# Heavy Majorana neutrino pair production from $Z^{\prime}$ at hadron and
lepton colliders

Arindam Das<EMAIL_ADDRESS>Institute for the Advancement of
Higher Education, Hokkaido University, Sapporo 060-0817, Japan  Department of
Physics, Hokkaido University, Sapporo 060-0810, Japan  Sanjoy Mandal
<EMAIL_ADDRESS>Korea Institute for Advanced Study, Seoul 02455, Korea
Takaaki Nomura<EMAIL_ADDRESS>College of Physics, Sichuan University,
Chengdu 610065, China  Sujay Shil<EMAIL_ADDRESS>Institute of Physics,
Sachivalaya Marg, Bhubaneswar 751005, India  Homi Bhabha National Institute,
BARC Training School Complex, Anushakti Nagar, Mumbai 400094, India  IIT
Gandhinagar, Palaj Campus, Gujarat 382355, India

###### Abstract

A gauged U$(1)$ extension of the Standard Model (SM) is a simple and anomaly
free framework where three generations of Majorana type right-handed neutrinos
(RHNs) are introduced to generate light neutrino mass and flavor mixings
through the seesaw mechanism. We investigate such models at different hadron
and lepton colliders via $Z^{\prime}$ induced Majorana type RHNs pair
production. We derive bounds on U$(1)$ gauge coupling $(g^{\prime})$ comparing
the model cross sections with experimentally observed data for different
$Z^{\prime}$ mass $(M_{Z^{\prime}})$ and RHNs mass $(M_{N})$. Using these
limits we estimate the allowed RHN pair production cross section which can be
manifested by lepton number violating signatures in association with fat-jets
at the hadron colliders depending on the mass of the RHNs. Hence we study
dilepton and trilepton modes with fat-jet/s of the signal for different
benchmark values of $M_{Z^{\prime}}$ and $M_{N}$. Using fat-jet signatures and
studying the signal and corresponding SM backgrounds, we estimate bounds on
$M_{N}-M_{Z^{\prime}}$ plane at different center of mass energies which could
be probed at different hadron colliders. In the context of the lepton
colliders we consider electron positron initial states where Majorana type
RHNs can be produced from $Z^{\prime}$ manifesting same sign dilepton plus
jets signature and trilepton plus jets in association with missing energy.
Studying the signal and corresponding SM backgrounds we estimate the bounds on
the $M_{N}-M_{Z^{\prime}}$ plane for different center of mass energies. In the
context of U$(1)$ extension of the SM there is an SM singlet BSM scalar which
couples with the RHNs. We can probe the Majorana nature of RHNs via this BSM
scalar production at electron positron colliders.

††preprint: EPHOU-21-016

##  I Introduction

The experimental observations of the SM have set it on a stable foundation,
however, the observations of the light neutrino mass and the flavor mixing [1]
give a strong indication of physics beyond the Standard Model (BSM). This
leads us to extend the SM. From a perspective of the low energy effective
theory, one can introduce a dimension-5 operator [2] involving the Higgs and
lepton doublets within the SM framework which violates lepton number by two
units. After the breaking of electroweak (EW) symmetry, the neutrinos acquire
tiny Majorana masses which are suppressed by the scale of the dimension-5
operator. In the context of a renormalizable theory, the dimension-5 operator
can be naturally generated by introducing SM singlet heavy Majorana RHNs. This
is the well known type-I seesaw mechanism [3, 4, 5, 6, 7, 8, 4].

RHNs in the TeV scale or lighter can be produced at the Large Hadron Collider
(LHC) or other hadron colliders with a distinctive same-sign di-lepton plus
jets final state which manifest the lepton number violation. The RHNs are SM-
singlet therefore they can only be produced at the colliders through the
mixings with the light neutrinos. The estimated light-heavy neutrino mixing
becomes naturally small ($\sim 10^{-6}$) when the TeV scale RHNs reproduce the
observed light neutrino mass around $0.1$ eV. This mixing parameter can be
comparatively large when the Dirac mass matrix is generally parametrized [9]
in order to satisfy the neutrino oscillation data, electroweak precision
measurements and the lepton flavor violating processes [10]. The study of the
heavy neutrinos have been a point of interest for a long period of time at
different high energy colliders from a variety of production modes to study
different final states to estimate bounds on light-heavy neutrino mixing as a
function of heavy neutrino mass [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21,
22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40,
41, 42, 43, 44, 45, 46, 47, 48, 49, 50].

Apart from the canonical seesaw scenario there is an interesting aspect,
namely B$-$L, which incorporates the seesaw mechanism for the neutrino mass
generation mechanism [51, 52, 53, 54, 55, 56, 57, 58, 59, 60] after the global
B$-$L symmetry breaking. The U$(1)$ extension introduces a neutral BSM gauge
boson, $Z^{\prime}$ which acquires mass after the B$-$L symmetry breaking. In
addition to the RHN productions through the light-heavy neutrino mixing [61,
62, 63, 64, 65, 66], this model provides a new mechanism for the RHN
production in pair through $Z^{\prime}$ at the colliders. Once produced these
RHNs can decay into the SM particles through the usual light-heavy neutrino
mixings.

In this paper we consider a general U$(1)_{X}$ scenario which includes three
generations of Majorana type RHNs to cancel the gauge and mixed gauge gravity
anomalies. After the U$(1)_{X}$ symmetry breaking, the light neutrino masses
are generated by the type-I seesaw mechanism. The $U(1)_{X}$ symmetry can be
identified as the linear combination of the $U(1)_{Y}$ in SM and the
$U(1)_{B-L}$ gauge groups, hence the $U(1)_{X}$ scenario is the generalization
of the $U(1)_{B-L}$ extension of the SM. A suitable choice of the U$(1)_{X}$
charges can even enhance the RHN pair production cross section from the
$Z^{\prime}$ compare to $U(1)_{B-L}$ [67, 68, 69], which can further increase
the discovery potential of the RHNs. In addition to that, there is an
alternative $U(1)_{X}$ scenario where two of the RHNs have U$(1)_{X}$ charge
as $-4$ and the third one has the U$(1)_{X}$ charge as $+5$ [70]. Note that,
although the RHNs production is light-heavy neutrino mixing independent, RHN
decays to SM final sates such as $\ell W,\nu Z$ and $\nu h$ through the
mixing. For small enough values of the mixing, the heavy neutrinos can be
long-lived, leading to displaced decays. The displaced decays of the RHNs in
this model have been studied in [71, 72].

If the $Z^{\prime}$ and Majorana type RHN masses reside in TeV scale, they can
be produced at high energy colliders. The U$(1)_{X}$ coupling can be
constrained from the existing and prospective experimental results. Using the
allowed parameters, pair production of RHNs can be possible from $Z^{\prime}$
at hadron colliders. Subsequently, each of the RHNs will dominantly decay into
a charged lepton and SM $W$ boson. The $W$ can decay dominantly into quarks
$(q\overline{q^{\prime}})$ or sub-dominantly into leptons $(\ell\nu_{\ell})$.
Depending on the mass of the RHNs, the $W$ boson produced can be sufficiently
boosted so that the hadrons produced from the $W$ can form a fat-jet. Hence
from each RHN we obtain $(\ell^{\pm})$ and a fat-jet $(J)$. As a result each
event contains two same sign charged leptons and two fat-jets
$(\ell^{\pm}\ell^{\pm}+2J)$ due to the Majorana nature of the RHNs. On the
other hand there is another possibility where one of the $W$ bosons can decay
leptonically which shows a trilepton plus one fat-jet accompanied by missing
momentum $(\ell^{\pm}\ell^{\pm}\ell^{\mp}+J+p_{T}^{\rm miss})$. In this paper,
analyzing these signals we investigate the discovery potential of the RHNs.
Majorana type RHN can be produced at the electron positron colliders from
$Z^{\prime}$ which can also produce $\ell W$ in pair. Hence same sign dilepton
signature can be produced in association with four jets
$(\ell^{\pm}\ell^{\pm}+4j)$. We can also study the trilepton mode in
association with two jets and missing energy when one of the $W$ from RHN
decays hadronically $(\ell^{\pm}\ell^{\pm}\ell^{\mp}+2j+\rm MET)$. General
U$(1)$ extension of the SM contains SM singlet scalars which has Yukawa
interaction withe Majorana type RHNs. In this article we consider production
of the BSM scalar in $e^{-}e^{+}$ collision by $Z$ association and vector
boson fusion process. The BSM scalar can decay into a pair of RHNs which can
further decay into same sign dilepton mode in association with jets and
missing energy.

We arrange this paper in the following way. In Sec. II we describe the
different gauged $U(1)$ extensions of the the SM and study the constraints on
the U$(1)$ coupling as a function of $M_{Z^{\prime}}$. We give the relation on
neutrino mass, mixing and partial widths in Sec. III. The bounds on
$g^{\prime}-M_{Z^{\prime}}$ plane has been discussed for two cases (general
$U(1)_{X}$ and alternative $U(1)_{X}$) in Sec. IV where we compare with the
existing and prospective limits. We estimate the production cross sections of
the heavy neutrino pair using the experimental bounds and propose
theoretically estimated density plots on $M_{N}-M_{Z^{\prime}}$ plane in Sec.
V at hadron colliders. In this section we choosing different benchmarks values
of $M_{Z^{\prime}}$ and $M_{N}$ to study the same sign dilepton and trilepton
modes in association with fat-jets. Simulating the signal, backgrounds and
applying kinematic cuts we postulate a 2$-\sigma$ exclusion contour on
$M_{N}-M_{Z^{\prime}}$ plane which could be probed at different hadron
colliders. In Sec. VI we study the RHN pair production at the electron
positron collider and study multilpeton modes in association with jets and
missing energy. We study the pair production of the RHNs from the BSM scalar
at the electron positron colliders in Sec VII. Finally we conclude the article
in Sec. VIII.

##  II Gauged U$(1)$ extension of the Standard Model

We consider a gauged U$(1)_{X}$ of the SM where three generations of the RHNs
are introduced to cancel all the gauge and mixed gauge-gravitational
anomalies. The U$(1)$ extension of the SM introduces neutral BSM gauge boson
which directly interact with the RHNs. In this paper we study two type of
$U(1)_{X}$ extensions which are described as below.

###  II.1 Case-I

| $SU(3)_{c}$ | $SU(2)_{L}$ | $U(1)_{Y}$ | $U(1)_{X}$
---|---|---|---|---
$q_{L_{i}}$ | 3 | 2 | $1/6$ | $(1/6)x_{H}+(1/3)x_{\Phi}$
$u_{R_{i}}$ | 3 | 1 | $2/3$ | $(2/3)x_{H}+(1/3)x_{\Phi}$
$d_{R_{i}}$ | 3 | 1 | $-1/3$ | $-(1/3)x_{H}+(1/3)x_{\Phi}$
$\ell_{L_{i}}$ | 1 | 2 | $-1/2$ | $(-1/2)x_{H}-x_{\Phi}$
$e_{R_{i}}$ | 1 | 1 | $-1$ | $-x_{H}-x_{\Phi}$
$N_{R_{i}}$ | 1 | 1 | $0$ | $-x_{\Phi}$
$H$ | 1 | 2 | $-1/2$ | $(-1/2)x_{H}$
$\Phi$ | 1 | 1 | $0$ | $+2x_{\Phi}$
Table 1:  Particle content of the minimal $U(1)_{X}$ model where $i$ is the
generation index.

The minimal particle content of $U(1)_{X}$ extension has been listed in Tab.
1. There are three RHNs ($N_{R_{i}}$) which are introduced to cancel the gauge
and the mixed gauge-gravitational anomalies. In this model we introduce a new,
SM singlet scalar field $\Phi$. This scalar field is introduced to break the
U$(1)_{X}$ gauge symmetry by its vacuum expectation value (VEV) which further
generates the Majorana mass term for the RHNs. The Yukawa Lagrangian for the
RHNs sector is given by

$\displaystyle-\mathcal{L}_{\rm int}\ \supset\
\sum_{i,j=1}^{3}Y_{1}^{ij}\overline{\ell_{L_{i}}}HN_{R_{j}}+\frac{1}{2}\sum_{i,j=1}^{3}Y_{2}^{ij}\overline{N_{R_{i}}^{C}}\Phi
N_{R_{j}}+\rm{H.c.}\,,$ (1)

where $C$ stands for the charge-conjugation. The Higgs potential of this model
is given by

$\displaystyle V\ =\
m_{H}^{2}(H^{\dagger}H)+\lambda_{h}(H^{\dagger}H)^{2}+m_{\Phi}^{2}(\Phi^{\dagger}\Phi)+\lambda_{\Phi}(\Phi^{\dagger}\Phi)^{2}+\lambda^{\prime}(H^{\dagger}H)(\Phi^{\dagger}\Phi)\,.$
(2)

In the limit where $\lambda^{\prime}$ is small we can analyze separately the
Higgs potential for $H$ and $\Phi$ as a good approximation. To break the
electroweak and the $U(1)_{X}$ gauge symmetries we consider the parameters of
the potential for the scalar fields $H$ and $\Phi$ to develop their VEVs

$\displaystyle\langle H\rangle\ =\ \frac{1}{\sqrt{2}}\begin{pmatrix}v_{h}\\\
0\end{pmatrix}\,,\quad{\rm and}\quad\langle\Phi\rangle\ =\
\frac{v_{\Phi}}{\sqrt{2}}\,,$ (3)

at the potential minimum where $v_{h}\simeq 246$ GeV is the electroweak scale
and $v_{\Phi}$ is a free parameter. After the symmetry breaking, the mass term
of the $U(1)_{X}$ gauge boson $(Z^{\prime})$ is generated

$\displaystyle M_{Z^{\prime}}\ $ $\displaystyle=$ $\displaystyle\
g^{\prime}\sqrt{4v_{\Phi}^{2}+\frac{1}{4}x_{H}^{2}v_{h}^{2}}\ \simeq\
2g^{\prime}v_{\Phi}.$ (4)

The process of $U(1)_{X}$ symmetry breaking also induces the Majorana mass
term for the RHNs from the second term of the Eq. 1

$\displaystyle M_{N_{i}}\ $ $\displaystyle=$ $\displaystyle\
\frac{Y^{i}_{2}}{\sqrt{2}}v_{\Phi},$ (5)

and followed by the electroweak symmetry breaking the neutrino Dirac mass term
is generated

$\displaystyle M_{D}^{ij}\ $ $\displaystyle=$ $\displaystyle\
\frac{Y_{1}^{ij}}{\sqrt{2}}v_{h}.$ (6)

In this model $x_{H}$ and $x_{\Phi}$ are real parameters and U$(1)_{X}$
coupling $g^{\prime}$ is a free parameter. Without the loss of generality we
consider the basis where $Y_{2}$ is a diagonal matrix. With the Majorana and
Dirac neutrino mass terms in Eqs. 5 and 6 respectively the seesaw mechanism
becomes accountable for the generation of the tiny Majorana masses of the
light neutrino mass eigenstates.

###  II.2 Case-II

| $SU(3)_{c}$ | $SU(2)_{L}$ | $U(1)_{Y}$ | $U(1)_{X}$
---|---|---|---|---
$q_{L_{i}}$ | 3 | 2 | $1/6$ | $(1/6)x_{H}+(1/3)$
$u_{R_{i}}$ | 3 | 1 | $2/3$ | $(2/3)x_{H}+(1/3)$
$d_{R_{i}}$ | 3 | 1 | $-1/3$ | $-(1/3)x_{H}+(1/3)$
$\ell_{L_{i}}$ | 1 | 2 | $-1/2$ | $(-1/2)x_{H}-1$
$e_{R_{i}}$ | 1 | 1 | $-1$ | $-x_{H}-1$
$N_{R_{1,2}}$ | 1 | 1 | $0$ | $-4$
$N_{R_{3}}$ | 1 | 1 | $0$ | $+5$
$H_{1}$ | 1 | 2 | $-1/2$ | $(-1/2)x_{H}$
$H_{2}$ | 1 | 2 | $-1/2$ | $(-1/2)x_{H}+3$
$\Phi_{1}$ | 1 | 1 | $0$ | $+8$
$\Phi_{2}$ | 1 | 1 | $0$ | $-10$
$\Phi_{3}$ | 1 | 1 | $0$ | $-3$
Table 2:  Particle content of the alternative $U(1)_{X}$ extension of the SM
where $i$ denotes the generation index.

There is another interesting $U(1)_{X}$ extension of the SM whose minimal
particle content is shown in Tab. 2. We call it an alternative U$(1)_{X}$
scenario. The U$(1)_{X}$ charge $x_{H}$ is a real parameter and the U$(1)_{X}$
coupling $g^{\prime}$ is a free parameter. The RHNs in this model are
differently charged under the U$(1)_{X}$. In this model first two generations
of RHNs have charge $-4$ whereas the third one has a charge $+5$. This non-
universal charge assignment is a unique choice in order to cancel all the
anomalies [73]. In this model we introduce two Higgs doublets $(H_{1},H_{2})$
and three additional SM-singlet scalars $(\Phi_{1,2,3})$. The Higgs doublet
$H_{2}$ is responsible for the generation of the Dirac mass term for
$N_{R_{1,2}}$. The SM-singlet scalar $\Phi_{1}$ is responsible for the
generation of the Majorana mass term of $N_{R_{1,2}}$ after the U$(1)_{X}$
breaking. The Majorana mass term of $N_{R_{3}}$ is generated from the VEV of
$\Phi_{2}$, however, there is no Dirac mass term for $N_{R_{3}}$ due to the
preservation of U$(1)_{X}$ symmetry. Hence $N_{R_{3}}$ does not participate in
the neutrino mass generation mechanism. The relevant part of the interaction
Lagrangian of the RHNs is given by

$\displaystyle-\mathcal{L}_{\rm int}$ $\displaystyle\ \supset\ $
$\displaystyle\sum_{i=1}^{3}\sum_{j=1}^{2}Y_{1}^{ij}\overline{\ell_{L_{i}}}H_{2}N_{R_{j}}+\frac{1}{2}\sum_{k=1}^{2}Y_{2}^{k}\overline{N_{R_{k}}^{C}}\Phi_{1}N_{R_{k}}+\frac{1}{2}Y_{3}\overline{N_{R_{3}}^{C}}\Phi_{2}N_{R_{3}}+\rm{H.c.}\,,$
(7)

where we have assumed a basis in which $Y_{2}$ is diagonal, without the loss
of generality. The scalar potential is given by

$\displaystyle V$ $\displaystyle\ =\ $ $\displaystyle
m_{H_{1}}^{2}(H_{1}^{\dagger}H_{1})+\lambda_{H_{1}}(H_{1}^{\dagger}H_{1})^{2}+m_{H_{2}}^{2}(H_{2}^{\dagger}H_{2})+\lambda_{H_{2}}(H_{2}^{\dagger}H_{2})^{2}$
(8)
$\displaystyle+m_{\Phi_{1}}^{2}(\Phi_{1}^{\dagger}\Phi_{1})+\lambda_{1}(\Phi_{1}^{\dagger}\Phi_{1})^{2}+m_{\Phi_{2}}^{2}(\Phi_{2}^{\dagger}\Phi_{2})+\lambda_{2}(\Phi_{2}^{\dagger}\Phi_{2})^{2}$
$\displaystyle+m_{\Phi_{3}}^{2}(\Phi_{3}^{\dagger}\Phi_{3})+\lambda_{3}(\Phi_{3}^{\dagger}\Phi_{3})^{2}+(\mu\Phi_{3}(H_{1}^{\dagger}H_{2})+{\rm
H.c.})$
$\displaystyle+\lambda_{4}(H_{1}^{\dagger}H_{1})(H_{2}^{\dagger}H_{2})+\lambda_{5}(H_{1}^{\dagger}H_{2})(H_{2}^{\dagger}H_{1})+\lambda_{6}(H_{1}^{\dagger}H_{1})(\Phi_{1}^{\dagger}\Phi_{1})$
$\displaystyle+\lambda_{7}(H_{1}^{\dagger}H_{1})(\Phi_{2}^{\dagger}\Phi_{2})+\lambda_{8}(H_{1}^{\dagger}H_{2})(\Phi_{3}^{\dagger}\Phi_{3})+\lambda_{9}(H_{2}^{\dagger}H_{2})(\Phi_{1}^{\dagger}\Phi_{1})$
$\displaystyle+\lambda_{10}(H_{1}^{\dagger}H_{1})(\Phi_{2}^{\dagger}\Phi_{2})+\lambda_{11}(H_{1}^{\dagger}H_{2})(\Phi_{3}^{\dagger}\Phi_{3})+\lambda_{12}(\Phi_{1}^{\dagger}\Phi_{1})(\Phi_{2}^{\dagger}\Phi_{2})$
$\displaystyle+\lambda_{13}(\Phi_{2}^{\dagger}\Phi_{2})(\Phi_{3}^{\dagger}\Phi_{3})+\lambda_{14}(\Phi_{3}^{\dagger}\Phi_{3})(\Phi_{1}^{\dagger}\Phi_{1}).$

We choose suitable parameters for the Higgs fields to develop their respective
VEVs:

$\displaystyle\langle H_{1}\rangle\ =\
\frac{1}{\sqrt{2}}\left(\begin{array}[]{c}v_{h_{1}}\\\
0\end{array}\right),\;\langle H_{2}\rangle\ =\
\frac{1}{\sqrt{2}}\left(\begin{array}[]{c}v_{h_{2}}\\\
0\end{array}\right),\langle\Phi_{1}\rangle\ =\
\frac{v_{1}}{\sqrt{2}},\;\langle\Phi_{2}\rangle\ =\
\frac{v_{2}}{\sqrt{2}},\;\langle\Phi_{3}\rangle\ =\
\frac{v_{3}}{\sqrt{2}},~{}~{}~{}~{}$ (13)

with the condition, $v_{h_{1}}^{2}+v_{h_{2}}^{2}=(246\,{\rm GeV})^{2}$. We
consider negligibly small mixed-quartic couplings between the Higgs doublets
and the SM singlets for simplicity, so that the Higgs singlet sector is
effectively separated from the Higgs doublets. It ensures that any higher-
order mixing effect between the three generations of the RHNs after $U(1)_{X}$
symmetry-breaking will be extremely suppressed. The singlet and doublet Higgs
sectors communicate only through the triple coupling
$\Phi_{3}(H_{1}^{\dagger}H_{2})+{\rm H.c.}$ Taking the collider constraints
into account $v_{1}^{2}+v_{2}^{2}+v_{3}^{2}\gg v_{h_{1}}^{2}+v_{h_{2}}^{2}$,
the triple coupling has no significant effect on determining the VEVs
$(v_{1,2,3})$ of the SM-singlet scalars $(\Phi_{1},\Phi_{2},\Phi_{3})$ when we
arrange the parameters in the scalar potential to have the VEVs of the SM-
singlet scalars almost same $(v_{1}\sim v_{2}\sim v_{3})$ and $\mu<v_{1}$. The
third SM-singlet scalar, $\Phi_{3}$, can be used as a spurion for the Higgs
doublet sector which generates the mixing between $H_{1}$ and $H_{2}$ through
the term $\mu\Phi_{3}(H_{1}^{\dagger}H_{2})+{\rm H.c.}$ Using
$\langle\Phi_{3}\rangle=\frac{v_{3}}{\sqrt{2}}$ the mixing mass term becomes
$m_{\rm{mix}}^{2}=\frac{\mu v_{3}}{\sqrt{2}}$. Hence the Higgs doublet sector
potential effectively becomes the Higgs potential of the two Higgs doublet
model. There is no mixing mass term among $\Phi_{1,2,3}$ due to the U$(1)_{X}$
symmetry. As a result there are two physical Nambu-Goldstone (NG) modes
present in our model. These NG modes are originated from the SM singlet
scalars and they are not phenomenologically dangerous. We consider the SM-
singlet scalars heavier than $Z^{\prime}$, so that $Z^{\prime}$ cannot decay
into the NG modes. After the $U(1)_{X}$ symmetry is broken the $Z^{\prime}$
boson acquires the mass term as

$\displaystyle
M_{Z^{\prime}}=g^{\prime}\sqrt{64v_{1}^{2}+100v_{2}^{2}+9v_{3}^{2}+\frac{1}{4}x_{H}^{2}v_{h_{1}}^{2}+\left(-\frac{1}{2}x_{H}+3\right)^{2}v_{h_{2}}^{2}}\simeq
g^{\prime}\sqrt{64v_{1}^{2}+100v_{2}^{2}+9v_{3}^{2}}~{}~{}~{}~{}~{}~{}~{}.$
(14)

and the Majorana masses of the RHNs are generated as

$\displaystyle
M_{N_{1,2}}=\frac{Y_{2}^{1,2}}{\sqrt{2}}v_{1},\;\;M_{N_{3}}=\frac{Y_{3}^{3}}{\sqrt{2}}v_{2}$
(15)

using the collider constraints to set
$(v_{1}^{2}+v_{2}^{2}+v_{3}^{2})\gg(v_{h_{1}}^{2}+v_{h_{2}}^{2})$. The Dirac
mass terms of the neutrinos are generated by $\langle H_{2}\rangle$:

$\displaystyle M_{D}^{ij}\ =\ \frac{Y_{1}^{ij}}{\sqrt{2}}\,v_{h_{2}}\,,$ (16)

after which the seesaw mechanism is implemented. Because of the $U(1)_{X}$
charges, only two RHNs ($N_{R_{1,2}}$) are involved in the minimal seesaw
mechanism [74, 75, 76, 77] while the third RHN ($N_{R_{3}}$) has no direct
interaction with the SM sector. Hence it can be a potential Dark Matter (DM)
candidate. Due to the $U(1)_{X}$ symmetry the Higgs doublet $H_{1}$ has no
coupling with the RHNs and the neutrino Dirac masses are generated by the VEV
of $H_{2}$ as mentioned in Eq. 16. This structure can be considered as a type
of the neutrinophilic two Higgs Doublet Model (2HDM) [78, 79, 80, 81, 82]. In
Eq. (8) we may consider $0<m_{\rm mix}^{2}=\frac{\mu v_{3}}{\sqrt{2}}\ll
m_{\Phi_{3}}^{2}$ which leads to $v_{h_{2}}\sim m_{\rm
mix}^{2}v_{h_{1}}/m_{\Phi_{3}}^{2}\ll v_{h_{1}}$ [78].

##  III Heavy neutrino interactions

After the breaking of the electroweak and $U(1)_{X}$ symmetry, the neutrino
mass matrix can be generated as

$\displaystyle m_{\nu}\ =\ \begin{pmatrix}0&&M_{D}\\\
M_{D}^{T}&&M_{N}\end{pmatrix}.$ (17)

without the loss of generality we consider that $M_{N}$ is a diagonal matrix.
The eigenvalues of this mass matrix $M_{N}$ have been written in the Eq. 5 for
Case-I and in the Eq. 15 for Case-II respectively. Similarly the Dirac mass
matrices for the Case-I and Case-II are written in Eqs. 6 and 16 respectively.
Diagonalizing Eq. 17 we get the light neutrino mass eigenvalue as
$m_{\nu}\simeq-M_{D}M_{N}^{-1}M_{D}^{T}$ which allows us to express the light
neutrino flavor eigenstate $(\nu_{\alpha})$ in terms of the light $(\nu_{i})$
and heavy $(N_{i})$ mass eigenstates

$\displaystyle\nu_{\alpha}\simeq U_{\alpha i}\nu_{i}+V_{\alpha i}N_{i},$ (18)

where $\alpha$ and $i$ are the generation indices, $U_{\alpha i}$ is the
$3\times 3$ light neutrino mixing matrix and can be written as $U_{\alpha
i}=(1-\frac{\epsilon}{2})U_{\rm PMNS}$ with $\epsilon=V^{\ast}V^{T}$, the non-
unitary parameter. Here

$\displaystyle V_{\alpha i}\simeq M_{D}M_{N}^{-1}$ (19)

is the mixing between the SM neutrinos and the heavy neutrinos which is
assumed to be much less than $1$. Hence the SM gauge singlet heavy neutrinos
interact with the $W$ and $Z$ bosons of the SM via mixing. $U_{\rm PMNS}$ is
the $3\times 3$ light neutrino mass matrix which diagonalizes the light
neutrino mass matrix as

$\displaystyle U_{\rm PMNS}^{T}~{}m_{\nu}~{}U_{\rm
PMNS}=diag(m_{1},m_{2},m_{3})$ (20)

in the presence of $\epsilon$ the mixing matrix $U$ is non-unitary. For the
Case-I, three generations of the RHNs are involved the seesaw mechanism,
however, in Case-II, only two generations are involved in the seesaw
mechanism. For simplicity, we assume that the heavy neutrinos are degenerate
in mass. Hence the light neutrino mass matrix can be given by [9]

$\displaystyle
m_{\nu}=\frac{1}{M_{N}}M_{D}M_{D}^{T}=U_{\rm{PMNS}}^{\ast}D_{\rm{NH/IH}}U_{\rm{PMNS}}^{\dagger},$
(21)

for the Normal Hierarchy (NH)/ Inverted Hierarchy (IH) cases of the light
neutrino mass eigenvalues.

The charged-current interactions can be expressed in terms of the neutrino
mass eigenstates

$\displaystyle{\mathcal{L}_{CC}\supset-\frac{g}{\sqrt{2}}W_{\mu}\bar{e}\gamma^{\mu}P_{L}V_{\alpha
i}N_{i}+\rm{h.c}.},$ (22)

where $e$ represents the three generations of the charged leptons, and
$P_{L}=\frac{1}{2}(1-\gamma_{5})$ is the projection operator. Similarly, in
terms of the mass eigenstates, the neutral-current interactions are written as

$\displaystyle{\mathcal{L}_{NC}\supset-\frac{g}{2c_{w}}Z_{\mu}\left[\overline{N}_{m}\gamma^{\mu}P_{L}(V^{\dagger}V)_{mi}N_{i}+\left\\{\overline{\nu}_{m}\gamma^{\mu}P_{L}(U^{\dagger}V)_{mi}N_{i}+\rm{h.c.}\right\\}\right],}$
(23)

where $c_{w}\equiv\cos\theta_{w}$ with $\theta_{w}$ being the weak mixing
angle.

From the Eqs. 22 and 23 we see that the heavy neutrinos $(N)$ decay into $\ell
W$, $\nu Z$ and $\nu h$ respectively. Here $h$ is the SM Higgs boson. For
sterile neutrinos heavier than $W$, $Z$ and $h$, the decays are on-shell,
i.e., tow body followed by the further decays of the SM bosons otherwise there
will be three-body decays of the heavy neutrinos, with the off-shell SM
bosons. For simplicity we assume that the $U(1)_{X}$ Higgs bosons in Cases-I
and II are heavier than the RHNs. In these models we consider heavy
$Z^{\prime}$ in the TeV scale. As a result, RHNs decay through the off-shell
$Z^{\prime}$ will be negligibly small due to the $Z^{\prime}$ mass
suppression. The coupling between the $Z^{\prime}$, light neutrino and RHN
arises after the $U(1)_{X}$ breaking, however, this is proportional to the
light-heavy mixing and suppressed. Therefore we neglect all the $Z^{\prime}$
mediated off-shell decays of the RHNs into the SM fermions. The allowed decay
modes of the RHNs are written in the following. When we consider the RHNs are
heavier than the SM bosons so that they can decay into $\ell W$,
$\nu_{\ell}Z$, and $\nu_{\ell}h$ on-shell modes. The corresponding partial
decay widths are

$\displaystyle\Gamma(N_{i}\rightarrow\ell_{\alpha}W)$ $\displaystyle=$
$\displaystyle\frac{|V_{\alpha
i}|^{2}}{16\pi}\frac{(M_{N_{i}}^{2}-M_{W}^{2})^{2}(M_{N_{i}}^{2}+2M_{W}^{2})}{M_{N_{i}}^{3}v_{h}^{2}},$
$\displaystyle\Gamma(N_{i}\rightarrow\nu_{\alpha}Z)$ $\displaystyle=$
$\displaystyle\frac{|V_{\alpha
i}|^{2}}{32\pi}\frac{(M_{N_{i}}^{2}-M_{Z}^{2})^{2}(M_{N_{i}}^{2}+2M_{Z}^{2})}{M_{N_{i}}^{3}v_{h}^{2}},$
$\displaystyle\Gamma(N_{i}\rightarrow\nu_{\alpha}h)$ $\displaystyle=$
$\displaystyle\frac{|V_{\alpha
i}|^{2}}{32\pi}\frac{(M_{N_{i}}^{2}-M_{h}^{2})^{2}}{M_{N_{i}}v_{h}^{2}}.$ (24)

When RHNs are lighter than the SM bosons they decay into three body modes
where partial decay widths of $N_{i}$ are approximately given by

$\displaystyle\Gamma(N_{i}\to\ell_{\alpha}^{-}\ell_{\beta}^{+}\nu_{\ell_{\beta}})$
$\displaystyle=\Gamma(N_{i}\to\ell_{\alpha}^{+}\ell_{\beta}^{-}\bar{\nu}_{\ell_{\beta}})\simeq|V_{\alpha
i}|^{2}\frac{G_{F}^{2}}{192\pi^{3}}M_{N_{i}}^{5}\quad(\alpha\neq\beta),$ (25)
$\displaystyle\Gamma(N_{i}\to\ell_{\beta}^{-}\ell_{\beta}^{+}\nu_{\ell_{\alpha}})$
$\displaystyle=\Gamma(N_{i}\to\ell_{\beta}^{+}\ell_{\beta}^{-}\bar{\nu}_{\ell_{\alpha}})$
(26) $\displaystyle\simeq|V_{\alpha
i}|^{2}\frac{G_{F}^{2}}{192\pi^{3}}M_{N_{i}}^{5}\left(\frac{1}{4}\cos^{2}2\theta_{W}+\sin^{4}\theta_{W}\right)\quad(\alpha\neq\beta),$
$\displaystyle\Gamma(N_{i}\to\ell_{\alpha}^{-}\ell_{\alpha}^{+}\nu_{\ell_{\alpha}})$
$\displaystyle=\Gamma(N_{i}\to\ell_{\alpha}^{+}\ell_{\alpha}^{-}\bar{\nu}_{\ell_{\alpha}})$
(27) $\displaystyle\simeq|V_{\alpha
i}|^{2}\frac{G_{F}^{2}}{192\pi^{3}}M_{N_{i}}^{5}\left(\frac{1}{4}\cos^{2}2\theta_{W}+\cos
2\theta_{W}+\sin^{4}\theta_{W}\right),$
$\displaystyle\Gamma(N_{i}\to\nu_{\beta}\bar{\nu}_{\beta}\nu_{\ell_{\alpha}})$
$\displaystyle=\Gamma(N_{i}\to\nu_{\beta}\bar{\nu}_{\beta}\bar{\nu}_{\ell_{\alpha}})\simeq|V_{\alpha
i}|^{2}\frac{1}{4}\frac{G_{F}^{2}}{192\pi^{3}}M_{N_{i}}^{5},$ (28)
$\displaystyle\Gamma(N_{i}\to\ell_{\alpha}^{-}q_{a}\bar{q}_{b})$
$\displaystyle=\Gamma(N_{i}\to\ell_{\alpha}^{+}\bar{q}_{a}q_{b})\simeq
N_{c}|V_{\alpha
i}|^{2}|V_{CKM}^{ab}|^{2}\frac{G_{F}^{2}}{192\pi^{3}}M_{N_{i}}^{5},$ (29)
$\displaystyle\Gamma(N_{i}\to q_{a}\bar{q}_{a}\nu_{\ell_{\alpha}})$
$\displaystyle=\Gamma(N_{i}\to
q_{a}\bar{q}_{a}\bar{\nu}_{\ell_{\alpha}})\simeq N_{c}|V_{\alpha
i}|^{2}\frac{G_{F}^{2}}{192\pi^{3}}M_{N_{i}}^{5}2\left(|g_{V}^{q}|^{2}+|g_{A}^{q}|^{2}\right),$
(30)

where

$\displaystyle
g_{V}^{u}=\frac{1}{2}-\frac{4}{3}\sin^{2}\theta_{W},\,\,g_{A}^{u}=-\frac{1}{2}$
$\displaystyle
g_{V}^{d}=-\frac{1}{2}+\frac{2}{3}\sin^{2}\theta_{W},\,\,g_{A}^{u}=\frac{1}{2}$
(31)

respectively which come from the $Z$ boson interaction with the quarks and
$N_{c}=3$ is the color factor. Eq. 27 is the interference between the $Z$ and
$W$ mediated channels where all flavors are same. We estimate the partial
decay widths in a limit neglecting the SM fermion masses.

##  IV $Z^{\prime}$ production and bounds on the U$(1)_{X}$ gauge coupling

In the U$(1)_{X}$ models, the RHNs have non-zero U$(1)_{X}$ charges like all
SM fermions. It ensures the production of the RHNs through the $Z^{\prime}$ at
the hadron collider. First $Z^{\prime}$ boson is resonantly produced and
subsequently decay into a pair of RHNs, if kinematically allowed. The
production cross section depends on the choice of the parameter which is
consistent with the dilepton [83, 84, 85], dijet [86, 87] searches from the
LHC and constraints obtained from LEP [88, 89, 90, 91]. To estimate the cross
section we calculate the partial decay widths into a pair of SM fermions
neglecting their masses and a pair of Majorna RHNs as

$\displaystyle\Gamma(Z^{\prime}\to\overline{f^{L(R)}}f^{L(R)})$
$\displaystyle=$ $\displaystyle
N_{c}\frac{{g^{\prime}}^{2}}{24\pi}{Q_{f}^{L(R)}}^{2}M_{Z^{\prime}},\,\,\,\,\,\,\Gamma(Z^{\prime}\to
N_{i}N_{i})=\frac{{g^{\prime}}^{2}}{24\pi}{Q_{N}}^{2}M_{Z^{\prime}}\Big{(}1-4\frac{M_{N_{i}}^{2}}{M_{Z^{\prime}}^{2}}\Big{)}^{\frac{3}{2}}~{}~{}~{}~{}~{}~{}$
(32)

respectively where $N_{c}=3(1)$ for SM quarks (leptons), $Q_{f_{L}(R)}$ are
the U$(1)_{X}$ charges of the left (right) handed SM fermions and $Q_{N}$ is
the U$(1)_{X}$ charge of the heavy neutrino.

Figure 1: The branching ratio of the $Z^{\prime}$ (top panel) into the charged
lepton (red), quarks (blue) and RHNs (green) for Case-I (II) in the left
(right) panel as a function of $x_{H}$. The quantity
$\frac{\Gamma(Z^{\prime}\to NN)}{\Gamma(Z^{\prime}\to\ell^{-}\ell^{+})}$
(bottom panel) shows the enhancement in the RHN decay over the dilepton for
Case-I (II) in the left (right) panel. We have considered $M_{Z^{\prime}}=3$
TeV (see text).

The branching ratios of $Z^{\prime}$ into a pair of charged fermions and the
heavy Majorana neutrinos for Case-I (II) are given in the left (right) side of
the top panel of Fig. 1 for $M_{Z^{\prime}}=3$ TeV as a function of $x_{H}$.
The $Z^{\prime}$ decays into a pair of RHNs when
$M_{N}<\frac{M_{Z^{\prime}}}{2}$. The solid lines correspond to the scenario
when $M_{N_{1,2,3}}=500$ GeV, the dotted line corresponds to the scenario when
$M_{N_{1,2}}=500$ GeV but $M_{N_{3}}>\frac{M_{Z^{\prime}}}{2}$ and the dashed
line corresponds to the scenario when $M_{N_{1}}=500$ GeV but
$M_{N_{2,3}}>\frac{M_{Z^{\prime}}}{2}$ from top to bottom. The lines in blue,
red and green represent the decay of $Z^{\prime}$ into $q\overline{q}$,
$\ell^{-}\ell^{+}$ and RHN pair respectively.

In the bottom panel of Fig. 1 we plot the ratio of the branching ratios of
$Z^{\prime}\to NN$ to $Z^{\prime}\to\ell^{-}\ell^{+}$ which is defined as

$\displaystyle\frac{\Gamma(Z^{\prime}\to
NN)}{\Gamma(Z^{\prime}\to\ell^{-}\ell^{+})}=\frac{Q_{N}^{2}}{2+3x_{H}+\frac{5}{4}x_{H}^{2}}\Big{(}1-4\frac{M_{N}^{2}}{M_{Z^{\prime}}^{2}}\Big{)}^{\frac{3}{2}}$
(33)

where $Q_{N}$ is the U$(1)_{X}$ charge of the RHNs under consideration from
the Cases-I and II respectively. The blue, green and red curves represent the
cases $M_{N_{1,2,3}}=500$ GeV, $M_{N_{1,2}}=500$ GeV but
$M_{N_{3}}>\frac{M_{Z^{\prime}}}{2}$ and $M_{N_{1}}=500$ GeV but
$M_{N_{2,3}}>\frac{M_{Z^{\prime}}}{2}$ respectively. In this analysis we have
considered the benchmarks as $M_{Z^{\prime}}=3$ TeV and $M_{N}=500$ GeV when
$M_{N_{i}}<\frac{M_{Z}^{\prime}}{2}$ in the bottom left panel of Fig. 1 for
Case-I. We study the Case-II in the bottom right panel where two generations
of the RHNs $(N_{1},N_{2})$ participate in the neutrino mass generation
mechanism. The dashed green line represents $M_{N_{1,2}}=500$ GeV and the
dotted red line represents the case $M_{N_{1}}=500$ GeV but
$M_{N_{2}}>\frac{M_{Z^{\prime}}}{2}$. We find that $\frac{\Gamma(Z^{\prime}\to
NN)}{\Gamma(Z^{\prime}\to\ell^{-}\ell^{+})}$ becomes maximum at $x_{H}=-1.2$
for $x_{\Phi}=1$.

From the bottom left panel of Fig 1 considering one generation of RHNs in
Case-I, for example with $M_{N_{1}}=500$ GeV, $\Gamma(Z^{\prime}\to NN)\simeq
4.22$ $\Gamma(Z^{\prime}\to\ell^{-}\ell^{+})$ which can be estimated from the
red dotted line. For two generations of RHNs with $M_{N_{1,2}}=500$ GeV the
situation is improved roughly by another factor of $2$ compared to the one
generation case. In case of three generations of RHNs with $M_{N_{1,2,3}}=500$
GeV, $\Gamma(Z^{\prime}\to NN)$ is nearly $13$ times greater than
$\Gamma(Z^{\prime}\to\ell^{-}\ell^{+})$. In Case-I we obtain the so-called
B$-$L scenario for $x_{H}=0$ and $x_{\Phi}=1$ where for $M_{N_{1,2,3}}=500$
GeV $\Gamma(Z^{\prime}\to NN)\simeq\Gamma(Z^{\prime}\to\ell^{-}\ell^{+})$
whereas one generation and two generation cases are suppressed compared to the
dilepton mode. In Case-II, $\Gamma(Z^{\prime}\to NN)$ is greater than
$\Gamma(Z^{\prime}\to\ell^{+}\ell^{-})$ for the choices $M_{N_{1,2}}=500$ GeV
and $M_{N_{1}}=500$ GeV, $M_{N_{2}}>\frac{M_{Z^{\prime}}}{2}$ at $x_{H}=-1.2$.
This property is valid for any RHN mass below $\frac{M_{Z^{\prime}}}{2}$ for
Cases-I and II.

To estimate the bounds on the U$(1)_{X}$ gauge coupling we calculate the
dilepton production cross section from the $Z^{\prime}$ at the $13$ TeV LHC.
We compare our cross section with the observed dilepton cross sections from
the LHC where sequential standard model (SSM) [92] was studied assumming
$\frac{\Gamma}{m}=3\%$. Using the following master equation

$\displaystyle
g^{\prime}=\sqrt{g_{\rm{Model}}^{2}\Big{(}\frac{\sigma_{\rm{ATLAS}}^{\rm{Observed}}}{\sigma_{\rm{Model}}}\Big{)}}$
(34)

we find the bounds on the U$(1)_{X}$ gauge coupling. The $Z^{\prime}$
production cross sections can be calculated using narrow width approximation
as

$\displaystyle\sigma(pp\to Z^{\prime})=2\sum_{q,\overline{q}}\int dx\int
dyq(x,Q)\overline{q}(y,Q)\hat{\sigma}(\hat{s})$ (35)

where $q(x,Q)$ and $\overline{q}(x,Q)$ are the parton distribution functions
of the quark and antiquark respectively and $\hat{s}=xys$ is the invariant
mass squared of the colliding quark at the center of mass energy $\sqrt{s}$.
Due to the narrow width approximation the cross section of the colliding
quarks to produce $Z^{\prime}$ boson is

$\displaystyle\hat{\sigma}=\frac{4\pi^{2}}{3}\frac{\Gamma(Z^{\prime}\to
q\overline{q})}{M_{Z^{\prime}}}\delta(\hat{s}-M_{Z^{\prime}})$ (36)

With a factorization scale set at $Q=M_{Z^{\prime}}$ and employing CTEQ6L
[93], one can estimate $\sigma_{\rm{Model}}=$ $\sigma(pp\to Z^{\prime})$
BR$(Z^{\prime}\to 2\ell)$ or $\sigma(pp\to Z^{\prime})$ BR$(Z^{\prime}\to 2j)$
for dilepton or dijet final states. For the dijet production cross section we
use $58\%$ and $70\%$ acceptance respectively to compare with the LHC results
at the ATLAS and CMS respectively. Finally the bounds are calculated from the
Eq. 34 for Case-I (II) are shown in the left (right) panel of Fig. 2.

To estimate these bounds from the LHC we have considered three scenarios: (i)
when all the RHNs are heavier than $\frac{M_{Z^{\prime}}}{2}$ which are shown
by the dashed lines (Red/ Green/ Blue) to represented the LHC dilepton
(ATLAS2l/ CMS2l/ ATLAS-TDR) bounds. Due to this fact the $Z^{\prime}$ can not
decay into the RHNs. Hence the other decay modes of $Z^{\prime}$ will be
enhanced providing the strongest bound on $g^{\prime}$; (ii) when two
generations of the RHN masses are below $\frac{M_{Z^{\prime}}}{2}$ leading to
the decay of the $Z^{\prime}$ into the kinematically allowed RHN pairs. We
have chosen two benchmark scenarios in this case for at least two generations
of the degenerate RHNs with $M_{N_{1,2}}=500$ GeV which are shown by the
dotted lines (Red/ Green/ Blue) to represent the LHC dilepton (ATLAS2l-2/
CMS2l-2/ ATLAS-TDR-2) bounds; (iii) similar bounds for $M_{N_{1,2}}=1$ TeV are
represented by dot-dashed lines (Red/ Green/ Blue) to represent the LHC
dilepton (ATLAS2l-3/ CMS2l-3/ ATLAS-TDR-3) bounds. The di-jet bounds are shown
by the magenta lines for the case (i). It is important to notice that the
dilepton bound is the strongest one with respect to the di-jet. Therefore we
did not extract the dijet bounds considering the RHN masses at $500$ GeV and
$1$ TeV. The lines with different RHN masses match with the case where RHNs
are heavier than $\frac{M_{Z^{\prime}}}{2}$ for the heavier $M_{Z^{\prime}}$
mass because of factor $(1-4\frac{M_{N}^{2}}{M_{Z^{\prime}}^{2}})$ factor in
the $Z^{\prime}\to NN$ partial mode. It becomes almost $1$ when
$M_{Z^{\prime}}>>M_{N}$. The bounds on the $g^{\prime}-M_{Z^{\prime}}$ plane
for Case-II are shown in the right panel of Fig. 2, the branching ratio of
$Z^{\prime}\to NN$ is more than one order of magnitude higher than the Case-I
due the U$(1)_{X}$ charges. Such a behavior has been reflected in the nature
of the constraints on $g^{\prime}$ for different $M_{Z^{\prime}}$. The bounds
from the LHC are estimated using narrow width approximation at 139(140) $\rm
fb^{-1}$ luminosity at the LHC where the $Z^{\prime}$ production cross section
is proportional to ${g^{\prime}}^{2}$. Hence we naively estimate the future or
prospective upper bound on $g^{\prime}$ scaling the luminosity following
$g^{\prime}\simeq g^{\prime}_{\rm current}\sqrt{\frac{139(140){\rm
fb}^{-1}}{\mathcal{L_{\rm future}{\rm fb^{-1}}}}}$ at a future luminosity of
$\mathcal{L_{\rm future}}$. At the High Luminosity LHC (HL-LHC) of 3000 fb-1
we scale the limits from ATLAS (CMS) considering
$M_{N}>\frac{M_{Z^{\prime}}}{2}$ and obtain that the limits get approximately
0.215 (0.216) times stronger and it is uniformly applicable. Similar scenario
could be obtained for the other choices of the heavy neutrino masses, however,
those bounds will be weaker than the case $M_{N}>\frac{M_{Z^{\prime}}}{2}$.
These prospective limits could be verified in the near future.

Figure 2: Upper bounds on the $g^{\prime}$ vs $M_{Z^{\prime}}$ plane for
Case-I (II) in the left (right) panel with $x_{H}=-1.2$. The bounds are
calculated from the dilepton channels at the ATLAS data [83], CMS [84, 94],
ATLAS-TDR [85]. The bounds from the LEP-II data have been calculated from
[90]. The bounds from the dijet have been estimated from the ATLAS [86] and
CMS [87] respectively. The shaded region is ruled out by the current
experimental data.

LEP-II bounds have been estimated from [88, 89, 90, 91] for $x_{H}=-1.2$ and
has been shown by the solid cyan line in Fig. 2. We estimated bounds on the
$g^{\prime}-M_{Z^{\prime}}$ plane from ILC at $\sqrt{s}=$250 GeV, 500 GeV and
1 TeV respectively from [91] considering $M_{Z^{\prime}}>\sqrt{s}$. Hence from
[91] for $x_{H}=-1.2$ we find, the limit is
$\frac{M_{Z^{\prime}}}{g^{\prime}}>2.68$ TeV from LEP-II and the prospective
limits on this quantity are 13.85 TeV, 23.74 TeV and 41.96 TeV respectively
from the ILC at $\sqrt{s}=$ 250 GeV, 500 GeV and 1 TeV respectively. Hence we
estimate the upper bounds on $g^{\prime}$-$M_{Z^{\prime}}$ plane from
$e^{-}e^{+}$ colliders which are represented by LEP-II (cyan), ILC250 (brown,
dashed), ILC500 (brown, dotted) and ILC1000 (brown, dot-dashed) respectively
in Fig. 2 for Case-I (II) in the left (right) panel. The LEP-II limit is
weaker compared to the current LHC bounds, however, the prospective bounds
obtained from the ILC are stronger than the existing LHC limits for heavier
$Z^{\prime}$. In Tab. 3 we show some allowed benchmark values of
$M_{Z^{\prime}}$, $M_{N}$ and $g^{\prime}$ which will be used in the RHN pair
production from $Z^{\prime}$.

$U(1)_{X}$ coupling | $M_{N}>\frac{M_{Z^{\prime}}}{2}$ |  $M_{N_{1,2}}=1000$ GeV |  $M_{N_{1,2}}=500$ GeV
---|---|---|---
$g^{\prime}$(Case-I) | 0.051 | 0.061 | 0.064
$g^{\prime}$(Case-II) | 0.046 | 0.072 | 0.095
Table 3: Allowed values of the $U(1)_{X}$ couplings considering all the
necessary constraints for $M_{Z^{\prime}}=3$ TeV with different benchmark
scenarios of the RHN masses for $x_{H}=-1.2$.

##  V Heavy neutrino pair production at $pp$ colliders

The RHNs can be produced in pair at $\sqrt{s}=$ 14 TeV, 27 TeV and 100 TeV
proton proton colliders respectively through the $Z^{\prime}$ production for
$x_{H}=-1.2$, $\sigma(pp\to Z^{\prime})$ BR$(Z^{\prime}\to NN)$. After the
RHNs are pair produced, they dominantly decay into $\ell W$ mode followed by
the hadronic decay of the $W$ boson. Hence due to the pair production of the
Majorana RHNs, same sign dilepton (SSDL) plus four jet signal is produced. We
show

Figure 3: The density plot representing SSDL plus four jet cross section on
$M_{N}-M_{Z^{\prime}}$ plane for Case-I (II) in the upper (lower) panel
considering $x_{H}=-1.2$ at different center of mass energies. The bar chart
represents the cross sections in fb.

the density plots on the $M_{N}-M_{Z^{\prime}}$ plane for $\sqrt{s}=$14 TeV,
27 TeV and 100 TeV in the upper (lower) panel of Fig 3 for Case-I (II)
satisfying the strongest upper limits on $g^{\prime}$ from the dilepton
results at the LHC for $M_{N}>\frac{M_{Z^{\prime}}}{2}$. The bar charts in the
density plots represent the cross sections from low to high (bottom to top)
values in fb. In this case we consider the $M_{Z^{\prime}}$ from 2 TeV to 10
TeV. For the Case-I at $14$ TeV we did not obtain any points for
$M_{Z^{\prime}}>6$ TeV, at 27 TeV did not obtain any point beyond
$M_{Z^{\prime}}>8$ TeV. However, at the 100 TeV we obtain larger cross
sections for the SSDL case from the RHN pair production. We show the density
plot for the Case-II in the lower panel of Fig. 3 for $\sqrt{s}=$14 TeV, 27
TeV and 100 TeV respectively. In this case we find the allowed points for
$M_{Z^{\prime}}$ up to 6 TeV at 14 TeV collider, 8 TeV at 27 TeV collider and
10 TeV at 100 TeV collider.

Apart from the SSDL plus four jet signal from the pair production of RHNs, we
can consider a trilepton final state plus two jet signature in association
with missing energy at $\sqrt{s}=$ 14 TeV, 27 TeV and 100 TeV hadron colliders
respectively for $x_{H}=-1.2$. Both the RHNs dominantly decay into $\ell W$
mode followed by the hadronic decay of one of the $W$ bosons whereas the other
one decays leptonically. Using the experimentally allowed strongest upper
bounds from the $g^{\prime}-M_{Z^{\prime}}$ plane for
$M_{N}>\frac{M_{Z^{\prime}}}{2}$ we show the density plots in the
$M_{N}-M_{Z^{\prime}}$ plane for 14 TeV, 27 TeV and 100 TeV are given in the
upper (lower) panel of Fig 4 for Case-I (II). The bar charts in the density
plots represent the cross sections from low to high (bottom to top) in fb. The
colored area shows the allowed cross sections of the trilepton pus two jet in
association with missing energy in $M_{N}-M_{Z^{\prime}}$ plane. In this case
we consider the $Z^{\prime}$ mass from 2 TeV to 10 TeV. For the Case-I at $14$
TeV we did not obtain any points for $M_{Z^{\prime}}>$ 6 TeV, at 27 TeV did
not obtain any data point for $M_{Z^{\prime}}>$ 8 TeV. However, at the 100 TeV
we obtain larger cross sections for the trilepton case from the RHN pair
production even for large $M_{Z^{\prime}}$. The density plots for the Case-II
are shown in the lower panel of Fig. 4. Similar behavior like the Case-I can
be observed, however, the cross sections increase due to the U$(1)_{X}$ charge
of the RHNs and the cross sections increase with center of mass energy.

Figure 4: The density plot representing trilepton plus two jet cross section
in association with missing energy on $M_{N}-M_{Z^{\prime}}$ plane for Case-I
(II) in the upper (lower) panel considering $x_{H}=-1.2$ at different center
of mass energies. The bar chart represents the cross sections in fb.

Another important motivation of this paper is to study the boosted objects
from the RHNs pair. After the RHNs are produced, they can decay through the
dominant mode $\ell W$, followed by the hadronic decay of the $W$. The
Majorana nature of the RHNs will manifest a distinct $\ell^{\pm}\ell^{\pm}$
signature along with the hadronic decay of the $W$ boson. We consider the RHNs
which are sufficiently heavy, e. g., $M_{N}\geq 500$ GeV which allow the $W$
boson to be boosted so that the hadronic decay mode of the $W$ can be
collimated to produce fat-jet. Hence SSDL plus two fat-jet and trilepton plus
one fat-jet shown in Fig. 5 could be interesting to study.

Figure 5: Heavy neutrino pair production processes at the hadron collider from
the $Z^{\prime}$. The heavy neutrinos $(N)$ decay into the SSDL plus two fat-
jets (left) and trilepton plus one fat-jet in association with missing energy
(right).

###  V.1 Same sign dilepton (SSDL) with fat-jets

The resonant production of the Majorana RHN pair from the $Z^{\prime}$ can
show a distinct signature of the lepton number violation at the collider. In
this case RHNs will produce same sign leptons and $W$ bosons. Due to the heavy
mass of the RHNs the $W$ boson will be boosted to make a fat-jet. The
production process is

$\displaystyle pp\to Z^{\prime}\to NN,N\to\ell^{\pm}W,N\to\ell^{\pm}W,W\to
j_{1}j_{2}(J_{1}),W\to j_{1}j_{2}(J_{2})$ (37)

where $J_{1}$ and $J_{2}$ are the fat-jets, shown in the left panel of Fig. 5.
In our scenarios there are three RHNs out of them we consider the first two
generations are degenerate. We consider two benchmark scenarios
$M_{N_{1,2}}=500$ GeV and $M_{N_{1,2}}=1$ TeV respectively for the U$(1)$
scenarios stated in Cases-I and II.

The dominant SM background in the case of SSDL scenario comes from the same
sign $W$ boson production in association with jets. The same sign $W$ will
decay leptonically to produce SSDL environment and the two jets will resemble
the $W^{\pm}$ like fat-jets. Another significant background will be
contributed from $W^{\pm}Z+$ jets where $W^{\pm}$ and $Z$ decay leptonically.
In addition to that, another important contribution will come from
$W^{\pm}W^{\mp}Z+$ jets background where one $W$ and $Z$ will decay
leptonically where as the remaining $W$ will decay hadronically. In these
cases there will be an SSDL pair in association with an opposite sign third
lepton. A third lepton veto will reduce these backgrounds. Other significant
contributions will come from $t\overline{t}W^{\pm}$ and $t\overline{t}Z$
channels. The SSDL events will come from either $t$ or $\overline{t}$ and the
leptonic decay of $W^{\pm}$. A similar scenario can be observed for
$t\overline{t}Z$ for the SSDL signature, however, there will be an additional
lepton. In this case, the choice of the two same sign lepton and third lepton
veto are very important to reduce the background. The $W$ like fat-jets can be
produced from the additional jets in the event or from the remaining $t$ or
$\overline{t}$. We use the veto on b-jets to reject events with b-jets.

Implementing the model111We have written the model file in FeynRules. The FR
and UFO files can be obtained in the category of Simple extensions of the SM
of the FeynRules model database General U(1) model in FeynRules [95, 96] we
generate the events using MadGraph [97, 98] and parton distribution function
CTEQ6L [93] fixing the factorization scale $\mu_{F}$ at the default MadGraph
option. The showering, fragmentation and hadronization of the signal and SM
backgrounds were performed by the PYTHIA8[99]. The detector simulation of the
showered events was performed using the detector simulation package Delphes
[100] equipped with the Cambridge-Achen (C/A) jet clustering algorithm [101,
102]. In our analysis, we have taken jet cone radius, $R=1.0$. The jets and
associated sub-jet variables are constructed using the softdrop procedure. The
softdrop procedure depends on two parameters, an asymmetry cut $Z_{cut}$, and
an angular exponents $\beta$ describe here [103, 104, 105]. For $\beta=0$, the
Softdrop algorithm to find sub-jets is the same as the modified mass-drop
tagger[105]. The algorithm goes as, (i) By undoing the last step of the C/A
clustering algorithm of fatjet we will get two subjets, for example, $j_{1}$
and $j_{2}$, (ii) Now, if these two subjets pass the asymmetry cut
$\frac{min(P_{T}^{j_{1}},P_{T}^{j_{2}})}{(P_{T}^{j_{1}}+P_{T}^{j_{2}})}>Z_{cut}$
(here we use $Z_{cut}=0.1$) we call the fat-jet as a softdrop jet. (iii) If
the last condition is not satisfied, we consider the largest $p_{T}$ jet as a
fat-jet and do the same procedure from step 1. In this algorithm, the
asymmetry cut helps us to discriminate the jets created from the decays of the
SM heavy resonance particles (top quark, $W$, $Z$ and Higgs bosons) with
respect to the quark and gluon jets which have pure QCD origin. This asymmetry
cut also removes contamination from ISR, FSR, etc. Because of that, the
softdrop jet mass is close to the heavy resonance particle mass if the source
of the fat-jet is a decay of heavy resonance particle.

Backgrounds processes | SSDL+jets | 14 TeV | 27 TeV | 100 TeV
---|---|---|---|---
|  | (fb) | (fb) | (fb)
$WWjj$ | $\mu^{\pm}\mu^{\pm}$ | 0.71 | 1.7 | 3.8
| $e^{\pm}e^{\pm}$ | 0.52 | 1.7 | 3.2
$WZjj$ | $\mu^{\pm}\mu^{\pm}$ | 4.84 | 16.2 | 39.7
| $e^{\pm}e^{\pm}$ | 3.6 | 16.2 | 39.8
$WWZ+$jets | $\mu^{\pm}\mu^{\pm}$ | 0.17 | 0.71 | 3.2
| $e^{\pm}e^{\pm}$ | 0.13 | 0.7 | 3.3
$t\overline{t}W$ | $\mu^{\pm}\mu^{\pm}$ | 1.12 | 3.6 | 7.2
| $e^{\pm}e^{\pm}$ | 0.82 | 2.5 | 7.0
$t\overline{t}W$ | $\mu^{\pm}\mu^{\pm}$ | 1.12 | 3.6 | 7.2
| $e^{\pm}e^{\pm}$ | 0.82 | 2.5 | 7.0
$t\overline{t}Z$ | $\mu^{\pm}\mu^{\pm}$ | 1.28 | 6.7 | 29.2
| $e^{\pm}e^{\pm}$ | 0.95 | 6.7 | 29.0
Table 4: SM backgrounds after imposing the basic cuts at different pp
colliders with $\sqrt{s}=14$ TeV, $27$ TeV and $100$ TeV for SSDL plus jets
signature.

Generating the $\mu^{\pm}\mu^{\pm}+2J$ and $e^{\pm}e^{\pm}+2J$ signals at $14$
TeV we study Cases-I and II. The corresponding SM backgrounds are $WWjj$,
$WZjj$, $WWZ+$jets, $t\overline{t}W$ and $t\overline{t}Z$ respectively. The
backgrounds are generated using the selection cuts $H_{T}>300$ GeV where
$H_{T}$ is the scalar sum of the transverse momentum of the jets
$(p_{T}^{j})$. The jets are selected with the transverse momentum
$p_{T}^{j}>10$ GeV, pseudorapidity of jets $|\eta^{j}|<2.5$. The transverse
momentum of the leptons $p_{T}^{\ell}>10$ GeV, pseudorapidity of the leptons
$|\eta^{\ell}|<2.5$, separation between the leptons in the $\eta-\phi$ plane
$\Delta R_{\ell\ell}>0.4$ and lepton and jets $\Delta R_{\ell j}>0.4$ are
among them. The partonic cross sections after the basic cuts are given in Tab.
4 for $\sqrt{s}=14$ TeV. To generate the $WWZ+\rm jets$, $t\bar{t}W$, and
$t\bar{t}Z$ backgrounds, we slightly change our generation level cuts. Here we
applied the same $H_{T}$ cut as above for all jets, including jets coming from
standard model heavy resonance decay. However we did not use any $p_{T}$,
$\eta$, and $\Delta_{R}$ cuts to leptons and jets coming from standard model
resonance decay. After the primary cuts the $p_{T}$ distributions of the fat-
jets and the leptons are shown in Fig. 6. The distributions of the jet masses
and missing energy are shown in Fig. 7. After selecting the signal events we
use advanced cuts for the signals and SM backgrounds. To study the SM
background from $WWZ+$ jets process we consider the leptonic decay of one $W$
and the hadronic decay of the other whereas the $Z$ boson decays into charged
leptons. Such leptonic events allow us to use a third lepton veto in the
advanced cuts. A similar procedure has been adopted for $t\overline{t}Z$. To
study SM backgrounds from $t\overline{t}Z$ process, we consider the leptonic
decay of one $W$ from the top quark and the hadronic decay of the other from
the remaining top quark, whereas the $Z$ boson decays leptonically.

Figure 6: Transverse momentum distribution of the leading (top, left), sub
leading (top, right) fat jets and the leading (bottom, left), subleading
(bottom, right) leptons for $M_{N}=500$ GeV (blue, solid) and $1$ TeV (red,
solid) respectively with the corresponding SM backgrounds at $\sqrt{s}=$ 14
TeV.
Figure 7: Jet mass distribution of the leading (top, left), subleading (top,
right) fat jets and the missing energy distribution (bottom) for $M_{N}=500$
GeV (blue, solid) and $1$ TeV (red, solid) respectively with the corresponding
SM backgrounds at $\sqrt{s}=$ 14 TeV.

From Fig. 6 we find the peak of the $p_{T}^{J_{1}}$ distribution situated in
the high $p_{T}$ region whereas the backgrounds are distributed mainly in the
low $p_{T}$ region. We consider this as the leading fat-jet and the remaining
fat-jet $(J_{2})$ as the sub-leading one. The sub-leading fat-jet has a peak
coinciding with the SM backgrounds. For the heavier RHNs, these jets are more
energetic, and their distribution shifted to the higher $p_{T}$ region. The
leptons from the RHNs are more energetic than those from SM backgrounds. Hence
leptons’ $p_{T}$ will be another important discriminator along with the
$p_{T}$ of the fat-jets. From the distribution of the jet masses in the upper
panel of Fig. 7, we find peaks from the signal and some SM backgrounds like
$WWZ$, $t\overline{t}W$ and $t\overline{t}Z$ around the $W$ boson mass;
however, the backgrounds peaks are not that prominent as the signal peak.
These allow us to choose a window of $15$ GeV around the $W$ boson mass to
suppress the SM backgrounds further and get rid of other low energy hadronic
effects below $60$ GeV. From the missing energy distributions shown in the
lower panel of Fig. 7 we find a conservative cut below 150 GeV. We have
noticed that many trilepton events coming from the SM processes $WWZ$,
$t\overline{t}W$ and $t\overline{t}Z$ where we use a criteria of at least two
leptons having the same sign. In addition to that we use a third lepton veto
which reduces such SM background events further. The backgrounds coming from
the top quark induced events contain b-jets. In the final selection we use a
b-veto to reject such events.

Selecting the signal and backgrounds with basic cuts we employ the post
selection cuts or advanced cuts. The transverse momentum of each fat jet has
been considered to be $p_{T}^{J}>180$ GeV (C-I). We select the events with
SSDL with transverse momenta $p_{T}^{\ell_{1}},p_{T}^{\ell_{2}}>100$ GeV
(C-II). Both of the jets have sub-jet number greater than 1 (C-III). We have
considered that the invariant mass of the two leading sub-jet is within a
window of $\pm 15$ GeV around the $W$ boson mass for the leading fat-jets
(C-IV). Finally we consider that both jets are not b-jets (C-V). We apply a
conservative cut on the missing energy as $E_{T}^{\rm miss}<150$ GeV (C-VI).
The cut flow has been shown in Tab. 5 with the corresponding efficiencies for
the signals and corresponding backgrounds for $\mu^{\pm}\mu^{\pm}$ and
$e^{\pm}e^{\pm}$ samples in the upper and lower halves of the table
respectively. The upper part of each row represents the $\ell^{+}\ell^{+}$
signal and the lower part represents the $\ell^{-}\ell^{-}$ signal
respectively. The combined significance of the SSDL mode has been calculated
using $\frac{S}{\sqrt{S+B}}$ for signal $(S)$ and background $(B)$ events.
Solving for a particular significance we estimate the required luminosity for
Cases-I and II. We find the significance is below 1$-\sigma$ up to the total
luminosity of the LHC for both of the benchmark RHN masses whereas an
achievable 5$-\sigma$ significance can be reached for Case-II around $1.88$
ab-1 for $M_{N}=1$ TeV and $1.52$ ab-1 luminosities for $M_{N}=500$ GeV
respectively.

$\mu^{\pm}\mu^{\pm}+2J$ | $W^{\pm}W^{\pm}jj$ | $W^{\pm}Zjj$ | $W^{\pm}W^{\mp}Zj$ | $t\bar{t}W^{\pm}$ | $t\bar{t}Z$ |  sig($500$ GeV) |  sig($1$ TeV) |  sig($500$ GeV) |  sig($1$ TeV)
---|---|---|---|---|---|---|---|---|---
|  |  |  |  |  | Case-I | Case-I | Case-II | Case-II
| (fb) | (fb) | (fb) | (fb) | (fb) | (fb) | (fb) | (fb) | (fb)
C-I | 0.1398 | 0.147 | 0.00562 | 0.0347 | 0.0224 | 0.00086 | 0.0012 | 0.028 | 0.025
| $[100\%]$ | $[100\%]$ | $[100\%]$ | $[100\%]$ | $[100\%]$ | $[100\%]$ | $[100\%]$ | $[100\%]$ | $[100\%]$
C-II | 0.01 | 0.0224 | 0.00093 | 0.0055 | 0.00114 | 0.00070 | 0.0011 | 0.021 | 0.024
| $[7.15\%]$ | $[15.23\%]$ | $[16.55\%]$ | $[15.85\%]$ | $[6.25\%]$ | $[81.39\%]$ | $[91.60\%]$ | $[75\%]$ | $[96\%]$
C-III | 0.0098 | 0.0221 | 0.00091 | 0.002 | 0.00112 | 0.00065 | 0.001 | 0.022 | 0.023
| $[7.01\%]$ | $[15.03\%]$ | $[16.19\%]$ | $[5.76\%]$ | $[5\%]$ | $[75.58\%]$ | $[83.30\%]$ | $[78.57\%]$ | $[92\%]$
C-IV | 0.00067 | 0.00206 | 0.00021 | 0.000354 | 0.00017 | 0.00043 | 0.00075 | 0.014 | 0.016
| $[0.48\%]$ | $[1.77\%]$ | $[3.74\%]$ | $[1.02\%]$ | $[0.76\%]$ | $[50\%]$ | $[62.50\%]$ | $[50\%]$ | $[64\%]$
C-V | 0.00061 | 0.00107 | 0.00019 | 0.000195 | 0.000063 | 0.00036 | 0.00062 | 0.012 | 0.013
| $[0.44\%]$ | $[0.73\%]$ | $[3.38\%]$ | $[0.56\%]$ | $[0.28\%]$ | $[38.37\%]$ | $[51.60\%]$ | $[42.85\%]$ | $[52\%]$
C-VI | 0.00043 | 0.00073 | 0.00014 | 0.000164 | 0.000063 | 0.00033 | 0.00048 | 0.011 | 0.01
| $[0.31\%]$ | $[0.50\%]$ | $[2.49\%]$ | $[0.47\%]$ | $[0.28\%]$ | $[38.37\%]$ | $[40\%]$ | $[39.28\%]$ | $[40\%]$
$e^{\pm}e^{\pm}+2J$ | $W^{\pm}W^{\pm}jj$ | $W^{\pm}Zjj$ | $W^{\pm}W^{\mp}Zj$ | $t\bar{t}W^{\pm}$ | $t\bar{t}Z$ |  sig($500$ GeV) |  sig($1$ TeV) |  sig($500$ GeV) |  sig($1$ TeV)
|  |  |  |  |  | Case-I | Case-I | Case-II | Case-II
| (fb) | (fb) | (fb) | (fb) | (fb) | (fb) | (fb) | (fb) | (fb)
C-I | 0.103 | 0.11 | 0.0041 | 0.026 | 0.0017 | 0.00061 | 0.0009 | 0.0204 | 0.018
| $[100\%]$ | $[100\%]$ | $[100\%]$ | $[100\%]$ | $[100\%]$ | $[100\%]$ | $[100\%]$ | $[100\%]$ | $[100\%]$
C-II | 0.0073 | 0.017 | 0.0007 | 0.0017 | 0.00090 | 0.00048 | 0.00082 | 0.0162 | 0.0173
| $[7.08\%]$ | $[15.45\%]$ | $[17.07\%]$ | $[6.54\%]$ | $[52.94\%]$ | $[78.69\%]$ | $[91.10\%]$ | $[79.41\%]$ | $[96.10\%]$
C-III | 0.00721 | 0.016 | 0.00066 | 0.00168 | 0.00082 | 0.00047 | 0.00081 | 0.016 | 0.015
| $[7\%]$ | $[14.54\%]$ | $[16.10\%]$ | $[6.46\%]$ | $[48.23\%]$ | $[77.05\%]$ | $[90\%]$ | $[78.43\%]$ | $[83.33\%]$
C-IV | 0.00050 | 0.0015 | 0.00016 | 0.000345 | 0.00013 | 0.00031 | 0.0006 | 0.01 | 0.012
| $[0.48\%]$ | $[1.36\%]$ | $[3.90\%]$ | $[1.33\%]$ | $[7.65\%]$ | $[50.82\%]$ | $[66.60\%]$ | $[49.02\%]$ | $[66.60\%]$
C-V | 0.00045 | 0.0014 | 0.00007 | 0.000234 | 0.000048 | 0.00025 | 0.0005 | 0.0084 | 0.01
| $[0.44\%]$ | $[1.27\%]$ | $[1.71\%]$ | $[0.90\%]$ | $[2.82\%]$ | $[40.98\%]$ | $[55.50\%]$ | $[41.18\%]$ | $[55.50\%]$
C-VI | 0.00032 | 0.001 | 0.00005 | 0.00018 | 0.000046 | 0.00023 | 0.00035 | 0.0080 | 0.007
| $[0.31\%]$ | $[0.90\%]$ | $[1.22\%]$ | $[0.69\%]$ | $[2.70\%]$ | $37.70[\%]$ | $[38.80\%]$ | $[39.21\%]$ | $[38.80\%]$
Table 5: Cut flow for SSDL plus two fat-jet signal. Upper block represents the
muon and lower block represents the electron signals. We consider
$M_{Z^{\prime}}=3$ TeV at $\sqrt{s}=14$ TeV.

We analyze the SSDL signal with two fat jets consisting
$\mu^{\pm}\mu^{\pm}+2J$ and $e^{\pm}e^{\pm}+2J$ signals at $\sqrt{s}=27$ TeV.
After that we generate the corresponding SM backgrund processes $WWjj$,
$WZjj$, $WWZ+$jets, $t\overline{t}W$ and $t\overline{t}Z$ to compare the
signals for $M_{N_{1,2}}=500$ GeV and $M_{N_{1,2}}=1$ TeV both for Case-I and
II. To generate the backgrounds we impose the selection cuts as $H_{T}>300$
GeV, $p_{T}^{j}>10$ GeV, $|\eta^{j}|<2.5$, $p_{T}^{\ell}>10$ GeV,
$|\eta^{\ell}|<2.5$, $\Delta R_{\ell\ell}>0.4$ and $\Delta R_{\ell j}>0.4$
respectively. The parton level cross sections of the SM backgrounds after the
selection cuts are given in Tab. 4. By studying the signal and the backgrounds
we finalize the advanced cuts for the signals and backgrounds. We do not show
the histograms, apart from the higher energy reaches of the kinematic
variables, their nature will remain almost the same as the previous case. The
advanced cuts are used selecting at least two fat-jets $(J)$ with transverse
momentum $p_{T}^{J}>180$ GeV (C-I). Only two same sign leptons with
$p_{T}^{\ell}>100$ GeV are selected (C-II) vetoing the third lepton. Both of
the jets have sub-jet number $>1$ (C-III). The Invariant mass of the leading
two sub-jets is within $\pm 15$ GeV window around the $W$ boson mass (C-IV).
We ensure that both jets are not b-jets (C-V). The missing energy is
conservatively considered as $\not{E_{T}}<150$ GeV (C-VI). The cut flow and
the corresponding efficiencies for $\mu^{\pm}\mu^{\pm}+2J$ and
$e^{\pm}e^{\pm}+2J$ signals and SM backgrounds are given in the top, right
panel of Tab. 6.

$\mu^{\pm}\mu^{\pm}+2J$ | $W^{\pm}W^{\pm}jj$ | $W^{\pm}Zjj$ | $W^{\pm}W^{\mp}Zj$ | $t\bar{t}W^{\pm}$ | $t\bar{t}Z$ |  sig($500$ GeV) |  sig($1$ TeV) |  sig($500$ GeV) |  sig($1$ TeV)
---|---|---|---|---|---|---|---|---|---
|  |  |  |  |  | Case-I | Case-I | Case-II | Case-II
| (fb) | (fb) | (fb) | (fb) | (fb) | (fb) | (fb) | (fb) | (fb)
C-I | $0.340$ | $0.40$ | $0.026$ | $0.125$ | $0.123$ | $0.008$ | $0.01$ | 0.37 | 0.31
| $[100\%]$ | $[100\%]$ | $[100\%]$ | $[100\%]$ | $[100\%]$ | $[100\%]$ | $[100\%]$ | $[100\%]$ | $[100\%]$
C-II | $0.037$ | $0.12$ | $0.0061$ | $0.01$ | $0.011$ | $0.006$ | $0.0097$ | 0.3 | 0.29
| $[10.88\%]$ | $[30\%]$ | $[23.46\%]$ | $[8\%]$ | $[8.94\%]$ | $[75\%]$ | $[97\%]$ | $[81.08\%]$ | $[93.55\%]$
C-III | $0.036$ | $0.117$ | $0.0051$ | $0.0093$ | $0.0103$ | $0.0056$ | $0.0093$ | 0.28 | 0.285
| $[10.59\%]$ | $[29.25\%]$ | $[19.61\%]$ | $[7.44\%]$ | $[8.37\%]$ | $[70\%]$ | $[93\%]$ | $[75.67\%]$ | $[91.93\%]$
C-IV | $0.0028$ | $0.01$ | $0.0011$ | $0.002$ | $0.0012$ | $0.0034$ | $0.0062$ | 0.17 | 0.19
| $[0.82\%]$ | $[2.5\%]$ | $[4.23\%]$ | $[1.6\%]$ | $[0.97\%]$ | $[42.5\%]$ | $[62\%]$ | $[45.94\%]$ | $[61.29\%]$
C-V | $0.0026$ | $0.007$ | $0.001$ | $0.001$ | $0.0007$ | $0.0028$ | $0.0051$ | 0.14 | 0.16
| $[0.76\%]$ | $[1.75\%]$ | $[3.85\%]$ | $[0.8\%]$ | $[0.57\%]$ | $[35\%]$ | $[51\%]$ | $[37.84\%]$ | $[51.61\%]$
C-VI | $0.0016$ | $0.006$ | $0.0006$ | $0.00071$ | $0.0006$ | $0.0025$ | $0.0038$ | 0.13 | 0.142
| $[0.47\%]$ | $[1.5\%]$ | $[2.31\%]$ | $[0.57\%]$ | $[0.49\%]$ | $[31.25\%]$ | $[38\%]$ | $[35.13\%]$ | $[45.81\%]$
$e^{\pm}e^{\pm}+2J$ | $W^{\pm}W^{\pm}jj$ | $W^{\pm}Zjj$ | $W^{\pm}W^{\mp}Zj$ | $t\bar{t}W^{\pm}$ | $t\bar{t}Z$ |  sig($500$ GeV) |  sig($1$ TeV) |  sig($500$ GeV) |  sig($1$ TeV)
|  |  |  |  |  | Case-I | Case-I | Case-II | Case-II
| (fb) | (fb) | (fb) | (fb) | (fb) | (fb) | (fb) | (fb) | (fb)
C-I | $0.19$ | $0.62$ | $0.0214$ | $0.08$ | $0.098$ | $0.0046$ | $0.0070$ | 0.15 | 0.134
| $[100\%]$ | $[100\%]$ | $[100\%]$ | $[100\%]$ | $[100\%]$ | $[100\%]$ | $[100\%]$ | $[100\%]$ | $[100\%]$
C-II | $0.084$ | $0.125$ | $0.0046$ | $0.0071$ | $0.0094$ | $0.0038$ | $0.0063$ | 0.12 | 0.125
| $[44.21\%]$ | $[20.16\%]$ | $[21.49\%]$ | $[8.87\%]$ | $[9.59\%]$ | $[82.61\%]$ | $[90\%]$ | $[80\%]$ | $[93.28\%]$
C-III | $0.025$ | $0.122$ | $0.0045$ | $0.007$ | $0.0092$ | $0.0036$ | $0.0060$ | 0.11 | 0.121
| $[13.16\%]$ | $[19.68\%]$ | $[21.03\%]$ | $[8.75\%]$ | $[9.39\%]$ | $[78.26\%]$ | $[85.71\%]$ | $[73.30\%]$ | $[90.30\%]$
C-IV | $0.002$ | $0.0114$ | $0.001$ | $0.0014$ | $0.0014$ | $0.0022$ | $0.0041$ | 0.07 | 0.083
| $[1.05\%]$ | $[1.84\%]$ | $[4.67\%]$ | $[1.75\%]$ | $[1.43\%]$ | $[47.82\%]$ | $[58.57\%]$ | $[46.60\%]$ | $[61.94\%]$
C-IV | $0.0018$ | $0.01$ | $0.0009$ | $0.00063$ | $0.00087$ | $0.002$ | $0.0036$ | 0.061 | 0.070
| $[0.95\%]$ | $[1.61\%]$ | $[4.20\%]$ | $[0.79\%]$ | $[0.89\%]$ | $[43.48\%]$ | $[51.43\%]$ | $[40.60\%]$ | $[52.24\%]$
C-VI | $0.0011$ | $0.0062$ | $0.0007$ | $0.00045$ | $0.00053$ | $0.0018$ | $0.0031$ | 0.054 | 0.061
| $[0.58\%]$ | $[1\%]$ | $[3.27\%]$ | $[0.56\%]$ | $[0.54\%]$ | $[39.13\%]$ | $[44.28\%]$ | $[36\%]$ | $[45.52\%]$
Table 6: Cut flow for SSDL plus two fat-jet signal. Upper block represents the
muon and lower block represents the electron signals. We consider
$M_{Z^{\prime}}=3$ TeV at $\sqrt{s}=27$ TeV.

Solving the signal and background relation for a particular significance we
can estimate the luminosity to achieve that significance. Hence we calculate
the significance for 1 TeV RHN from Case-I around $5-\sigma$ at 14.23 ab-1
luminosity and that for 1 TeV reaches up to $3.5-\sigma$ at 15 ab-1
luminosity. The results for Case-II are also improved compared to the previous
case. A significance of $5-\sigma$ can be attained at $144$ fb-1 luminosity
for 500 GeV RHN whereas that can be attained at $130.5$ fb-1 luminosity for 1
TeV RHN respectively.

Finally, we study the SSDL signal from Cases-I and II at $\sqrt{s}=100$ TeV
proton proton collider considering $M_{Z^{\prime}}=3$ TeV. All primary
selection cuts are the same as the previous two cases except $H_{T}$. Here we
use $H_{T}>500$ GeV for all the background processes. The partonic cross
sections of the SM backgrounds are given in Tab. 4. After selecting the signal
events we use the advanced cuts for the signals and SM backgrounds. The
advanced cuts are used by selecting at least two fat-jets $(J)$ with
transverse momentum $p_{T}^{J}>250$ GeV (C-I). The other advanced cuts are the
same as the previous two cases. In this case we have only considered two
degenerate RHNs with mass $M_{N}=1$ TeV. The cut flow table for SSDL scenario
is given in Tab. 7 with the efficiencies after each level of cuts for the
$\mu^{\pm}\mu^{\pm}+2J$ and $e^{\pm}e^{\pm}+2J$ channels. We find that the
$5-\sigma$ significance could be achievable for the Case-I around $534$ fb-1
luminosity and for the Case-II, the same benchmark point can be achievable
with $22.2$ fb-1 luminosity.

$\mu^{\pm}\mu^{\pm}+2J$ | $W^{\pm}W^{\pm}jj$ | $W^{\pm}Zjj$ | $W^{\pm}W^{\mp}Zj$ | $t\bar{t}W^{\pm}$ | $t\bar{t}Z$ |  sig($1$ TeV) |  sig($1$ TeV)
---|---|---|---|---|---|---|---
| (fb) | (fb) | (fb) | (fb) | (fb) | (fb) | (fb)
|  |  |  |  |  | Case-I | Case-II
C-I | $0.811$ | $2.221$ | $0.122$ | $0.202$ | $0.476$ | $0.102$ | 2.0
| $[100\%]$ | $[100\%]$ | $[100\%]$ | $[100\%]$ | $[100\%]$ | $[100\%]$ | $[100\%]$
C-II | $0.141$ | $0.674$ | $0.0384$ | $0.034$ | $0.065$ | $0.1$ | 1.94
| $[17.38\%]$ | $[30.35\%]$ | $[31.47\%]$ | $[16.83\%]$ | $[13.65\%]$ | $[98.04\%]$ | $[97\%]$
C-III | $0.138$ | $0.664$ | $0.0224$ | $0.033$ | $0.064$ | $0.097$ | 1.92
| $[17.02\%]$ | $[29.90\%]$ | $[18.36\%]$ | $[16.34\%]$ | $[13.44\%]$ | $[95.10\%]$ | $[96\%]$
C-IV | $0.011$ | $0.0484$ | $0.0065$ | $0.0028$ | $0.0085$ | $0.0542$ | 1.1
| $[1.36\%]$ | $[2.18\%]$ | $[5.33\%]$ | $[1.39\%]$ | $[1.78\%]$ | $[53.14\%]$ | $[55\%]$
C-V | $0.009$ | $0.042$ | $0.0059$ | $0.002$ | $0.0038$ | $0.047$ | 0.91
| $[1.11\%]$ | $[1.89\%]$ | $[4.84\%]$ | $[0.99\%]$ | $[0.80\%]$ | $[46.08\%]$ | $[45.50\%]$
C-VI | $0.0038$ | $0.0224$ | $0.0031$ | $0.0006$ | $0.0018$ | $0.033$ | 0.61
| $[0.47\%]$ | $[1\%]$ | $[2.54\%]$ | $[0.30\%]$ | $[0.38\%]$ | $[32.35\%]$ | $[30.50\%]$
$e^{\pm}e^{\pm}+2J$ | $W^{\pm}W^{\pm}jj$ | $W^{\pm}Zjj$ | $W^{\pm}W^{\mp}Zj$ | $t\bar{t}W^{\pm}$ | $t\bar{t}Z$ |  sig($1$ TeV) |  sig($1$ TeV)
| (fb) | (fb) | (fb) | (fb) | (fb) | (fb) | (fb)
|  |  |  |  |  | Case-I | Case-II
C-I | $0.42$ | $1.94$ | $0.0933$ | $0.0944$ | $0.34$ | $0.065$ | 1.3
| $[100\%]$ | $[100\%]$ | $[100\%]$ | $[100\%]$ | $[100\%]$ | $[100\%]$ | $[100\%]$
C-II | $0.087$ | $0.65$ | $0.0161$ | $0.0184$ | $0.060$ | $0.063$ | 1.25
| $[20.71\%]$ | $[33.50\%]$ | $[17.26\%]$ | $[19.49\%]$ | $[17.65\%]$ | $[96.92\%]$ | $[96.15\%]$
C-III | $0.085$ | $0.64$ | $0.008$ | $0.0181$ | $0.056$ | $0.063$ | 1.24
| $[20.24\%]$ | $[32.99\%]$ | $[8.57\%]$ | $[19.17\%]$ | $[16.47\%]$ | $[96.92\%]$ | $[95.38\%]$
C-IV | $0.031$ | $0.054$ | $0.0055$ | $0.0021$ | $0.0067$ | $0.038$ | 0.74
| $[7.38\%]$ | $[2.78\%]$ | $[5.90\%]$ | $[2.22\%]$ | $[1.97\%]$ | $[58.46\%]$ | $[56.92\%]$
C-V | $0.0052$ | $0.046$ | $0.0045$ | $0.00143$ | $0.0047$ | $0.031$ | 0.614
| $[1.24\%]$ | $[2.37\%]$ | $[4.82\%]$ | $[1.51\%]$ | $[1.38\%]$ | $[47.69\%]$ | $[47.23\%]$
C-VI | $0.0024$ | $0.024$ | $0.0025$ | $0.00072$ | $0.0023$ | $0.028$ | 0.541
| $[0.57\%]$ | $[1.24\%]$ | $[2.68\%]$ | $[0.76\%]$ | $[0.68\%]$ | $[43.10\%]$ | $[41.61\%]$
Table 7: Cut flow for SSDL plus two fat-jet signal. Upper block represents the
muon and lower block represents the electron signals. We consider
$M_{Z^{\prime}}=3$ TeV at $\sqrt{s}=100$ TeV.

###  V.2 Trilepton plus fat-jet in association with missing energy

After the pair production of the RHNs from $Z^{\prime}$, each of the RHNs will
dominantly decay into $\ell W$ mode. One of the $W$ can decay hadronically
producing a fat-jet and the other $W$ can decay leptonically. Hence a
trilepton signal will be generated with a fat-jet in association with missing
energy. In the leptonic decay of the $W$ boson we do not consider tau lepton
in this analysis. In this case the selection and advanced cuts will remain the
same as the SSDL case. Hence we do not show any histograms. The fat-jet will
be created from the boosted $W$ produced from the decay of one of the RHNs. We
consider $M_{Z^{\prime}}=$3 TeV and RHNs are degenerate with two benchmark
masses at $M_{N_{1,2}}=500$ GeV and $M_{N_{1,2}}=1$ TeV. The trilepton signal
consists of two same sign leptons and the other one must have the opposite
sign. Therefore the combination of three lepton system has either $`+1$’ or
$`-1$’ charge taking all possible combinations of the leptons.

We generate SM backgrounds with pre-selection cuts including the transverse
momentum of the jets $p_{T}^{j}>10$ GeV and pseudo-rapidity of the jets
$|\eta^{j}|<2.5$. We consider the separation cuts in the $\eta-\phi$ plane
between jets and leptons as $\Delta R_{j\ell}>0.4$ and lepton-lepton as
$\Delta R_{\ell\ell}>0.4$. At the 27 TeV and 100 TeV we consider the
transverse momentum of the jets as $p_{T}^{j}>20$ GeV with other pre-selection
cuts.

We generate the backgrounds $W^{\pm}h+$ jets, $W^{\pm}Z+$jets, $ZZ+$jets,
$W^{\pm}W^{\pm}W^{\mp}+$jets, $ZZW^{\pm}$, $W^{\pm}W^{\mp}Z$, $t\overline{t}W$
and $t\overline{t}Z$, to study trilepton final state plus one fat-jet in
association with missing energy. For these backgrounds we consider leptonic
decay modes of the gauge bosons. From these backgrounds we consider
$\mu^{\pm}\mu^{\pm}\mu^{\pm}(e^{\pm})$ and $e^{\pm}e^{\pm}e^{\pm}(\mu^{\pm})$
modes respectively in association with a fat-jet and missing energy which will
mimic the trilepton plus missing energy and fat-jet signal from the RHN pair.
The SM background cross sections are given in Tab. 8 at proton colliders for
different $\sqrt{s}$ after the application of the basic cuts.

Backgrounds processes | Signals | 14 TeV | 27 TeV | 100 TeV
---|---|---|---|---
|  | (fb) | (fb) | (fb)
$t\overline{t}W^{\pm}$ | $\mu^{\pm}\mu^{\pm}\mu^{\pm}(e^{\pm})$ | 7.3 | 21.33 | 127.5
| $e^{\pm}e^{\pm}e^{\pm}(\mu^{\pm})$ | 5.36 | 15.67 | 93.7
$t\overline{t}Z$ | $\mu^{\pm}\mu^{\pm}\mu^{\pm}(e^{\pm})$ | 23.84 | 112.9 | 1510.0
| $e^{\pm}e^{\pm}e^{\pm}(\mu^{\pm})$ | 17.51 | 82.9 | 1110.0
$W^{\pm}hj$ | $\mu^{\pm}\mu^{\pm}\mu^{\pm}(e^{\pm})$ | 6.8 | 15.1 | 87.9
| $e^{\pm}e^{\pm}e^{\pm}(\mu^{\pm})$ | 5.0 | 11.1 | 64.6
$W^{\pm}Zj$ | $\mu^{\pm}\mu^{\pm}\mu^{\pm}(e^{\pm})$ | 634.37 | 1248.2 | 11000.0
| $e^{\pm}e^{\pm}e^{\pm}(\mu^{\pm})$ | 466.07 | 916.9 | 8050.0
$W^{\pm}W^{\pm}W^{\mp}j$ | $\mu^{\pm}\mu^{\pm}\mu^{\pm}(e^{\pm})$ | 2.55 | 8.5 | 64.7
| $e^{\pm}e^{\pm}e^{\pm}(\mu^{\pm})$ | 1.88 | 6.22 | 47.5
$ZZj$ | $\mu^{\pm}\mu^{\pm}\mu^{\pm}(e^{\pm})$ | 94.75 | 256.9 | 1500.0
| $e^{\pm}e^{\pm}e^{\pm}(\mu^{\pm})$ | 69.61 | 188.8 | 1300.0
$ZZW^{\pm}$ | $\mu^{\pm}\mu^{\pm}\mu^{\pm}(e^{\pm})$ | 1.043 | 11.9 | 7.582
| $e^{\pm}e^{\pm}e^{\pm}(\mu^{\pm})$ | 0.78 | 6.21 | 10.35
$W^{\pm}W^{\mp}Z$ | $\mu^{\pm}\mu^{\pm}\mu^{\pm}(e^{\pm})$ | 3.39 | 9.1 | 48.04
| $e^{\pm}e^{\pm}e^{\pm}(\mu^{\pm})$ | 2.5 | 6.7 | 35.3
Table 8: SM backgrounds after imposing the basic cuts at different pp
colliders with $\sqrt{s}=14$ TeV, $27$ TeV and $100$ TeV for trilepton plus
jets in association with missing energy process.

In this $\rm trilepton+jet+MET$ final state, the leading two leptons in
$p_{T}$ order are expected to come directly from the decay of the RHN ($N\to
W\ell$) and the third lepton is mostly coming from further decays of one of
the $W$ boson (we call it a trailing lepton). As the RHN is very heavy, the
leading two leptons have larger $p_{T}$ than the third lepton. Here we use a
large $p_{T}$ cut for leading and sub-leading leptons and a low $p_{T}$ cut
for the third lepton. Another $W$ boson comes from one of the RHN if decays
hadronically form collimated jet or fat-jet as this $W$ boson is coming from
heavy RHN decay, so it is boosted. This fat-jet can have a large $p_{T}$.
Using softdrop algorithm, we estimate the jet mass of the fat-jet for the
signal and background. The nature of the distributions regarding the leading
and subleading leptons and fat-jet are almost the same as those of the SSDL
case. Therefore we do not add the histograms for trilepton mode.

At the $14$ TeV and $27$ TeV hadron colliders we select the events with at
least one fat-jet with $p_{T}^{J}>180$ GeV (C-I). Three leptons are selected
as leading, sub-leading and trailing respectively with $p_{T}^{\ell_{1}}>200$
GeV, $p_{T}^{\ell_{2}}>100$ GeV and $p_{T}^{\ell_{3}}>20$ GeV (C-II). Finally
the softdrop jet mass is selected within a window of $\pm 15$ GeV around the
$W$ boson mass (C-III). At the $100$ TeV collider along with C-I and C-III we
add different lepton $p_{T}$ cuts. At the 100 TeV three leptons are selected
with $p_{T}^{\ell_{1}}>200$ GeV, $p_{T}^{\ell_{2}}>200$ GeV and
$p_{T}^{\ell_{3}}>20$ GeV respectively (C-II). The cut flow with the
efficiencies for $\mu^{\pm}\mu^{\pm}\ell^{\mp}$ and $e^{\pm}e^{\pm}\ell^{\mp}$
signals and backgrounds are shown in the upper and lower panels of Tab. 9 for
$\sqrt{s}=14$ TeV. We study 500 GeV and 1 TeV RHNs as benchmark points for
Cases-I and II. Like the SSDL case here also we consider the degenerate RHNs.
The leading background is originated from $WZ+$jets process where as the sub-
leading contribution from the background is coming from $WWZ$ process. The
other backgrounds are 2-3 orders of magnitude smaller than the leading
background.

$\mu^{\pm}\mu^{\pm}\mu^{\mp}/e^{\mp}$ | $t\bar{t}W^{\pm}$ | $t\bar{t}Z$ | $W^{\pm}hj$ | $W^{\pm}Zj$ | $W^{\pm}W^{\pm}W^{\mp}j$ | $ZZj$ | $ZZW^{\pm}$ | $W^{\pm}W^{\mp}Z$ |  Sig($500$ GeV) |  Sig($1$ TeV) |  Sig($500$ GeV) |  Sig($1$ TeV)
---|---|---|---|---|---|---|---|---|---|---|---|---
|  |  |  |  |  |  |  |  | Case-I | Case-I | Case-II | Case-II
|  |  |  |  |  |  |  |  | (fb) | (fb) | (fb) | (fb)
C-I | 0.0344 | 0.674 | 0.0084 | 6.461 | 0.0274 | 0.28 | 0.015 | 0.114 | 0.0016 | 0.00134 | 0.051 | 0.028
| $[100\%]$ | $[100\%]$ | $[100\%]$ | $[100\%]$ | $[100\%]$ | $[100\%]$ | $[100\%]$ | $[100\%]$ | $[100\%]$ | $[100\%]$ | $[100\%]$ | $[100\%]$
C-II | 0.0015 | 0.0464 | 0.000302 | 0.464 | 0.003 | 0.024 | 0.005 | 0.0064 | 0.00142 | 0.0013 | 0.047 | 0.027
| $[4.36\%]$ | $[6.88\%]$ | $[3.60\%]$ | $[7.18\%]$ | $[10.95\%]$ | $[8.57\%]$ | $[33.30\%]$ | $[5.61\%]$ | $[88.75\%]$ | $[97.01\%]$ | $[92.16\%]$ | $[96.43\%]$
C-III | 0.00025 | 0.01 | 0.00013 | 0.0323 | 0.00035 | 0.003 | 0.002 | 0.003 | 0.0008 | 0.0009 | 0.0254 | 0.019
| $[0.73\%]$ | $[1.48\%]$ | $[1.55\%]$ | $[0.50\%]$ | $[1.277\%]$ | $[1.07\%]$ | $[13.30\%]$ | $[2.63\%]$ | $[50\%]$ | $[67.16\%]$ | $[49.80\%]$ | $[67.85\%]$
$e^{\pm}e^{\pm}e^{\mp}/\mu^{\mp}$ | $t\bar{t}W^{\pm}$ | $t\bar{t}Z$ | $W^{\pm}hj$ | $W^{\pm}Zj$ | $W^{\pm}W^{\pm}W^{\mp}j$ | $ZZj$ | $ZZW^{\pm}$ | $W^{\pm}W^{\mp}Z$ |  Sig($500$ GeV) |  Sig($1$ TeV) |  Sig($500$ GeV) |  Sig($1$ TeV)
|  |  |  |  |  |  |  |  | Case-I | Case-I | Case-II | Case-II
|  |  |  |  |  |  |  |  | (fb) | (fb) | (fb) | (fb)
C-I | 0.0253 | 0.5 | 0.0036 | 4.75 | 0.02 | 0.207 | 0.011 | 0.09 | 0.0012 | 0.001 | 0.0374 | 0.013
| $[100\%]$ | $[100\%]$ | $[100\%]$ | $[100\%]$ | $[100\%]$ | $[100\%]$ | $[100\%]$ | $[100\%]$ | $[100\%]$ | $[100\%]$ | $[100\%]$ | $[100\%]$
C-II | 0.0011 | 0.034 | 0.00022 | 0.333 | 0.00217 | 0.0174 | 0.004 | 0.005 | 0.001 | 0.00096 | 0.0345 | 0.012
| $[4.35\%]$ | $[6.80\%]$ | $[6.10\%]$ | $[7.01\%]$ | $[10.85\%]$ | $[8.40\%]$ | $[36.36\%]$ | $[5.50\%]$ | $[83.30\%]$ | $[96\%]$ | $[92.24\%]$ | $[92.31\%]$
C-III | 0.0002 | 0.007 | 0.0001 | 0.024 | 0.00027 | 0.0021 | 0.00161 | 0.0022 | 0.0006 | 0.00063 | 0.0183 | 0.011
| $[0.79\%]$ | $[1.40\%]$ | $[2.70\%]$ | $[0.50\%]$ | $[1.35\%]$ | $[1.01\%]$ | $[14.63\%]$ | $[2.40\%]$ | $[50\%]$ | $[63\%]$ | $[48.93\%]$ | $[84.61\%]$
Table 9: Cut flow for trilepton plus fat-jet signal with missing energy. Upper
block represents the muon and lower block represents the electron signals. We
consider $M_{Z^{\prime}}=3$ TeV at $\sqrt{s}=14$ TeV.

Solving the signal and background relation for a particular significance
applied in Sec. V.1, we estimate the luminosity to achieve that significance.
We find that the significance of the two benchmarks in Case-I are very close
and roughly below $1-\sigma$. In Case-II, $5-\sigma$ significance can be
attained at $3$ ab-1 luminosity for $M_{N}=500$ GeV and for $M_{N}=1$ TeV
$5-\sigma$ significance could be attained at $1.5$ ab-1.

$\mu^{\pm}\mu^{\pm}\mu^{\mp}/e^{\mp}$ | $t\bar{t}W^{\pm}$ | $t\bar{t}Z$ | $W^{\pm}hj$ | $W^{\pm}Zj$ | $W^{\pm}W^{\pm}W^{\mp}j$ | $ZZj$ | $ZZW^{\pm}$ | $W^{\pm}W^{\mp}Z$ |  Sig($1$ TeV) |  Sig($1$ TeV)
---|---|---|---|---|---|---|---|---|---|---
|  |  |  |  |  |  |  |  | Case-I | Case-II
|  |  |  |  |  |  |  |  | (fb) | (fb)
C-I | 0.107 | 3.82 | 0.0171 | 22.94 | 0.112 | 0.931 | 0.042 | 0.207 | 0.012 | 0.252
| $[100\%]$ | $[100\%]$ | $[100\%]$ | $[100\%]$ | $[100\%]$ | $[100\%]$ | $[100\%]$ | $[100\%]$ | $[100\%]$ | $[100\%]$
C-II | 0.007 | 0.634 | 0.0008 | 3.35 | 0.0203 | 0.11 | 0.006 | 0.044 | 0.011 | 0.24
| $[6.54\%]$ | $[16.60\%]$ | $[4.68\%]$ | $[14.60\%]$ | $[18.12\%]$ | $[11.81\%]$ | $[14.28\%]$ | $[21.26\%]$ | $[91.60\%]$ | $[95.24\%]$
C-III | 0.005 | 0.133 | 0.0004 | 0.455 | 0.0014 | 0.01 | 0.0023 | 0.02 | 0.008 | 0.17
| $[4.67\%]$ | $[3.48\%]$ | $[2.34\%]$ | $[1.98\%]$ | $[1.25\%]$ | $[1.07\%]$ | $[5.48\%]$ | $[9.66\%]$ | $[66.60\%]$ | $[67.46\%]$
$e^{\pm}e^{\pm}e^{\mp}/\mu^{\mp}$ | $t\bar{t}W^{\pm}$ | $t\bar{t}Z$ | $W^{\pm}hj$ | $W^{\pm}Zj$ | $W^{\pm}W^{\pm}W^{\mp}j$ | $ZZj$ | $ZZW^{\pm}$ | $W^{\pm}W^{\mp}Z$ |  Sig($1$ TeV) |  Sig($1$ TeV)
|  |  |  |  |  |  |  |  | Case-I | Case-II
|  |  |  |  |  |  |  |  | (fb) | (fb)
C-I | 0.08 | 2.81 | 0.0125 | 16.87 | 0.083 | 0.685 | 0.024 | 0.153 | 0.008 | 0.18
| $[100\%]$ | $[100\%]$ | $[100\%]$ | $[100\%]$ | $[100\%]$ | $[100\%]$ | $[100\%]$ | $[100\%]$ | $[100\%]$ | $[100\%]$
C-II | 0.0054 | 0.466 | 0.00057 | 2.5 | 0.016 | 0.0984 | 0.0042 | 0.025 | 0.007 | 0.15
| $[6.75\%]$ | $[16.58\%]$ | $[4.56\%]$ | $[14.82\%]$ | $[19.28\%]$ | $[14.36\%]$ | $[17.50\%]$ | $[16.34\%]$ | $[87.50\%]$ | $[83.30\%]$
C-III | 0.00035 | 0.103 | 0.0003 | 0.34 | 0.001 | 0.008 | 0.0017 | 0.0072 | 0.0062 | 0.12
| $[0.44\%]$ | $[3.66\%]$ | $[2.40\%]$ | $[2.01\%]$ | $[1.20\%]$ | $[1.17\%]$ | $[7.08\%]$ | $[4.70\%]$ | $[77.50\%]$ | $[66.60\%]$
Table 10: Cut flow for trilepton plus fat-jet signal with missing energy.
Upper block represents the muon and lower block represents the electron
signals. We consider $M_{Z^{\prime}}=3$ TeV at $\sqrt{s}=27$ TeV.

The cut flow and the efficiencies for $\mu^{\pm}\mu^{\pm}\ell^{\mp}$ signal
and backgrounds at $\sqrt{s}=27$ TeV are shown in the upper panel and those
for the $e^{\pm}e^{\pm}\ell^{\mp}$ events are shown in the lower panel of Tab.
10. We study 1 TeV RHN as a benchmark for Cases-I and II. The leading
background is originated from $WZ+$jets processes, whereas the sub-leading
contribution to the background comes from $WWZ$ process, respectively. The
other backgrounds are 2-3 orders of magnitude smaller than the leading
background. We find that significance is below $2-\sigma$ up to a high
luminosity for Case-I, however, that for Case-II can be $5-\sigma$ at $200$
fb-1 luminosity.

$\mu^{\pm}\mu^{\pm}\mu^{\mp}/e^{\mp}$ | $t\bar{t}W^{\pm}$ | $t\bar{t}Z$ | $W^{\pm}hj$ | $W^{\pm}Zj$ | $W^{\pm}W^{\pm}W^{\mp}j$ | $ZZj$ | $ZZW^{\pm}$ | $W^{\pm}W^{\mp}Z$ |  Sig($1$ TeV) |  Sig($1$ TeV)
---|---|---|---|---|---|---|---|---|---|---
|  |  |  |  |  |  |  |  | Case-I | Case-II
|  |  |  |  |  |  |  |  | (fb) | (fb)
C-I | 0.455 | 44.7 | 0.0811 | 133.9 | 0.771 | 6.854 | 0.14 | 0.746 | 0.124 | 2.54
| $[100\%]$ | $[100\%]$ | $[100\%]$ | $[100\%]$ | $[100\%]$ | $[100\%]$ | $[100\%]$ | $[100\%]$ | $[100\%]$ | $[100\%]$
C-II | 0.0083 | 1.12 | 0.00302 | 7.3 | 0.0622 | 0.338 | 0.11 | 0.0502 | 0.113 | 2.38
| $[1.82\%]$ | $[2.50\%]$ | $[3.72\%]$ | $[5.45\%]$ | $[8.07\%]$ | $[4.93\%]$ | $[78.57\%]$ | $[6.73\%]$ | $[91.13\%]$ | $[93.70\%]$
C-III | 0.00151 | 0.272 | 0.0064 | 0.66 | 0.0057 | 0.09 | 0.034 | 0.0149 | 0.065 | 1.36
| $[0.33\%]$ | $[0.61\%]$ | $[7.90\%]$ | $[0.49\%]$ | $[0.74\%]$ | $[1.31\%]$ | $[24.28\%]$ | $[1.99\%]$ | $[52.42\%]$ | $[53.54\%]$
$e^{\pm}e^{\pm}e^{\mp}/\mu^{\mp}$ | $t\bar{t}W^{\pm}$ | $t\bar{t}Z$ | $W^{\pm}hj$ | $W^{\pm}Zj$ | $W^{\pm}W^{\pm}W^{\mp}j$ | $ZZj$ | $ZZW^{\pm}$ | $W^{\pm}W^{\mp}Z$ |  Sig($1$ TeV) |  Sig($1$ TeV)
|  |  |  |  |  |  |  |  | Case-I | Case-II
|  |  |  |  |  |  |  |  | (fb) | (fb)
C-I | 0.3344 | 32.84 | 0.06 | 98.42 | 0.57 | 5.035 | 0.101 | 0.525 | 0.106 | 1.83
| $[100\%]$ | $[100\%]$ | $[100\%]$ | $[100\%]$ | $[100\%]$ | $[100\%]$ | $[100\%]$ | $[100\%]$ | $[100\%]$ | $[100\%]$
C-II | 0.0034 | 0.82 | 0.0022 | 5.343 | 0.046 | 0.772 | 0.01 | 0.084 | 0.098 | 1.74
| $[1.02\%]$ | $[2.50\%]$ | $[3.60\%]$ | $[5.43\%]$ | $[8.07\%]$ | $[15.33\%]$ | $[9.90\%]$ | $[16\%]$ | $[92.45\%]$ | $[95.08\%]$
C-III | 0.002 | 0.12 | 0.0017 | 0.483 | 0.005 | 0.076 | 0.0021 | 0.025 | 0.060 | 0.93
| $[0.60\%]$ | $[0.36\%]$ | $[2.83\%]$ | $[0.49\%]$ | $[0.88\%]$ | $[1.51\%]$ | $[2.08\%]$ | $[4.76\%]$ | $[56.60\%]$ | $[50.82\%]$
Table 11: Cut flow for trilepton plus fat-jet signal with missing energy.
Upper block represents the muon and lower block represents the electron
signals. We consider $M_{Z^{\prime}}=3$ TeV at $\sqrt{s}=100$ TeV.

The cut flow and the efficiencies for $\mu^{\pm}\mu^{\pm}\ell^{\mp}$ and
$e^{\pm}e^{\pm}\ell^{\mp}\ell^{\mp}$ events at $\sqrt{s}=100$ TeV are shown in
the upper and lower panels of Tab. 11. We notice that a $5-\sigma$
significance can be attained at $2$ ab-1 in Case-I whereas the same can be
achieved at the $15$ fb-1 luminosity in Case-II.

###  V.3 Bounds on $M_{N}-M_{Z^{\prime}}$ plane

Figure 8: $2-\sigma$ exclusion at different hadron colliders on
$M_{N}-M_{Z^{\prime}}$ plane from SSDL+2$J$ signature. We consider $3$ ab-1
luminosity for $14$ TeV and $27$ TeV colliders and $30$ fb-1 luminosity for
$100$ TeV respectively.

Applying the constraints on the $g^{\prime}-M_{Z^{\prime}}$ plane we produce
the RHN pair from $Z^{\prime}$ using Cases-I and II. We study the boosted
objects from the RHNs. Considering the SSDL final state manifesting the
Majorana nature of the RHNs we perform a scan over $3$ TeV $\leq
M_{Z^{\prime}}<7$ TeV considering $500$ GeV $\leq
M_{N}<\frac{M_{Z^{\prime}}}{2}$. We consider the leading decay mode $\ell W$
from each RHN and the heavy mass will boost each $W$ boson from the RHN to
produce fat-jet. Hence we study the SSDL+ 2 J signature at hadron colliders
with $\sqrt{s}=14$ TeV, $27$ TeV and $100$ TeV respectively. Following the
selection and advanced cuts described in Sec. V.1, we produce the signal and
study the backgrounds to produce a $2-\sigma$ exclusion limit in the
$M_{N}-M_{Z^{\prime}}$ plane. The contours are produced using $3$ ab-1
luminosity for $14$ TeV and $27$ TeV colliders and $30$ fb-1 luminosity for
$100$ TeV collider respectively. These contours for Case-I (II) are shown in
the left (right) panel of Fig. 8. Hence we infer SSDL plus two fat-jets
provides an interesting handle to study the RHN pair production from
$Z^{\prime}$. In this context, we mention that trilepton plus single fat-jet
signature is another interesting aspect that could be tested from the RHN pair
production, however, to test this channel we require a high luminosity at $27$
TeV and $100$ TeV colliders. In that case, we could probe a relatively smaller
parameter space compared to the SSDL scenario in the proton colliders.

##  VI Heavy neutrino pair production at the $e^{+}e^{-}$ collider

Pair production of heavy neutrinos at the electron positron collider is
another important aspect of this model which can be studied at $\sqrt{s}=250$
GeV, $500$ GeV and $1$ TeV for Cases-I and II. We calculate the heavy neutrino
pair production at $e^{-}e^{+}$ collider in terms of $M_{Z^{\prime}}$, $N_{N}$
and $x_{H}$ as

$\displaystyle\sigma(e^{-}e^{+}\to{Z^{\prime}}^{\ast}\to
N_{i}N_{i})=\frac{{g^{\prime}}^{4}Q_{N}^{2}s}{192\pi}\frac{(8+12x_{H}+5x_{H}^{2})}{(s-M_{Z^{\prime}}^{2})^{2}+M_{Z^{\prime}}^{2}\Gamma_{Z^{\prime}}^{2}}\Big{(}1-4\frac{M_{N}^{2}}{s}\Big{)}^{\frac{3}{2}}$
(38)

where $Q_{N}$ is the $U(1)_{X}$ charge of the heavy neutrino under the
$U(1)_{X}$ gauge group and $\Gamma_{Z^{\prime}}$ is the total decay width of
$Z^{\prime}$ in Cases-I and II respectively. In the limit of
$M_{Z^{\prime}}>\sqrt{s}$ the pair production of the heavy neutrinos from Eq.
38 reduces to

$\displaystyle\sigma(e^{-}e^{+}\to{Z^{\prime}}^{\ast}\to
N_{i}N_{i})\simeq\Big{(}\frac{{g^{\prime}}^{4}}{{M_{Z^{\prime}}}^{4}}\Big{)}\frac{Q_{N}^{2}s(8+12x_{H}+5x_{H}^{2})}{192\pi}\Big{(}1-4\frac{M_{N}^{2}}{s}\Big{)}^{\frac{3}{2}}$
(39)

We note that the cross section is directly proportional to
$\frac{g^{\prime}}{M_{Z^{\prime}}}$. Before estimating the cross sections at
different $\sqrt{s}$ we calculate bounds on $g^{\prime}$ depending on
$M_{Z^{\prime}}$ using the dilepton cross sections from ATLAS [83], CMS [84]
and ATLAS technical design report [85] and dijet cross sections from ATLAS
[86] and CMS [87] using Eq. 34 considering $M_{N}>\frac{M_{Z^{\prime}}}{2}$
which exerts strongest limit on the $U(1)_{X}$ coupling. The constraints from
LEP [88, 89, 90, 91] are obtained using $M_{Z^{\prime}}>\sqrt{s}$. Hence we
obtain the limits on the quantity $\frac{M_{Z^{\prime}}}{g^{\prime}}$ in terms
of $M_{Z^{\prime}}$ for different $x_{H}$ which measures the VEV of the
$U(1)_{X}$ theory. The $Z^{\prime}$ phenomenology at the $e^{-}e^{+}$ collider
been studied in [91] where it has been shown that for $x_{H}=-2$ there is no
interaction between the left handed fermion doublet and $Z^{\prime}$ which is
$U(1)_{R}$ scenario. It can also be noted that for $x_{H}=-1$ the interaction
between the right handed electron and $Z^{\prime}$ vanishes. Therefore we
consider these two charges as they directly affect the interaction between
electron and $Z^{\prime}$ for the heavy neutrino pair production. In addition
to that we consider $x_{H}=1$ where left handed lepton doublet and right
handed electron interact with $Z^{\prime}$, however, right handed down type
quark does not have coupling with $Z^{\prime}$ and it has no direct impact in
the heavy neutrino pair production process at the $e^{-}e^{+}$ collider. The
bounds on $\frac{M_{Z^{\prime}}}{g^{\prime}}$ for $x_{H}=-2$, $-1$ and $1$
depending on $M_{Z^{\prime}}$ from LEP and the $e^{-}e^{+}$ colliders
considering $M_{Z^{\prime}}>\sqrt{s}$ are shown as the parallel lines with
respect to the horizontal axis in Fig. 9. The shaded regions in this figure
are excluded by dilepton and dijet searches from LHC and limits on the
effective scale form LEP-II experiment respectively. These lower bounds show
the scale of the VEV could be probed in these experiments. We estimate the
bounds on $\frac{M_{Z^{\prime}}}{g^{\prime}}$ at the HL-LHC scaling the ATLAS
$(\mathcal{L}_{\rm current}=139$ fb${}^{-1})$ and CMS $(\mathcal{L}_{\rm
current}=140$ fb${}^{-1})$ results for $M_{N}>\frac{M_{Z^{\prime}}}{2}$ using
$g^{\prime}\simeq g^{\prime}_{\rm current}\sqrt{\frac{\mathcal{L}_{\rm
current}}{\mathcal{L}_{\rm future}}}$ where $\mathcal{L}_{\rm future}=3000$
fb-1. These prospective bounds estimated by scaling the ATALS (CMS) bounds are
shown by the orange dashed (dot-dashed) line in Fig. 9.

We calculate the cross section for the heavy neutrino pair production being
normalized by the $U(1)_{X}$ gauge coupling for $M_{Z^{\prime}}=7.5$ TeV (left
panel) and 10 TeV (right panel) depending on $x_{H}$ for different $\sqrt{s}$
in Fig. 10 fixing $M_{N}=100$ GeV. We find that the cross sections attain
certain values at $x_{H}=-2$ and decreases with the increase in $x_{H}$
attaining a minimum at $x_{H}=-1.2$. Finally the cross section increases with
the increase in $x_{H}$ and attains a maximum at $x_{H}=1$ and maintains a
constant value with $x_{H}\geq 1$. We show the Case-I (II) in the upper
(lower) panel of Fig. 10 for $\sqrt{s}=250$ GeV, 500 GeV and 1 TeV in each
panel from bottom to top respectively. In this calculation we consider only
one generation of the heavy neutrino for simplicity. Production of more than
one generation of heavy neutrino pair can be possible, however, due to the
universal $U(1)_{X}$ gauge coupling of our model, the corresponding cross
sections can be multiplied by the number of generations.

Figure 9: Lower limits on $M_{Z^{\prime}}/g^{\prime}$ as a function of
$M_{Z^{\prime}}$ for $x_{H}=-2$ (top, left), $-1$ (top, right) and $1$
(bottom) respectively from dilepton searches at ATLAS [83], CMS [84] and
ATLAS-TDR [85], dijet searches from ATLAS [86] and CMS [87] respectively.
Considering $M_{Z^{\prime}}>>\sqrt{s}$ we estimate the limits on
$M_{Z^{\prime}}/g^{\prime}$ as a function of $M_{Z^{\prime}}$ from LEP-II [88]
and prospective ILC [106]. The shaded regions are excluded by respective
experiments.

Figure 10: Heavy neutrino pair production cross section normalized by the
U$(1)_{X}$ gauge coupling at electron positron colliders for $M_{N}=100$ GeV
fixing $M_{Z^{\prime}}=7.5$ TeV (left panel) and 10 TeV (right panel)
depending on $x_{H}$ for Cases-I (upper panel) and II (lower panel). We
consider three different center of mass energies $\sqrt{s}=250$ GeV, 500 GeV
and 1 TeV from bottom to top respectively in each panel.

###  VI.1 SSDL+2j signal

In electron positron colliders we can produce the heavy neutrinos in pair and
each heavy neutrino decay into the leading mode following $N\to\ell W$. In
this analysis we consider SSDL plus four jet signal. The jets are coming from
the hadronic decay of one of the $W$ bosons. To study this signal we consider
$x_{H}=-2$ and 1 for $\sqrt{s}=250$ GeV, 500 GeV and 1 TeV respectively for
$5$ TeV $\leq M_{Z^{\prime}}\leq 20$ TeV. From Fig. 10 we find that heavy
neutrino pair production cross section is small at $x_{H}=-1$ compared to
$x_{H}=-2$ and 1. Therefore in further analyses we proceed with $x_{H}=-2$ and
$1$. We considered $100$ GeV $\leq M_{N}\leq 125$ GeV for $\sqrt{s}=250$ GeV,
$100$ GeV $\leq M_{N}\leq 250$ GeV for $\sqrt{s}=500$ GeV, and $100$ GeV $\leq
M_{N}\leq 500$ GeV for $\sqrt{s}=1$ TeV respectively to produce the density
plots for Cases-I and II in Figs. 11 and 12 respectively. The bar chart in the
right of each panel represents corresponding cross sections at different
$\sqrt{s}$ for different $x_{H}$. The results in Case-II is higher than the
Case-I due to different $U(1)_{X}$ charges of the heavy neutrinos. Due to the
heavy neutrino pair production process, the cross section becomes negligibly
small at the threshold $M_{N}\sim\frac{\sqrt{s}}{2}$. The cross section almost
remains the same before the vicinity of this threshold beyond which the cross
section sharply drops when $M_{N}\sim\frac{\sqrt{s}}{2}$. The density plots
also resemble the nature of the production cross section depending on $x_{H}$
as shown in Fig. 10. According to the choice of $x_{H}$ we obtain the maximum
cross section at $x_{H}=1$. The cross section increases with the increase in
$\sqrt{s}$. In this analysis we consider the strongest bounds on $g^{\prime}$
for different $M_{Z^{\prime}}$ at $x_{H}=-2$ and 1 respectively which can be
easily estimated form Fig. 9. The results for $x_{H}=-2$ (1) is shown in the
upper (lower) panel of each figure for $\sqrt{s}=250$ GeV, $500$ GeV and 1 TeV
from left to right.

Figure 11: The density plot representing dilepton plus four jet cross section
on $M_{N}-M_{Z^{\prime}}$ plane for Case-I considering $x_{H}=-2$ (upper
panel) and 1 (lower panel) at different center of mass energies such as 250
GeV, 500 GeV and 1TeV from left to right. The bar chart represents the cross
sections in fb.

Figure 12: The density plot representing dilepton plus four jet cross section
on $M_{N}-M_{Z^{\prime}}$ plane for Case-II considering $x_{H}=-2$ (upper
panel) and 1 (lower panel) at different center of mass energies such as 250
GeV, 500 GeV and 1TeV from left to right. The bar chart represents the cross
sections in fb.

###  VI.2 3$\ell$+2j+missing energy signal

In electron positron colliders we can produce the heavy neutrinos in pair and
each heavy neutrino decay into the leading mode following $N\to\ell W$. In
this analysis we consider a trilepton signal in association with two jets and
missing energy. The jets are coming from the hadronic decay of one of the $W$
bosons whereas the third lepton and missing momentum is coming the leptonic
decay of the other $W$ boson. In this case we consider two generations of the
heavy neutrinos comprising the trilepton signal where the leptons include all
possible combinations with electron and muon with trilepton charge combination
as $+1$ and $-1$. To study this signal we consider $x_{H}=-2$ and 1 for
$\sqrt{s}=250$ GeV, 500 GeV and 1 TeV respectively for $5$ TeV $\leq
M_{Z^{\prime}}\leq 20$ TeV. We considered $100$ GeV $\leq M_{N}\leq 125$ GeV
for $\sqrt{s}=250$ GeV, $100$ GeV $\leq M_{N}\leq 250$ GeV for $\sqrt{s}=500$
GeV, and $100$ GeV $\leq M_{N}\leq 500$ GeV for $\sqrt{s}=1$ TeV respectively
to produce the density plots for Cases-I and II in Figs. 13 and 14
respectively. The results in Case-II is roughly more than one order of
magnitude higher than the results of Case-I due to different $U(1)_{X}$
charges of the heavy neutrinos. Due to the heavy neutrino pair production
process, the cross section becomes negligibly small at the threshold
$M_{N}\sim\frac{\sqrt{s}}{2}$. The cross section almost remains the same
before the vicinity of $M_{N}\sim\frac{\sqrt{s}}{2}$ after that it drops
sharply to zero. According to the choice of $x_{H}$ we obtain the maximum
cross section at $x_{H}=1$. The cross section increases with the increase in
$\sqrt{s}$.

Figure 13: The density plot representing trilepton plus two jet cross section
in association with missing energy on $M_{N}-M_{Z^{\prime}}$ plane for Case-I
considering $x_{H}=-2$ (upper panel) and 1 (lower panel) at different center
of mass energies such as 250 GeV, 500 GeV and 1TeV from left to right. The bar
chart represents the cross sections in fb.

Figure 14: The density plot representing trilepton plus two jet cross section
in association with missing energy on $M_{N}-M_{Z^{\prime}}$ plane for Case-II
considering $x_{H}=-2$ (upper panel) and 1 (lower panel) at different center
of mass energies such as 250 GeV, 500 GeV and 1TeV from left to right. The bar
chart represents the cross sections in fb.

###  VI.3 Bounds on $M_{N}-M_{Z^{\prime}}$ plane

We simulate the SSDL plus four jet signal the generic backgrounds using
MadGraph [97, 98], hadronizing the events by PYTHIA8 [99] followed by the
detector simulation using the ILD card in Delphes [100] for different $x_{H}$
and $\sqrt{s}$ varying $M_{N}$ and $M_{Z^{\prime}}$ according to Figs. 11 and
12 for Cases-I and II respectively to prepare a 2$-\sigma$ limit plot in the
$M_{N}-M_{Z^{\prime}}$ plane. We apply $p_{T}^{j}>20$ GeV, $p_{T}^{\ell}>10$
GeV, $|\eta^{j,l}|<2.5$ to estimate the SSDL background from $W^{\pm}W^{\pm}$
and $4Z$ channels. The SSDL plus four jet process can be generated from
$W^{\pm}W^{\pm}$ process in association with missing energy where two same
sign $W$ will decay leptonically into same flavor and the remaining ones will
decay hadronically. The $e^{\pm}e^{\pm}+4j$ background has extremely small
cross section at 250 GeV, however, the cross section becomes 0.0015 fb and
0.0082 fb at 500 GeV and 1 TeV respectively. The $\mu^{\pm}\mu^{\pm}+4j$
background also has extremely small cross section at 250 GeV, however, the
cross section becomes 0.0015 fb and 0.0082 fb at 500 GeV and 1 TeV
respectively. From $4Z$ we consider two of the $Z$ bosons decay leptonically
and rest of the two decay hadronically. The cross section for this process at
500 GeV is $3\times 10^{-6}$ fb and $1.2\times 10^{-5}$ fb respectively with
the electrons. Similar cross sections can be obtained for muons. We also
produce the $ZZ$+jets background where each $Z$ decays into $e^{-}e^{+}$
$(\mu^{-}\mu^{+})$ modes giving rise to SSDL pairs. The cross sections for the
$e^{-}e^{+}$ modes at 250 GeV, 500 GeV and 1 TeV are $1.5\times 10^{-5}$ fb,
0.00126 fb and 0.00132 fb respectively. The cross sections for the
$\mu^{-}\mu^{+}$ modes at 250 GeV, 500 GeV and 1 TeV are $1.5\times 10^{-5}$
fb, 0.00127 fb and 0.00133 fb respectively. We estimate the $t\overline{t}$
backgrounds at $\sqrt{s}=$ 500 GeV and 1 TeV applying $p_{T}^{j}>20$ GeV,
$p_{T}^{\ell}>10$ GeV, $|\eta^{j,l}|<2.5$. We can not generate this background
at $\sqrt{s}=250$ GeV which is energetically disallowed. We consider the final
state $4j2b$ which has the cross section 222.4 fb (68.0 fb) at $\sqrt{s}=$ 500
GeV (1 TeV). The cross section of $2e2\nu 2b$ final state at $\sqrt{s}=$ 500
GeV (1 TeV) is 6.23 fb (1.88 fb). We find that a final state of $2\mu 2\nu 2b$
has a cross section of 6.23 fb (1.9 fb) at $\sqrt{s}=500$ GeV (1 TeV). Finally
we consider another combination of final state $e\mu 2\nu 2b$ form
$t\overline{t}$ process having cross section of 12.4 fb (3.76 fb) at
$\sqrt{s}=$ 500 GeV (1 TeV). We ensure that the final signal and SM
backgrounds have SSDL pair only and impose the azimuthal angular cut on the
leptons as $|\cos\theta_{\ell}|<0.95$ defining
$\theta_{\ell}=\tan^{-1}\Big{(}\frac{p_{T}^{\ell}}{p_{Z}^{\ell}}\Big{)}$ where
$p_{T}^{\ell}$ as the transverse momentum and $p_{z}$ is the $z-$ component of
the three momentum of the lepton respectively. We impose a missing energy cut
for the events such that $E_{T}^{\rm miss}<80$ GeV. The backgrounds coming
from the top quark pair production do not survive after the application of
these cuts. Considering the signals and backgrounds we estimate the $2-\sigma$
significance limit using $\frac{S}{\sqrt{S+B}}$ where $S$ stands for signal
events and $B$ stands for backgrounds using luminosities as 2 ab-1, 4 ab-1 and
8 ab-1 at 250 GeV, 500 GeV and 1 TeV $e^{-}e^{+}$ colliders following Ref.
[107]. Corresponding limit plots are shown in Fig. 15 where the Case-I (II) is
given in the upper (lower) panel for $x_{H}=-2$ (1) in the left (right) panel
combining the electron and muon events. At $\sqrt{s}=250$ GeV we predict that
$6.2$ TeV $\leq M_{Z^{\prime}}<9.5$ TeV with $0.095$ TeV $\leq M_{N}\leq
0.115$ TeV could be probed at 2$-\sigma$ level for $x_{H}=-2$ in Case-I.
Similarly we predict that $6.4$ TeV $\leq M_{Z^{\prime}}<12.25$ TeV with
$0.095$ TeV $\leq M_{N}\leq 0.12$ TeV could be probed at 2$-\sigma$ level for
$x_{H}=1$ in Case-I. Due to the improved charges in case of Case-II we predict
that at 2$-\sigma$ level $5.7$ TeV $\leq M_{Z^{\prime}}\leq 16$ TeV with $0.1$
TeV $\leq M_{N}\leq 0.12$ TeV could be probed at 250 GeV for $x_{H}=-2$. This
range for the heavy $Z^{\prime}$ could be extended up to 20 TeV at 250 GeV
keeping the range of $M_{N}$ almost same for Case-II considering $x_{H}=1$.
These ranges widen in case of $M_{Z^{\prime}}$ and as well as $M_{N}$ for 500
GeV and 1 TeV colliders respectively.

Figure 15: 2$-\sigma$ contour plot on $M_{N}-M_{Z^{\prime}}$ plane in the
upper (lower) panel for Case-I(II) considering $x_{H}=-2$ (left) and $1$
(right) at 2 ab-1, 4 ab-1 and 8 ab-1 luminosities in 250 GeV, 500 GeV and 1
TeV $e^{-}e^{+}$ colliders respectively for the combined SSDL plus four jet
signal.

We simulate the trilepton plus two jet signal in association with missing
energy and the generic backgrounds using MadGraph [97, 98], hadronizing the
events by PYTHIA8 [99] followed by the detector simulation using the ILD card
in Delphes [100] for different $x_{H}$ and $\sqrt{s}$ varying $M_{N}$ and
$M_{Z^{\prime}}$ according to Figs. 13 and 14 for Cases-I and II respectively
to prepare a 2$-\sigma$ limit plot in the $M_{N}-M_{Z^{\prime}}$ plane. We
apply $p_{T}^{j}>20$ GeV, $p_{T}^{\ell}>10$ GeV, $|\eta^{j,l}|<2.5$ to
estimate the three electron plus two jets in association with missing energy
generic SM background and the cross sections are 0.111 fb at $\sqrt{s}=250$
GeV, $1.05$ fb at $\sqrt{s}=500$ GeV and $3.53$ fb at $\sqrt{s}=1$ TeV
respectively. Using the same cuts we simulate the three muon plus two jet
background in association with missing energy and obtain the corresponding
generic SM backgrounds cross sections as 0.1 fb at $\sqrt{s}=250$ GeV, 0.2 fb
at $\sqrt{s}=500$ GeV and 0.2 fb at $\sqrt{s}=1$ TeV respectively. Using the
mentioned cuts we produce the two electron and one muon generic SM background
in association with two jets and missing momentum. The cross sections at
$\sqrt{s}=250$ GeV, 500 GeV and 1 TeV are obtained as 0.061 fb, 0.6 fb and
1.63 fb respectively. Similarly we generate the two muon and one electron
generic SM background events in association with two jets and missing energy
at $\sqrt{s}=250$ GeV, 500 GeV and 1 TeV. We checked that $t\overline{t}Z$
channel gives extremely low cross sections $\mathcal{O}(10^{-4}\rm fb)$ after
the application of the kinematic cuts. Hence we do not consider this process
in further analysis. After applying the cuts we obtain the cross sections as
0.05 fb, 0.18 fb and 0.352 fb respectively. To estimate the SM backgrounds we
considered charge combinations of the three charged leptons as $+1$ and $-1$.
Using the azimuthal angular cut on the leptons as $|\cos\theta_{\ell}|<0.95$.
To estimate the signal and generic SM background events we consider the
luminosities as 2 ab-1, 4 ab-1 and 8 ab-1 at 250 GeV, 500 GeV and 1 TeV
$e^{-}e^{+}$ colliders following [107]. We estimate the 2$-\sigma$ contours on
the $M_{N}-M_{Z^{\prime}}$ plane in the upper (lower) panel of Fig. 16 using
$\frac{S}{\sqrt{S+B}}$ using the maximum reach of the luminosities at
different $e^{-}e^{+}$ colliders for Case-I (II) where result for $x_{H}=-2$
(1) is shown in the left (right) panel. We find that combined trilepton final
states with electron and muon flavors can provide 2-$\sigma$ exclusion limits
on $M_{N}-M_{Z^{\prime}}$ plane. The 2$-\sigma$ exclusion limit for Case-I at
$x_{H}=-2$ is shown in the top left panel of Fig. 16 for 6.4 TeV $\leq
M_{Z^{\prime}}\leq 7.6$ TeV for 0.095 TeV $\leq M_{N}\leq 0.117$ TeV at
$\sqrt{s}=500$ GeV whereas the range increases up to 6.1 TeV $\leq
M_{Z^{\prime}}\leq 10.5$ TeV for 0.095 TeV $\leq M_{N}\leq 0.34$ TeV at
$\sqrt{s}=1$ TeV. On the other hand for $x_{H}=1$, the cross section is higher
and we can probe 6.5 TeV $\leq M_{Z^{\prime}}\leq 6.95$ TeV for 0.095 TeV
$\leq M_{N}\leq 0.195$ TeV at $\sqrt{s}=500$ GeV and 6.25 TeV $\leq
M_{Z^{\prime}}\leq 13.75$ TeV for 0.095 TeV $\leq M_{N}\leq 0.45$ TeV at
$\sqrt{s}=1$ TeV which is shown in the top right panel of Fig. 16. Due to the
change in the $U(1)_{X}$ charge of the heavy neutrinos in Case-II we find
enhancement in the Majorana heavy neutrino pair production cross section which
further allows us to obtain 2$-\sigma$ exclusion contours analyzing the signal
and the backgrounds which are shown in the lower panel of Fig. 16 for
$x_{H}=-2$ (left) and 1 (right) respectively. In this case we can probe the
$M_{N}-M_{Z^{\prime}}$ plane at 250 GeV for 6.1 TeV $\leq M_{Z^{\prime}}\leq
9.0$ TeV and 0.095 TeV $\leq M_{N}\leq 0.165$ TeV in case of $x_{H}=-2$ and
that goes to 6.4 TeV $\leq M_{Z^{\prime}}\leq 11.2$ TeV and 0.095 TeV $\leq
M_{N}\leq 0.12$ TeV in case of $x_{H}=1$. However, the reaches in
$M_{Z^{\prime}}$ and $M_{N}$ increase with the increase in $\sqrt{s}$.

Figure 16: 2$-\sigma$ contour plot on $M_{N}-M_{Z^{\prime}}$ plane in the
upper (lower) panel for Case-I(II) considering $x_{H}=-2$ (left) and $1$
(right) at 2 ab-1, 4 ab-1 and 8 ab-1 luminosities in 250 GeV, 500 GeV and 1
TeV $e^{-}e^{+}$ colliders respectively for combined trilepton plus two jet in
association with missing momentum.

##  VII Heavy neutrino pair production from BSM scalar at $e^{-}e^{+}$
colliders

There is another interesting aspect of the Majorana heavy neutrino pair
production from scalar involved in a general $U(1)_{X}$ scenario. In Case-I
for simplicity, applying the stationary conditions on the scalar potential in
Eq. 2 we find that the mass matrix of the scalars as

$\displaystyle
M^{2}=\begin{pmatrix}2\lambda_{h}v_{h}^{2}&\lambda^{\prime}v_{h}v_{\Phi}\\\
\lambda^{\prime}v_{h}v_{\Phi}&2\lambda_{\Phi}v_{\Phi}^{2}\\\ \end{pmatrix}$
(40)

Hence diagonalizing the mass matrix in Eq. 40 we obtain the mass eigenvalues
of the physical scalars as

$\displaystyle
m_{h_{1}}^{2}=\lambda_{h}v_{h}^{2}+\lambda_{\Phi}v_{\Phi}^{2}-\sqrt{(\lambda_{h}v_{h}^{2}-\lambda_{\Phi}v_{\Phi}^{2})^{2}+(\lambda^{\prime}v_{h}v_{\Phi})^{2}},\,\,m_{h_{2}}^{2}=\lambda_{h}v_{h}^{2}+\lambda_{\Phi}v_{\Phi}^{2}+\sqrt{(\lambda_{h}v_{h}^{2}-\lambda_{\Phi}v_{\Phi}^{2})^{2}+(\lambda^{\prime}v_{h}v_{\Phi})^{2}}~{}~{}~{}~{}$
(41)

and the scalar quartic couplings in Eq. 2 of Case-I can be written as

$\displaystyle\lambda_{h}=\frac{m_{h_{2}}^{2}}{4v_{h}^{2}}(1-\text{cos}~{}2\alpha)+\frac{m_{h_{1}}^{2}}{4v_{h}^{2}}(1+\text{cos}~{}2\alpha),\,\,\lambda_{\Phi}=\frac{m_{h_{1}}^{2}}{4v_{\Phi}^{2}}(1-\text{cos}~{}2\alpha)+\frac{m_{h_{2}}^{2}}{4v_{\Phi}^{2}}(1+\text{cos}~{}2\alpha),$
(42)
$\displaystyle\lambda^{\prime}=\text{sin}~{}2\alpha\left(\frac{m_{h_{1}}^{2}-m_{h_{2}}^{2}}{2v_{\Phi}v_{h}}\right)$
(43)

where $\alpha$ is the mixing angle between the two scalars required to
diagonalize the mass matrix given in Eq. 40. The BSM scalar $h_{2}$ can decay
into SM particles or gauge bosons through the scalar mixing and the partial
decay width of $h_{2}$ can be written as

$\displaystyle\Gamma(h_{2}\to X_{\rm SM}X_{\rm
SM})=\sin^{2}\alpha~{}\Gamma_{h\to X_{\rm SM}X_{\rm SM}}(m_{h}\to m_{h_{2}})$
(44)

where $\Gamma_{h\to X_{SM}X_{SM}}$ is the partial decay widths of the SM
Higgs. The BSM scalar can decay into SM Higgs $(m_{h_{1}}=125~{}\rm GeV)$
through the interaction

$\displaystyle\mathcal{L}=C_{h_{2}h_{1}h_{1}}h_{2}h_{1}h_{1}$ (45)

where
$C_{h_{2}h_{1}h_{1}}\equiv\frac{1}{2}[(-s_{\alpha}^{3}v_{h}+c_{\alpha}^{3}v_{\Phi}+2c_{\alpha}^{2}s_{\alpha}v_{h}-2c_{\alpha}s_{\alpha}^{2}v_{\Phi})\lambda^{\prime}-6c_{\alpha}^{2}s_{\alpha}v_{h}\lambda_{h}+6c_{\alpha}s^{2}_{\alpha}v_{\Phi}\lambda_{\Phi}]$
and $s_{\alpha}(c_{\alpha})=\sin\alpha(\cos\alpha)$. The decay width for
$h_{2}\to h_{1}h_{1}$ is given by

$\Gamma(h_{2}\to h_{1}h_{1})=\frac{C_{h_{2}h_{1}h_{1}}^{2}}{8\pi
m_{h_{2}}}\sqrt{1-4\frac{m_{h_{1}}^{2}}{m_{h_{2}}^{2}}}.$ (46)

The BSM scalar interacts with the pair of heavy neutrinos through the Yukawa
interaction in Eq. 1. The partial decay process of $h_{2,1}\to NN$ can be
written as

$\displaystyle\Gamma(h_{2}\to
NN)=\frac{3Y_{2}^{2}\cos^{2}\alpha}{16\pi}m_{h_{2}}\left(1-4\frac{M_{N_{i}}^{2}}{m_{h_{2}}^{2}}\right)^{\frac{3}{2}},\,\,\Gamma(h_{1}\to
NN)=\frac{3Y_{2}^{2}\sin^{2}\alpha}{16\pi}m_{h_{1}}\left(1-4\frac{M_{N_{i}}^{2}}{m_{h_{2}}^{2}}\right)^{\frac{3}{2}}$
(47)

where $Y_{2}=\frac{M_{N_{i}}\sqrt{2}}{v_{\Phi}}$ with
$v_{\Phi}=\frac{M_{Z^{\prime}}}{2g^{\prime}}$. Hence we find a relation
between the $Z^{\prime}$ and heavy neutrino mass as
$g^{\prime}=\frac{1}{2\sqrt{2}}\frac{Y_{2}M_{Z^{\prime}}}{M_{N_{i}}}$.

The complete form of the partial decay widths of $h_{1}$ and $h_{2}$ are given
in the Appendix. We show the branching ratios of $h_{2}$ into different modes
including a pair of RHNs in Fig. 17 as a function of $m_{h_{2}}$ considering
$M_{Z^{\prime}}=5$ TeV, $g^{\prime}=0.1$ and $M_{N_{i}}=$ 50 GeV. The allowed
benchmark value of $g^{\prime}$ is taken from [91] satisfying LHC dilepton,
dijet and LEP-II constraints. We consider three benchmarks of
$\sin\alpha=0.1$, 0.05 and 0.025 respectively which is taken from [108] which
are allowed benchmark points after the application of LEP and LHC limits. In
this case we consider $x_{H}=0$ which is the B$-$L case.

Figure 17: Branching ratios of $h_{2}$ as functions of $m_{h_{2}}$ where
$m_{Z^{\prime}}=5$ TeV, $g^{\prime}=0.1$ and $M_{N_{i}}=50$ GeV.

We consider two different modes of the $h_{2}$ production one at the electron
positron collider. One is $e^{-}e^{+}\to Zh_{2}$ and the other is
$e^{+}e^{-}\to h_{2}\nu\overline{\nu}$. In Fig. 18 we show the cross sections
of $Zh_{2}$ (upper panel) and $h_{2}\nu\overline{\nu}$ (lower panel)
respectively considering $\sin\alpha=0.1$ and $m_{h_{2}}=150$ GeV and 300 GeV
respectively as a function of $\sqrt{s}$. In this study we consider three
choices of the polarizations $P_{e^{-}}=P_{e^{+}}=0$,
$P_{e^{-}}=-0.3,P_{e^{+}}=0.8$ and $P_{e^{-}}=0.3,P_{e^{+}}=-0.8$
respectively.

Figure 18: Cross section for $e^{+}e^{-}\to Zh_{2}$ (upper panel) and
$e^{+}e^{-}\to h_{2}\nu\overline{\nu}$ (lower panel) process as functions of
$\sqrt{s}$ for different values of $m_{h_{2}}$ and different electron and
positron polarizations.

producing $h_{2}$ in the above modes we consider the heavy neutrino pair
production followed by the decay $h_{2}\to N_{i}N_{i}$. Note that $h_{2}$ can
be produced via $e^{+}e^{-}\to e^{+}e^{-}h_{2}$ via $Z$ boson fusion but its
production cross section is smaller compared to the other modes. As a result
we did not consider it in this analysis. In our following analysis masses of
extra scalar and heavy neutrinos are fixed to be $m_{h_{2}}=150$ GeV,
$M_{N_{1}}=M_{N_{2}}=50$ GeV for illustration. We also choose
$M_{Z^{\prime}}=6.5$ TeV and $g^{\prime}=1$ to enhance $h_{2}\to N_{i}N_{i}$
decay mode. We consider two generations of the heavy neutrinos where $N_{1}$
dominantly decays into electron and $N_{2}$ dominantly decays into muon. Here
we consider the dominant decay modes of the heavy neutrinos where
$N_{i}\to\ell jj$ where two jets come from the hadronic decay mode of
${W^{\pm}}^{\ast}$.

###  VII.1 Signal from $e^{+}e^{-}\to Zh_{2}$

Here we consider associated production process $e^{+}e^{-}\to Zh_{2}$ for
$\sqrt{s}=250$ GeV since the process provides the largest $h_{2}$ production
cross section at the center of mass energy. In this case our signal process is
$e^{+}e^{-}\to h_{2}Z\to NNjj\to\ell^{\pm}\ell^{\pm}+6j$ where two of these
jets are coming from the hadronic decay of the $Z$ boson. We assume that the
Yukawa coupling involved in this process is diagonal. The signal indicates
lepton number violation and number of SM background (BG) events can be reduced
by same sign charged lepton tagging. For the BG from SM processes at the LHC
250 GeV, we consider following processes $\ell^{+}\ell^{-}W^{+}W^{-}$ (BG1),
$\ell^{+}\ell^{-}ZZ$ (BG2), $\ell^{\pm}\nu W^{\pm}Z$ (BG3, BG4) and the cross
sections of these processes are 0.33 fb, 0.094 fb and 2.5 fb respectively at
$\sqrt{s}=250$ GeV. These backgrounds can mimic our signals providing same
sign charged lepton events. We consider $M_{N}=50$ GeV so that heavy neutrinos
can decay off-shell.

We generate signal and SM background events using MadGraph [98]. Then PYTHIA8
[109] is applied to deal with hadronization effects, the initial-state
radiation (ISR) and final-state radiation (FSR) effects and the decays of SM
particles, and Delphes [100] is used for detector level simulation using
$\\{P_{e^{-}},P_{e^{+}}\\}=\\{0,0\\}$. At the detector level, event selection
is imposed with kinematical cuts demanding an SSDL pair for electrons and
muons as: (i) transverse momenta of leptons: $p_{T}^{\ell^{\pm}}>10$ GeV and
those of the jets are: $p_{T}^{j}>20$ GeV, (ii) pseudo-rapidity:
$\eta^{\ell,j}<2.5$ and (iii) missing energy: $\not{E}_{T}<20\ {\rm GeV}$. We
evaluate number of signals and SM baground events imposing selection cuts
where $N_{\rm event}=\mathcal{L}_{\rm int}\sigma\frac{N_{\rm Selected}}{N_{\rm
Generated}}$, $N_{\rm Select}$ is number of events after selection, $N_{\rm
Generated}$ is number of generated events, $\sigma$ is a cross section for
each process and $\mathcal{L}_{\rm int}=$2 ab-1is integrated luminosity. We
combine the electron and muon events claiming at least 3 jets $n_{j}\geq 3$ in
the events. After the application of the selection cuts we evaluate the
significance if the process using

$S=\sqrt{2\left[(N_{S}+N_{BG})\ln\left(1+\frac{N_{S}}{N_{BG}}\right)-N_{S}\right]},$
(48)

In Table 12, we summarize number of signal and SM background events after
selection choosing $\sin\alpha=0.1$ and $0.2$ as benchmark values. Hence we
notice that a sizable significance can be observed at the 250 GeV electron
positron collider.

signal/BGs | signal [$s_{\alpha}=0.1$] | signal [$s_{\alpha}=0.2$] | BG1 | BG2 | BG3 | BG4
---|---|---|---|---|---|---
$N_{\rm ev}$ (without selection) | $6.6\times 10^{2}$ | $1.2\times 10^{3}$ | $6.6\times 10^{2}$ | $1.9\times 10^{2}$ | $5.0\times 10^{3}$ | $5.0\times 10^{3}$
$N_{\rm ev}$ (with selection) | 28 | 51 | 0.81 | 0.30 | 16 | 16
$S[s_{\alpha}=0.1]$ | 4.4
$S[s_{\alpha}=0.2]$ | 7.4
Table 12: Number of signal and SM background events for $\sqrt{s}=250$ GeV and
integrated luminosity 2 ab-1 before and after selection for SSDL mode.

###  VII.2 Signal from $e^{+}e^{-}\to h_{2}\bar{\nu}\nu$

Here we consider $W$ boson fusion for $h_{2}$ production, $e^{+}e^{-}\to
h_{2}\bar{\nu}\nu$, with $\sqrt{s}=1$ TeV since the cross section for $h_{2}$
production increases with $\sqrt{s}$. We use polarization of the $e^{+}$ and
$e^{-}$ beam as $\\{P_{e^{-}},P_{e^{+}}\\}=\\{-0.8,0.3\\}$, since this gives
the largest cross section as shown in the lower panel of Fig. 18. Our signal
process is $e^{+}e^{-}\to h_{2}\nu\bar{\nu}\to
NN\nu\bar{\nu}\to\ell^{\pm}\ell^{\pm}+4j+\nu\bar{\nu}$. We simulate the signal
and the SM backgrounds events using MadGraph [98] followed by hadronization
using PYTHIA8 [109] and detector simulation using Delphes [100]. The SM
backgrounds $\ell^{+}\ell^{-}W^{+}W^{-}$ (BG1), $\ell^{+}\ell^{-}ZZ$ (BG2),
$\ell^{\pm}\nu W^{\pm}Z$ (BG3, BG4) have the cross sections 44.0 fb, 0.48 fb
and 39.0 fb at $\sqrt{s}=1$ TeV respectively. We use the selection cuts for
the signal and SM background events as: $p_{T}(\ell^{\pm})>10\ {\rm GeV}$,
$\eta^{\ell,j}<2.5$, $p_{T}^{j}>20\ {\rm GeV}$ and $\not{E}_{T}>30\ {\rm GeV}$
respectively. As in the previous case we summarize number of events after the
selection cuts and significance in Table 13 where we write the events
combining the signals with electron and muon. We find that the significance
can be sizable at 1 TeV electron positron collider using 2 ab-1 luminosity.

signal/BGs | signal [$s_{\alpha}=0.1$] | signal [$s_{\alpha}=0.2$] | BG1 | BG2 | BG3 | BG4
---|---|---|---|---|---|---
$N_{\rm ev}$ (no selection) | $2.6\times 10^{3}$ | $4.4\times 10^{3}$ | $8.8\times 10^{4}$ | $9.6\times 10^{2}$ | $7.8\times 10^{4}$ | $7.8\times 10^{4}$
$N_{\rm ev}$ (with selection) | $1.1\times 10^{2}$ | $2.0\times 10^{2}$ | $5.7\times 10^{2}$ | 7.4 | $4.6\times 10^{2}$ | $4.5\times 10^{2}$
$S[s_{\alpha}=0.1]$ | 3.0
$S[s_{\alpha}=0.2]$ | 5.1
Table 13: Number of signal and SM background events after selection cuts at
$\sqrt{s}=1$ TeV with $\\{P_{e^{-}},P_{e^{+}}\\}=\\{-0.8,0.3\\}$ and
integrated luminosity 2 ab-1 for SSDL plus four jet process.

##  VIII Conclusions

We have considered two general U$(1)$ extensions of the SM where neutrino mass
can be generated by the seesaw mechanism. To cancel the gauge and mixed gauge-
gravity anomalies, these models include three generations of the Majorana type
heavy neutrinos which could be produced at the high energy colliders. Studying
the existing constraints on the U$(1)_{X}$ gauge coupling as a function of the
$M_{Z^{\prime}}$ for different $M_{N}$, we produce the RHNs in pair from
$Z^{\prime}$ in hadron colliders at different center of mass energies. We
study the SSDL and trilepton modes manifesting the Majorana nature of the RHNs
considering the leading decay mode of the RHNs. Considering sufficiently heavy
RHNs we find that the $W$ boson from the RHN decay can be sufficiently
boosted. Selecting the signal and backgrounds and passing through advanced
cuts we find that these signals can be obtained with reasonable significance
for different benchmark scenarios of $M_{N}$ and $M_{Z^{\prime}}$ at hadron
colliders. Hence scaning over a range of $M_{Z^{\prime}}$ and $M_{N}$ we
estimate $2-\sigma$ exclusion contours at hadron colliders with different
luminosities for the SSDL signal. Finally we conclude that the SSDL signal
with two fat-jets can be probed at hadron colliders with different
luminosities in the near future. Majorana heavy neutrinos can be produced in
pair at the electron positron colliders where heavy $Z^{\prime}$ can be probed
from the SSDL and trilepton signature. To do this we estimate the limits on
the scale of the vacuum expectation value of the U$(1)$ breaking from LHC and
LEP. Using those limits we simulate the SSDL and trilepton events. Applying
the kinematic cuts on the signal and SM backgrounds on the combined electron
and muon events with jets, we estimate a 2$-\sigma$ contours on the
$M_{N}-M_{Z^{\prime}}$ plane at 250 GeV, 500 GeV and 1 TeV respectively
depending on the choice of the U$(1)$ charges of the particles. In addition to
the $Z^{\prime}$ induced Majorana type heavy neutrino pair production we
consider the production of a BSM scalar under the general U$(1)_{X}$ scenario
considering the current bounds on the scalar mixing angle form LHC and LEP at
different center of mass energies and polarizations. This BSM scalar can decay
into a pair of Majorana type heavy neutrinos through the Yukawa interaction
which can further decay into SSDL modes in association with jets and missing
energy. Studying the signals and corresponding SM backgrounds we find that
such process can also be probed at the electron positron colliders with
sizable significance in the near future.

## Appendix

The partial decay widths of the SM like Higgs boson of mass $m_{h_{1}}$ into
various modes are given below from [110]:

  * (i)

SM fermions (f):

$\displaystyle\Gamma_{h_{1}\to
f{\bar{f}}}=\cos^{2}{\alpha}\times\frac{3N_{c}m_{h_{1}}m_{f}^{2}}{8\pi
v_{h}^{2}}\left(1-\frac{4m_{f}^{2}}{m_{h}^{2}}\right)^{\frac{3}{2}},$ (49)

$N_{c}=1$ and $3$ for SM leptons and quarks respectively.

  * (ii)

on-shell gauge bosons (V $=W^{\pm}$ or $Z$ ):

$\displaystyle\Gamma_{h_{1}\to
VV}=\cos^{2}{\alpha}\times\frac{C_{V}}{32\pi}\frac{m_{h_{1}}^{3}}{v_{h}^{2}}\left(1-4\frac{m_{V}^{2}}{m_{h_{1}}^{2}}\right)^{\frac{1}{2}}\left(1-4\frac{m_{V}^{2}}{m_{h_{1}}^{2}}+12\frac{m_{V}^{4}}{m_{h_{1}}^{4}}\right),$
(50)

where $C_{V}=1$ and $2$ for $V=Z$ or $W^{\pm}$ gauge boson, respectively.

  * (iii)

gluon (g) via top-quark loop:

$\displaystyle\Gamma_{h_{1}\to
gg}=\cos^{2}{\alpha}\times\frac{\alpha_{s}^{2}m_{h_{1}}^{3}}{128\pi^{3}v_{h}^{2}}(F_{1/2}(m_{h_{1}}))^{2},$
(51)

where

$\displaystyle
F_{1/2}(m_{h_{1}})=-2\frac{4m_{t}^{2}}{m_{h_{1}}^{2}}\left[1-\left(1-\frac{4m_{t}^{2}}{m_{h_{1}}^{2}}\left({\rm
sin^{-1}}\left(\frac{m_{h_{1}}}{2m_{t}}\right)\right)^{2}\right)\right].$ (52)

  * (iv)

one off-shell gauge boson :

$\displaystyle\Gamma_{h_{1}\to
W^{\pm}{W^{\pm}}^{\ast}}=\cos^{2}{\alpha}\times\frac{3m_{W}^{4}m_{h_{1}}}{32\pi^{3}v_{h}^{4}}G\left(\frac{m_{W}^{2}}{m_{h_{1}}^{2}}\right)$
(53) $\displaystyle\Gamma_{h_{1}\to
ZZ^{*}}=\cos^{2}{\alpha}\times\frac{3m_{Z}^{4}m_{h_{1}}}{32\pi^{3}v_{h_{1}}^{4}}\left(\frac{7}{12}-\frac{10}{9}\sin^{2}\theta_{W}+\frac{40}{9}\sin^{4}\theta_{W}\right)G\left(\frac{m_{W}^{2}}{m_{h_{1}}^{2}}\right),$
(54)

where $\sin^{2}\theta_{W}=0.231$ and the loop functions can be represented as

$\displaystyle
G(x)=3\frac{1-8x+20x^{2}}{\sqrt{4x-1}}{\cos^{-1}}\left(\frac{3x-1}{2x^{3/2}}\right)-\frac{|1-x|}{2x}(2-13x+47x^{2})-\frac{3}{2}(1-6x+4x^{2})\log(\sqrt{x}),$
(55)

where $1/4<x<1$, for energetically allowed decays. For the $U(1)_{X}$ scalar
$h_{2}$, $\cos{\alpha}$ will be replaced by $\sin\alpha$.

###### Acknowledgements.

The work of S.M. is supported by KIAS Individual Grants (PG086001) at Korea
Institute for Advanced Study.

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