# FIMP dark matter and low scale leptogenesis in an extended neutrino two
Higgs doublet model($\nu$2HDM)

Lavina Sarma<EMAIL_ADDRESS>Bichitra Bijay Boruah<EMAIL_ADDRESS>Mrinal Kumar Das<EMAIL_ADDRESS>Department of Physics, Tezpur
University, Tezpur 784028, India

###### Abstract

We have studied dark matter (DM) phenomenology, neutrinoless double beta decay
(NDBD) and realised low scale leptogenesis in a simple extension of Standard
Model(SM) with three neutral fermions, a scalar doublet and a dark sector
incorporating a singlet scalar and a Dirac singlet fermion. A generic model
based on $A_{4}\times Z_{4}$ flavor symmetry is used to explain both normal
and inverted hierarchy mass pattern of neutrino and also to accommodate the
dark matter mass. In this extension of the $\nu$2HDM, the effective neutrino
mass observed in $0\nu\beta\beta$ is well within the experimental limit
provided by KamLAND-ZEN. In order to validate DM within this model, we have
checked relic abundance and free streaming length of the dark sector
component, i.e. a Dirac singlet fermion constraining its mass in keV range.
More importantly we have also realised low scale leptogenesis simultaneously
within this framework and also the Dirac CP phase gets constrained with the
results. Co-relation among the observable and model parameters are also
carried out within this framework.

Standard Model, Two higgs doublet model , Dark matter, NDBD, Leptogenesis

###### pacs:

12.60.-i,14.60.Pq,14.60.St

##  I Introduction

The significance of physics Beyond the Standard Model(BSM) can be mainly
verified by the existence of neutrino mass deSalas:2017kay , Dark
Matter(DM)bertone2005particle ; Moore:1999nt  and Baryon Asymmetry of the
Universe(BAU)leptogenesis ; Hugle:2018qbw . The mass of neutrinos to be of the
order of sub-eV Lattanzi:2016rre  got its confirmation from the neutrino
oscillationsneuOsc ; neuOsc2 . Neutrino oscillation was first suggested by
Pontecorvo in the 1960’s. Whereas, its experimental discovery was made by
Super-Kamiokande Observatory and Sudbury Neutrino Observatories which was
recognised by the 2015 Noble Prize shared by Takaaki Kajita and Arthur B.
McDonald. Neutrino physics has come up with some benchmark evidences for
developing new physics for elementary particles thereby probing into the
evolution of the Universe. The recent Neutrino experiments MINOSMINOS
,RENORENO ,T2KT2K ,Double-ChoozDCHOOZ  have not only confirmed but also
measured the neutrino oscillation parameters more accurately. Incase of Type I
seesaw Minkowski ; Mohapatra  , the Standard Model is extended with the help
of three right handed neutrinos which results in the generation of neutrino
masses at sub eV scale whereas the right handed neutrino(RHN) mass is related
to the Grand Unified Theory(GUT) scale. Though the heavy RHN can generate
leptogenesisFukugita  and simulataneously the small neutrino mass, a
naturalness problem arisesnaturalness . Some possible pathways to overcome the
high scale leptogenesis are through resonant leptogenesisResLEP , ARS
mechanism via neutrino oscillationARS ; ARS1 , or via Higgs decayHiggsDECAY ;
HiggsDECAY1 . Such mechanisms however demand for a degenerate RHN mass, which
further is another naturalness problem. Whereas, we do have models like the
$\nu$2HDMnu2HDM  and scotogenic modelScotogenic ; Lavina ; Borah:2018rca
which accounts an explaination for these discripencies by considering
hierarchical masses of RHN and thereby producing leptogenesis at TeV scale. In
this work, we have taken the $\nu$2HDM and studied various phenomenologies
persisting in it.

There have been numerous lines of evidence of DM including observations from
galaxy rotation curveRubin:1970zza , cosmic microwave
backgroundcosmicmicrowave , galaxy clusterZwicky:1933gu , gravitational
lensingTreu:2012sn  , etc. However, the most recent cosmological constraints
on DM comes from the Planck satelliteAde:2015fva . It acertains 27$\%$ of the
present Universe to constitue of DM, which accounts for five times the
abundance of the baryonic matter. Depending upon the internal structure i.e.
the constituent particles and symmetries, the history of thermalisation of the
dark sector take different paths. This gives rise to different types of DM on
the basis of its mass rangeFIMP . We basically have the FIMP, hot and cold
type dark matter followed from the above mentioned aspect. Though the lightest
of the RHN can be considered to be a sterile DM in keV scaleN1DM ; N1DM1 , it
faces stringent bounds from various experimentsSterile , thereby leaving a
very small viable region for it to be a probable dark matter candidate. So, we
introduce another scenario in our work by incorporating a dark sector to the
$\nu$2HDM. It includes a scalar singlet($\eta$) and a Dirac singlet
fermion($\xi$), which are charged under $Z_{2}$ symmetrydarksector .
Interestingly, the bounds from X-ray experiment can be ignored as the
stability of DM candidate $\xi$ is maintained by the $Z_{2}$ symmetrySterile .
Due to the fact that we have null results from the Direct
detectionAprile:2017iyp  as well as indirect detectionFermi-LAT , the nature
of the DM candidate is taken to be of FIMP type.

The basic idea followed in our work comprises of the realisation of the
$\nu$2HDM by the virtue of the flavor symmetries $A_{4}\otimes Z_{4}$. Along
with the constituent particles of the generic $\nu$2HDM, we have added a
singlet scalar and a Dirac singlet fermion to the model. The newly added
particles incorporate the dark sector, by which the DM phenomenology can be
explained. As already mentioned, the DM considered in our work is of FIMP
type. We have analysed leptogenesis in our work with the consideration of a
mass hierarchy of the RHN as mentioned earlier. On finding the numerical
values of the Yukawa coupling matrix, we further carry out a detailed analysis
of leptogenesis as well as neutrinoless double beta decay($0\nu\beta\beta$).
We emphasize on a relative study due to the variation in the values of the
rotational matrix angles used for parametrization of the Yukawa coupling
matrix. This yields variation in decay parameter value K, which is used in
determining baryon asymmetry of the Universe and also explains the various DM
phenomenologies. Thus, we see how the choice of the angles has an impact on
neutrino phenomenology and related cosmology carried out in this work.

We have further categorised the paper into six sections which are as follows.
Sec.(II) and sec.(III) includes the basic model and the flavor symmetric model
respectively. Phenomenologies such as leptogenesis and dark matter are
mentioned in sec.(IV) and sec.(V) respectively. We finally show the numerical
analysis and the results obtained in sec.(VI), followed by the conclusion in
sec.(VII).

##  II Extended neutrino two higgs doublet model($\nu$2HDM)

In spite of being the most successful theory of particle physics , Standard
model(SM) fails to explain quite a few experimental and theoritical phenomena
such as smallness of neutrino mass, Dark matter(DM), Baryon asymmetry of the
universe(BAU) etc. In SM neutrino remains massless, although it can accquire a
small Majorana mass through dimension five operatorWeinberg1 ; Ma11 .

$\frac{1}{\Lambda}(\nu_{i}H^{0}-l_{i}H^{+})(\nu_{j}H^{0}-l_{j}H^{+})$ (1)

where $\Lambda$ is the effective large mass scale and H is usual Higss
doublet. This operator can be commonly realized in the framework of canonical
seesawMohapatra , where SM is extended with three right handed
neutrino($N_{i}$) such that-

$m_{\nu}=\frac{m_{D}^{2}}{m_{N}}$ (2)

where $m_{D}=fv$ is the Dirac neutrino mass considering vaccum expection
value(vev) of Higgs to be $<H^{0}>=v$ and $m_{N}=\Lambda f$. Such kind of
mechanisms doest have a direct test in the experiment beacuse of the large
mass scale of $m_{N}$.

$\rm\nu 2HDM$ is one such framework where Eq(2) can be realised naturally with
mass of the right handed neutrino to be in experimental reach (order of 1 TeV)
by extending the SM with three right handed neutrino and a scalar doublet
having a very small vaccum expectation value (vev)nu2HDM . In this model the
smallness of Dirac neutrino mass is achieved through this additional scalar
doublet. Particle content of the $\rm\nu 2HDM$ with charge assignment under
$SU(3)_{C}\otimes SU(2)_{L}\otimes U(1)_{Y}$ are given below-

$\left[\begin{array}[]{cc}\nu_{i}\\\
l_{i}\end{array}\right]_{L}=(1,2,-1/2);\left[\begin{array}[]{c}l_{iR}\end{array}\right]=(1,1,-1),\left[\begin{array}[]{c}N_{iR}\end{array}\right]=(1,1,0),$
(3)

Because of the this particular charge assignment given to the scalar doublet
the following term -

$\frac{1}{2}M_{i}N_{iR}^{2}+y_{ij}\bar{N_{iR}}(\nu_{jL}\phi^{0}-l_{jL}\phi^{+})$
(4)

appears in the Lagrangian. Considering the vev of the neutral component of the
scalar doublet to be small ($<\phi^{0}>=u$) one can conclude that $M_{i}$
appearing in the Eq.(4) can be of the order of 1 TeV as the cut off scale no
longer remains large. To accomodate dark matter candidate in this model, we
introduce a dark sector which includes a singlet scalar $\eta$ and a Dirac
singlet fermion $\xi$lowscale ; FIMP . They are charged under $Z_{2}$
symmetry. Now the scalar doublets can be denoted as-

$H=\left[\begin{array}[]{cc}H^{+}\\\
\frac{v+H^{0,r}+iH^{0,i}}{\sqrt{2}}\end{array}\right]\\\
,\phi=\left[\begin{array}[]{cc}\phi^{+}\\\
\frac{u+\phi^{0,r}+i\phi^{0,i}}{\sqrt{2}}\end{array}\right]$ (5)

The corresponding Higgs potential will be-

$\begin{split}V=&m_{H}^{2}H^{\dagger}H+m_{\phi}^{2}\phi^{\dagger}\phi+m_{\eta}^{2}\eta^{\dagger}\eta+\frac{1}{2}\lambda_{1}(H^{\dagger}H)^{2}+\frac{1}{2}\lambda_{2}(\phi^{\dagger}\phi)+\lambda_{3}(H^{\dagger}H)(\phi^{\dagger}\phi)+\\\
&\lambda_{4}(H^{\dagger}\phi)(\phi^{\dagger}H)-\mu_{12}^{2}(H^{\dagger}\phi+h.c)+\frac{1}{2}\lambda_{5}(\eta^{\dagger}\eta)^{2}+\lambda_{6}(\eta^{\dagger}\eta)(H^{\dagger}H)+\lambda_{7}(\eta^{\dagger}\eta)(\phi^{\dagger}\phi)\end{split}$
(6)

where the lepton symmetry is explicitly and softly broken by the $\mu_{12}$
term. In order for the unbroken $Z_{2}$ symmetry to accomodate the dark sector
requires the condition $<\eta>=0$. The minimization condition given by the
equation -

$v\big{[}m_{H}^{2}+\frac{\lambda_{1}}{2}v^{2}+\frac{\lambda_{3}+\lambda_{4}}{2}u^{2}\big{]}-\mu_{12}^{2}u=0$
(7)

$u\big{[}m_{\phi}^{2}+\frac{\lambda_{1}}{2}u^{2}+\frac{\lambda_{3}+\lambda_{4}}{2}v^{2}\big{]}-\mu_{12}^{2}v=0$
(8)

helps in expressing the vev of Higgs doublet in terms of parameter present in
the Higgs potential. Considering the parameter space as given below-

$m_{H}^{2}<0,m_{\phi}^{2}>0,|\mu_{12}|^{2}\ll m_{\phi}^{2}$ (9)

we have the vev of Higgs doublet as -

$v=\sqrt{\frac{-2m_{H}^{2}}{\lambda_{1}}},u=\frac{\mu_{12}^{2}v}{m_{\phi}^{2}+(\lambda_{3}+\lambda_{4})/2}$
(10)

By taking $m_{\phi}\simeq 100$GeV and $\mu\simeq 10$ GeV one can obtain
$u\simeq 1$ GeV. Radiative corrections to the term $\mu^{2}$ is proportional
to $\mu^{2}$ itself and it is logarithimically sensitive to the cut off scale
as it is the only source of lepton number violationLeptonV .

The effective Lagrangian can be written as-

$\mathcal{L}\subset y\bar{L}\tilde{\phi}N+\lambda\bar{\xi}\eta
N+\frac{1}{2}\bar{N^{c}}m_{N}N+m_{\xi}\xi\bar{\xi}+h.c$ (11)

From eq(11), the light neutrino mass can derived asMinkowski -

$m_{\nu}=-\frac{v^{2}}{2}ym^{-1}_{N}y^{T}=U_{PMNS}m^{d}_{\nu}U^{T}_{PMNS}$
(12)

where $m^{d}_{\nu}=diag(m_{1},m_{2},m_{3})$ is the diagonal neutrino mass
matrix and $U_{PMNS}$ is the PMNS(Pontecorvo-Maki-Nakagawa-Sakata) matrix.

##  III Flavor symmetric model

In this section we have realized this simple extension of the $\nu$2HDM with
the help of dicrete flavor symmetry group $A_{4}\otimes Z_{4}$. With the
generic $\nu$2HDM , a singlet scalar and a Dirac singlet fermion is added as
dark sector. To maintain the stability of the dark sector, a discrete $Z_{2}$
symmetry is also introduced under which the extended particles are charged.
Particle content of the model with its charge assignment are given in the
table(1).

Field | $l_{L}$ | $l_{R_{1}}$ | $l_{R_{2}}$ | $l_{R_{3}}$ | $N_{1}$ | $N_{2}$ | $N_{3}$ | H | $\phi$ | $\rho$ | $\rho^{{}^{\prime}}$ | $\rho^{{}^{\prime\prime}}$ | $\zeta$ | $\zeta^{{}^{\prime}}$
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
$A_{4}$ | $3$ | $1$ | $1^{{}^{\prime\prime}}$ | $1^{{}^{\prime}}$ | $1$ | $1^{{}^{\prime}}$ | $1$ | $1$ | $1$ | $3$ | $3$ | $3$ | $1$ | $1^{{}^{\prime}}$
$Z_{4}$ | $1$ | $1$ | $1$ | $1$ | $1$ | $-i$ | $-1$ | $1$ | $1$ | $1$ | $i$ | $-1$ | $1$ | $-1$
Table 1: Fields and their respective transformations under the symmetry group
of the model.

Now, the Lagrangian for charge lepton sector can be written as-

$\mathcal{L}=\frac{y_{e}}{\Lambda}(\bar{l_{L}}H\rho)l_{R_{1}}+\frac{y_{\mu}}{\Lambda}(\bar{l_{L}}H\rho)l_{R_{2}}+\frac{y_{\tau}}{\Lambda}(\bar{l_{L}}H\rho)l_{R_{3}}+\frac{y_{1}}{\Lambda}(\bar{l_{L}}\tilde{H}\rho)\nu_{R_{1}}+\frac{y_{2}}{\Lambda}(\bar{l_{L}}\tilde{H}\rho^{{}^{\prime}})\nu_{R_{2}}+\frac{y_{3}}{\Lambda}(\bar{l_{L}}\tilde{H}\rho^{{}^{\prime\prime}})\nu_{R_{3}}+h.c$
(13)

where $\Lambda$ denotes the cut off scale. The term $\bar{L}H\rho$ in the
Eq(13) transforms as $1$, $1^{{}^{\prime}}$ and $1^{{}^{\prime\prime}}$
respectively under $A_{4}$ for initial three terms. Additionally $Z_{4}$
included to constrain some term in the Lagrangian.

For neutrino sector the Lagrangain is given by-

$\mathcal{L}=\frac{1}{2}\lambda_{1}\zeta\bar{N_{1}^{c}}N_{1}+\frac{1}{2}\lambda_{2}\zeta^{{}^{\prime}}\bar{N_{2}^{c}}N_{2}+\frac{1}{2}\lambda_{3}\zeta\bar{N_{3}^{c}}N_{3}+h.c$
(14)

The flavon alignmments of considred in the model are given below-

$\displaystyle<\rho>=(\omega,0,0),<\rho^{{}^{\prime}}>=(\omega,\omega,\omega),<\rho^{{}^{\prime\prime}}>=(0,-\omega,\omega),<\zeta>=<\zeta^{{}^{\prime}}>=\omega$
(15)

Using this alignment we can have diagonal charge lepton matrix given by-

$m_{l}=\frac{<H>\omega}{\Lambda}\left[\begin{array}[]{ccc}y_{e}&0&0\\\
0&y_{\mu}&0\\\ 0&0&y_{\tau}\end{array}\right]\\\ ,$ (16)

and the Dirac mass matrix will have form given by-

$M_{D}=\frac{<H>\omega}{\Lambda}\left[\begin{array}[]{ccc}y_{1}&y_{2}&0\\\
0&y_{2}&y_{3}\\\ 0&y_{2}&-y_{3}\end{array}\right]\\\ ,$ (17)

here $y_{1}$,$y_{2}$ and $y_{3}$ stands for Yukawa couplings. Because of the
additional $Z_{4}$ symmetry the right handed neutrino mass matrix will be
diagonal one given as-

$M_{R}=\left[\begin{array}[]{ccc}\lambda_{1}\omega&0&0\\\
0&\lambda_{2}\omega&0\\\ 0&0&\lambda_{3}\omega\end{array}\right]\\\ ,$ (18)

where $\lambda_{1}$,$\lambda_{2}$ and $\lambda_{3}$ are the Majorana coupling.

##  IV Leptogenesis

We study leptogenesis in this work, which is a consequence of the out-of-
equilibrium decay of $N_{1}\rightarrow l_{L}\phi_{2}^{*},\bar{l_{L}}\phi_{2}$
leptogenesis . The lepton asymmetry produced only by the decay of $N_{1}$ is
converted into the baryon asymmetry of the Universe(BAU) by the electro-weak
sphaleron phase transitionsDine:2003ax . In case of vanilla leptogenesis,
there exists an absolute lower bound on the mass of the lightest RHN to be
$M_{1}\simeq 10^{9}$ GeV Davidson:2002qv ; Buchmuller:2002rq . However, the
limit on the lightest RHN mass scale can be lowered to 10 TeV Hugle:2018qbw ;
Borah:2018rca  in certain scenarios. With the consideration of a hierarchical
mass spectrum $m_{\phi}<<M_{1}<<M_{2,3}$ and the incorporation of the Yukawa
couplings from the model, the CP-asymmetry term is given by:

$\epsilon_{1}=\frac{1}{8\pi(h^{\dagger}h)_{11}}\sum_{j\neq
1}Im[(h^{\dagger}h)^{2}]_{1j}\left[f(r_{j1},\eta_{1})-\frac{\sqrt{r_{j1}}}{r_{j1}-1}(1-\eta_{1})^{2}\right],$
(19)

where,

$f(r_{j1},\eta_{1})=\sqrt{r_{j1}}\left[1+\frac{(1-2\eta_{1}+r_{j1})}{(1-\eta_{1})^{2}}ln(\frac{r_{j1}-\eta_{1}^{2}}{1-2\eta_{1}+r_{j1}})\right],$
(20)

$r_{j1}=\big{(}\frac{M_{N_{j}}}{M_{N_{1}}}\big{)}^{2}$ and
$\eta_{1}\equiv\big{(}\frac{m_{\phi^{0}}}{M_{N_{1}}}\big{)}^{2}$ .

The decay rate equation for $N_{1}$ which is given by,

$\Gamma_{1}=\frac{M_{N_{1}}}{8\pi}(h^{\dagger}h)_{11}\left[1-\Big{(}\frac{m_{\phi^{0}}}{M_{N_{1}}}\Big{)}^{2}\right]^{2}=\frac{M_{N_{1}}}{8\pi}(h^{\dagger}h)_{11}(1-\eta_{1})^{2}$
(21)

Meanwhile, the washout effect is checked by the decay parameter

$K_{1}=\frac{\Gamma_{1}}{H(z=1)},$ (22)

where, $\Gamma_{1}$ is the decay width of $N_{1}$, H is the Hubble parameter
and $z=M_{1}/T$ with T being the temperature of the thermal bath. We can
express $H$ in terms of $T$ and the corresponding equation is given by:

$H=\sqrt{\frac{8\pi^{3}g_{*}}{90}}\dfrac{T^{2}}{M_{Pl}}.$ (23)

In Eq.(23), $g_{*}$ stands for the effective number of relativistic degrees of
freedom and $M_{Pl}\simeq 1.22\times 10^{19}$ GeV is the Planck mass.
Using the parametrization for Yukawa coupling, i.e.

$Y=\frac{\sqrt{2}}{u}U_{PMNS}(m^{diag}_{\nu})^{1/2}R(M^{diag}_{N})^{1/2},$
(24)

we can verify the following relation:

$Y^{\dagger}Y=\frac{2}{u}(M^{diag}_{N})^{1/2}R^{\dagger}(m^{diag}_{\nu})^{1/2}R(M^{diag}_{N})^{1/2}$
(25)

Here, R matrix is parametrized as:

$R=\begin{pmatrix}cos\omega_{12}&sin\omega_{12}&0\\\
sin\omega_{12}&cos\omega_{12}&0\\\ 0&0&1\\\
\end{pmatrix}\begin{pmatrix}cos\omega_{13}&0&-sin\omega_{13}\\\ 0&1&0\\\
sin\omega_{13}&0&cos\omega_{13}\\\ \end{pmatrix}\begin{pmatrix}1&0&0\\\
0&cos\omega_{23}&-sin\omega_{23}\\\ 0&sin\omega_{23}&cos\omega_{23}\\\
\end{pmatrix}$ (26)

where, $\omega_{12,13,23}$ represents the arbitrary complex angles. Thus, we
can now express the decay parameter K interms of the arbitrary angles by the
following equation:

$K_{1}\simeq
897\big{(}\frac{v}{\phi_{v}}\big{)}^{2}\frac{((m_{\nu}^{diag})^{R})_{11}}{eV}$
(27)

In the above equation, $(m_{\nu}^{diag})^{R}=R^{\dagger}m_{\nu}^{diag}R$,
which further gives:

$(m_{\nu}^{diag})^{R})_{11}=m_{1}|cos\omega_{12}|^{2}|cos\omega_{13}|^{2}+m_{2}|sin\omega_{12}|^{2}|cos\omega_{13}|^{2}+|sin\omega_{13}|^{2}$
(28)

From Eq(27) and Eq(28), we can see that $K_{1}$ has dependency on the
arbitrary complex angles $\omega_{12,13}$ but is independent of $\omega_{23}$.
This relationship implies that by proper variation of these arbitrary angles,
we can have a comparative analysis of the different phenomenologies associated
with it. In our work, we have choosen some benchmark values for
$\omega_{13,23}$ for which we have varied $\omega_{12}$. We showcase the
variations in BAU as well as DM phenomenology for two different values of
$\omega_{12}$, however, keeping $\omega_{13,23}$ fixed. Also, the vev of the
scalar doublet $\phi$ plays a vital role in the determination of K, which
further has significant impact on baryogenesis. As mentioned in various
literatureslepto ; lepto1 , the $\Delta L$= 0 washout processes are crucial in
context of small values $\phi_{v}$. Thereby, the condition
$\Gamma_{1}/M_{1}<<1$ for low scale seesaw is satisfied. Thus, the Boltzmann
equations for the evolution of the abundance of $N_{1}$ and $N_{B-L}$ are
given by Davidson:2002qv ,

$\frac{dn_{N_{1}}}{dz}=-D_{1}(n_{N_{1}}-n_{N_{1}}^{eq}),$ (29)
$\frac{dn_{B-L}}{dz}=-\epsilon_{1}D_{1}(n_{N_{1}}-n_{N_{1}}^{eq})-W_{1}n_{B-L},$
(30)

respectively. $n_{N_{1}}^{eq}=\frac{z^{2}}{2}K_{1}(z)$ is the equilibrium
number density of $N_{1}$, where $K_{i}(z)$ is the modified Bessel function of
$i^{th}$ type and

$D_{1}\equiv\frac{\Gamma_{1}}{Hz}=K_{N_{1}}z\frac{K_{1}(z)}{K_{2}(z)}$ (31)

gives the measure of the total decay rate with respect to the Hubble rate, and
$W_{1}=\frac{\Gamma_{W}}{Hz}$ is the total washout rate. Again,
$W_{1}=W_{1D}+W_{\Delta L=2}$ Hugle:2018qbw , i.e the total washout term is
the sum of the washout due to inverse decays $l\eta,\bar{l}\eta^{*}\rightarrow
N_{1}$ ($W_{1D}=\frac{1}{4}K_{N_{1}}z^{3}K_{1}(z)$) and the washout due to the
$\Delta L=2$ scatterings
$l\eta\leftrightarrow\bar{l}\eta^{*},ll\leftrightarrow\eta^{*}\eta^{*}$ which
is given by,

$W_{\Delta
L=2}\simeq\dfrac{18\sqrt{10}M_{Pl}}{\pi^{4}g_{l}\sqrt{g_{*}}z^{2}v^{4}}(\frac{2\pi^{2}}{\lambda_{5}})^{2}M_{N_{1}}\bar{m}^{2}.$
(32)

Here, $g_{l}$ is the internal degrees of freedom for the SM leptons, and
$\bar{m}$ is the absolute neutrino mass scale, defined by:

$\bar{m}^{2}=m_{1}^{2}+m_{2}^{2}+m_{3}^{2}$ (33)

The final B-L asymmetry $n_{B-L}^{f}$ is evaluated by numerically calculating
Eq.(29) and Eq.(30) before the sphaleron freeze-out. This is converted into
the baryon-to-photon ratio given by:

$n_{B}=\frac{3}{4}\frac{g_{*}^{0}}{g_{*}}a_{sph}n_{B-L}^{f}\simeq 9.2\times
10^{-3}n_{B-L}^{f},$ (34)

In Eq.(34), $g_{*}=110.75$ is the effective relativistic degrees of freedom at
the time when final lepton asymmetry was produced, $g_{*}^{0}=\frac{43}{11}$
is the effective degrees of freedom at the recombination epoch and
$a_{sph}=\frac{8}{23}$ is the sphaleron conversion factor taking two Higgs
doublet into consideration. The Planck limit 2018 gives a bound on the
observed BAU($n_{B}^{obs}$) to be $(6.04\pm 0.08)\times 10^{-10}$
Aghanim:2018eyx . Therefore, in our work, we have chosen the free parameters
such that we can generate the observed BAU for NH/IH. As mentioned earlier, we
have fixed the arbitrary angles $\omega_{13,23}$ bearing values
$10^{-3}+10^{-3}i$ and $10^{-1}+10^{-1}i$ respectively. Whereas, we have
chosen two benchmark values for $\omega_{12}$, i.e $10^{-2}+10^{-2}i$ and
$10^{-12}+10^{-12}i$ which shows quite a significant change in the
cosmological phenomenologies carried out in this work. In addition to this, we
have taken a certain parameter space for $\phi_{v}=0.1-30$ GeV to show its
dependence on BAU for both NH/IH.

##  V DARK MATTER

In this work, we consider a FIMP type dark matter with the real Yukawa
coupling, $\lambda_{DM}<<1$FIMP  between the right handed neutrino N with the
singlet fermion $\xi$ and singlet scalar $\chi$. As of the DM type taken into
account, it is known that the coupling is so weak that it never reaches
thermalization even within the dark sector. Due to the feeble interaction of
the DM candidate, its relic abundance is generated via the freeze-in
mechanismFIMP2 . We can obtain the expression for relic abundance by solving
the Boltzman equation:

$\frac{dY_{\xi}}{dz}=DY_{N_{1}}\rm BR_{\xi}$ (35)

where, $Y_{\xi}$ and $Y_{\xi}$ are the abundance of DM candidate $\xi$ and
right handed neutrino $N_{1}$ respectively, and $\rm BR_{\xi}$ is the
branching ratio of $N_{1}\longrightarrow\xi\chi$. The inverse decay process,
i.e. $\xi\chi\longrightarrow N_{1}$ is neglected and a hierarchical criteria
of $\rm BR_{\xi}\equiv\rm BR(N_{1}\longrightarrow\xi\chi)<<\rm
BR_{SM}\equiv\rm BR(N_{1}\longrightarrow SM)\simeq 1$ is considered due to the
FIMP nature of $\xi$. Alongside, for the decay $N_{1}\longrightarrow\xi\chi$
to obey out of equilibrium condition, it is crucial that $\rm
BR_{\xi}<10^{-2}$ and the decay parameter to be $K\lesssim 10$. In our work,
as we have solved the model parameters so as to find the value of $K$, we see
that it is consistent with the above criteria. From Eq.35, an asymptotic
abundance of the FIMP DM $\xi$ can be approximated by the relationFIMP :

$Y_{\xi}(\infty)\simeq Y_{N_{1}}(0)\rm
BR_{\xi}\big{(}1+\frac{15\pi\zeta(5)}{16\zeta(3)}K\big{)}$ (36)

Therefore, the expression for relic abundance in agreement with the asymtotic
abundance is given by:

$\Omega_{\xi}^{\rm
FIMP}h^{2}=\frac{m_{\xi}s_{0}Y_{\xi}(\infty)}{\rho_{c}}h^{2}\simeq
0.12\times\big{(}\frac{m_{\xi}}{keV}\big{)}(\rm BR_{\xi}\times
10^{3})(0.009+\frac{K}{44}))$ (37)

where, $\rho_{c}=1.05371\times 10^{-5}h^{2}\rm GeV\rm cm^{-3}$ is the critical
density of the Universe and $s_{0}=2891.2\rm cm^{-3}$FIMP3  is the current
entropy density and $h=0.72$ is the Hubble parameter. One of the frequently
and significantly considered cosmological constraints that comes into play
while considering light DM candidate is the free streaming limit. It provides
stringent bounds on the FIMP DM mass. Due to non- trivial velocity dispersion,
the free streaming of dark matter particles strike outs the matter density
perturbations and consequently the structure formation on scales
$\lambda_{DM}<\lambda_{FS}$. Subsequently, small structure formation gives the
most robust constraint on the free streaming length, viz.
$\lambda_{FS}<\mathcal{O}(0.1)$MpcFIMP4 . The free streaming length is defined
as the average distance covered by a particle without confronting
collisionFIMP ;

$\lambda_{FS}=\int_{a_{rh}}^{a_{eq}}\frac{<v_{\xi}>}{a^{2}H}da\simeq\frac{a_{NR}}{H_{0}\sqrt{\Omega_{R}}}\big{(}0.62+ln\big{(}\frac{a_{e}q}{a_{NR}}\big{)}\big{)}$
(38)

where, $<v_{\xi}>$ is the average velocity at given time of the FIMP DM $\xi$,
$a_{rh}$ and $a_{eq}$ are the scale factors at reheating and equilibrium
respectively. The values of the cosmological parameters used are
$H_{0}=67.3\rm km\rm s^{-1}\rm Mpc^{-1}$, $\Omega_{R}=9.3\times 10^{-5}$ and
$a_{eq}=2.9\times 10^{-4}$ FIMP5 . Again, the non- relativistic scale factor
for FIMP DM is expressed as:

$a_{NR}=\frac{T_{0}}{2m_{\xi}}\big{(}\frac{g_{*,0}}{g_{*,rh}}\big{)}K^{-1/2}$
(39)

with $g_{*,0}=3.91$, $g_{*,rh}=106.75$ and $T_{0}=2.35\times 10^{-4}\rm eV$.
There are different decoupling and production mechanism of DM and thereby, the
free streaming length is different for each of the relativistic decoupling
scenarios such as hot, FIMP DM. As the free streaming length is highly
dependent on the production mechanism, thus, the specific free streaming
length for FIMP DM can be given by:

$\lambda_{FS}\simeq 2.8\times
10^{-4}\big{(}\frac{keV}{m_{\xi}}\big{)}\big{(}\frac{50}{K}\big{)}^{1/2}\times\big{(}1+0.09ln\big{[}\big{(}\frac{m_{\xi}}{keV}\big{)}\big{(}\frac{K}{50}\big{)}^{1/2}\big{]}\big{)}\rm
Mpc$ (40)

From the above equation, we can clearly say that the free streaming length is
relied upon the decay parameter and the DM mass. Hence, we do a detailed
analysis on these factors in our work for both NH/IH.

##  VI Analysis and results

Figure 1:  Baryon asymmetry of the Universe as a function of lightest
RHN($N_{1}$) is shown in the first row. The second and third row depicts BAU
as a function of $K_{1}$ and vev of $\phi_{1}$ respectively. And in the fourth
row we see a plot between BAU and delta phase($\delta$) of neutrino
oscillation. We take two different values of arbitrary angle $\omega_{12}$ and
benchmark values of $\omega_{13}=10^{-3}+10^{-3}i$ and
$\omega_{23}=10^{-2}+10^{-2}i$ for NH/IH .

Figure 2:  Yukawa coupling term $YY^{\dagger}_{11}$ as a function of BAU for
two different values of arbitrary angle $\omega_{12}$ and benchmark values of
$\omega_{13}=10^{-3}+10^{-3}i$ and $\omega_{23}=10^{-2}+10^{-2}i$ for NH/IH .

Figure 3:  Baryon asymmetry of the Universe as a function of relic abundance
of the dark matter candidate $\xi$ for NH/IH with the consideration of two
different values of arbitrary angle $\omega_{12}$ and benchmark values of
$\omega_{13}=10^{-3}+10^{-3}i$ and $\omega_{23}=10^{-2}+10^{-2}i$.

Figure 4:  Decay parameter $K_{1}$ as a function of lightest neutrino mass for
two different values of arbitrary angle $\omega_{12}$ with benchmark values
$\omega_{13}=10^{-3}+10^{-3}i$ and $\omega_{23}=10^{-2}+10^{-2}i$ for NH/IH .

Figure 5:  Effective mass of the neutrinos w.r.t the lightest active neutrino
is being showed for NH/IH with two different values of arbitrary angle
$\omega_{12}$ and benchmark values of $\omega_{13}=10^{-3}+10^{-3}i$ and
$\omega_{23}=10^{-2}+10^{-2}i$ . The horizontal black line signify the
KamLAND-Zen upper limit on the effective mass of neutrinos.

Figure 6: The plot shows variation of relic abundance w.r.t the dark matter
mass for NH/IH. Two different values of arbitrary angle $\omega_{12}$ and
benchmark values of $\omega_{13}=10^{-3}+10^{-3}i$ and
$\omega_{23}=10^{-2}+10^{-2}i$ for NH/IH is chosen .

Figure 7: The plot shows variation of relic abundance w.r.t the decay
parameter $K_{1}$ for NH/IH for two different values of arbitrary angle
$\omega_{12}$ and benchmark values of $\omega_{13}=10^{-3}+10^{-3}i$ and
$\omega_{23}=10^{-2}+10^{-2}i$ .

Figure 8: The plot shows variation of relic abundance w.r.t the branching
ratio of the decay $N_{1}\longrightarrow\xi\chi$ for two different values of
arbitrary angle $\omega_{12}$ and benchmark values of
$\omega_{13}=10^{-3}+10^{-3}i$ and $\omega_{23}=10^{-2}+10^{-2}i$ for NH/IH.

Figure 9: The plot shows variation of free streaming length w.r.t the decay
parameter $K_{1}$ for NH/IH. Here, the variation is shown for two different
values of arbitrary angle $\omega_{12}$ and benchmark values of
$\omega_{13}=10^{-3}+10^{-3}i$ and $\omega_{23}=10^{-2}+10^{-2}i$.

Figure 10: The plot shows variation of free streaming length w.r.t the relic
abundance of $\xi$ for two different values of arbitrary angle $\omega_{12}$
and benchmark values of $\omega_{13}=10^{-3}+10^{-3}i$ and
$\omega_{23}=10^{-2}+10^{-2}i$ for NH/IH.

In the flavor symmetric realisation of the extended $\nu$2HDMlowscale , we
have explicitly studied the implication of the Yukawa coupling matrix and the
rotational matrix used for parametrization. By numerically solving the model
parameters which constitute the Yukawa coupling matrix, we eventually find the
range of the coupling matrix Y. Incorporating the parameter space of the
Yukawa coupling matrix in Eq.(24), we can deduce the active neutrino mass
matrix. It has been found from the model we have worked upon that the
summation of the active neutrino is consistent with the Planck limit, i.e.
$\sum_{i=1,2,3}m_{i}<0.11eV$Lattanzi:2016rre . Consecutively, we have also
analysed the phenomenology of $0\nu\beta\beta$Mohapatra:1986su ; Barry:2013xxa
; Borgohain:2017inp  in our work. It is indeed a well known and significant
experimental technique of detecting neutrino mass, thereby, making it
relatable with observations in the on-going experiments. The various
experiments in this context are KamLAND-ZenKamland2 ; kamland , KATRINkatrin2
; KATRIN , GERDAgerda ; GERDA2 , etc. The phenomenon that is measured in these
experiment is the effective mass of the active neutrinos which is given by the
expression:howing

$|m_{\beta\beta}|=|c_{12}^{2}c_{13}^{2}m_{1}+s_{12}^{2}c_{13}^{2}m_{2}e^{2i\alpha}+s_{13}^{2}m_{3}e^{2i\beta}|$
(41)

where, $c_{ij}$= $\cos\theta_{ij}$ and $s_{ij}$= $\sin\theta_{ij}$ are the
elements of the UPMNS matrix. It is crucial to obey the effective mass
constraint so as to make the model sensitive to the on-going as well as future
collider signatures. Throughout the work, we have considered a specific range
of the right handed neutrinos, viz. $M_{N_{1}}=10^{4}$ GeV - $10^{6}$ GeV,
$M_{N_{2}}=10^{7}$ GeV - $5\times 10^{7}$ GeV and $M_{N_{3}}=10^{8}$ GeV -
$5\times 10^{8}$ GeV. Also, the lightest active neutrino mass is taken in the
span of $m_{l}=10^{-11}-10^{-13}$ eV incase of both NH/IH which is a crucial
parameter required to achieve baryogenesis in TeV scaleHugle:2018qbw . In our
study, we have shown the variations that may occur due to the choice of
arbitrary angles of the rotational matrix given in Eq.(28). Since, the
dependency of BAU and dark matter phenomenology are influenced by the decay
parameter $K_{1}$, thus, we can generate desired values of $K_{1}$ by fine
tuning the the arbitrary angles $\omega_{12}$ and $\omega_{13}$ lowscale . In
our case, we have kept $\omega_{13,23}$ fixed at $10^{-3}+10^{-3}i$ and
$10^{-2}+10^{-2}i$ respectively and simultaneously choosen two different
values of $\omega_{12}$ as $10^{-1}+10^{-1}i$ and $10^{-12}+10^{-12}i$. In
Fig.(1), we have shown co-relation plots between BAU and parameters such as
$M_{1}$, $K_{1}$, vev of $\phi$ (v) and $\delta$ respectively for NH and IH.
From the first row of Fig.(1), we see that the observed baryogenesis is
generated for $M_{1}=5\times 10^{4}-10^{6}$ GeV for NH, however, the choice of
the arbitrary angle $\omega_{12}=10^{-1}+10^{-1}i$ gives large number of
points satisfying the Planck limit compared to that of
$\omega_{12}=10^{-12}+10^{-12}i$. For $\omega_{12}=10^{-12}+10^{-12}i$ , a
small region of $M_{1}=5\times 10^{5}-10^{6}$ is successful in producing the
desired BAU in case of NH. We donot observe much change in the variation of
BAU w.r.t $M_{1}$, except for a slight shift in the mass range, i.e.
$10^{5}-10^{6}$ GeV satisfying the Planck bound for IH. Also the variation due
to the arbitrary angles are almost similar for both the heirarchies. In the
second row of Fig.(1), an interesting result is seen which constraints the
parameter space of $K_{1}$ satisfying the Planck limit depending upon the
choice of arbitrary angle $\omega_{12}$. The points satisfying BAU for
$\omega_{12}=10^{-12}+10^{-12}i$ falls on the very weak washout regime,
whereas for $\omega_{12}=10^{-1}+10^{-1}i$ it is seen that the washout is
higher though it falls in the weak washout region serving the DM candidate to
be a FIMP type for both NH/IH lowscale ; FIMP . Again from third row of
Fig.(1), we have obtained a clear distinction between the range of the vev
satisfying the Planck limit for BAU w.r.t the choice of $\omega_{12}$. We see
that for $\omega_{12}=10^{-1}+10^{-1}i$, the entire range of vev is seen to
produce the observed BAU, whereas for $\omega_{12}=10^{-12}+10^{-12}i$ the
region of vev satisfying the BAU limit is constrained from 5-30 GeV for NH. In
case of IH, though the entire range of v satisfies BAU for
$\omega_{12}=10^{-1}+10^{-1}i$, but for $\omega_{12}=10^{-12}+10^{-12}i$ we
obtain a more narrow range of v, i.e. 10-30 GeV obeying Planck limit for BAU.
We have shown a variation of BAU with the delta phase ($\delta$) in the fourth
row of Fig.(1), wherein the region for which we get the observed baryon
asymmetry is $10^{-1}-2\pi$ for $\omega_{12}=10^{-1}+10^{-1}i$. But for
$\omega_{12}=10^{-12}+10^{-12}i$, the allowed region is limited to $1-2\pi$
for both NH and IH. We have obtained a crucial result from Fig.(2) which
remains almost the same for both the mass heirarchies. Here, the parameter
space of the Yukawa coupling element $(Y^{\dagger}Y)_{11}$ which has a
significant contribution in the CP asymmetry is constrained for
$\omega_{12}=10^{-12}+10^{-12}i$ in a small region obeying the BAU limit.
However, we have quite a wide range of $(Y^{\dagger}Y)_{11}$ corresponding to
$\omega_{12}=10^{-12}+10^{-12}i$ satisfying the Planck bound for BAU. Fig.(3)
is a co-relation plot between the relic abundance and the BAU. Here, we see
that there are points for both the choice of $\omega_{12}$ satisfying both the
bounds simultaneously, though we have little larger number number of points
obeying the bounds for higher value of $\omega_{12}$ compared to that for
lower value of $\omega_{12}$.

From Fig.(4), we see a divergence in the parameter space of decay parameter
w.r.t the lightest active neutrino mass. For $\omega_{12}=10^{-1}+10^{-1}i$,
the decay parameter $K_{1}$ falls in the weak washout region for the light
neutrino mass obtained from the model. This range of $K_{1}$ corresponds to
warm DM samples. Whereas, for $\omega_{12}=10^{-12}+10^{-12}i$, we also obtain
very small value of $K_{1}$ corresponding to hot DM samples which are
disfavored by small structure formationFIMP . Thus, we can draw a conclusion
that larger value of the arbitrary angle $\omega_{12}$ is more prominent in
satisfying experimental bounds for small structure formation compared to
smaller values incase of both NH/IH. However, there is hardly any change in
the effective mass curve depicted in Fig.(5) irrespective of the two different
choice of $\omega_{12}$ for NH as well as IH. Typically for a warm DM, the
branching ratio $\rm Br_{\xi}>10^{-4}$, whereas for a cold DM it is $\rm
Br_{\xi}\lesssim 10^{-3}$. For further deduction of the relic abundance
numerically, we choose the branching ratio in the range $\rm
Br_{\xi}=10^{-2}-10^{-7}$. From Fig.(6), we see that for
$\omega_{12}=10^{-1}+10^{-1}i$, we get observed relic abundance for DM mass in
the range 5-100 GeV which corresponds to warm DM, whereas for
$\omega_{12}=10^{-12}+10^{-12}i$, smaller mass of $\xi$ signifying hot DM also
satisfies the relic abundance limit. In a plot between relic abundance and
decay parameter($K_{1}$) as shown in Fig.(7), we observe that the parameter
space for $K_{1}$ satisfying the Planck limit is larger for
$\omega_{12}=10^{-12}+10^{-12}$ compared to $\omega_{12}=10^{-1}10^{-1}$.
However, for satifying the small structure formation constraint, $K_{1}$ must
not fall in the very weak washout region, therefore,
$\omega_{12}=10^{-12}+10^{-12}$ is less favorable for both NH/IH. Also as
mentioned earlier the preferable branching ratio range for warm DM satisfying
the relic abundance bound is obeyed for $\omega_{12}=10^{-1}+10^{-1}$ as
depicted in Fig.(8). We know that the most stringent bound on $r_{FS}$ comes
from the small structure formation, $r_{FS}<0.1$. A relationship between
$r_{FS}$ and $K_{1}$ is shown in Fig.(9), where we have maximum points falling
on the allowed region for $\omega_{12}=10^{-1}+10^{-1}$. As we know that
$K_{1}$ is inversely proportional to $r_{FS}$, thus, very small values of
$K_{1}$ donot satisfy the small structure formation bound. An analysis on the
points satisfying both relic abundance and small structure formation is been
shown in Fig.(10), wherein we see that almost all the points corresponding to
$\omega_{12}=10^{-1}+10^{-1}$ satisfy both the bounds, though a small range of
parameter space for $\omega_{12}=10^{-12}+10^{-12}$ seems to fall on the
disallowed regime for both NH and IH.

##  VII Conclusion

Our work basically showcases a comparative analysis of different
phenomenological consequences corresponding to variation in the arbitrary
angles of the rotational matrix. As mentioned in the earlier sections, we have
realised the extended $\nu$2HDM with the help of flavor symmetries
$A_{4}\otimes Z_{4}$. It is an interesting model as it can accomodate both
neutrino as well as DM phenomenologies. Also, we can generate the light
neutrino mass at the tree level by consideration of an extra scalar doublet
which is assigned L=-1nu2HDM . From the model, we numerically calculate the
Yukawa coupling matrix, which plays the source for the various phenomenologies
that we have studied. Also, we have chosen the rotational matrix R and the
values of the arbitrary angles such that we can draw crucial conclusions of
its impact on the cosmological phenomenologies we have discussed. Following
various literaturesHugle:2018qbw ; FIMP5 , we have chosen the values of the
RHN masses, lightest active neutrino mass, vev of the scalar doublet($\phi$)
as already mentioned in sec.(VI) so as to achieve TeV scale leptogenesis. From
Fig.(1),(2) and (3) we can conclude that the choice
$\omega_{12}=10^{-1}+10^{-1}$ is more preferable in satisfying the BAU limit
given by Planck data. Also, we have obtained a constraint range of the free
parameters as well as the Yukawa couplings corresponding to the two choices of
the arbitrary angles. In plot (3) we obtain points which abide by the
constraint from both BAU and relic abundance simultaneously which is a vital
part of the analysis. We have also generated some important results in context
with the decay parameter. Depending on $\omega_{12}$ values, we can generalise
the transition in the relation between $K_{1}$ and $m_{l}$ for both NH and IH.
Considering the DM scenario, it is viable to have a warm DM candidate as it
obeys the constraint coming from small structure formation in comparison to
hot DM. Simultaneously, we can get a warm DM source when the decay parameter
donot fall in the very weak washout region. Therefore, it is seen from Fig.(4)
that for $\omega_{12}=10^{-1}+10^{-1}i$ we obtain the preferable range of
$K_{1}$ which further explains the warm DM. For warm DM, $m_{\xi}\sim 10$ KeV
gives significant observed relic abundance, however, small DM mass correspond
to hot DM which again is constrained by small structure formation. Again, a
difference in the branching ratio range is obtained depending on the two
different values of $\omega_{12}$. So, we can say that $\rm Br_{\xi}\simeq
10^{-5}-10^{-3}$ is the prefered range for $\omega_{12}=10^{-1}+10^{-1}i$
obeying the Planck limit for relic abundance, whereas for
$\omega_{12}=10^{-12}+10^{-12}i$, $\rm Br_{\xi}\simeq 10^{-6}-10^{-3}$
produces the observed relic abundance. Summarizing the above results, it can
be said that for the choice of $\omega_{12}=10^{-1}+10^{-1}i$, the value of
$K_{1}$ falls in the weak washout region which further is successful in
generating the desired BAU, relic abundance and also the small structure
formation for both NH and IH. Thus, the model can explain the neutrino mass,
leptogenesis and dark matter for the choice of the free parameters considered
and the values of model parameters obtained.

##  VIII Acknowledgement

The research work of MKD is supported by the Department of Science and
Technology, Government of India, under the project grant EMR/2017/001436. LS
and BBB would like to acknowledge Tezpur University institutional grant and
Research and Innovation grant DoRD/RIG/10-73/ 1592-A for funding their
research work.

## References

  * (1) P. de Salas, D. Forero, C. Ternes, M. Tortola, and J. Valle, “Status of neutrino oscillations 2018: 3$\sigma$ hint for normal mass ordering and improved CP sensitivity,” Phys. Lett. B 782 (2018) 633–640, arXiv:1708.01186 [hep-ph].
  * (2) G. Bertone, D. Hooper, and J. Silk, “Particle dark matter: Evidence, candidates and constraints,” Phys. Rept. 405 (2005) 279–390, arXiv:hep-ph/0404175.
  * (3) B. Moore, S. Ghigna, F. Governato, G. Lake, T. R. Quinn, J. Stadel, and P. Tozzi, “Dark matter substructure within galactic halos,” Astrophys. J. Lett. 524 (1999) L19–L22, arXiv:astro-ph/9907411.
  * (4) M. Fukugita and T. , “Baryogenesis Without Grand Unification,” Phys. Lett. B 174 (1986) 45–47.
  * (5) T. Hugle, M. Platscher, and K. Schmitz, “Low-Scale Leptogenesis in the Scotogenic Neutrino Mass Model,” Phys. Rev. D 98 no. 2, (2018) 023020, arXiv:1804.09660 [hep-ph].
  * (6) Planck Collaboration, M. Lattanzi, “Planck 2015 constraints on neutrino physics,” J. Phys. Conf. Ser. 718 no. 3, (2016) 032008.
  * (7) Super-Kamiokande Collaboration, Y. Fukuda et al., “Evidence for oscillation of atmospheric neutrinos,” Phys. Rev. Lett. 81 (1998) 1562–1567, arXiv:hep-ex/9807003.
  * (8) SNO Collaboration, Q. R. Ahmad et al., “Direct evidence for neutrino flavor transformation from neutral current interactions in the Sudbury Neutrino Observatory,” Phys. Rev. Lett. 89 (2002) 011301, arXiv:nucl-ex/0204008.
  * (9) MINOS Collaboration, P. Adamson et al., “Improved search for muon-neutrino to electron-neutrino oscillations in MINOS,” Phys. Rev. Lett. 107 (2011) 181802, arXiv:1108.0015 [hep-ex].
  * (10) RENO Collaboration, J. K. Ahn et al., “Observation of Reactor Electron Antineutrino Disappearance in the RENO Experiment,” Phys. Rev. Lett. 108 (2012) 191802, arXiv:1204.0626 [hep-ex].
  * (11) T2K Collaboration, K. Abe et al., “Indication of Electron Neutrino Appearance from an Accelerator-produced Off-axis Muon Neutrino Beam,” Phys. Rev. Lett. 107 (2011) 041801, arXiv:1106.2822 [hep-ex].
  * (12) Double Chooz Collaboration, Y. Abe et al., “Indication of Reactor $\bar{\nu}_{e}$ Disappearance in the Double Chooz Experiment,” Phys. Rev. Lett. 108 (2012) 131801, arXiv:1112.6353 [hep-ex].
  * (13) P. Minkowski, “$\mu\to e\gamma$ at a Rate of One Out of $10^{9}$ Muon Decays?,” Phys. Lett. B 67 (1977) 421–428.
  * (14) R. N. Mohapatra and G. Senjanovic, “Neutrino Mass and Spontaneous Parity Nonconservation,” Phys. Rev. Lett. 44 (1980) 912.
  * (15) M. Fukugita and T. Yanagida, “Baryogenesis Without Grand Unification,” Phys. Lett. B 174 (1986) 45–47.
  * (16) F. Vissani, “Do experiments suggest a hierarchy problem?,” Phys. Rev. D 57 (1998) 7027–7030, arXiv:hep-ph/9709409.
  * (17) A. Pilaftsis and T. E. J. Underwood, “Resonant leptogenesis,” Nucl. Phys. B 692 (2004) 303–345, arXiv:hep-ph/0309342.
  * (18) E. K. Akhmedov, V. A. Rubakov, and A. Y. Smirnov, “Baryogenesis via neutrino oscillations,” Phys. Rev. Lett. 81 (1998) 1359–1362, arXiv:hep-ph/9803255.
  * (19) T. Asaka and M. Shaposhnikov, “The $\nu$MSM, dark matter and baryon asymmetry of the universe,” Phys. Lett. B 620 (2005) 17–26, arXiv:hep-ph/0505013.
  * (20) T. Hambye and D. Teresi, “Higgs doublet decay as the origin of the baryon asymmetry,” Phys. Rev. Lett. 117 no. 9, (2016) 091801, arXiv:1606.00017 [hep-ph].
  * (21) T. Hambye and D. Teresi, “Baryogenesis from L-violating Higgs-doublet decay in the density-matrix formalism,” Phys. Rev. D 96 no. 1, (2017) 015031, arXiv:1705.00016 [hep-ph].
  * (22) E. Ma, “Naturally small seesaw neutrino mass with no new physics beyond the TeV scale,” Phys. Rev. Lett. 86 (2001) 2502–2504, arXiv:hep-ph/0011121.
  * (23) E. Ma, “Verifiable radiative seesaw mechanism of neutrino mass and dark matter,” Phys. Rev. D 73 (2006) 077301, arXiv:hep-ph/0601225.
  * (24) L. Sarma, P. Das, and M. K. Das, “Scalar dark matter and leptogenesis in the minimal scotogenic model,” arXiv:2004.13762 [hep-ph].
  * (25) D. Borah, P. B. Dev, and A. Kumar, “TeV scale leptogenesis, inflaton dark matter and neutrino mass in a scotogenic model,” Phys. Rev. D 99 no. 5, (2019) 055012, arXiv:1810.03645 [hep-ph].
  * (26) V. C. Rubin and W. K. Ford, Jr., “Rotation of the Andromeda Nebula from a Spectroscopic Survey of Emission Regions,” Astrophys. J. 159 (1970) 379–403.
  * (27) R. Durrer, “The cosmic microwave background: the history of its experimental investigation and its significance for cosmology,” Class. Quant. Grav. 32 no. 12, (2015) 124007, arXiv:1506.01907 [astro-ph.CO].
  * (28) F. Zwicky, “Die Rotverschiebung von extragalaktischen Nebeln,” Helv. Phys. Acta 6 (1933) 110–127. [Gen. Rel. Grav.41,207(2009)].
  * (29) T. Treu, P. Marshall, and D. Clowe, “Resource Letter: Gravitational Lensing,” Am. J. Phys. 80 (2012) 753, arXiv:1206.0791 [astro-ph.CO].
  * (30) Planck Collaboration, P. Ade et al., “Planck 2015 results. XXIV. Cosmology from Sunyaev-Zeldovich cluster counts,” Astron. Astrophys. 594 (2016) A24, arXiv:1502.01597 [astro-ph.CO].
  * (31) A. Falkowski, E. Kuflik, N. Levi, and T. Volansky, “Light Dark Matter from Leptogenesis,” Phys. Rev. D 99 no. 1, (2019) 015022, arXiv:1712.07652 [hep-ph].
  * (32) S. Dodelson and L. M. Widrow, “Sterile-neutrinos as dark matter,” Phys. Rev. Lett. 72 (1994) 17–20, arXiv:hep-ph/9303287.
  * (33) M. Drewes et al., “A White Paper on keV Sterile Neutrino Dark Matter,” JCAP 01 (2017) 025, arXiv:1602.04816 [hep-ph].
  * (34) A. Boyarsky, M. Drewes, T. Lasserre, S. Mertens, and O. Ruchayskiy, “Sterile neutrino Dark Matter,” Prog. Part. Nucl. Phys. 104 (2019) 1–45, arXiv:1807.07938 [hep-ph].
  * (35) M. Chianese, B. Fu, and S. F. King, “Minimal Seesaw extension for Neutrino Mass and Mixing, Leptogenesis and Dark Matter: FIMPzillas through the Right-Handed Neutrino Portal,” JCAP 03 (2020) 030, arXiv:1910.12916 [hep-ph].
  * (36) XENON Collaboration, E. Aprile et al., “First Dark Matter Search Results from the XENON1T Experiment,” Phys. Rev. Lett. 119 no. 18, (2017) 181301, arXiv:1705.06655 [astro-ph.CO].
  * (37) Fermi-LAT Collaboration, M. Ackermann et al., “Searching for Dark Matter Annihilation from Milky Way Dwarf Spheroidal Galaxies with Six Years of Fermi Large Area Telescope Data,” Phys. Rev. Lett. 115 no. 23, (2015) 231301, arXiv:1503.02641 [astro-ph.HE].
  * (38) S. Weinberg, “Baryon and Lepton Nonconserving Processes,” Phys. Rev. Lett. 43 (1979) 1566–1570.
  * (39) E. Ma, “Pathways to naturally small neutrino masses,” Phys. Rev. Lett. 81 (1998) 1171–1174, arXiv:hep-ph/9805219.
  * (40) A. Liu, Z.-L. Han, Y. Jin, and F.-X. Yang, “Leptogenesis and dark matter from a low scale seesaw mechanism,” Phys. Rev. D 101 no. 9, (2020) 095005, arXiv:2001.04085 [hep-ph].
  * (41) S. M. Davidson and H. E. Logan, “Dirac neutrinos from a second Higgs doublet,” Phys. Rev. D 80 (2009) 095008, arXiv:0906.3335 [hep-ph].
  * (42) M. Dine and A. Kusenko, “The Origin of the matter - antimatter asymmetry,” Rev. Mod. Phys. 76 (2003) 1, arXiv:hep-ph/0303065.
  * (43) S. Davidson and A. Ibarra, “A Lower bound on the right-handed neutrino mass from leptogenesis,” Phys. Lett. B 535 (2002) 25–32, arXiv:hep-ph/0202239.
  * (44) W. Buchmuller, P. Di Bari, and M. Plumacher, “Cosmic microwave background, matter - antimatter asymmetry and neutrino masses,” Nucl. Phys. B 643 (2002) 367–390, arXiv:hep-ph/0205349. [Erratum: Nucl.Phys.B 793, 362 (2008)].
  * (45) J. D. Clarke, R. Foot, and R. R. Volkas, “Natural leptogenesis and neutrino masses with two Higgs doublets,” Phys. Rev. D 92 no. 3, (2015) 033006, arXiv:1505.05744 [hep-ph].
  * (46) N. Haba and O. Seto, “Low scale thermal leptogenesis in neutrinophilic Higgs doublet models,” Prog. Theor. Phys. 125 (2011) 1155–1169, arXiv:1102.2889 [hep-ph].
  * (47) Planck Collaboration, N. Aghanim et al., “Planck 2018 results. VI. Cosmological parameters,” arXiv:1807.06209 [astro-ph.CO].
  * (48) L. J. Hall, K. Jedamzik, J. March-Russell, and S. M. West, “Freeze-In Production of FIMP Dark Matter,” JHEP 03 (2010) 080, arXiv:0911.1120 [hep-ph].
  * (49) Particle Data Group Collaboration, M. Tanabashi et al., “Review of Particle Physics,” Phys. Rev. D 98 no. 3, (2018) 030001.
  * (50) A. Berlin and N. Blinov, “Thermal Dark Matter Below an MeV,” Phys. Rev. Lett. 120 no. 2, (2018) 021801, arXiv:1706.07046 [hep-ph].
  * (51) Planck Collaboration, P. A. R. Ade et al., “Planck 2015 results. XIII. Cosmological parameters,” Astron. Astrophys. 594 (2016) A13, arXiv:1502.01589 [astro-ph.CO].
  * (52) R. Mohapatra, “New Contributions to Neutrinoless Double beta Decay in Supersymmetric Theories,” Phys. Rev. D 34 (1986) 3457–3461.
  * (53) J. Barry and W. Rodejohann, “Lepton number and flavour violation in TeV-scale left-right symmetric theories with large left-right mixing,” JHEP 09 (2013) 153, arXiv:1303.6324 [hep-ph].
  * (54) H. Borgohain and M. K. Das, “Neutrinoless double beta decay and lepton flavour violation in broken $\mu-\tau$ symmetric neutrino mass models,” Int. J. Theor. Phys. 56 no. 9, (2017) 2911–2934, arXiv:1705.00922 [hep-ph].
  * (55) KamLAND-Zen Collaboration, Y. Gando, “Neutrinoless double beta decay search with liquid scintillator experiments,” in Prospects in Neutrino Physics.  4, 2019.  arXiv:1904.06655 [physics.ins-det].
  * (56) KamLAND-Zen Collaboration, A. Li, “A Bayesian Approach to Neutrinoless Double Beta Decay Analysis in KamLAND-Zen,” J. Phys. Conf. Ser. 1468 no. 1, (2020) 012201.
  * (57) KATRIN Collaboration, F. Fraenkle, “The neutrino mass experiment KATRIN,” J. Phys. Conf. Ser. 1342 no. 1, (2020) 012024.
  * (58) KATRIN Collaboration, L. Schlüter and T. Lasserre, “First neutrino mass measurement with the KATRIN experiment,” J. Phys. Conf. Ser. 1468 no. 1, (2020) 012180.
  * (59) GERDA Collaboration, M. Agostini et al., “Final Results of GERDA on the Search for Neutrinoless Double-$\beta$ Decay,” Phys. Rev. Lett. 125 (2020) 252502, arXiv:2009.06079 [nucl-ex].
  * (60) GERDA Collaboration, V. D’Andrea, “Neutrinoless double beta decay search with the GERDA experiment,” Nuovo Cim. C 43 no. 2-3, (2020) 24.