first illustration depicts a forbidden freeze-in (FFI) picture where gauge
boson and dark matter are heavier than the complex scalar residing in the
thermal bath. Hence, the decay is kinematically disallowed, and consequently,
such a picture is utterly missing in the SFI framework. The appealing feature
here is that the production of the DM can take place in two steps: first from
the decay of the scalar due to the thermal corrections to its mass and then
subsequently from the late time decay of the gauge boson. Our example explores
the synergy between these two processes depending upon parameters in the
model.

For further clarity, in our second case study, we choose a particular mass
hierarchy ($M_{Z_{BL}}>m_{s}>M_{1}$) to mark the role of FFI over SFI
scenarios. Here the production of $Z_{BL}$ is kinematically forbidden in the
SFI case, while the dark matter is produced only from $B-L$ scalar’s decay.
Unlike the standard scenario, with the help of a large thermally corrected
mass in FFI, the scalar can produce the gauge boson together with the dark
matter in the early Universe. Similar to the first scenario, the DM production
again happens in two steps which makes the distinction of FFI with SFI
noticeable. Finally, due to the involvement of a large $B-L$ breaking scale,
the model can be tested indirectly in the GW search experiments. Such a scale
leads to the formation of cosmic string, which further oscillates and radiates
its energy in the form of gravitational waves.

## Acknowledgments

This work is supported by the Physical Research Laboratory (PRL), Department
of Space, Government of India. Computational work was performed using the HPC
resources (Vikram-100 HPC) and TDP project at PRL. Authors gratefully thank H.
Mishra for stimulating discussions during the project.

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