# Light Dirac neutrino portal dark matter with
gauged $U(1)_{B-L}$ symmetry

Debasish Borah<EMAIL_ADDRESS>Department of Physics, Indian Institute of
Technology Guwahati, Assam 781039, India  Nayan Das<EMAIL_ADDRESS>Department of Physics, Indian Institute of Technology Guwahati, Assam 781039,
India

###### Abstract

We propose a gauged $U(1)_{B-L}$ version of the light Dirac neutrino portal
dark matter. The $U(1)_{B-L}$ symmetry provides a UV completion by naturally
accommodating three right handed neutrinos from anomaly cancellation
requirements which, in combination with the left handed neutrinos, form the
sub-eV Dirac neutrinos after electroweak symmetry breaking. The particle
content and the gauge charges are chosen in such a way that light neutrinos
remain purely Dirac and dark matter, a gauge singlet Dirac fermion, remain
stable. We consider both thermal and non-thermal production possibilities of
dark matter and correlate the corresponding parameter space with the one
within reach of future cosmic microwave background (CMB) experiments sensitive
to enhanced relativistic degrees of freedom $\Delta N_{\rm eff}$. The
interplay of dark matter, CMB, structure formation and other terrestrial
constraints keep the scenario very predictive leading the $U(1)_{B-L}$
parameter space into tight corners.

##  I Introduction

The observations of dark matter (DM) in astrophysics and cosmology related
experiments together with non-zero neutrino mass and mixing provide strong
evidence for beyond standard model (BSM) physics Zyla:2020zbs ;
Aghanim:2018eyx . Just like the particle nature of DM is not yet known, there
are several unknowns in neutrino physics, including the origin of neutrino
mass. The nature of neutrinos: Dirac or Majorana, is one of them. While there
exist several BSM proposals for particle DM, the weakly interacting massive
particle (WIMP) and feebly interacting massive particle (FIMP) scenarios have
been studied extensively in the literature. In a typical WIMP scenario111A
recent review of WIMP type scenarios can be found in Arcadi:2017kky ., a
particle DM candidate having mass and interaction strength with standard model
(SM) particles typically around the electroweak ballpark can give rise to the
observed DM abundance after thermal freeze-out. On the other hand, in FIMP
paradigm222A recent review of FIMP can be found in Bernal:2017kxu ., the DM
can never enter equilibrium with the SM bath in the early universe due to its
feeble interactions with the latter. Such a DM candidate, with negligible
initial abundance, freezes in by virtue of decay or scattering from other
particles in the bath.

In this work, we consider a scenario where the origin of DM is related to the
Dirac nature of light neutrinos, known as the light Dirac neutrino portal DM
Biswas:2021kio ; Biswas:2022vkq  scenario. In such a setup, light Dirac
neutrinos take the role of mediating the interactions between DM and the SM
bath. In Biswas:2021kio  and Biswas:2022vkq , the DM was assumed to be of WIMP
and FIMP type respectively. In addition to linking the origin of neutrino mass
and nature with DM, this also offers additional discovery prospects due to
right chiral part of Dirac neutrinos, contributing to the effective
relativistic degrees of freedom ${\rm N_{eff}}$. Measurement related to the
cosmic microwave background (CMB) puts tight constraints ${\rm
N_{eff}=2.99^{+0.34}_{-0.33}}$ at $2\sigma$ or $95\%$ CL including baryon
acoustic oscillation (BAO) data Planck:2018vyg . Similar bound also exists
from big bang nucleosynthesis (BBN) $2.3<{\rm N}_{\rm eff}<3.4$ at $95\%$ CL
Cyburt:2015mya . Both of these cosmological bounds are consistent with the SM
predictions ${\rm N^{SM}_{eff}}=3.045$ Mangano:2005cc ; Grohs:2015tfy ;
deSalas:2016ztq 333A very recent paper Cielo:2023bqp  reports $N_{\rm
eff}^{\rm SM}=3.043$ by incorporating next-to-leading order correction to
$e^{+}e^{-}\leftrightarrow\nu_{L}\bar{\nu}_{L}$ interactions along with finite
temperature QED corrections to the electromagnetic plasma density and effect
of neutrino oscillations. . Future CMB experiment CMB Stage IV (CMB-S4) is
expected reach a much better sensitivity of $\Delta{\rm N}_{\rm eff}={\rm
N}_{\rm eff}-{\rm N}^{\rm SM}_{\rm eff}=0.06$ Abazajian:2019eic , taking it
closer to the SM prediction. Thus, Dirac neutrino scenarios can be probed in
future CMB experiments if right handed neutrinos (RHN) can be sufficiently
produced in the early universe via thermal or non-thermal processes. Light
Dirac neutrino models often lead to enhanced $\Delta{\rm N}_{\rm eff}$, some
recent works on which can be found in Abazajian:2019oqj ;
FileviezPerez:2019cyn ; Nanda:2019nqy ; Han:2020oet ; Luo:2020sho ;
Borah:2020boy ; Adshead:2020ekg ; Luo:2020fdt ; Mahanta:2021plx ; Du:2021idh ;
Biswas:2021kio ; Borah:2022obi ; Borah:2022qln ; Li:2022yna ; Biswas:2022fga ;
Borah:2023dhk ; Borah:2022enh ; Das:2023oph . In earlier works on light Dirac
neutrino portal DM, discrete symmetry like $Z_{4}$ was considered to get the
desired couplings, mass terms as well as the stability of DM. Here, we
consider a UV completion with a gauged $B-L$ framework. The right handed
neutrinos are naturally part of the model providing the minimal anomaly-free
setup. While RHNs are thermally produced due to gauged $B-L$ interactions, DM
can be of WIMP or FIMP type depending upon Dirac neutrino portal couplings.
The model not only gives rise to the desired DM phenomenology with observable
$\Delta{\rm N}_{\rm eff}$, but also leads to new constraints in the gauged
$B-L$ parameter space not obtained previously.

This paper is organised as follows. In section II, we briefly discuss the
minimal gauged $U(1)_{B-L}$ model of light Dirac neutrino portal dark matter
followed by separate discussions of FIMP and WIMP type DM in section III and
IV respectively. We finally conclude in section V.

##  II The Model

Gauged $B-L$ extension of the SM Davidson:1978pm ; Mohapatra:1980qe ;
Marshak:1979fm ; Masiero:1982fi ; Mohapatra:1982xz ; Buchmuller:1991ce  has
been a popular BSM framework studied in the context of neutrino mass among
others. The three right handed neutrinos $\nu_{R}$ having $B-L$ charge -1 each
not only keep the model anomaly free but also lead to massive Dirac neutrinos
in combination with $\nu_{L}$ after electroweak symmetry breaking. A singlet
fermion $\psi$ is considered to be the DM candidate while two singlet scalars
$\phi_{1}$ and $\phi_{2}$ with non-zero $B-L$ charges help in realising light
Dirac neutrino portal DM and spontaneous $B-L$ symmetry breaking respectively.
The relevant particle content is shown in table 1.

| $L$ | $\Phi$ | $\nu_{R}$ | $\psi$ | $\phi_{1}$ | $\phi_{2}$
---|---|---|---|---|---|---
$SU(2)$ | 2 | 2 | 1 | 1 | 1 | 1
$U(1)_{Y}$ | $-\frac{1}{2}$ | $\frac{1}{2}$ | 0 | 0 | 0 | 0
$U(1)_{B-L}$ | $-1$ | $0$ | -1 | 0 | 1 | 3
Table 1: Relevant particle content of the model with respective quantum
numbers under the symmetry group.

The scalar Lagrangian for the model can be written as

$\mathcal{L_{\text{s}}}=(D^{\mu}\Phi)^{\dagger}D_{\mu}\Phi+(D^{\mu}\phi_{1})^{\dagger}D_{\mu}\phi_{1}+(D^{\mu}\phi_{2})^{\dagger}D_{\mu}\phi_{2}-V(\Phi,\phi_{1},\phi_{2}),$
(1)

where

$D_{\mu}\Phi=\left(\partial_{\mu}-i\frac{g}{2}\tau_{a}W^{a}_{\mu}-i\frac{g^{\prime}}{2}B_{\mu}\right)\Phi,\,D_{\mu}\phi_{1}=\left(\partial_{\mu}-ig_{BL}B^{\prime}_{\mu}\right)\phi_{1},\,D_{\mu}\phi_{2}=\left(\partial_{\mu}-i3g_{BL}B^{\prime}_{\mu}\right)\phi_{2},$
(2)

denote the respective covariant derivatives and

$\displaystyle V(\Phi,\phi_{1},\phi_{2})$ $\displaystyle=$
$\displaystyle-\mu^{2}(\Phi^{\dagger}\Phi)+\mu_{1}^{2}(\phi_{1}^{\dagger}\phi_{1})-\mu_{2}^{2}(\phi_{2}^{\dagger}\phi_{2})+\lambda(\Phi^{\dagger}\Phi)^{2}+\lambda_{1}(\phi^{\dagger}_{1}\phi_{1})^{2}+\lambda_{2}(\phi^{\dagger}_{2}\phi_{2})^{2}$
(3) $\displaystyle+$
$\displaystyle\lambda_{H\phi_{1}}(\Phi^{\dagger}\Phi)(\phi_{1}^{\dagger}\phi_{1})+\lambda_{H\phi_{2}}(\Phi^{\dagger}\Phi)(\phi_{2}^{\dagger}\phi_{2})+\lambda_{\phi_{1}\phi_{2}}(\phi_{1}^{\dagger}\phi_{1})(\phi_{2}^{\dagger}\phi_{2})$

denotes the scalar potential. The relevant part of the fermion Lagrangian is

$-\mathcal{L_{\text{Y}}}\supset
Y_{\nu}\overline{L}\tilde{\Phi}\nu_{R}+y_{\phi_{1}}\overline{\psi}\phi_{1}\nu_{R}+m_{\psi}\overline{\psi}\psi.$
(4)

The singlet scalar field $\phi_{2}$ acquires a non-zero vacuum expectation
value (VEV) denoted by $\langle\phi_{2}\rangle=v_{2}$, leading to the
spontaneous breaking of $U(1)_{B-L}$ gauge symmetry. The singlet field
$\phi_{1}$ does not acquire any VEV and remains heavier than DM $\psi$,
ensuring latter’s stability. We also consider the Higgs portal coupling
$\lambda_{H\phi_{2}}$ to be negligible for simplicity.

Considering the the free parameters in the model to be
$\mu_{1},\mu_{2},\lambda_{1},\lambda_{2},\lambda_{H\phi_{1}},g_{BL},y_{\phi_{1}},m_{\psi}$,
the physical masses of $\phi_{1},\phi_{2}$ and $Z^{\prime}$ (the $U(1)_{B-L}$
gauge boson) can be written as

$\displaystyle m_{\phi_{1}}^{2}$ $\displaystyle=$
$\displaystyle\mu_{1}^{2}+\frac{1}{2}\lambda_{H\phi_{1}}v^{2},$ (5)
$\displaystyle m_{\phi_{2}}^{2}$ $\displaystyle=$ $\displaystyle
2\lambda_{2}v_{2}^{2},$ (6) $\displaystyle m_{Z^{\prime}}^{2}$
$\displaystyle=$ $\displaystyle 9g_{B-L}^{2}v_{2}^{2},$ (7)

where $v\left(=\sqrt{\frac{\mu^{2}}{\lambda}}\right)$ and
$v_{2}\left(=\sqrt{\frac{\mu_{2}^{2}}{\lambda_{2}}}\right)$ are the VEVs of
the neutral component of the SM Higgs $\Phi$ and $\phi_{2}$ respectively. As a
result, the parameters $\mu_{1},\mu_{2}$ and $\lambda_{2}$ can be traded for
the parameters $m_{\phi_{1}},m_{\phi_{2}}$ and $m_{Z^{\prime}}$. This leads to
the free parameters as
$m_{\phi_{1}},m_{\phi_{2}},m_{Z^{\prime}},\lambda_{1},\lambda_{H\phi_{1}},g_{BL},y_{\phi_{1}},m_{\psi}$,
out of which the relevant parameters for the phenomenology to be discussed are
$m_{\phi_{1}},m_{Z^{\prime}},m_{\psi},\lambda_{H\phi_{1}},g_{BL},y_{\phi_{1}}$.

While we have a gauged $B-L$ symmetry, DM is neutral under $U(1)_{B-L}$ and
its relic depends on the Dirac neutrino portal couplings in the spirit of
light Dirac neutrino portal DM Biswas:2021kio ; Biswas:2022vkq . Now,
depending on the value of this Dirac neutrino portal Yukawa coupling
$y_{\phi_{1}}$, the DM analysis can be broadly divided into two categories
namely, (a) FIMP ($y_{\phi_{1}}<10^{-7}$) and (b) WIMP
($y_{\phi_{1}}>10^{-7}$). In the first case, due to small Yukawa coupling, the
DM $\psi$ is produced non-thermally and dominantly from $\phi_{1}$ decay. In
the second case, DM can attain equilibrium in the early universe due to
sizeable interactions. We now discuss these two broad cases one by one.

##  III FIMP type dark matter

In this scenario, the Yukawa coupling among $\nu_{R}$, DM and singlet scalar
is small $y_{\phi_{1}}<10^{-7}$ keeping the associated processes out-of-
equilibrium throughout. While DM never thermalises, the RHNs can thermalise
with the SM bath by virtue of $U(1)_{B-L}$ gauge interactions. Similar to the
active neutrinos, the RHNs also decouple from the bath at a temperature
$T_{\nu_{R}}$ and contributes to the effective relativistic degrees of freedom
defined as

$N_{\rm
eff}\equiv\frac{8}{7}\left(\frac{11}{4}\right)^{4/3}\left(\frac{\rho_{\rm
rad}-\rho_{\gamma}}{\rho_{\gamma}}\right),$

where $\rho_{\rm rad}$ is the net radiation content of the universe. As
mentioned earlier, the SM prediction is $N^{\rm SM}_{\rm eff}=3.045$
Mangano:2005cc ; Grohs:2015tfy ; deSalas:2016ztq . In our model, $\Delta
N_{\rm eff}$ can be estimated by finding the decoupling temperature
$T_{\nu_{R}}$ of right handed neutrinos ($\nu_{R}$) using

$\displaystyle\Gamma(T_{\nu_{R}})=\mathcal{H}(T_{\nu_{R}})$ (8)

where $\Gamma(T)$ is the interaction rate and $\mathcal{H}(T)$ is the
expansion rate of the universe.

Let us consider the themalisation of $\nu_{R}$ with the SM bath via the
$Z^{\prime}$ portal interactions. Sufficiently large $Z^{\prime}$ portal
interactions can lead to thermalisation of $\nu_{R}$ via the s-channel process
$f\bar{f}\leftrightarrow\nu_{R}\bar{\nu}_{R}$ (here $f$ denotes SM fermions).
As a result, $\nu_{R}$ enters the thermal bath and then, after a certain
period, it decouples from the bath. This generates the thermal contribution of
$\nu_{R}$ to $\Delta N_{\rm eff}$ Abazajian:2019oqj . Fig. 1 shows the thermal
$\Delta N_{\rm eff}$ as a function of coupling $g_{BL}$ and gauge boson mass
$m_{Z^{\prime}}$. Thus for $g_{BL}\gtrsim 10^{-8}$, $\Delta N_{\rm eff}$
receives a thermal contribution from light Dirac neutrinos which we denote as
$\Delta N^{\rm th}_{\rm eff}$. The region corresponding to large $Z^{\prime}$
portal interactions, labelled as $\Delta N_{\rm eff}>0.28$ is disfavoured by
Planck 2018 limits at $2\sigma$ CL.

Figure 1: $\rm\Delta N_{\rm eff}$ as a function of $m_{Z^{\prime}}$ and
$g_{BL}$.

On the other hand, due to the tiny dark sector Yukawa coupling $y_{\phi_{1}}$,
it is also possible to get a non-thermal contribution to $\Delta N_{\rm eff}$
Luo:2020fdt ; Biswas:2022vkq , which we denote by $\Delta N_{\rm eff}^{\rm
non-th}$. In addition to the Yukawa coupling $y_{\phi_{1}}$ involved in
$\phi_{1}-\nu_{R}-\psi$ coupling, the $U(1)_{B-L}$ portal coupling and scalar
portal coupling $\lambda_{H\phi_{1}}$ can also play a crucial role in deciding
the strength of this non-thermal contribution. While the total contribution to
$\Delta N_{\rm eff}$, in general, is a combination of thermal contribution and
non-thermal contribution, the latter can occur only if the decoupling of RHNs
precedes the non-thermal or freeze-in production.

To study this non-thermal contribution $\Delta N_{\rm eff}^{\rm non-th}$ in
details, let us first consider the situation where $\phi_{1}$ remains in
thermal equilibrium initially and then freezes out from the bath. The frozen
out $\phi_{1}$ later decays into $\psi$ and $\nu_{R}$, leading to non-thermal
production of both dark matter and $\nu_{R}$. The non-thermal contribution to
$\Delta N_{\rm eff}$ from $\nu_{R}$ can be calculated from the total energy
density of frozen-in Dirac neutrinos.

The Boltzmann equation for comoving number density of $\phi_{1}$ can be
written as

$\frac{dY_{\phi_{1}}}{dx}=\frac{\beta s}{\mathcal{H}x}\left(-\langle\sigma
v\rangle_{{}_{\phi_{1}\phi^{\dagger{}}_{1}\to
X\bar{X}}}\left((Y_{\phi_{1}})^{2}-(Y_{\phi_{1}}^{\rm
eq})^{2}\right)-\frac{\Gamma_{\phi_{1}}}{s}\frac{K_{1}(x)}{K_{2}(x)}Y_{\phi_{1}}\right),$
(9)

where $K_{1}$ and $K_{2}$ are the modified Bessel functions of first and
second kind respectively. The comoving abundance of species $i$ is defined as
$Y_{i}=n_{i}/s$ with $n_{i},s$ being number density of species $i$ and entropy
density of the universe respectively. $\mathcal{H}$ denotes the Hubble
parameter. The variable $x$ is defined as $x=m_{\phi_{1}}/T$ and
$\beta=1+\frac{T}{3g^{s}_{*}}\frac{dg^{s}_{*}}{dT}$ with $g^{s}_{*}$ being the
relativistic entropy degrees of freedom. Here $\langle{\sigma
v}\rangle_{{}_{\phi_{1}\phi_{1}^{\dagger{}}\to X\bar{X}}}$ is the thermally
averaged annihilation cross-section of $\phi_{1}$ into the all allowed final
state particles and $\Gamma_{\phi_{1}}$ denotes the decay rate of $\phi_{1}$
into $\nu_{R}$ and $\psi$. The annihilation cross-section of $\phi_{1}$
depends upon Higgs portal coupling $\lambda_{H\phi_{1}}$ as well as
$U(1)_{B-L}$ portal coupling. Depending upon the strength of these individual
couplings, the freeze-out abundance of $\phi_{1}$ is either determined by
scalar portal or gauge portal interactions. The freeze-in abundance of $\psi$
and $\nu_{R}$ (from the decay of frozen out $\phi_{1}$) can be obtained by
solving the relevant Boltzmann equations given by

$\frac{dY_{\psi}}{dx}=\frac{\beta}{x\mathcal{H}}\Gamma_{\phi_{1}}\frac{K_{1}(x)}{K_{2}(x)}Y_{\phi_{1}},$
(10) $\frac{d\widetilde{Y}}{dx}=\frac{\beta}{\mathcal{H}s^{1/3}x}\langle
E\Gamma\rangle_{\phi_{1}}Y_{\phi_{1}},$ (11)

with

$\displaystyle\Gamma_{\phi_{1}}=\frac{1}{32\pi}m_{\phi_{1}}y^{2}_{\phi_{1}}\left(1-\frac{m^{2}_{\psi}}{m^{2}_{\phi_{1}}}\right)^{2};\hskip
28.45274pt\langle
E\Gamma\rangle_{\phi_{1}}=\frac{1}{64\pi}m^{2}_{\phi_{1}}y^{2}_{\phi_{1}}\left(1-\frac{m^{2}_{\psi}}{m^{2}_{\phi_{1}}}\right)^{3}.$

The term $\tilde{Y}=\frac{\rho_{\nu_{R}}}{s}$ represents the comoving energy
density of RHNs. The preferred choice of energy density instead of number
density of $\nu_{R}$ is based on the fact that the calculation of $\Delta
N_{\rm eff}$ requires comoving energy density of $\nu_{R}$.

Figure 2: Evolution of comoving number density of $\phi_{1}$, $\psi$ and
comoving energy density of $\nu_{R}$. The left (right) panel shows the
variation with respect to $g_{BL}$ ($y_{\phi_{1}}$).

The non-thermal contribution to $\Delta N_{\rm eff}$ namely, $\Delta N^{\rm
non-th}_{\rm eff}$ depends on the parameters $m_{\phi_{1}},m_{Z^{\prime}},$
$\lambda_{H\phi_{1}},g_{BL}$ and $y_{\phi_{1}}$. While the parameters
$m_{\phi_{1}},m_{Z^{\prime}},\lambda_{H\phi_{1}}$ and $g_{BL}$ determine the
freeze-out abundance of $\phi_{1}$, the parameters that determine the decay
width of $\phi_{1}$ are $m_{\phi_{1}},y_{\phi_{1}}$. Fig. 2 shows the
evolution of comoving densities for chosen benchmark points clearly indicating
the roles of $g_{BL}$ and $y_{\phi_{1}}$. While we only show the evolution of
non-thermal contribution to $\Delta N_{\rm eff}$ in this figure, the total
$\Delta N_{\rm eff}$ for the chosen benchmark values of parameters include
both thermal and non-thermal contributions. In the left panel plot of Fig. 2,
the solid line for $\widetilde{Y}$ corresponds to an asymptotic non-thermal
contribution $\Delta N^{\rm non-th}_{\rm eff}=0.10$ while the same choice of
parameters generates a thermal contribution $\Delta N^{\rm th}_{\rm
eff}=0.18$. Therefore, the total contribution of $\Delta N_{\rm eff}=0.28$, as
indicated in the legends for the solid line. For the dashed line,
corresponding to a larger value of $g_{BL}$, the thermal contribution also
comes out to be slightly larger $\Delta N^{\rm th}_{\rm eff}=0.19$, as
expected. However, a significant decrease in non-thermal contribution is
observed for this benchmark point leading to $\Delta N^{\rm non-th}_{\rm
eff}=0.03$. Thus, for the dashed line in left panel plot of Fig. 2, we have
total $\Delta N_{\rm eff}=0.22$. For the right panel plot of the same figure,
the thermal contribution remains same $\Delta N^{\rm th}_{\rm eff}=0.18$ as
$g_{BL}$ and $m_{Z^{\prime}}$ are kept fixed and variation in Yukawa coupling
$y_{\phi_{1}}$ in the non-thermal ballpark does not alter thermal contribution
to $\Delta N_{\rm eff}$. The dashed line, with a larger $y_{\phi_{1}}$, have a
smaller $\Delta N^{\rm non-th}_{\rm eff}=0.02$ and vice versa. As a result,
for the dashed line in right panel plot, the total contribution is $\Delta
N_{\rm eff}=0.20$. In both the left and right panel plots, the values of
$m_{\psi}$ are taken in such a way that it gives correct FIMP type dark matter
relic. Also, we have seen that the value of $m_{\psi}$ does not affect $\Delta
N_{\rm eff}$ as long as $m_{\psi}\ll m_{\phi_{1}}$.

Figure 3: Scan plot showing the total $\Delta N_{\rm eff}$ versus
$m_{Z^{\prime}}$ for different $\lambda_{H\phi_{1}}$, $g_{BL}$ and $m_{\psi}$.
The magenta coloured region is excluded by Planck 2018 bounds at $2\sigma$ CL.
The region in grey color is within the reach of future experiments.

After studying the evolution of comoving densities for different benchmark
points, we perform a numerical scan over key model parameters to find out the
parameter space consistent with $\Delta N_{\rm eff}$ and DM properties. We
have kept $m_{\phi}$ fixed at $1000$ GeV and $y_{\phi_{1}}$ at $10^{-10}$. The
rest of the parameters are varied in the following range:

$\displaystyle 250\,\text{GeV}$ $\displaystyle<m_{Z^{\prime}}$
$\displaystyle<1000\,\text{GeV}$ (12) $\displaystyle 10^{-5}$
$\displaystyle<\lambda_{H\phi_{1}}$ $\displaystyle<5\times 10^{-5}$
$\displaystyle 10^{-4}$ $\displaystyle<g_{BL}$ $\displaystyle<10^{-2}$
$\displaystyle 10^{-5}\,\text{GeV}$ $\displaystyle<m_{\psi}$
$\displaystyle<10^{-3}\,\text{GeV}.$

The resulting parameter space in terms of $\Delta N_{\rm eff}$ and
$m_{Z^{\prime}}$ is shown in Fig. 3. All the points shown in this figure
satisfy the requirements of correct DM relic abundances. The magenta shaded
region denotes the region excluded by Planck 2018 bounds at $2\sigma$ CL while
the grey shaded region remains within the reach of future experiments like
CMB-S4. The colour codes in the left, middle and right panel plots of Fig. 3
show the variation in $\lambda_{H\phi_{1}},g_{BL},m_{\psi}$ respectively. As
the left panel plot shows, decrease in $\lambda_{H\phi_{1}}$, while keeping
$m_{\phi_{1}}$ constant, increases $\Delta{N_{\rm eff}}$. A smaller value of
Higgs portal coupling $\lambda_{H\phi_{1}}$ leads to a larger freeze-out
abundance of $\phi_{1}$ followed by enhanced production of $\nu_{R}$ from
$\phi_{1}$ decay. Since the same decay is also responsible for freeze-in
production of DM, we require smaller DM masses in order to keep its relic
abundance within Planck limits, as seen from the right panel plot of Fig. 3.
As the middle panel plot shows, small values of $\Delta N_{\rm eff}$ typically
correspond to smaller $g_{BL}$ as the corresponding thermal contribution
$\Delta N^{\rm th}_{\rm eff}$ decreases.

While we have incorporated the constraints from cosmological observations on
$\Delta N_{\rm eff}$ and DM relic abundance, there can be strong constraints
on light dark matter from astrophysical structure formation. Such bounds can
be imposed on a particular DM scenario by calculating the free-streaming
length (FSL) of DM. While hot DM is already ruled out, warm DM with FSL
$\lambda_{\rm FSL}<0.1$ Mpc is still allowed, and can be favourable over cold
DM of FSL $\lambda_{\rm FSL}<0.01$ Mpc due to the small-scale structure
problems associated with the latter Drewes:2016upu . Dark matter free-
streaming length can be estimated from matter power spectrum inferred from the
Lyman-$\alpha$ forest data Croft:2000hs ; Kim:2003qt ; Viel:2005qj ;
Irsic:2017ixq  and also from Quasar data Hsueh:2019ynk . Such estimates are
also supported by theoretical and simulation based results Colombi:1995ze ;
Boyarsky:2008xj ; deVega:2009ku ; Schneider:2011yu . For some recent
discussions on structure formation constraints on DM production mechanisms,
please see Merle:2013wta ; Decant:2021mhj ; Ballesteros:2020adh  and
references therein. The detailed calculations related to FSL of DM are given
in appendix A.

As shown in appendix A, the FSL of DM depends primarily on the production
temperature, DM mass and the production mechanism of DM or distribution
function of DM. A higher production temperature gives a smaller FSL due to
high momentum redshift making DM non-relativistic earlier and vice-versa. A
different production mechanism also gives different FSL. In our case, the DM
is produced due to decay of a frozen out scalar. Fig. 4 shows the average
velocity of DM for four different benchmark parameters. The parameters are
shown in table 2 along with total $\Delta N_{\rm eff}$ and FSL. The benchmark
point (BP) II corresponds to a smaller $\phi_{1}$ mass than BP I. Hence, the
freeze-out abundance of $\phi_{1}$ for BP II is smaller than that of BP I.
This implies that a larger $\psi$ mass is required to satisfy DM abundance. As
a result, we obtain a smaller FSL for BP II. However, for both BP I and BP II,
the computed FSL keeps DM in hot DM category and hence ruled out from
structure formation constraints. The FSL can be reduced by increasing the
production temperature as well as increasing DM mass. BP III and BP IV have a
larger $y_{\phi_{1}}$ giving a higher production temperature. Similarly it has
a larger $\lambda_{H\phi_{1}}$ coupling, giving smaller $\phi_{1}$ freeze-out
abundance. Hence for both BP III and BP IV, DM masses can be large while being
consistent with relic abundance criteria. Combining these two effects, we get
warm dark matter for BP III and BP IV. From the resulting on $\Delta N_{\rm
eff}$, it can be seen that both BP III and BP IV remain within the sensitivity
of future CMB experiments like CMB-S4.

Figure 4: Thermal average velocity of FIMP dark matter as a function of temperature for four different benchmark points. | Parameters | $\Omega_{\rm DM}{\rm h}^{2}$ | $\Delta{\rm N_{eff}}$ | FSL(Mpc)
---|---|---|---|---
$m_{\phi_{1}}$(GeV) | $\lambda_{H\phi_{1}}$ | $y_{\phi_{1}}$ |  $m_{Z^{\prime}}$(GeV) | $g_{BL}$ |  $m_{\psi}$(keV)
BP I | $1000$ | $5\times 10^{-5}$ | $10^{-10}$ | $500$ | $0.001$ | $126$ | $0.12$ | $0.251$ | $3.09$
BP II | $500$ | $5\times 10^{-5}$ | $10^{-10}$ | $500$ | $0.001$ | $233$ | $0.12$ | $0.210$ | $1.32$
BP III | $1000$ | $2\times 10^{-4}$ | $10^{-9}$ | $500$ | $0.001$ | $970$ | $0.12$ | $0.184$ | $0.06$
BP IV | $500$ | $2\times 10^{-4}$ | $10^{-9}$ | $500$ | $0.001$ | $938$ | $0.12$ | $0.184$ | $0.05$
Table 2: Benchmark parameters for FIMP DM

Figure 5: Parameter space in $m_{Z^{\prime}}$ versus $g_{BL}$ plane for the
FIMP scenario considering three different values of DM mass. The other
parameters are kept fixed as $m_{\phi_{1}}=500$ GeV,
$\lambda_{H\phi_{1}}=10^{-5}$, $y_{\phi_{1}}=10^{-9}$.

Fig. 5 shows the parameter space in $m_{Z^{\prime}}$ versus $g_{BL}$ parameter
space for the FIMP scenario. The left, middle and right panel plots correspond
to dark matter mass of $400$ keV, $500$ keV and $1300$ keV respectively. The
other relevant parameters are fixed as $m_{\phi_{1}}=500$ GeV,
$\lambda_{H\phi_{1}}=10^{-5}$, $y_{\phi_{1}}=10^{-9}$. For the region below
the solid brown line in each of these plots, $U(1)_{B-L}$-portal coupling does
not play any role in FIMP DM production. In that region, the freeze-out
abundance of $\phi_{1}$, responsible for DM production, is determined by the
Higgs-portal coupling $\lambda_{H\phi_{1}}$. In other words, the cross section
$\langle{\sigma v}\rangle_{{}_{\phi_{1}\phi_{1}^{\dagger{}}\to
SM\overline{SM}}}$ dominates over $\langle{\sigma
v}\rangle_{{}_{\phi_{1}\phi_{1}^{\dagger{}}\to Z^{\prime}Z^{\prime}}}$. In the
region above the solid brown line, the freeze-out abundance of $\phi_{1}$ is
determined by the $U(1)_{B-L}$-portal coupling. The dashed line in the left
and middle panel plots, separating the pink and cyan shaded regions, represent
the parameter space satisfying correct DM relic for $m_{\psi}=400$ keV and
$m_{\psi}=500$ keV respectively. However, for DM mass $m_{\psi}=400$ keV, the
free-streaming length turns out to be very large $\sim 0.1$ Mpc, keeping it in
hot DM ballpark and hence ruled out. In the middle panel plot, due to a higher
DM mass $m_{\psi}=500$ keV, the dashed line corresponds to an intermediate FSL
keeping it in the warm DM regime which is still allowed by structure formation
constraints. In the region below the dashed line, the freeze-out abundance of
$\phi_{1}$ is more than that for the dashed line, leading to a larger FIMP DM
abundance $\Omega_{\rm DM}h^{2}>0.12$, produced from $\phi_{1}$ decay. On the
other hand, the region above the dashed line have $\Omega_{\rm DM}h^{2}<0.12$
due to smaller freeze-out abundance of $\phi_{1}$. This is understood from the
fact that a larger $U(1)_{B-L}$-portal interaction leads to smaller freeze-out
abundance of $\phi_{1}$ and vice versa. While the total $\Delta N_{\rm eff}$
can have both thermal and non-thermal contributions, for the choice of
parameters in Fig. 5, the non-thermal contribution is suppressed compared to
the thermal contribution. The magenta solid line corresponds to $\Delta N_{\rm
eff}=0.28$. The region above this line has $\Delta N_{\rm eff}>0.28$ and hence
ruled out from Planck 2018 limits. In the right panel plot, the dashed line
corresponds to relic satisfying region for DM mass $m_{\psi}=1300$ keV. The
region below the dashed line correspond to over-abundance like before.
However, the region above the dashed line is no longer under-abundant, but
satisfies the current DM abundance. For this region of parameter space and
chosen DM mass, the production of FIMP DM while $\phi_{1}$ is in equilibrium
dominates compared to the production after $\phi_{1}$ freeze-out. This can
also be understood from the evolution plots shown in Fig. 2. From the left
panel plot of Fig. 2, we see that there is a non-zero yield of DM $Y_{\psi}$
from the decay of $\phi_{1}$ when the latter is in equilibrium. Additionally,
the freeze-out abundance of $\phi_{1}$ ($Y_{\phi_{1}}(x\rightarrow\infty)$)
decreases as we increase $g_{BL}$. Combining these two, we can have a
situation where for sufficiently large $U(1)_{B-L}$-portal couplings,
$Y_{\psi}$ from the decay of $\phi_{1}$ in equilibrium is larger compared to
the post freeze-out production. Hence, $U(1)_{B-L}$-portal couplings do not
determine the comoving abundance of $Y_{\psi}$ and as a result, the region
above the dashed line satisfies the correct DM relic in the right panel plot
of Fig. 5. In all the plots, the region below the magenta solid line has
$0.28>\Delta N_{\rm eff}>0.14$ keeping it within reach of future CMB
experiments like CMB-S4. The blue shaded region towards the upper right corner
indicates the region ruled out from the large hadron collider (LHC) bounds,
for specifically from ATLAS experiments at 13 TeV centre of mass energy
Aaboud:2017buh ; Aad:2019fac ; ATLAS:2017fih . Similar bounds also exist from
the CMS experiment at the LHC Sirunyan:2018exx . A relatively weaker bound
exists from the large electron positron (LEP) collider disfavouring the region
$m_{Z^{\prime}}/g_{BL}<7$ TeV Carena:2004xs ; Cacciapaglia:2006pk . The
parameter space shown in Fig. 5 already satisfies the LEP bound.

### When $\phi_{1}$ is always in equilibrium:

Before moving onto the WIMP DM scenario, we briefly comment on the possibility
of FIMP DM production from $\phi_{1}$ when the latter remains in equilibrium
throughout the production. When $\phi_{1}$ is in equilibrium during the
production of DM, the comoving number density of $\psi$ and comoving energy
density of $\nu_{R}$, respectively, are given as

$\frac{dY_{\psi}}{dx}=\frac{\beta}{x\mathcal{H}}\Gamma_{\phi_{1}}\frac{K_{1}(x)}{K_{2}(x)}Y^{\rm
eq}_{\phi_{1}},$ (13)
$\frac{d\widetilde{Y}}{dx}=\frac{\beta}{\mathcal{H}s^{1/3}x}\langle
E\Gamma\rangle_{\phi_{1}}Y^{\rm eq}_{\phi_{1}}.$ (14)

These two equations can be analytically solved to get the asymptotic
abundances (in the limit $m_{\psi}\ll m_{\phi_{1}}$)

$\displaystyle Y_{\psi}(x\rightarrow\infty)$ $\displaystyle=$
$\displaystyle\frac{135\,}{16\pi\times 1.66\times
8\pi^{3}g^{s}_{*}\sqrt{g^{\rho}_{*}}}M_{\rm
pl}\,\frac{y^{2}_{\phi_{1}}}{m_{\phi}}\,,$
$\displaystyle\widetilde{Y}(x\rightarrow\infty)$ $\displaystyle=$
$\displaystyle\frac{675\,}{32\pi\times 1.66\times
8\pi^{3}g^{s}_{*}\sqrt{g^{\rho}_{*}}}\left(\frac{45}{2\pi^{2}\,g^{s}_{*}}\right)^{1/3}M_{\rm
pl}\,\frac{y^{2}_{\phi_{1}}}{m_{\phi}}\,,$ (15)

where $M_{\rm pl}$ is the Planck mass. With this, the DM abundance and
contribution to extra radiation energy density can be calculated as -

$\displaystyle\Omega_{\rm DM}h^{2}$ $\displaystyle=$ $\displaystyle
2\frac{m_{\psi}s^{0}}{\rho^{0}_{c}}Y_{\psi}(x\rightarrow\infty)h^{2}$
$\displaystyle\Delta N^{\rm non-th}_{\rm eff}$ $\displaystyle=$ $\displaystyle
2\times 3\times\left(\frac{s^{4/3}}{\rho_{\nu_{L}}}\right)_{\rm
T=10\,MeV}\widetilde{Y}(x\rightarrow\infty)\simeq
0.61\widetilde{Y}(x\rightarrow\infty),$ (16)

where $s^{0}$ and $\rho^{0}_{c}$ denote the entropy density and critical
energy density, respectively, at the present epoch. $\rho_{\nu_{L}}$ denotes
energy density of one species of SM neutrino and $h$ represents $H_{0}/100$
with $H_{0}$ being the current expansion rate of the universe. Here,
$m_{\phi_{1}},y_{\phi_{1}}$ and $m_{\psi}$ determine the DM abundance and
$\Delta N^{\rm non-th}_{\rm eff}$ is determined by $m_{\phi_{1}}$ and
$y_{\phi_{1}}$ only. The thermal contribution to $N_{\rm eff}$ is determined
by the parameters $m_{Z^{\prime}}$ and $g_{BL}$. When DM abundance is
satisfied, it turns out that the non-thermal contribution to $\Delta N_{\rm
eff}$ is way below the minimum thermal contribution. So, the total $\Delta
N_{\rm eff}$ is only determined by the thermal contribution (shown in Fig. 1).

Figure 6: Evolution of comoving number densities of $\phi_{1}$ and $\psi$ and comoving energy density of $\nu_{R}$ for two different values of $y_{\phi_{1}}$ (left panel) and $m_{\phi_{1}}$ (right panel). | Parameters | $\Omega_{\rm DM}{\rm h}^{2}$ | $\Delta{\rm N_{eff}}$ | FSL(Mpc)
---|---|---|---|---
$m_{\phi_{1}}$(GeV) | $y_{\phi_{1}}$ |  $m_{Z^{\prime}}$(GeV) | $g_{BL}$ |  $m_{\psi}$(keV)
BP I | $10$ | $10^{-10}$ | $500$ | $0.001$ | $2030$ | $0.12$ | $0.184$ | $0.0009$
BP II | $10$ | $10^{-9}$ | $500$ | $0.001$ | $20$ | $0.12$ | $0.184$ | $0.06$
BP III | $100$ | $10^{-10}$ | $500$ | $0.001$ | $23500$ | $0.12$ | $0.184$ | $0.0001$
BP IV | $100$ | $10^{-9}$ | $500$ | $0.001$ | $235$ | $0.12$ | $0.184$ | $0.0064$
Table 3: Table for FIMP DM (equilibrium $\phi_{1}$)

Fig. 6 shows evolutions of comoving abundances of $\phi_{1}$, $\psi$ and
$\nu_{R}$ for two different values of $y_{\phi_{1}}$ (left panel) and
$m_{\phi_{1}}$ (right panel). For $m_{\psi}\ll m_{\phi_{1}}$ the evolutions
for comoving number density of $\psi$ and comoving energy density of $\nu_{R}$
are independent of $m_{\psi}$. So in all the evolution plots, the dark matter
masses are chosen in such a way that observational DM abundance is satisfied.
The solid lines in the left plot have $y_{\phi_{1}}=10^{-10}$ and DM mass
$m_{\psi}=2030$ keV. The non-thermal contribution to $\Delta N_{\rm eff}$ from
frozen-in $\nu_{R}$ is $6.5\times 10^{-6}$. For the dashed lines, the Yukawa
coupling is larger by a factor of 10. Consequently, we have higher
$Y_{\psi}(x\rightarrow\infty)$ and $\tilde{Y}(x\rightarrow\infty)$. From Eq.
(III), we get DM mass to be $20$ keV and $\Delta N^{\rm non-th}_{\rm
eff}=6.5\times 10^{-4}$. The solid lines in the right plot have the same
parameters as those of in the left panel plot. The dashed lines in the right
panel plot have $m_{\phi_{1}}=100$ GeV. Due to higher mass of $\phi_{1}$, the
Boltzmann suppression occurs early on, leading to smaller
$Y_{\psi}(x\rightarrow\infty)$ and $\tilde{Y}(x\rightarrow\infty)$. As a
result DM mass for the solid lines is about $23.5$ MeV and $\Delta N^{\rm non-
th}_{\rm eff}=5.5\times 10^{-7}$. We have kept $B-L$ gauge coupling and mass
fixed for all the plots, $g_{BL}=10^{-3}$ and $m_{Z^{\prime}}=500$ GeV. This
gives a thermal contribution to $\Delta N_{\rm eff}=0.184$. As in all the
plots non-thermal contribution is way smaller than the thermal counterpart, so
the total $\Delta N_{\rm eff}$ is determined by the thermal contribution as
shown in the plot. Since $U(1)_{B-L}$-portal parameters do not decide DM
abundance in this case, we do not show any summary plot like before.

Table 3 shows four benchmark points (BP) and their corresponding contribution
to effective number of relativistic species and structure formation. BP I and
BP II have same $m_{\phi_{1}}=10$ GeV whereas BP III and BP IV have
$m_{\phi_{1}}=100$ GeV. As the Yukawa coupling is ten times larger in BP II
compared to BP I, the corresponding DM mass is 100 times smaller in BP II (see
Eq. (III)). The similar trend is followed between BP III and BP IV. Due to
smaller DM mass, $\lambda_{\rm FSL}$ is larger in both the BP II (compared to
BP I) and BP IV (compared to BP III). Again from equation (III), we get that
$m_{\phi_{1}}$ and $m_{\psi}$ are correlated for constant DM abundance. Hence
a larger $m_{\phi_{1}}$ gives a larger $m_{\psi}$ (BP III, BP IV vs BP I, BP
II). The DM production temperature ($T\sim m_{\phi_{1}}$) is also larger for
larger $m_{\psi_{1}}$. Combining these two, we get a smaller $\lambda_{\rm
FSL}$ for BP III (BP IV) compared to BP I (BP II). Among the four benchmark
points mentioned in the table, only BP II gives warm DM whereas rest points
give cold DM. As all the points have same $B-L$ gauge coupling and mass, we
have same $\Delta N_{\rm eff}$.

##  IV WIMP type dark matter

If the Yukawa coupling among the dark sector particles i.e. $\phi_{1}$, $\psi$
and $\nu_{R}$ is large ($y_{\phi_{1}}>10^{-7}$), the dark sector can be in
thermal equilibrium among themselves even after their decoupling from the SM
bath. Since the thermalisation of $\phi_{1},\nu_{R}$ with the SM bath relies
on Higgs portal and $U(1)_{B-L}$ portal couplings instead of Yukawa
$y_{\phi_{1}}$, we have two different sub-cases depending upon whether the
Yukawa interactions in dark sector go out of equilibrium before or after the
dark sector decouples from the thermal bath. Among the dark sector particles,
$\phi_{1}$ and $\nu_{R}$ can have interactions with other particles in the
bath bath via the processes - $\phi_{1}X\to\phi_{1}X$,
$\phi_{1}Z^{\prime}\to\phi_{1}Z^{\prime}$ and $\nu_{R}X\to\nu_{R}X$, here $X$
denotes SM particles. Since $Z^{\prime}$ has sizeable interactions with all
the SM fermions, we consider it to be in the bath while dark sector particles
decouple. While the first two processes can arise via contact interactions
too, the $\nu_{R}$ scattering arises via mediation of heavy $Z^{\prime}$ boson
only. Unlike other interactions, $\nu_{R}$ interactions with the bath rely
significantly on resonant enhancement and hence we consider
$\nu_{R}\bar{\nu}_{R}\rightarrow X\bar{X}$ via resonantly enhanced s-channel
mediation of $Z^{\prime}$ to calculate its decoupling Abazajian:2019oqj .
Therefore, the ratio of interaction rate of dark sectors with SM bath to the
expansion rate, $\mathcal{H}$, can be written as Gondolo:2012vh ;
Heeck:2014zfa

$\displaystyle\frac{\Gamma_{\rm
total}}{\mathcal{H}}=\frac{1}{\mathcal{H}}\left[n^{\rm
eq}_{X}\left(\langle\sigma v\rangle_{\phi_{1}X\to\phi_{1}X}+\langle\sigma
v\rangle_{\phi_{1}Z^{\prime}\to\phi_{1}Z^{\prime}}\right)+n^{\rm
eq}_{\nu_{R}}\langle\sigma v\rangle_{\nu_{R}\bar{\nu}_{R}\to
X\bar{X}}\right].$ (17)

The decoupling temperature of dark sector is approximately calculated by
comparing the total interaction rate with the Hubble expansion rate i.e.
$\frac{\Gamma_{\rm total}}{\mathcal{H}}\approx 1$. Fig. 7 shows interaction
rates of various processes responsible for keeping the dark sector in bath
with the SM as a function of temperature. In the left panel plot, a large
scalar portal coupling $\lambda_{H\phi}=10^{-2}$ dictates the decoupling
temperature whereas in the right panel plot, the resonantly enhanced process
$\nu_{R}\bar{\nu}_{R}\to X\bar{X}$ determines the decoupling temperature due
to smaller scalar portal coupling $\lambda_{H\phi}=10^{-4}$.

Figure 7: Interaction rate as a function of temperature for processes
responsible for keeping dark sector in equilibrium with the SM bath. The
benchmark parameters chosen are $\lambda_{H\phi_{1}}=10^{-2}\,(\rm
left\,panel),10^{-4}\,(\rm right\,panel)$, $m_{\phi_{1}}=100$ GeV,
$m_{Z^{\prime}}=100$ GeV, $g_{BL}=10^{-4}$.

Figure 8: Evolution of dark sector temperature after its decoupling from SM
bath ($T<T_{\rm dec}$) for different values of $m_{Z^{\prime}}$ (left panel),
dark sector Yukawa coupling $y_{\phi_{1}}$ (middle panel) and dark matter mass
$m_{\psi}$ (right panel). The rest of the parameters are kept fixed and are
shown in respective plots.

To track the evolution of dark sector, one has solve the Boltzmann equation in
terms of comoving number densities $Y_{\phi_{1}}$ and $Y_{\psi}$. Using the
approximation $\frac{Y_{i}}{Y}\approx\frac{Y^{\rm eq}_{i}}{Y^{\rm eq}}$
($i=\phi_{1},\psi$), with $Y=Y_{\phi_{1}}+Y_{\psi}$, we can write a single
Boltzmann equation as

$\displaystyle\frac{dY}{dx}=\frac{1}{2}\frac{\beta
s}{\mathcal{H}x}\langle\sigma v\rangle_{\rm eff}\left((Y^{\rm
eq})^{2}-Y^{2}\right),$ (18)

where

$\displaystyle\langle\sigma v\rangle_{\rm
eff}=\frac{(Y^{eq}_{\phi_{1}})^{2}\langle\sigma
v\rangle_{\phi_{1}\phi^{\dagger}_{1}\to
X\bar{X},\nu_{R}\bar{\nu}_{R},Z^{\prime}Z^{\prime}}+(Y^{eq}_{\psi})^{2}\langle\sigma
v\rangle_{\psi\bar{\psi}\to\nu_{R}\bar{\nu}_{R}}}{(Y^{eq}_{\phi_{1}}+Y^{eq}_{\psi})^{2}}.$
(19)

The equation (18) can be solved numerically from a sufficiently high
temperature to the decoupling temperature, $T_{\rm dec}$. If the Yukawa
coupling $y_{\phi_{1}}$ is large enough, even after $T<T_{\rm dec}$, the dark
sector particles $\phi_{1}$, $\psi$ and $\nu_{R}$ can be in dark sector
equilibrium among themselves. We can track the dark sector temperature
$T_{\nu_{R}}$($\equiv T_{\rm DS}$) by solving the following equation.

$\displaystyle\frac{d\xi}{dx}$ $\displaystyle=$
$\displaystyle\frac{1}{x}\left(-\frac{1}{2}\frac{\beta
x^{4}s^{2}}{4\alpha\xi^{3}Hm^{4}_{\phi_{1}}}\langle E\sigma v\rangle_{\rm
eff}\left((Y^{\rm eq})^{2}-Y^{2}\right)-(\beta-1)\xi\right),$ (20)

where

$\displaystyle\langle E\sigma v\rangle_{\rm eff}=\frac{(Y^{\rm
eq}_{\nu_{R}})^{2}\langle E\sigma
v\rangle_{\nu_{R}\bar{\nu}_{R}\to\psi\bar{\psi}}+(Y^{\rm
eq}_{\nu_{R}})^{2}\langle E\sigma
v\rangle_{\nu_{R}\bar{\nu}_{R}\to\phi_{1}\phi^{\dagger}_{1}}}{(Y^{\rm
eq}_{\phi_{1}}+Y^{\rm eq}_{\psi})^{2}},$ (21)

and $\xi$ is the ratio of dark sector temperature to that of the SM bath,
$\frac{T_{\nu_{R}}}{T}$ and $\alpha=6\times\frac{7}{8}\frac{\pi^{2}}{30}$.
After decoupling, $\langle\sigma v\rangle_{\rm eff}$ in Eq. (18) can depend
upon both $x$ as well as $\xi$. We solve Eq. (18) and Eq. (20) simultaneously
from $T=T_{\rm dec}$ to the onset of the BBN era namely, $T\sim 10$ MeV and
estimate the corresponding DM relic and $\Delta N_{\rm eff}$. Fig. 8 shows the
evolution of $T_{\nu_{R}}$ for different benchmark parameters. Different
$T_{\nu_{R}}$ can lead to changes in $\Delta N_{\rm eff}$. The first term on
the RHS of Eq. (20) is a positive term as $Y\geq Y^{\rm eq}$ for $T<T_{\rm
dec}$. Thus, this term indicates the increase in $T_{\nu_{R}}$ over $T$ due to
annihilation of heavy dark sector particles $\phi_{1}$ and $\psi$ into
$\nu_{R}$. On the other hand, the second term on the RHS of the same equation
indicates the decrease in $T_{\nu_{R}}$ over $T$ due to changes in entropy
degrees of freedom $g^{s}_{*}$. The combined effects of these two terms can be
seen in Fig. 8. For an analytical description of the behaviour of the plots
shown in Fig. 8, please refer to appendix B.

The left panel plot in Fig. 8 shows the variation of $\xi^{4}$ with respect to
bath temperature $T$ for different choices of $m_{Z^{\prime}}$. The rest of
the parameter are kept fixed as shown in the plot. Increasing $m_{Z^{\prime}}$
gives a higher decoupling temperature, $T_{\rm dec}$. As long as $T_{\rm
dec}\lesssim m_{\phi_{1}}$, a higher decoupling temperature leads to a larger
conversion of $\phi_{1}$ and $\psi$ to $\nu_{R}$ resulting in a larger
$\frac{T_{\nu_{R}}}{T}$. Another way to understand it is from entropy
conservation (see appendix B). For a higher $T_{\rm dec}$, we have a larger
dark sector entropy degrees of freedom $g^{\rm s,DS}_{*}$ resulting in a
larger $\frac{T_{\nu_{R}}}{T}$. From an analytical estimation, we get a
maximum value of $\xi^{4}$ to be 2.6 provided $g^{s}_{*}$ remains same from
the epoch of dark sector decoupling to DM freeze-out, as shown in appendix B.
This is consistent with the numerical results shown in left panel of Fig. 8.
The reason why maximum value of $\xi^{4}$ has not reached $2.6$ in the figure
is that $g^{s}_{*}$ changes from the period of kinetic decoupling of dark
sector to DM freeze-out. The middle panel plot shows the results for different
values of dark sector Yukawa coupling. As the dark sector Yukawa coupling does
not affect the DM-SM decoupling temperature, so all the coloured lines
correspond to the same decoupling temperature. For a larger Yukawa coupling,
conversion of $\phi_{1}$ and $\psi$ to $\nu_{R}$ is possible and as a result,
temperature of $\nu_{R}$ with respect to SM bath increases. For the green line
in the middle panel plot of Fig. 8, the dark sector freeze-out occurs before
$T_{\rm dec}$ due to small Yukawa coupling. Hence the first term on the RHS of
Eq. (20) ceases and due to the second term, the ratio $\frac{T_{\nu_{R}}}{T}$
decreases. To see the effect of $m_{\psi}$, in the right panel plot, we vary
$m_{\psi}$ keeping the other parameter constant. For a smaller $m_{\psi}$,
freeze-out occurs late, hence we get a higher $\frac{T_{\nu_{R}}}{T}$ ratio.

Figure 9: Scan plot showing $\Delta N_{\rm eff}$ vs $m_{\phi_{1}}$ for
different dark matter mass $m_{\psi}$ and different $m_{Z^{\prime}}$ and
$g_{BL}$.

To see the complete picture in WIMP dark matter scenario, we perform a
numerical scan by varying the following parameters as

$\displaystyle 50\text{GeV}$ $\displaystyle<m_{\phi_{1}}<150\text{GeV},\hskip
28.45274pt100\text{GeV}$ $\displaystyle<m_{Z^{\prime}}<500\text{GeV},$ (22)
$\displaystyle 10\text{GeV}$ $\displaystyle<m_{\psi}<100\text{GeV},\hskip
42.67912pt10^{-5}$ $\displaystyle<\lambda_{H\phi_{1}}<10^{-2},$ $\displaystyle
10^{-5}$ $\displaystyle<g_{BL}<10^{-3},\hskip 56.9055pt0.2$
$\displaystyle<y_{\phi_{1}}<0.3.$

The corresponding results are shown in Fig. 9 with y-axis representing $\Delta
N_{\rm eff}$ whereas x-axis showing the mass of $\phi_{1}$. The relevant
parameters $m_{\psi}$, $m_{Z^{\prime}}$ and $g_{BL}$ are shown in colour bars
of left, middle and right panel plots respectively of Fig. 9. All the points
shown in this figure satisfy the requirements of correct WIMP DM relic
abundance. The magenta shaded region denotes the region excluded by Planck
2018 bounds at $2\sigma$ CL while the grey shaded region remains within the
reach of future experiments like CMB-S4. From the left panel plot, we can see
that for some points with dark matter mass $m_{\psi}\lesssim 50$ GeV are
already excluded by Planck 2018 data at $2\sigma$ CL. Similarly, from the
middle and right panel plots, we get some points having $m_{Z^{\prime}}$
between $100$ GeV and $500$ GeV and $g_{BL}$ between $10^{-3}$ and $10^{-5}$
GeV, that are already excluded. All the others points in these plots can be
probed in the future CMB experiments like CMB-S4 keeping the detection
prospects promising.

Figure 10:  Bound on $m_{Z^{\prime}}$ vs $g_{BL}$ for the WIMP scenario with
different values of $\lambda_{H\phi_{1}}$. The solid black line separates
regime 1 (below the solid black line) and regime 2 (above the solid black
line). In all the plots, $m_{\phi_{1}}=60$ GeV, $m_{\psi}=15$ GeV,
$y_{\phi_{1}}=0.28$.

As we have discussed above, the kinetic decoupling of the dark sector
($\phi,\psi$ and $\nu_{R}$) from its bath is determined by both the Higgs-
portal coupling and B-L gauge coupling. After the kinetic decoupling, due to
large Yukawa coupling, dark sector maintain a dark equilibrium. An increase in
$\lambda_{H\phi_{1}}$ coupling or $g_{BL}$ coupling keeps the dark sector in
bath for longer. Increasing the $g_{BL}$ coupling to a very high value gives
us a scenario where the dark matter freezes out before the decoupling of
$\nu_{R}$. In this case, we do not have a separate dark sector evolution
different from the SM bath. Hence, we have two regimes namely, regime 1: DM
freezes out when $\xi_{\rm fo}\neq 1$ (i.e. separate dark sector); regime 2:
DM freezes out when $\xi_{\rm fo}=1$ (no separate dark sector). Depending upon
whether kinetic decoupling temperature is determined by gauged B-L coupling
(sub-case 1) or Higgs-poral coupling (sub-case 2) we can have two further sub-
cases.

In the plots shown in Fig. 10, we show the parameter space in $m_{Z^{\prime}}$
versus $g_{BL}$ plane for WIMP type DM. Each of these plots correspond to a
fixed value of $\lambda_{H\phi_{1}}$. The other parameters fixed in all the
plots are, $m_{\phi_{1}}=60$ GeV, $m_{\psi}=15$ GeV, $y_{\phi_{1}}=0.28$. The
solid black line separates regime 1 (upper left triangular region) from regime
2. The solid brown line separates the regime where kinetic decoupling
temperature is determined by Higgs portal coupling from the one where it is
determined by the gauge portal coupling. In the grey coloured region below the
solid brown line, $g_{BL}$ and $m_{Z^{\prime}}$ have no role in the kinetic
decoupling of DM. In the upper panel plots, the region below the magenta line
is where we have $\Delta N_{\rm eff}>0.28$, disfavoured by Planck 2018 limits.
In the grey coloured region, due to constant decoupling temperature, we have
constant $\Delta N_{\rm eff}$. Due to smaller Higgs-portal coupling in the
upper right panel plot compared to the upper left one, the grey coloured
region shrinks. In the lower left panel plot, we increase the Higgs-portal
coupling compared to upper left panel plot to $2.5\times 10^{-4}$. For this
particular choice of $\lambda_{H\phi_{1}}$, the region between the black and
brown line has $\Delta N_{\rm eff}<0.28$ and the region below the brown line
has $\Delta N_{\rm eff}$=0.28, which is the maximum allowed value from Planck
2018 limits. In the lower right panel plot, $\lambda_{H\phi_{1}}$ is increased
even further to $10^{-3}$ giving $\Delta N_{\rm eff}<0.28$ in the whole regime
1. For regime 2, as DM freezes out before RHN decoupling, the bound from
$\Delta N_{\rm eff}$ (denoted by region above the green line) is same as the
bound shown in Fig. 1. The entire parameter space shown in these plots satisfy
the criteria of DM relic abundance. This is because DM abundance has very weak
dependence on the kinetic decoupling temperature. For the WIMP type DM
scenario, we also check the constraints from direct detection and find them to
be very weak due to radiative suppression. The details are given in appendix
C.

##  V Conclusion

We have studied a possible UV completion of the light Dirac neutrino portal
dark matter scenario. In such a scenario, right chiral parts of light sub-eV
Dirac neutrinos act like a portal between dark and visible sectors responsible
for the production of dark matter. A gauged $U(1)_{B-L}$ symmetry provides one
possible UV completion by naturally accommodating right chiral parts of
neutrinos from anomaly cancellation requirements while also preventing direct
coupling of DM, a gauge singlet Dirac fermion with the SM required for its
stability. Keeping $U(1)_{B-L}$ symmetry breaking scale upto a few TeV
ballpark, we study the details of dark matter production together with
additional relativistic degrees of freedom $\Delta N_{\rm eff}$ brought in by
right chiral parts of light Dirac neutrinos. While the chosen values of
$U(1)_{B-L}$ gauge couplings ensures a non-zero thermal contribution to
$\Delta N_{\rm eff}$, DM production can be either purely thermal or non-
thermal depending upon the light Dirac neutrino portal Yukawa coupling.
Although dark matter does not face stringent direct detection bounds due to
loop-suppressed couplings with the SM quarks and charged leptons, the
parameter space can be tightly constrained from other constraints related to
structure formation, CMB constraints on $\Delta N_{\rm eff}$, collider
constraints on $U(1)_{B-L}$ gauge bosons. We show interesting correlations in
the parameter space from simultaneous requirement of correct DM phenomenology
and $\Delta N_{\rm eff}$ indicating the region within reach of future CMB
experiments.

## Acknowledgements

The work of DB is supported by the Science and Engineering Research Board
(SERB), Government of India grant MTR/2022/000575. ND would like to thank
Anirban Biswas, Dibyendu Nanda, Pritam Das, Sahabub Jahedi, and Suruj Jyoti
Das for useful discussions. The work of ND is supported by the Ministry of
Education, Government of India via the Prime Minister’s Research Fellowship
(PMRF) December 2021 scheme.

##  Appendix A Calculation of FSL

The free-streaming length (FSL) can be quantified as 2012MNRAS.424..684S-

$\displaystyle\lambda_{\rm FSL}=\bigintss_{t_{\rm prod}}^{t_{\rm
eq}}\dfrac{\langle v\rangle}{a}\,dt=\bigintss_{T_{\rm prod}}^{T_{\rm
eq}}\dfrac{\langle v(T)\rangle}{a(T)}\dfrac{dt}{dT}\,dT.$ (23)

Here $t_{\rm prod}$ ($T_{\rm prod}$) is the time (temperature) when maximum
production of dark matter occurs. $t_{\rm eq}$ ($T_{\rm eq}$) is the time
(temperature) of matter-radiation equality after which the structure formation
starts. In terms of distribution function $f_{\psi}(q_{1},T)$, the average
velocity $\langle v(T)\rangle$ can be written as

$\displaystyle\langle{v(T)}\rangle=\dfrac{\bigintss\frac{q_{1}}{E_{1}}\,\frac{d^{3}q_{1}}{(2\pi)^{3}}\,f_{\psi}(q_{1},T)}{\bigintss\frac{d^{3}q_{1}}{(2\pi)^{3}}\,f_{\psi}(q_{1},T)}.$
(24)

Here, $q_{1}$ and $E_{1}$ denotes the momentum and energy of dark matter
respectively. To calculate the distribution function $f_{\psi}(q_{1},T)$, it
is convenient to change these variables to $r$ and $\xi_{\psi}$ given by

$\displaystyle r$ $\displaystyle=$ $\displaystyle\frac{m_{0}}{T},$ (25)
$\displaystyle\xi_{\psi}$ $\displaystyle=$
$\displaystyle\left(\frac{g^{s}_{*}(T_{0})}{g^{s}_{*}(T)}\right)^{1/3}\frac{q_{1}}{T},$

where $m_{0}$ and $T_{0}$ is some reference mass and temperature respectively.
For calculating $f_{\psi}(q_{1},T)$, one also needs to calculate the
distribution function of $\phi_{1}$, $f_{\phi_{1}}(p,T)$.

For the process $\phi_{1}(p_{1})\to\psi(q_{1})+\nu_{R}(q_{2})$, the Boltzmann
equation for the distribution function of $\psi$ can be written as

$\displaystyle\frac{\partial f_{\psi}}{\partial
t}-\mathcal{H}q_{1}\frac{\partial f_{\psi}}{\partial q_{1}}=$
$\displaystyle\frac{1}{16\pi\,E_{q_{1}}\,q_{1}}\int_{p^{\rm min}_{1}}^{p^{\rm
max}_{1}}\dfrac{p_{1}dp_{1}}{E_{p_{1}}}\rvert\mathcal{M}\rvert^{2}_{\phi\to\bar{\nu}_{R}\psi}\,f_{\phi_{1}}(p_{1}).$
(26)

The expressions for $p_{1}^{\rm max}$ and $p_{1}^{\rm min}$ for
$m_{\phi_{1}}\gg m_{\psi}$ are

$\displaystyle p^{\rm min}_{1}$ $\displaystyle\simeq$
$\displaystyle\dfrac{m^{2}_{\phi_{1}}}{2\,m^{2}_{\psi}}\left(-q_{1}+\sqrt{q_{1}^{2}-4\frac{m^{2}_{\psi}}{m_{\phi_{1}}^{2}}q_{1}^{2}+m^{2}_{\psi}}\right)\,,$
(27) $\displaystyle p^{\rm max}_{1}$ $\displaystyle\simeq$
$\displaystyle\dfrac{m^{2}_{\phi_{1}}}{2\,m^{2}_{\psi}}\left(q_{1}+\sqrt{q_{1}^{2}-4\frac{m^{2}_{\psi}}{m_{\phi_{1}}^{2}}q_{1}^{2}+m^{2}_{\psi}}\right)\,.$
(28)

The distribution function for $\phi_{1}$ after its freeze-out can be obtained
from

$\displaystyle\frac{\partial f_{\phi_{1}}}{\partial
t}-\mathcal{H}p_{1}\frac{\partial f_{\phi_{1}}}{\partial
p_{1}}=-\frac{1}{2E_{p_{1}}}$
$\displaystyle\int\frac{d^{3}q_{1}}{2E_{q_{1}}(2\pi)^{3}}\frac{d^{3}q_{2}}{2E_{q_{2}}(2\pi)^{3}}(2\pi)^{4}\delta^{4}(P_{1}-Q_{1}-Q_{2})$
$\displaystyle\times\lvert\mathcal{M}\rvert^{2}_{\phi_{1}\to\psi\bar{\nu}_{R}}f_{\phi_{1}}(p_{1})\,$
(29) $\displaystyle=-f_{\phi_{1}}$
$\displaystyle\frac{m_{\phi_{1}}}{\sqrt{p_{1}^{2}+m_{\phi_{1}}^{2}}}\Gamma_{\phi_{1}\to\psi\bar{\nu}_{R}}.$

Here $P_{1},Q_{1}$ and $Q_{2}$ are the four momenta corresponding to
$\phi_{1}$, $\psi$ and $\nu_{R}$ respectively. With these above equations and
using the transformation
$r=\frac{m_{0}}{T},\xi_{\psi}=\left(\frac{g^{s}_{*}(T_{0})}{g^{s}_{*}(T)}\right)^{1/3}\frac{q_{1}}{T}$,
we obtain the distribution function $f_{\psi}(\xi_{\psi},r)$ Konig:2016dzg ;
Biswas:2016iyh ; Biswas:2022vkq . With these, the average velocity of DM and
free-streaming length can be re-expressed in terms of new variables as

$\displaystyle\langle
v(r)\rangle=\dfrac{\mathcal{A}(r)}{\bigintss\xi^{2}_{\psi}\,f_{\psi}(\xi_{\psi},r)\,d\xi_{\psi}}\times{\bigintss\dfrac{\xi^{3}_{\psi}\,f_{\psi}(\xi_{\psi},r)\,d\xi_{\psi}}{\sqrt{(\mathcal{A}(r)\xi_{\psi})^{2}+(\frac{r}{m_{0}}\,m_{\psi})^{2}}}}\,$
(30) $\displaystyle\lambda_{\rm
FSL}=\left(\dfrac{11}{43}\right)^{1/3}\,r_{0}\int_{r_{\rm prod}}^{r_{\rm
eq}}{\langle{v(r)}\rangle\,g_{s}^{1/3}}\frac{\beta}{\mathcal{H}(r)}\frac{dr}{r^{2}},$
(31)

where
$\mathcal{A}(r)=\left(\frac{g_{*}^{s}(m_{0}/r)}{g_{*}^{s}(m_{0}/T_{0})}\right)^{1/3}$.

##  Appendix B Estimation of dark sector temperature

Here we provide an estimation of dark sector temperature using the entropy
conservation. Let us assume that $g^{\rm s,DS}_{*}$ and $g^{\rm s}_{*}$ are
the effective number of relativistic degree of freedom of dark sector and
standard model respectively. At the epoch of decoupling of dark sector from
the standard model bath, the scale factor is taken as $a_{\rm dec}$. Using the
conservation of entropy, we get

$\displaystyle g^{\rm s}_{\rm*,dec}a^{3}_{\rm dec}(T_{\rm dec})^{3}=g^{\rm
s}_{*,\rm aft}a^{3}_{\rm aft}(T_{\rm aft})^{3}$ $\displaystyle g^{\rm
s,DS}_{\rm*,dec}a^{3}_{\rm dec}(T^{\rm DS}_{\rm dec})^{3}=g^{\rm s,DS}_{*,\rm
aft}a^{3}_{\rm aft}(T^{\rm DS}_{\rm aft})^{3},$ (32)

where $a_{\rm aft}$ denotes any period after decoupling and $T^{\rm DS}$ and
$T$ denote temperature of dark sector and SM bath respectively. At the epoch
of decoupling, both the dark sector and SM bath have same temperature, $T_{\rm
dec}=T^{\rm DS}_{\rm dec}$. Hence, from the above equations, we get -

$\displaystyle\left(\frac{T^{\rm DS}_{\rm aft}}{T_{\rm
aft}}\right)^{4}=\left(\frac{g^{\rm s,DS}_{\rm*,dec}}{g^{\rm s,DS}_{\rm*,\rm
aft}}\frac{g^{\rm s}_{\rm*,\rm aft}}{g^{\rm s}_{\rm*,dec}}\right)^{4/3}.$ (33)

For the situation where dark sector consists of $\phi_{1},\psi$ and $\nu_{R}$,
we have $g^{\rm s,DS}_{*,\rm
dec}=2+6\times\frac{7}{8}+4\times\frac{7}{8}=10.75$. Taking $a_{\rm aft}$ to a
period when both $\phi_{1}$ and $\psi$ annihilate or become non-relativistic,
we get $g^{\rm s,DS}_{*,\rm aft}=6\times\frac{7}{8}=5.25$. If during this
period, $g^{\rm s}_{*}$ remains constant, we get

$\displaystyle\left(\frac{T^{\rm DS}_{\rm aft}}{T_{\rm
aft}}\right)^{4}=\xi^{4}=\left(\frac{10.75}{5.25}\right)^{4/3}=2.6.$ (34)

So, at maximum, we can expect a 2.6 times increase in dark sector temperature
from the SM bath. The numerical results shown in Fig. 8 are consistent with
this analytical estimation.

However, we can have a value of $g^{\rm s,DS}_{\rm*,dec}<10.75$ at the epoch
of decoupling. This is because before decoupling, some fraction of dark sector
particles can become non-relativistic. For example, let us consider
$m_{\phi_{1}}=60$ GeV, $m_{\psi}=15$ GeV. Decoupling above $T\sim 60$ GeV
gives $g^{s,\rm DS}_{*,\rm dec}<10.75$ whereas decoupling after $T\sim 60$ GeV
(e.g. $10$ GeV) gives $g^{\rm s,DS}_{*,\rm dec}<10.75$. This results in a
smaller increase in dark sector temperature from SM bath. The same behaviour
can also been seen from Fig. 8.

##  Appendix C Direct Detection of WIMP type DM

Figure 11: The DM-nucleon cross-section vs DM mass as a function of effective
number of relativistic species. The blue color region shows the sensitivity of
DARWIN experiment DARWIN:2016hyl .

In this setup, dark matter $\psi$ does not interact with the nucleus via tree
level diagram. However, via one loop diagram, it can scatter off the nucleus.
We get two diagrams, one where the mediator is SM Higgs and the other where
the mediator is B-L gauged boson. For the Higgs mediated diagram, the relevant
interaction vertices are $g_{\psi\bar{\psi}h}\psi\bar{\psi}h$ and
$\frac{m_{q}}{v}q\bar{q}$. Here $q$ represents the SM quarks. The effective
interaction term for Higgs mediated diagram can be written as

$\displaystyle\mathcal{L}^{h}_{eff}=\frac{m_{q}}{v}\frac{1}{m_{h}^{2}}g_{\psi\bar{\psi}h}\psi\bar{\psi}q\bar{q}.$
(35)

Similarly for the $Z^{\prime}$ mediated diagram, the effective interaction
term can be written as

$\displaystyle\mathcal{L}^{Z^{\prime}}_{eff}=-\frac{g_{BL}}{3}\frac{1}{m_{Z^{\prime}}^{2}}g_{\psi\bar{\psi}Z^{\prime}}\psi\gamma^{\mu}\bar{\psi}q\gamma_{\mu}\bar{q}.$
(36)

The interaction vertices $g_{\psi\bar{\psi}h}$ and
$g_{\psi\bar{\psi}Z^{\prime}}$ for one loop diagram are calculated using
Package-XPatel:2016fam  and their expressions are given as follows (for zero
momentum transfer)444We consider only one of the two different one-loop
diagrams to calculate the effective $g_{\psi\bar{\psi}Z^{\prime}}$ vertex for
simplicity.

$\displaystyle g_{\psi\bar{\psi}h}$ $\displaystyle=$
$\displaystyle\frac{i}{16\pi^{2}}y^{2}_{\phi_{1}}\lambda_{H\phi_{1}}v\frac{1}{m_{\psi}}\left[1+\left(\frac{m^{2}_{\phi_{1}}}{m^{2}_{\psi}}-1\right)\text{ln}\left(1-\frac{m^{2}_{\psi}}{m^{2}_{\phi_{1}}}\right)\right]$
$\displaystyle g_{\psi\bar{\psi}Z^{\prime}}$ $\displaystyle=$
$\displaystyle\frac{i}{16\pi^{2}}y^{2}_{\phi_{1}}g_{BL}\left[\frac{3}{2}\frac{m^{2}_{\phi_{1}}}{m^{2}_{\psi}}+\left(\frac{m^{2}_{\phi_{1}}}{m^{2}_{\psi}}-1\right)\text{ln}\left(1-\frac{m^{2}_{\psi}}{m^{2}_{\phi_{1}}}\right)\left(\frac{3}{2}\frac{m^{2}_{\phi_{1}}}{m^{2}_{\psi}}+\frac{1}{2}\right)\right].$
(37)

The total DM-nucleon cross-section can be written as Biswas:2016ewm ;
Bhattacharya:2022vxm ; Jungman:1995df  -

$\displaystyle\sigma_{\rm
tot}=\frac{1}{\pi}\frac{m_{N}^{2}m_{\psi}^{2}}{(m_{N}+m_{\psi})^{2}}\left[\frac{m_{N}}{v}\frac{1}{m_{h}^{2}}g_{\psi\bar{\psi}h}f_{N}+\frac{g_{BL}}{3}\frac{1}{m_{Z^{\prime}}^{2}}g_{\psi\bar{\psi}Z^{\prime}}f_{Z^{\prime}}\right]^{2},$
(38)

where $f_{N}\sim 0.3$, $f_{Z^{\prime}}=3$ and $m_{N}$ is the nucleon mass. In
Fig. 11, we show the DM-nucleon cross-section as a function of $m_{\psi}$. To
calculate the cross-section, we take the values of different parameters same
as for the resultant points we obtained in Fig. 9. All the points satisfy
correct DM relic abundance and range of different parameters for these points
are given by Eq. (22). We find that the total cross-section is way below the
direct detection limit.

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