#  Searching for the minimal Seesaw models at the LHC and beyond

Arindam<EMAIL_ADDRESS>ORCID-ID:orcid.org/0000-0002-8934-2300
School of Physics, KIAS, Seoul 130-722, Korea

###### Abstract

The existence of the tiny neutrino mass and the flavor mixing can be naturally
explained by type-I Seesaw model which is probably the simplest extension of
the Standard Model (SM) using Majorana type SM gauge singlet heavy Right
Handed Neutrinos (RHNs). If the RHNs are around the Electroweak(EW)-scale
having sizable mixings with the SM light neutrinos, they can be produced at
the high energy colliders such as Large Hadron Collider (LHC) and future $100$
TeV proton-proton (pp) collider through the characteristic signatures with
same sign di-lepton introducing lepton number violations(LNV). On the other
hand Seesaw models, namely inverse Seesaw, with small LNV parameter can
accommodate EW-scale pseudo-Dirac neutrinos with sizable mixings with SM light
neutrinos while satisfying the neutrino oscillation data. Due to the smallness
of the LNV parameter of such models, the ‘smoking-gun’ signature of same-sign
di-lepton is suppressed where as the RHNs in the model will manifest at the
LHC and future $100$ TeV pp collider dominantly through the Lepton number
conserving (LNC) trilepton final state with Missing Transverse Energy (MET).
Studying various production channels of such RHNs we give an updated upper
bound on the mixing parameters of the light-heavy neutrinos at the 13 TeV LHC
and future $100$ TeV pp collider.

type-I Seesaw, right handed neutrinos, bounds

###### Contents

  1. I Introduction
  2. II Production cross sections
  3. III Decay of the RHNs at the LHC
  4. IV Bounds on the mixing angle
    1. IV.1 Same-sign di-lepton plus di-jet signal
    2. IV.2 Trilepton signal
  5. V Conclusions
  6. VI Conflicts of Interest

##  I Introduction

The experimental evidence of the neutrino oscillation and flavor mixings from
neutrino oscillation experiments Neut1 ; Neut2 ; Neut3 ; Neut4 ; Neut5 ; Neut6
indicates that the SM is not enough to explain the existence of the tiny
neutrino mass and flavor mixing. After the pioneering discovery of the $d=5$
operator Weinberg:1979sa  within the SM using the SM leptons, SM Higgs
doublets and $\Delta L=2(L=\rm{Lepton~{}number})$ unit of LNV, it turned out
that the Seesaw mechanism Seesaw0 ; Seesaw1 ; Seesaw2 ; Seesaw3 ; Seesaw4 ;
Seesaw5 ; Seesaw6  could be the simplest idea to explain the small neutrino
mass and flavor mixing where the SM can be extended by the SM-gauge singlet
Majorana type RHNs. After the Electroweak (EW) symmetry breaking, the light
Majorana neutrino masses are generated after the so-called type-I Seesaw
mechanism.

In type-I Seesaw, we introduce SM gauge-singlet Majorana RHNs $N_{R}^{\beta}$
where $\beta$ is the flavor index. $N_{R}^{\beta}$ couples with the SM lepton
doublets $\ell_{L}^{\alpha}$ and the SM Higgs doublet $H$. The relevant part
of the Lagrangian is

$\displaystyle\mathcal{L}\supset-
Y_{D}^{\alpha\beta}\overline{\ell_{L}^{\alpha}}HN_{R}^{\beta}-\frac{1}{2}m_{N}^{\alpha\beta}\overline{N_{R}^{\alpha
C}}N_{R}^{\beta}+H.c..$ (1)

where $Y_{D}^{\alpha\beta}$ is the Yukawa coupling, $\ell_{L}^{\alpha}$ is the
SM $SU(2)_{L}$ lepton doublet and $m_{N}$ is the Majorana mass term. After the
spontaneous EW symmetry breaking by the vacuum expectation value (VEV),
$H=\begin{pmatrix}\frac{v}{\sqrt{2}}\\\ 0\end{pmatrix}$, we obtain the Dirac
mass matrix as $M_{D}=\frac{Y_{D}v}{\sqrt{2}}$ suppressing the indices. Using
the Dirac and Majorana mass matrices we can write the neutrino mass matrix as

$\displaystyle M_{\nu}=\begin{pmatrix}0&&M_{D}\\\
M_{D}^{T}&&m_{N}\end{pmatrix}$ (2)

where $M_{D}$ is the Dirac mass matrix and $m_{N}$ is the Majorana mass term.
Diagonalizing $M_{\nu}$ we obtain the Seesaw formula for the light Majorana
neutrinos as

$\displaystyle m_{\nu}\simeq-M_{D}m_{N}^{-1}M_{D}^{T}.$ (3)

For $m_{N}\sim 100$ GeV, we may find $Y_{D}\sim 10^{-6}$ with $m_{\nu}\sim
0.1$ eV. However, in the general parameterization for the Seesaw formula
Casas:2001sr , $Y_{D}$ can be large and sizable $(\mathcal{O}(1))$, and this
is the case we consider in this paper.

The searches of the Majorana RHNs can be performed by the ‘smoking-gun’ of the
same-sign di-lepton plus di-jet signal which is suppressed by the square of
the light-heavy neutrino mixing parameter $|V_{\ell
N}|^{2}\simeq|m_{D}m_{N}^{-1}|^{2}$. A comprehensive general study on the
parameters of $|V_{\ell N}|^{2}$ has been given in Das:2017nvm  using the data
from the neutrino oscillation experiments Neut1 ; Neut2 ; Neut3 ; Neut4 ;
Neut5 ; Neut6 , bounds from the Lepton Flavor Violation (LFV) experiments
Adam:2011ch ; Aubert:2009ag ; OLeary:2010hau , Large Electron-Positron (LEP)
and Electroweak Precision test deBlas:2013gla ; Adriani:1992pq ;
Acciarri:1999qj ; Achard:2001qv ; Abreu:1996pa ; Blondel:2014bra ;
delAguila:2008pw ; Akhmedov:2013hec  experiments using the non-unitarity
effects Antusch:2006vwa ; Abada:2007ux  applying the Casas- Ibarra conjecture
Casas:2001sr ; Ibarra:2010xw ; Ibarra:2011xn ; Dinh:2012bp ; Penedo:2017knr .
At this point we mention that Das:2017nvm  has a good agreement with a
previous analysis Rasmussen:2016njh  on the sterile neutrinos. The bounds in
Das:2017nvm  has been compared with the existing results in Achard:2001qv ;
delAguila:2008pw ; Akhmedov:2013hec ; BhupalDev:2012zg ; Aad:2015xaa ;
CMS8:2016olu  considering the degenerate Majorana RHNs. In case of Seesaw
mechanism the Dirac Yukawa matrix $(Y_{D})$ can carry the flavors where the
RHN mass matrix is considered to be diagonal. This case is favored by the
neutrino oscillation data as studied in Das:2017nvm ; Kersten:2007vk ;
Xing:2009in ; He:2009ua ; Ibarra:2010xw ; Deppisch:2010fr ; Dev:2013oxa . Such
a scenario for the Seesaw scenario is called the Flavor Non-Diagonal (FND).
The other possibility of considering both of the diagonal $Y_{D}$ and $m_{N}$
is not supported by the neutrino oscillation data.

Since any number of singlets can be added in a gauge theory without
contributing to anomalies, one can utilize such freedom to find a natural
alternative of the low-scale realization of the Seesaw mechanism. Simplest
among such scenarios is commonly known as the inverse Seesaw mechanism
Mohapatra:1986aw ; Mohapatra:1986bd  where a small Majorana neutrino mass
originates from tiny LNV parameters rather than being suppressed by the RHN
mass as done in the case of conventional Seesaw mechanism. In the inverse
Seesaw model two sets of SM-singlet RHNs are introduced which are pseudo-Dirac
by nature and their Dirac Yukawa couplings can be even order one, while
reproducing the neutrino oscillation data. Therefore at the high energy
colliders such pseudo-Dirac neutrinos can be produced through a sizable mixing
with the SM light neutrinos Das:2012ze ; Das:2014jxa ; Das:2015toa ;
Das:2016akd ; Das:2016hof ; Das:2017pvt ; Das:2017zjc ; Das:2017rsu .

In the inverse Seesaw mechanism the relevant part of the Lagrangian is given
by

$\displaystyle\mathcal{L}\supset-
Y_{D}^{\alpha\beta}\overline{\ell_{L}^{\alpha}}HN_{R}^{\beta}-m_{N}^{\alpha\beta}\overline{S_{L}^{\alpha}}N_{R}^{\beta}-\frac{1}{2}\mu_{\alpha\beta}\overline{S_{L}^{\alpha}}S_{L}^{\beta^{C}}+H.c.,$
(4)

where $N_{R}^{\alpha}$ and $S_{L}^{\beta}$ are two SM-singlet heavy neutrinos
with the same lepton numbers, $m_{N}$ is the Dirac mass matrix, and $\mu$ is a
small Majorana mass matrix violating the lepton numbers. After the EW symmetry
breaking we obtain the neutrino mass matrix as

$\displaystyle M_{\nu}=\begin{pmatrix}0&&M_{D}&&0\\\
M_{D}^{T}&&0&&m_{N}^{T}\\\ 0&&m_{N}&&\mu\end{pmatrix}.$ (5)

Diagonalizing this mass matrix we obtain the light neutrino mass matrix

$\displaystyle M_{\nu}\simeq M_{D}m_{N}^{-1}\mu(m_{N}^{-1})^{T}M_{D}^{T}.$ (6)

Note that the smallness of the light neutrino mass originates from the small
LNV term $\mu$. The smallness of $\mu$ allows the $m_{D}m_{N}^{-1}$ parameter
could be on the order one even for an EW scale RHNs. In the inverse Seesaw
case we can consider $Y_{D}$ as non-diagonal when $\mu$ and $m_{N}$ are
diagonal which is called the Flavor Non-Diagonal (FND) case. On the other hand
we can also consider the diagonal $Y_{D}$, $m_{N}$ when $\mu$ will be non-
diagonal. This situation is called the Flavor Democratic scenario. For the
inverse Seesaw both of the FND and FD cases are supported by the neutrino
oscillation data. In this article we will consider the FND case from the
Seesaw and the FD case from the inverse Seesaw mechanisms.

Through the Seesaw mechanism, a flavor eigenstate ($\nu$) of the SM neutrino
is expressed in terms of the mass eigenstates of the light ($\nu_{m}$) and
heavy ($N_{m}$) Majorana neutrinos such as

$\displaystyle\nu\simeq\nu_{m}+V_{\ell N}N_{m},$ (7)

Using the mass eigenstates, the charged current (CC) interaction for the heavy
neutrino is given by

$\displaystyle\mathcal{L}_{CC}\supset-\frac{g}{\sqrt{2}}W_{\mu}\bar{e}\gamma^{\mu}P_{L}V_{\ell
N}N_{m}+H.c.,$ (8)

where $e$ denotes the three generations of the charged leptons in the vector
form, and $P_{L}=\frac{1}{2}(1-\gamma_{5})$ is the projection operator.
Similarly, the neutral current (NC) interaction is given by

$\displaystyle\mathcal{L}_{NC}\supset-\frac{g}{2c_{w}}Z_{\mu}\left[\overline{N_{m}}\gamma^{\mu}P_{L}|V_{\ell
N}|^{2}N_{m}+\left\\{\overline{\nu_{m}}\gamma^{\mu}P_{L}V_{\ell
N}N_{m}+H.c.\right\\}\right],$ (9)

where $c_{w}=\cos\theta_{w}$ with $\theta_{w}$ being the weak mixing angle and
$W_{\mu},Z_{\mu}$ are the SM gauge bosons.

The main decay modes of the heavy neutrino are $N\rightarrow\ell W$,
$\nu_{\ell}Z$, $\nu_{\ell}h$. The corresponding partial decay widths
Das:2012ze  are respectively given by

$\displaystyle\Gamma(N\rightarrow\ell W)$ $\displaystyle=$
$\displaystyle\frac{g^{2}|V_{\ell
N}|^{2}}{64\pi}\frac{(m_{N}^{2}-m_{W}^{2})^{2}(m_{N}^{2}+2m_{W}^{2})}{m_{N}^{3}m_{W}^{2}},$
$\displaystyle\Gamma(N\rightarrow\nu_{\ell}Z)$ $\displaystyle=$
$\displaystyle\frac{g^{2}|V_{\ell N}|^{2}}{128\pi
c_{w}^{2}}\frac{(m_{N}^{2}-m_{Z}^{2})^{2}(m_{N}^{2}+2m_{Z}^{2})}{m_{N}^{3}m_{Z}^{2}},$
$\displaystyle\Gamma(N\rightarrow\nu_{\ell}h)$ $\displaystyle=$
$\displaystyle\frac{|V_{\ell N}|^{2}(m_{N}^{2}-m_{h}^{2})^{2}}{32\pi
m_{N}}\left(\frac{1}{v}\right)^{2}.$ (10)

The decay width of heavy neutrino into charged gauge bosons being twice as
large as neutral one owing to the two degrees of freedom $(W^{\pm})$. We plot
the branching ratios $BR_{i}\left(={\Gamma_{i}}/{\Gamma_{\rm total}}\right)$
of the respective decay modes $\left(\Gamma_{i}\right)$ with respect to the
total decay width $\left(\Gamma_{\rm total}\right)$ of the heavy neutrino into
$W$, $Z$ and Higgs bosons in Fig. 1 as a function of the heavy neutrino mass
$\left(m_{N}\right)$.

Figure 1: Heavy neutrino branching ratios ($BR_{i}$) for different decay modes
are shown with respect to its mass $\left(m_{N}\right)$ Das:2016hof .

Note that for larger values of $m_{N}$ such as $m_{N}\geq 1500$ GeV, the
branching ratios can be obtained as

$\displaystyle BR\left(N\rightarrow\ell W\right):BR\left(N\rightarrow\nu
Z\right):BR\left(N\rightarrow\nu h\right)\simeq 2:1:1.$ (11)

In this paper we study the RHN production from various initial states such as
the quark-quark $(qq)$, quark-gluon$(qg)$ and gluon-gluon$(gg)$ at the 13 TeV
LHC and future $100$ TeV pp collider. We consider the photon mediated
processes as well as the gluon-gluon fusion (ggF), photon-proton interactions
and Vector Boson Fusion (VBF) processes to produce the RHNs. For different RHN
masses $(m_{N})$, we calculate the cross section of each of the processes
normalized by the square of the light-heavy neutrino mixing angles.

This paper is is organized as follows. In Sec. II we calculate the production
cross sections at the 13 TeV LHC and future $100$ TeV pp collider Arkani-
Hamed:2015vfh  with a variety of initial states. In Sec. III we study the
multilepton decay modes of the RHNs at the $13$ TeV LHC and future $100$ TeV
pp collider. For both of the cases we take the luminosity as $3000$ fb-1. In
Sec. IV we put the bounds on the mixing angle as function of the RHN mass.
Sec. V is dedicated for conclusions.

##  II Production cross sections

The RHNs can be produced at the 13 TeV LHC and future $100$ TeV pp collider
from various initial states. We consider the combined production of the heavy
neutrinos from various initial states including quark-(anti)quark
$(q\bar{q}^{\prime})$, quark-gluon $(qg)$ and gluon-gluon $(gg)$ interactions.
We also study contributions coming from the gluon-gluon fusion (ggF), photon-
proton interactions and Vector Boson Fusion (VBF) processes to produce the
RHNs at the LHC and beyond. In this section we consider the RHN productions in
association with SM leptons and jets. To obtain the cross sections and
generate the events we implement our model in MadGraphAlwall:2011uj  bundled
with PYTHIASjostrand:2006za  and DELPHESdeFavereau:2013fsa . The production
modes of the heavy neutrinos in association with leptons and jets are
proportional to the square of the mixing angle, $|V_{\ell N}|^{2}$. In this
section we use $|V_{\ell N}|^{2}=10^{-4}$ to estimate the cross sections
222The EWPD bounds $|V_{\ell N}|^{2}$ have been discussed in deBlas:2013gla ;
delAguila:2008pw ; Akhmedov:2013hec . The universal bound has been considered
to be $9\times 10^{-4}$ which rules out the possibility of the mixing angles
above this value. Therefore we used a value of $|V_{\ell N}|^{2}<9\times
10^{-4}$ to calculate the cross sections..

  * (1)

Charged current interaction mediated by $W$:

> generate $pp\rightarrow N\ell$
> add process $pp\rightarrow N\ell j$
> add process $pp\rightarrow N\ell jj$
along with the charge conjugates where $\tt j$ stands for the jets.

In this process the the $s$-channel quark anti-quark pair of different flavors
$(q\overline{q^{\prime}})$ will interact through the $W$ exchange and finally
follow Eq. 8 to produce the RHN $(N)$ in association with a lepton $(\ell)$.
The corresponding Feynman diagram for $q\overline{q^{\prime}}\rightarrow
N\ell$ is given in Fig. 2.

The contributions from the quark-quark interaction $(q\overline{q^{\prime}})$
to one-jet $(N\ell j)$ are given in Fig. 3 for the $t$ -channel process and
additional contributions to the one-jet process from the quark-gluon $(qg)$
interaction as shown in Fig. 4 form the $s,~{}t$-channel processes. In both of
the cases the $W$ is radiated form the quarks/ anti-quarks and hence follow
the CC to to produce $N$ in association with $\ell$.

two-jet contributions $(N\ell jj)$ coming from the $q\overline{q^{\prime}}$
process are shown in Fig. 5. In this case, the $s,~{}t$ channel contributions
between the quarks are involved to produce $N$ with $\ell$ in association with
two jets following the CC interactions at the $N$ production vertex. The
quark-gluon $(qg)$ processes contributing in the two-jet case are shown in
Fig. 6 where the $s,~{}t$ channel contributions between the quarks and gluons
are involved to produce $N$ with $\ell$ in association with two jets following
the CC interactions at the $N$ production vertex. In addition to that gluon-
gluon $(gg)$ processes contributed in $N\ell jj$ are shown in Fig. 7 in the
$s,~{}t$-channel followed by the $N$ production from the CC interactions with
$\ell$ in association with two jets.

Figure 2: $q\overline{q^{\prime}}(q\overline{q})$ process to produce
$N\ell(N\nu)$ from the CC(NC) interaction. Figure 3: 1-jet process to produce
$N$ with $\ell(\nu)$ in association with a jet from the CC(NC) interaction
from $q\overline{q^{\prime}}(q\overline{q})$ initial state. Figure 4: 1-jet
process to produce $N$ with $\ell(\nu)$ in association with a jet from the
CC(NC) interaction from $qg$ initial state. Figure 5: 2-jet process to produce
$N$ with $\ell(\nu)$ in association with a jet from the CC(NC) interaction
from $q\overline{q^{\prime}}(q\overline{q})$ initial state. Figure 6: 2-jet
process to produce $N$ with $\ell(\nu)$ in association with a jet from the
CC(NC) interaction from $qg$ initial state. Figure 7: 2-jet process to produce
$N$ with $\ell(\nu)$ in association with a jet from the CC(NC) interaction
from $gg$ initial state.

We combine all these processes to compute the production cross section of the
RHNs through the charged current interaction for $N\ell X$ final state where
$X$ stands for $n$-jet and $n=0,1,2,3$. To produce such processes we use the
following trigger cuts

    * (a)

transverse momentum of the jets, $p_{T}^{j}>30$ GeV,

    * (b)

transverse momentum of the leptons, $p_{T}^{\ell}>10$ GeV,

    * (c)

pseudo-rapidity of the jets, $|\eta^{j}|<2.5$,

    * (d)

pseudo-rapidity of the leptons, $|\eta^{\ell}|<2.5$

with the Multi-Leg Matching (MLM) scheme. In the MLM matching scheme one can
obtain $n$-parton events by MadEvent generator under the $k_{T}$ clustering
algorithm. After the rewight, each final state parton can find the node where
it was generated. This was $n$-parton shower can be generated $(\mu_{g}^{2})$
in the directions of the primary partons so that the initial condition for
each parton shower is kept at $\mu_{g}^{2}$. Hence running the $k_{T}$
clustering algorithm one can find all the jets at the resolution scale
$(\tau_{r})$. If all the jets match with the primary partons then the event is
accepted otherwise rejected. For elaborate discussions on this mechanism for
the event generators, see Matching ; Matching1 ; Matching2 ; Matching3 ;
Matching31 ; Matching32 ; Matching33 ; Matching34 ; Mangano:2002ea ;
Mangano:2006rw ; Cooper:2011gk ; Alwall:2007fs ; Catani:2001cc . In MadGraph
the switch ${\tt ickkw=1}$ stands for the MLM scheme in MadGraph Alwall:2011uj
using

    * (i)

${\tt ickkw}=1$

    * (ii)

${\tt xqcut}=p_{T}^{j}$

    * (iii)

${\tt QCUT=Max\\{xqcut+5GeV,1.2\ast xqcut\\}}$

using ${\tt MSTP(81)=21}$. The MSTP switches modifies the generation
procedure. In this case to switch off or on the Initial State Radiation (ISR),
Final State Radiation (FSR), Multiple Interactions (MI) among the beam jets
and fragmentation. It is also used to give the ‘parton skeleton’ of the hard
process. MSTP(81)=21 is the master switch in PYTHIA 6.4 which switches on the
Multi-Interactions (MI) for the new model at the time of showering and
hadronization. The ${\tt MSTP=21}$ is used for the $p_{T}$ ordered shower
switching on SHOWER-KT Alwall:2008qv ; Matching4 ; Matching5 ; Hoche:2006ph
in PYTHIA Sjostrand:2006za . The trigger cuts used on the basis of the
anomalous multi-lepton searches done by the CMS CMS-trilep . The use of cone
jets or $k_{T}$ jets is decided by whether the parameter ${\tt xqcut}$
(specifying the minimum $k_{T}$ jet measure between jets, i.e. gluons or
quarks (except top quarks) which are connected in Feynman diagrams) in the
${\tt MadGraph}$ is set to be $0$ or not. If ${\tt xqcut=0}$, cone jets are
used, while if ${\tt xqcut>0}$, $k_{T}$ jet matching is assumed. In this case,
transverse momentum of jet $(p_{T}^{j})$ and separation between the jets
$(\Delta R_{jj})$ should be set to zero. For most processes, the generation
speed can be improved by setting ${\tt p_{T}^{j}=xqcut}$ which is done
automatically if the switch ${\tt auto-ptj-mjj}$ is set to ‘T’ at the time of
event generation using ${\tt MadGraph}$ to control the transverse momentum of
the jets $(p_{T}^{j})$ and the invariant mass of the jets $(m_{jj})$. If some
jets should not be restricted this way (as in single top or Vector Boson
Fusion (VBF) production, where some jets are not radiated from QCD) in that
case the switch ${\tt auto-pTj-mjj}$ should be set to ‘F’ in the MagGraph at
the time of event generation. ${\tt QCUT}$ is used for the matching with the
$k_{T}$ scheme, this is case the jet measure cutoff is used by Pythia. If the
value is not given, it will be set to ${\tt max(xqcut+5,xqcut*1.2)}$, where
${\tt xqcut}$ is taken from the ${\tt MadGraph}$ Matching ; Matching1 . We use
such cuts to keep the analysis collinear safe. The production cross sections
are shown in Figs. 11 and 12 respectively as a function of $m_{N}$.

  * (2)

Neutral current interaction mediated by $Z$:

> generate $pp\rightarrow N\nu$
> add process $pp\rightarrow N\nu j$
> add process $pp\rightarrow N\nu jj$
followed by the same production procedure as we discussed in (1). Using quark
anti-quark pair of same flavor $(q\overline{q})$ at the $s$-channel by Eq. 9
$N$ can be produced in association with a light neutrino $(\nu)$ as shown in
Fig. 2. The $N\nu j$ (in Figs. 3, 4) and $N\nu jj$ (in Figs. 5, 6, 7)
processes are same as those have been discussed in (1). The only difference is
the production vertex of $N\nu$ comes from the NC from Eq. 9. Combining these
final states we can denote as $N\nu X$ where $X$ stands for $n$-jet with
$n=0,1,2,3$. In this case we use the following trigger cuts

    * (a)

transverse momentum of the jets, $p_{T}^{j}>30$ GeV,

    * (b)

pseudo-rapidity of the jets, $|\eta^{j}|<2.5$

and the same matching scheme used in (1). Such $p_{T}^{j}$ cuts will keep the
analysis collinear safe specially in (1) and (2) to calculate the cross
sections using MLM matching Matching ; Matching1 ; Matching2 ; Matching3 ;
Mangano:2002ea ; Mangano:2006rw ; Cooper:2011gk ; Alwall:2007fs ;
Catani:2001cc  schemes.

Figure 8: ggF production of the heavy neutrinos from Higgs (h) and Z bosons.
$h\rightarrow N\nu$ can be prompt when $m_{h}>m_{N}$ giving an enhancement to
the $N\nu$ production from the ggF channel. For $m_{h}<m_{N}$, off-shell Higgs
$(h^{\ast})$ produces $N\nu$ final state. Similarly off-shell $Z$ $(Z^{\ast})$
can also produce $N\nu$ final state when $m_{N}>m_{Z}$. For the Higgs mediated
case, top quark loop will contribute dominantly whereas for the $Z^{\ast}$
case all contributions coming from the quarks will be counted according to its
coupling with the quarks.

The production cross sections at the $13$ TeV LHC and future $100$ TeV pp
collider are shown in Figs. 11 and 12 respectively as a function of $m_{N}$.

  * (3)

Gluon-fusion channel (ggF):
In the ggF top loop produces the SM Higgs and Higgs can decay into $N$ in
association with $\nu$ Das:2017zjc ; Das:2017rsu ; Hessler:2014ssa . The
corresponding Feynman diagram is given in the left panel of Fig. 8. There is
another complementary production channel of $N$ in association with $\nu$
which comes from the $Z$ Das:2017zjc ; Ruiz:2017yyf . The corresponding
Feynman diagram is given in the right panel of Fig. 8 The production process
depends upon $m_{N}$, one can produce them promptly from the Higgs decay when
$m_{N}<m_{h}$ where $m_{h}$ is the Higgs mass and also from the off-shell $Z$
$(Z^{\ast})$. When $m_{N}<m_{h}$, Higgs can decay into RHNs and the partial
decay width can be written as

$\displaystyle\Gamma_{h}^{\rm{new}}=\frac{Y_{D}^{2}m_{H}}{8\pi}(1-\frac{m_{N}^{2}}{m_{h}^{2}})^{2}.$
(12)

This has additive contribution to the total decay width of the SM Higgs boson
keeping $Y_{D}$ as a free parameter. Constraints on $Y_{D}$ and $|V_{\ell
N}|^{2}$ have recently been studied in BhupalDev:2012zg ; Das:2017zjc ;
Das:2017rsu . The cross section goes down at the 13 TeV LHC for $m_{h}<m_{N}$
due to the off-shell decay. However, at the future $100$ TeV pp collider the
cross section again rises around $m_{N}=250$ GeV due to large contributions
coming from the gluon PDFs. In this channel we are testing $m_{N}=100$GeV $-$
$1$ TeV. The production cross sections are shown in Figs. 11 and 12
respectively with respect to $m_{N}$. At the future $100$ TeV, the large gluon
PDF starts contributing to the RHN production resulting a rise in the ggF
curve compared to the 13 TeV result.

  * (4)

Photon initiated processes:
The photon (denoted by ${\tt a}$ in MadGraph) initiated processes also have
important contributions in the RHNs production which have been studied in
Williams:1934ad ; vonWeizsacker:1934nji ; Budnev:1974de ; Dev:2013wba ;
Alva:2014gxa ; Das:2015toa ; Deppisch:2015qwa . In case of the the photon
initiated process, the photon can be radiated from the proton and also from a
parton. Those production channels will give additive contributions at the
colliders once the RHNs decay into multi-lepton modes. The corresponding
Feynman diagrams for these processes are given in Figs. 9. The $N\ell j$
production process is possible as $\gamma\rightarrow WW$ vertex is present
which will make the production cross section for $N\ell j$ higher in
comparison to that of $N\nu j$ where $\gamma\rightarrow ZZ$ vertex is not
present under the SM gauge group. $N\nu j$ will be produced from the
$Z\rightarrow N\nu$ vertex under the NC interaction according to Eq. 9. For
the photon initiated production processes we use
> generate ${\tt pa\rightarrow N\ell j~{}QED=3~{}QCD=0}$333QED (Quantum
> Electrodynamic) processes switches the QED interactions with photon, $W$ and
> $Z$ bosons, QED$=3$ means the order of the interaction whereas QCD stands
> for the Quantum Chromodynamic processes with QCD$=0$ is the order of the
> interaction.
with $p_{T}^{j}>30$ GeV, $p_{T}^{\ell}>10$ GeV, $|\eta^{\ell,j}|<2.5$. To
generate the $N\nu j$ final state with photon initiated process we use
> generate ${\tt pa\rightarrow N\nu j~{}QED=3~{}QCD=0}$
with $p_{T}^{j}>30$ GeV, $|\eta^{j}|<2.5$.

Figure 9: The photon initiated $N\ell j$ final state. The $N\nu j$ final state
can also be obtained, however, the contribution will be low as
$\gamma\rightarrow ZZ$ vertex is absent due to the gauge invariance. Figure
10: VBF production of the $N\ell jj$ final state from the $W\rightarrow N\ell$
vertex. The $N\nu jj$ finals state can also be obtained using the
$Z\rightarrow N\nu$ vertex. The two jets can be obtained at the forward-
backward regions.

  * (5)

Vector Boson Fusion (VBF) channels:
The VBF processes are shown in Fig. 10 where the $N\ell$ and $N\nu$ are
produced with two widely separated jets and the corresponding final states
will be $N\ell jj(N\nu jj)$ depending upon the CC (NC) vertex. In the VBF case
the $\gamma WW$ and $ZWW$ vertices will come into effect to produce the $N$ in
association with $\ell(\nu)$ depending upon the CC (NC) between $N-W(N-Z)$. To
generate the $N\ell jj(N\nu j)$ final state from the VBF channels we use
>generate ${\tt pp\rightarrow N\ell jj~{}QED=4~{}QCD=0}$
with $p_{T}^{j}>30$ GeV, $p_{T}^{\ell}>10$ GeV, $|\eta^{\ell}|<2.5$,
$|\eta^{j_{1}}-\eta^{j_{2}}|>4.2$,
$p_{T}^{j_{1},^{\rm{leading}}}=p_{T}^{j_{2},^{\rm{trailing}}}>50$ GeV,
$m_{jj}^{min}=250$ GeV.
>generate ${\tt pp\rightarrow N\nu jj~{}QED=4~{}QCD=0}$
with $p_{T}^{j}>30$ GeV, $p_{T}^{\ell}>10$ GeV, $|\eta^{\ell}|<2.5$,
$|\eta^{j_{1}}-\eta^{j_{2}}|>4.2$,
$p_{T}^{j_{1},^{\rm{leading}}}=p_{T}^{j_{2},^{\rm{trailing}}}>50$ GeV,
$m_{jj}^{min}=250$ GeV. The corresponding production cross sections for the 13
TeV LHC and future $100$ TeV pp collider are shown in Figs. 11 and 12
respectively with respect to $m_{N}$.

Figure 11: Production cross sections of the RHNs in association with leptons
from different modes at the 13 TeV LHC with $|V_{\ell N}|^{2}=10^{-4}$. ‘a’
stands for photon $(\gamma)$ as it is defined by MadGraph. Figure 12:
Production cross sections of the RHNs in association with leptons from
different modes at the future $100$ TeV pp collider with $|V_{\ell
N}|^{2}=10^{-4}$.‘a’ stands for photon $(\gamma)$ as it is defined by
MadGraph.

##  III Decay of the RHNs at the LHC

Implementing our model in the event generator MadGraph, we produce the RHNs
and allow it to decay in various multi-lepton modes. We use kinematic cuts in
this adopting from the CMS analysis in Ref.CMS-trilep . According to our
choice we consider two models such as type-I Seesaw and Inverse Seesaw. In
type-I Seesaw the RHN is Majorana type and in inverse Seesaw the RHN is
pseudo-Dirac type. From the Majorana RHN we get LNV same-sign di-lepton plus
di-jet signal. Whereas from the pseudo-Dirac type RHNs we get LNC trilepton
plus MET $(E_{T}^{miss}/\not{E}_{T})$ final state. Considering the leading
decay mode of the RHNs $(N\rightarrow W\ell)$ and depending upon the choice of
the final states, the $W$ will decay either leptonically or hadronically. We
consider the 13 TeV LHC to study such processes. We consider $m_{N}=100$ GeV
in this section for different production processes. We use the data files
after the hadronization using PYTHIA Sjostrand:2006za  and detector simulation
using DELPHES deFavereau:2013fsa  bundled with MadGrpah Alwall:2011uj  and use
the MLM matched results as prescribed in the previous section. In this
analysis the leading particle is the particle having longest transverse
momentum $(p_{T})$ distribution where as the trailing particle is that which
has shorter transverse momentum $(p_{T})$ distribution. We have plotted the
transverse momentum distributions of the leptons $(p_{T}^{\ell})$ in Fig. 13
and the Missing Transverse Energy (MET/$E_{T}^{miss}/\not{E}_{T}$)
distribution in the Fig. 14 from the trilepton plus MET $(3\ell+$MET$+X)$
signal. We notice that the leading lepton transverse momentum, $p_{T}^{\ell}$
for $\ell_{1}$ peaks around $40$ GeV where as the trailing leptons mostly stay
between $10$ GeV$-40$ GeV. Looking at the distribution of the leading lepton
and the peak of the distribution we suggest that the leading lepton is coming
from the leptonic decay of the $W$. The MET distribution peaks around $40$ GeV
which also backs the ides of having the leading lepton of the $W$. Therefore
for the trilepton analysis at $m_{N}=100$ GeV using $E_{T}^{miss}<50$ GeV will
be reasonable choice.

Figure 13: Transverse momenta distributions for the three leptons from the
$\tt{3\ell+MET}$ final state from $pp\rightarrow N\ell X,N\rightarrow\ell
W,W\rightarrow\ell\nu$ at the 13 TeV LHC. Figure 14: MET distribution from the
$\tt{3\ell+MET}$ final state from $pp\rightarrow N\ell X,N\rightarrow\ell
W,W\rightarrow\ell\nu$ at the 13 TeV LHC.

We study the same sign di-lepton plus di-jet $(\ell^{\pm}\ell^{\pm}jj+X)$
where $X$ stands for the radiated jets. This signal is an important signature
to test the Majorana nature of the RHNs produced from the type-I Seesaw model.
The transverse momentum of the leptons $(p_{T}^{\ell})$ and the jets
$(p_{T}^{j})$ are plotted in Figs. 15 and 16, respectively. The invariant mass
distribution of the two jets are plotted in Fig. 17. The distribution shows a
peak around $m_{W}$. The invariant mass distribution of the lepton plus two-
jet system has been shown in Fig. 18. The figure show the distributions
including $m_{\ell_{1}jj}$ and $m_{\ell_{2}jj}$. Among them $m_{\ell_{1}jj}$
shows a peak around the assumed $m_{N}$ value around $100$ GeV.

Figure 15: Transverse momentum distributions of leptons $(p_{T}^{\ell})$ from
the same-sign di-lepton plus di-jet final state events. Figure 16: Transverse
momentum distributions of the jets $(p_{T}^{j})$ from the same-sign di-lepton
plus di-jet final state.  Figure 17: The invariant mass distributions of two-
jet $(m_{jj})$ system from the same-sign di-lepton plus di-jet final state
showing a peak around the $W$ mass $(m_{W})$. Figure 18: The invariant mass
distributions of one lepton plus two-jet $(m_{ljj})$ system showing a peak
around $m_{N}\sim 100$ GeV (blue) from the same sign di-lepton plus di-jet
signal.

We study the VBF processes to produce the heavy neutrinos and hence study the
trilepton plus MET final state. We have used the VBF prescriptions written in
the previous section where the jets $j_{1}$ and $j_{2}$ are widely separated.
The other two jets are initial state radiations (ISR) which also are not
populating the central region. The rapidity distributions of the jets are
shown in Fig. 19. This is a striking feature of the VBF process. The leading
lepton coming out of the $W$ decay after the $W$ boson is produced from the
$N\rightarrow W\ell$ decay. The leading leptons are also slightly away form
the central region. We use a lepton pseudo-rapidity cut, $|\eta^{\ell}|<2.5$.
The rapidity distributions of the leptons are shown in the Fig. 20.

Figure 19: Rapidity distributions of the associated jets in the trilepton mode
from VBF process. Figure 20: Rapidity distributions of the the leptons in the
trilepton mode from VBF process.

Another important contribution is coming from the $N\nu jj+X$ process from the
NC interaction between the RHNs and $Z$ bosons. Here $X$ are the radiated
jets. The final state consists of a single lepton, two jets and MET. The
produced jets can be used to reconstruct the $W$ boson. The distribution shows
a peak around the $W$ mass. Similarly from the RHN decay we get
$N\rightarrow\ell jj$ and the invariant mass distribution of the $\ell jj$
system can show a peak around the RHN mass $(m_{N})$. The invariant mass
distribution of $m_{jj}$ is shown in Fig. 21 and that of $m_{\ell jj}$ are
shown in Fig. 22.

Figure 21: Invariant mass distribution of the two jet $(m_{jj})$ from the
$\ell+MET$$+2j+X$ final state from the $pp\rightarrow N\nu X$ process. The two
jet system shows a peak around the $W$ mass $(m_{W})$. Figure 22: Invariant
mass distribution of the lepton plus two jet systems $(m_{\ell jj})$ from the
$\ell+MET$$+2j+X$ final state from the $pp\rightarrow N\nu X$ process. The
lepton plus two jet system shows a peak around $m_{N}\sim 100$ GeV.

We produced the RHNs from the ggF process where the top loop becomes dominant
to produce the Higgs which can promptly decay into the RHN when $m_{N}<m_{H}$.
In our case we consider the $m_{N}=100$ GeV and the Higgs will decay promptly
into $N\nu$ and the RHN will decay into $W$ followed by its hadronic decay.
The invariant mass distribution of the two jet system $(m_{jj})$ is shown in
Fig. 23 where as that of the lepton and two jet system are shown in Fig. 24.
The $m_{jj}$ distribution peaks around the $W$ mass where as the $m_{\ell jj}$
distribution peaks around the RHN mass. For detailed results regarding the
Higgs decaying into RHNs can be found in BhupalDev:2012zg ; Das:2017zjc ;
Das:2017rsu  with pervious and updated limits.

Figure 23: Invariant mass distribution of the two jet $(m_{jj})$ from the
$\ell+MET+2j$ final state coming from ggF process. The two jet system shows a
peak around the $W$ mass $(m_{W})$. Figure 24: Invariant mass distribution of
the lepton plus two jet systems $(m_{\ell jj})$ from the $\ell+MET+2j$ final
state coming from ggF process. The lepton plus two jet system shows a peak
around the RHN mass $(m_{N})$ at the 13 TeV LHC.

##  IV Bounds on the mixing angle

For $m_{N}<M_{Z}$, the RHN can be produced from the on-shell $Z$-decay through
the NC interaction in association with missing energy, however, if
$m_{N}>M_{Z}$ off-shell production will take place with the same final state.
The heavy neutrino can decay according to the CC and NC interactions.
Different production processes, different decay modes of the heavy neutrinos
and various phenomenologies have been discussed in Drewes:2013gca ;
Dube:2017jgo ; Das:2017hmg ; Das:2017gke ; Bhardwaj:2018lma ; Das:2017flq ;
Das:2017pvt ; Das:2016hof ; Das:2017deo ; Alva:2014gxa ; Mitra:2016kov ;
Das:2017kkm ; Drewes:2016jae ; Tang:2017plx . RHN production from various
initial states and as well as scale dependent production cross sections at the
Leading Order (LO) and Next-to-Leading-Oder QCD (NLO QCD) of $N\nu$ have been
studied at the 14 TeV LHC and future $100$ TeV pp collider Das:2016hof ;
Das:2017pvt . The L3 collaboration Adriani:1992pq  has performed a search on
such heavy neutrinos directly from the LEP data and found a limit on
$\mathcal{B}(Z\rightarrow\nu N)<3\times 10^{-5}$ at the $95\%$ CL for the mass
range up to 93 GeV. The exclusion limits from L3 are given in Fig. 25 where
the red dot-dashed line stands for the limits obtained from electron (L3-$e$)
and the red dashed line stands for the exclusion limits coming from $\mu$
(L3-$\mu$). The corresponding exclusion limits on $|V_{(\ell=e)N}|$ at the
$95\%$ CL Acciarri:1999qj ; Achard:2001qv  have been drawn from the LEP2 data
which have been denoted by the dark magenta line. In this analysis they
searched for $80~{}\rm{GeV}\leq M_{\it N}\leq 205~{}\rm{GeV}$ with a center of
mass energy between 130 GeV to 208 GeV Achard:2001qv .

The DELPHI collaboration Abreu:1996pa  had also performed the same search from
the LEP-I data which set an upper limit for the branching ratio
$\mathcal{B}(Z\rightarrow N\nu)$ about $1.3\times 10^{-6}$ at $95\%$ CL for
$3.5$ GeV $\leq m_{N}\leq 50$ GeV. Outside this range the limit starts to
become weak with the increase in $m_{N}$. Here it has been considered that
$N\rightarrow W\ell$ and $N\rightarrow Z\nu$ decays after the production of
the heavy neutrino was produced. The exclusion limits for $\ell=e,\mu,\tau$
are depicted by the blue dotted, dashed and dot-dashed lines in Fig. 25.

The search of the sterile neutrinos can be made at high energy lepton
colliders with a very high luminosity such as Future Circular Collider (FCC)
for the Seesaw model. A design of such collider has been launched recently
where nearly 100 km tunnel will be used to study high luminosity $e^{+}e^{-}$
collision (FCC-$ee$) with a center-of-mass energy around 90 GeV to 350 GeV
Blondel:2014bra . According to this report, a sensitivity down to $|V_{\ell
N}|^{2}\sim 10^{-11}$ could be achieved from a range of the heavy neutrino
mass, $10$ GeV $\leq m_{N}\leq 80$ GeV. The darker cyan-solid line in Fig. 25
shows the prospective search reaches by the FCC-$ee$. A sensitivity down to a
mixing of $|V_{\ell N}|^{2}\sim 10^{-12}$ can be obtained in FCC-$ee$
Blondel:2014bra , covering a large phase space for $m_{N}$ from $10$ GeV to
$80$ GeV.

The heavy neutrinos can participate in many electroweak (EW) precision tests
due to the active-sterile couplings. For comparison, we also show the $95\%$
CL indirect upper limit on the mixing angle, $|V_{\ell
N}|<0.030,0.041~{}\rm{and}~{}0.065$ for $\ell=\mu,e,\tau$ respectively derived
from a global fit to the electroweak precision data (EWPD), which is
independent of $m_{N}$ for $m_{N}>M_{Z}$, as shown by the horizontal pink
dash, solid and dolled lines respectively in Fig. 25. The bounds from the EWPD
have been studied in Refs.deBlas:2013gla ; delAguila:2008pw ; Akhmedov:2013hec
. For the mass range, $m_{N}<M_{Z}$, it is shown in Deppisch:2015qwa  that the
exclusion limit on the mixing angle remains almost unaltered, however, it
varies drastically at the vicinity of $m_{N}=1~{}\rm{GeV}.$ For the flavor
universal case the bound on the mixing angle is given as $|V_{\ell
N}|^{2}=0.025$ from deBlas:2013gla  which has been depicted in Fig. 25 with a
pink dot-dashed line.

The CMS limits from the $8$ TeV LHC are represented by the Plink lines. The
dashed Pink line represents the $\mu$ and solid pink line represents the $e$.
For both of the flavors CMS has tested the mass range, $40$ GeV $\leq
m_{N}\leq 200$ GeV. For electron it has probed down to $10^{-4}$ and for muon
it has probed down to $10^{-5}$ for $|V_{eN}|^{2}$ and $|V_{\mu N}|^{2}$
respectively. The ATLAS bounds from the $8$ TeV LHC are weaker than the CMS
bounds. The ATLAS Aad:2015xaa  bounds are represented by the brown dashed
lines for $\mu$ $(|V_{\mu N}|^{2})$ and brown dotted line represents the
bounds from $e$ $(|V_{eN}|^{2})$ in Fig. 25.

The relevant existing upper limits at the $95\%$ CL are also shown to compare
with the experimental bounds using the LHC Higgs boson data in
BhupalDev:2012zg ; Das:2014jxa  using Chatrchyan:2012ty ; CMS:2012xwa ;
Aad:2012uub ; Chatrchyan:2012ft ; Aad:2012ora . The darker green dot-dashed
line named Higgs boson shows the relevant bounds on the mixing angle. In this
analysis we will compare our results taking this line as one of the
references. We have noticed that the $|V_{\ell N}|^{2}$ can be as low as
$4.86\times 10^{-4}$ while $m_{N}=60$ GeV and the bound becomes stronger at
$m_{N}=100$ GeV as $3.73\times 10^{-4}$. When $m_{N}>100$ GeV, the bounds on
$|V_{\ell N}|^{2}$ become weaker.

###  IV.1 Same-sign di-lepton plus di-jet signal

For simplicity we consider the case that only one generation of the heavy
neutrino is light and accessible to the LHC which couples to only the second
generation of the lepton flavor. To generate the events in the MadGraph we use
the CTEQ6L1 PDF CTEQ  with xqcut$=p_{T}^{j}=20$ GeV and QCUT$=25$ GeV. We
calculate the cross sections for the combined processes $N\ell
X,N\rightarrow\ell jj$ as functions of $m_{N}$ from various initial states as
described in Fig. 11 and 12. Comparing our generated events with the recent
ATLAS results Aad:2015xaa  at the $8$ TeV LHC with the luminosity $20.3$ fb-1,
we obtain an upper limit on the mixing angles between the Majorana type heavy
neutrino and the SM leptons as a function of $m_{N}$. In the ATLAS analysis
the upper bound of the production cross section ($\sigma^{\rm{ATLAS/CMS}}$) is
obtained for the final state with the same-sign di-muon plus di-jet as a
function of $m_{N}$. For $\sigma^{\rm{ATLAS}}$ results we use the cross
sections given in Aad:2015xaa  and for $\sigma^{\rm{CMS}}$ we use CMS8 .
Implementing our model in MadGraph we generate the signal event and compare
the experimental cross sections from ATLAS and CMS using

$\displaystyle|V_{\mu
N}|^{2}\lesssim\frac{\sigma^{\rm{ATLAS/CMS}}}{(pp\rightarrow N\ell
X,N\rightarrow\ell jj)+\rm{All~{}same~{}sign~{}di-lepton~{}modes}}.$ (13)

Our upper bounds for the $13$ TeV LHC with $3000$ fb-1 luminosity on the
mixing angles are shown in Fig. 25 along with the exclusion limits obtained
from the different experimental results shaded in gray as described in the
previous subsection. We can see that a significant improvement on the bounds
can be made adding the jets and various initial states as described in this
paper. Applying the ATLAS bound at $8$ TeV with $20.3$ fb-1 luminosity
Aad:2015xaa , we put a prospective upper bound on the mixing angles at the
$13$ TeV LHC with $3000$ fb-1 luminosity. Recently the CMS has performed the
same-sign di-lepton plus di-jet search CMS8 . Using this result and adopting
the same procedure for the ATLAS result we calculate the upper bound on the
mixing angles at the $13$ TeV LHC with $3000$ fb-1 luminosity using Eq. 13.
The results are shown in Fig. 25.

Figure 25: Bounds on the square of the mixing angle from the same sign di-
lepton plus di-jet final state as a function of the heavy neutrino mass
$(m_{N})$. The shaded region is ruled out by DELPHI, EWPD and Higgs data.
Prospective bounds from the Higgs data had been calculated in Das:2017zjc  for
the LHC at the high luminosity $(3000$ fb${}^{-1})$ and future $100$ TeV pp
collider.

A significant improvement for the upper bound on the mixing angle is obtained
by combining all the processes with jets and various initial states as
described in this paper. We can see that at $91.2$ GeV$\leq m_{N}\leq 200$
GeV, the bounds obtained using the ATLAS $(\rm{ATLAS}13-\mu@3000fb^{-1})$
analysis is better than the recent exclusions limits shaded in gray. Using the
$8$ TeV CMS result CMS8  we obtain prospective bound at the 13 TeV LHC for the
luminosity $3000$ fb-1$(\rm{CMS}13-\mu@3000$ fb${}^{-1})$ in Fig. 25. The
results are also shown in Fig. 25. In this analysis we have only chosen the
$\mu$ flavor. The Purple lines represent the $13$ TeV LHC and black lines
represent the future $100$ TeV pp collider. The corresponding improved bounds
at the future $100$ TeV pp collider with $3000$ fb-1 luminosity are shown in
Fig. 25 with the black dashed and dot-dashed lines corresponding to the ATLAS
and CMS data marked as $(100$TeV$-\mu@3000$fb${}^{-1})$, respectively in Fig.
25. The prospective bounds calculated from Eq. 13 at the $13$ TeV LHC are
represented by the purple and those for the future $100$ TeV are represented
by Black lines.

We find that at the HL-LHC, using the same-sign di-muon and di-jet final state
LHC can probe the squared of the mixing angle down to $1.95\times 10^{-5}$ for
the $\mu$ flavor where as the future $100$ TeV pp collider can probe the
square of the mixing angle down to $5.83\times 10^{-6}$ for the $\mu$ flavor.

###  IV.2 Trilepton signal

In this analysis we consider two cases. One is the Flavor Diagonal case (FD),
where we introduce three generations of the degenerate heavy neutrinos and
each generation couples with the single, corresponding lepton flavor. The
other one is the Single Flavor case (SF) where only one heavy neutrino is
light and accessible to the LHC which couples to only the first or second
generation of the lepton flavor using the CTEQ6L PDF CTEQ . In this analysis
we consider $N\ell X$ final state followed by
$N\rightarrow\ell\ell\nu_{\ell}$.

We generate the combined parton level events using MadGraph and then gradually
hadronize and perform detector simulations with xqcut$=p_{T}^{j}=30$ GeV and
QCUT$=36$ GeV for the hadronization. In our analysis we use the matched cross
section after the detector level analysis. After the signal events are
generated we adopt the following basic criteria, used in the CMS trilepton
analysis CMS-trilep ,

  * (i)

the transverse momentum of each lepton: $p^{\ell}_{T}>10$ GeV;

  * (ii)

the transverse momentum of at least one lepton: $p^{\ell,{\rm
leading}}_{T}>20$ GeV;

  * (iii)

the jet transverse momentum: $p_{T}^{j}>30$ GeV;

  * (iv)

the pseudo-rapidity of leptons: $|\eta^{\ell}|<2.4$ and of jets:
$|\eta^{j}|<2.5$;

  * (v)

the lepton-lepton separation: $\Delta R_{\ell\ell}>0.1$ and the lepton-jet
separation: $\Delta R_{\ell j}>0.3$;

  * (vi)

the invariant mass of each OSSF (opposite-sign same flavor) lepton pair:
$m_{\ell^{+}\ell^{-}}<75$ GeV or $>105$ GeV to avoid the on-$Z$ region which
was excluded from the CMS search. Events with $m_{\ell^{+}\ell^{-}}<12$ GeV
are rejected to eliminate background from low-mass Drell-Yan processes and
hadronic decays;

  * (vii)

the scalar sum of the jet transverse momenta: $H_{T}<200$ GeV;

  * (viii)

the missing transverse energy: $\not{E}_{T}<50$ GeV.

The additional trilepton contributions come from $N\rightarrow Z\nu,h\nu$,
followed by the $Z$, $h$ decays into a pair of OSSF leptons. However, the $Z$
contribution is rejected after the implementation of the invariant mass cut
for the OSSF leptons to suppress the SM background and the $h$ contribution is
suppressed due to very small Yukawa couplings of electrons and muons. Using
different values of $\not{E}_{T}$ and $H_{T}$, the CMS analysis provides
different number of observed events and the corresponding SM background
expectations. For our analysis the set of cuts listed above are the most
efficient ones as implemented by the CMS analysis CMS-trilep . To derive the
limits on $|V_{\ell N}|^{2}$, we calculate the signal cross section normalized
by the square of the mixing angle as a function of the heavy neutrino mass
$m_{N}$ for both SF and FD cases, by imposing the CMS selection criteria
listed above. The corresponding number of signal events passing all the cuts
is then compared with the observed number of events at the 19.5 fb-1
luminosity CMS-trilep . For the selection criteria listed above, the CMS
experiment observed:

  * (a)

510 events with the SM background expectation of 560$\pm$87 events for
$m_{\ell^{+}\ell^{-}}<$ 75 GeV which is called below the $Z-$pole.

  * (b)

178 events with the SM background expectation of 200$\pm$35 events for
$m_{\ell^{+}\ell^{-}}>$ 105 GeV which is called above the $Z-$ pole.

In case (a) we have an upper limit of 37 signal events, while in case (b)
leads to an upper limit of 13 signal events. Using the limits obtained in case
(b), we can set an upper bound on $|V_{\ell N}|^{2}$ for a given value of
$m_{N}$. In this analysis we use above the $Z$ pole situation with $3000$ fb-1
luminosity for the 13 TeV LHC and future $100$ TeV pp collider. (Exactly the
same procedure can be adopted for the situation with $m_{\ell^{+}\ell^{-}}<$
$75$ GeV.)

Figure 26: Bounds on the square of the mixing angle from the trilepton plus
MET final state as a function of the heavy neutrino mass $(m_{N})$. The shaded
region is ruled out by the EWPD and Higgs data. Prospective bounds from the
Higgs data had been calculated in Das:2017zjc  for the LHC at the high
luminosity $(3000$ fb${}^{-1})$ and future $100$ TeV pp collider.

The mixing-squared vs $m_{N}$ contours shown in Fig. 26 are allowed by the
EWPD with in the range $91.2$ GeV$\leq m_{N}\leq 163$ GeV for the SF case at
the 13 TeV LHC at 3000 fb-1 luminosity. At the same time we analyze the FD
case through which one can probe the range of $91.2$ GeV$\leq m_{N}\leq 195$
GeV. The limits for the FD case are roughly twice stronger than the SF case.
However, at the future $100$ TeV pp collider with $3000$ fb-1 luminosity the
upper bounds on $m_{N}$ goes up to $200$ GeV and $275$ GeV for the SF and FD
cases respectively keeping lower mass bound at the $91.2$ GeV. As the coupling
between the Higgs and RHNs can not distinguish between the Majorana and
pseudo-Dirac neutrinos, we use the same limits for both of the cases.
Similarly the EWPD can measure the deviation from the SM couplings so we case
use the same bounds in both of the cases.

##  V Conclusions

In this paper we have studied both of the type-I and inverse Seesaw models
where SM singlet RHNs are involved. The RHNs involved in the type-I Seesaw
mechanism are Majorana type where as those are present in the inverse Seesaw
model are pseudo-Dirac in nature. We have studied the production mechanisms of
the RHNs from various initial states at the 13 TeV LHC and future $100$ TeV pp
collider considering the square of the mixing angle as $10^{-4}$ when the RHNs
are produced in association with charged lepton, neutral lepton along with the
jets. It has been shown due to the prompt decay of Higgs boson $(m_{N}<m_{H})$
into the RHNs and SM light neutrino, the cross section dominates over all the
remaining production modes. Such heavy neutrinos can dominantly decay into the
$\ell W$ mode at the colliders. Depending upon the nature of the model we pick
up two distinguishing signals from the type-I Seesaw and inverse Seesaw in the
form of same sign di-lepton plus di-jet and trilepton plus MET respectively.

Using the recent searches by the CMS and ATLAS for the type-I Seesaw with
same-sign di-muon and di-jet final state we put a prospective upper bound on
the $|V_{\mu N}|^{2}$ for the 13 TeV LHC and future $100$ TeV pp collider at
$3000$ fb-1 luminosity. Applying the cuts used in the anomalous multi-lepton
search done by the LHC, we can put upper bounds on the square of the mixing
angle between the SM light and of the degenerate RHNs. We consider the SF and
FD cases where the limits in FD case is twice stronger than the SF case.

We have noticed that our currently given projected limits on the square of the
mixing angles are better than those obtained from different experiments and
even at the current stage of the LHC. We expect improvements in the status at
the HL-LHC and future $100$ TeV pp collider when multi-lepton final states
will be studied from the decays of the RHNs which will lead us to a more
optimistic conclusion.

##  VI Conflicts of Interest

The author declares that there is no conflict of interest regarding the
publication of this paper.

###### Acknowledgements.

The Author would like to thank E. J. Chun, P. S. Bhupal Dev, Y. Gao, T. Kamon,
P. Konar, R. N. Mohapatra, N. Okada, A. Thalapillil, U. K. Yang for useful
discussions at different times. The author would like to thank A. G. Hessler,
A. Ibarra, E. Molinaro and S. Vogl for useful information and discussions
regarding Hessler:2014ssa .

## References

  * (1) K. Abe et. al. [T2K Collaboration] Phys. Rev. Lett. 107, 041801 (2011).
  * (2) P. Adamson et al. [MINOS Collaboration], Phys. Rev. Lett. 107, 181802 (2011).
  * (3) Y. Abe et al. [DOUBLE-CHOOZ Collaboration], Phys. Rev. Lett. 108, 131801 (2012).
  * (4) F. P. An et al. [DAYA-BAY Collaboration], Phys. Rev. Lett. 108, 171803 (2012).
  * (5) J. K. Ahn et al. [RENO Collaboration], Phys. Rev. Lett. 108, 191802 (2012).
  * (6) C. Patrignani et al. [Particle Data Group], “Review of Particle Physics,” Chin. Phys. C 40, no. 10, 100001 (2016). doi:10.1088/1674-1137/40/10/100001
  * (7) S. Weinberg, “Baryon and Lepton Nonconserving Processes,” Phys. Rev. Lett. 43, 1566 (1979). doi:10.1103/PhysRevLett.43.1566
  * (8) P. Minkowski, “$\mu\rightarrow e\gamma$ at a Rate of One Out of $10^{9}$ Muon Decays?,” Phys. Lett. 67B, 421 (1977). doi:10.1016/0370-2693(77)90435-X
  * (9) T. Yanagida, “Horizontal Symmetry and Masses of Neutrinos,” Prog. Theor. Phys. 64, 1103 (1980).
  * (10) J. Schechter and J. W. F. Valle, “Neutrino Masses in SU(2) $\otimes$ U(1) Theories,” Phys. Rev. D 22, 2227 (1980).
  * (11) T. Yanagida, in Proceedings of the Work- shop on the Unified Theory and the Baryon Number in the Universe (O. Sawada and A. Sugamoto, eds.), KEK, Tsukuba, Japan, 1979, p. 95.
  * (12) M. Gell-Mann, P. Ramond, and R. Slansky, Supergravity (P. van Nieuwenhuizen et al. eds.), North Holland, Amsterdam, 1979, p. 315.
  * (13) S. L. Glashow, The future of elementary particle physics, in Proceedings of the 1979 Carg‘ese Summer Institute on Quarks and Leptons (M. Levy et al. eds.), Plenum Press, New York, 1980, p. 687.
  * (14) R. N. Mohapatra and G. Senjanovic, “Neutrino Mass and Spontaneous Parity Violation,” Phys. Rev. Lett. 44, 912 (1980).
  * (15) J. A. Casas and A. Ibarra, “Oscillating neutrinos and muon$--->e$, gamma,” Nucl. Phys. B 618, 171 (2001) doi:10.1016/S0550-3213(01)00475-8 [hep-ph/0103065].
  * (16) A. Ibarra, E. Molinaro and S. T. Petcov, “TeV Scale See-Saw Mechanisms of Neutrino Mass Generation, the Majorana Nature of the Heavy Singlet Neutrinos and $(\beta\beta)_{0\nu}$-Decay,” JHEP 1009, 108 (2010) doi:10.1007/JHEP09(2010)108 [arXiv:1007.2378 [hep-ph]].
  * (17) A. Das and N. Okada, “Bounds on heavy Majorana neutrinos in type-I Seesaw and implications for collider searches,” arXiv:1702.04668 [hep-ph].
  * (18) J. Adam et al. [MEG Collaboration], “New limit on the lepton-flavour violating decay $\mu^{+}\rightarrow e^{+}\gamma$,” Phys. Rev. Lett. 107, 171801 (2011) doi:10.1103/PhysRevLett.107.171801 [arXiv:1107.5547 [hep-ex]].
  * (19) B. Aubert et al. [BaBar Collaboration], “Searches for Lepton Flavor Violation in the Decays tau$+---->$ e$+-$ gamma and tau$+---->$ mu$+-$ gamma,” Phys. Rev. Lett. 104, 021802 (2010) doi:10.1103/PhysRevLett.104.021802 [arXiv:0908.2381 [hep-ex]].
  * (20) B. O’Leary et al. [SuperB Collaboration], “SuperB Progress Reports – Physics,” arXiv:1008.1541 [hep-ex].
  * (21) F. del Aguila, J. de Blas and M. Perez-Victoria, “Effects of new leptons in Electroweak Precision Data,” Phys. Rev. D 78, 013010 (2008) doi:10.1103/PhysRevD.78.013010 [arXiv:0803.4008 [hep-ph]].
  * (22) E. Akhmedov, A. Kartavtsev, M. Lindner, L. Michaels and J. Smirnov, “Improving Electro-Weak Fits with TeV-scale Sterile Neutrinos,” JHEP 1305, 081 (2013) doi:10.1007/JHEP05(2013)081 [arXiv:1302.1872 [hep-ph]].
  * (23) S. Antusch, C. Biggio, E. Fernandez-Martinez, M. B. Gavela and J. Lopez-Pavon, “Unitarity of the Leptonic Mixing Matrix,” JHEP 0610, 084 (2006) doi:10.1088/1126-6708/2006/10/084 [hep-ph/0607020].
  * (24) A. Abada, C. Biggio, F. Bonnet, M. B. Gavela and T. Hambye, “Low energy effects of neutrino masses,” JHEP 0712, 061 (2007) doi:10.1088/1126-6708/2007/12/061 [arXiv:0707.4058 [hep-ph]].
  * (25) A. Ibarra, E. Molinaro and S. T. Petcov, “Low Energy Signatures of the TeV Scale See-Saw Mechanism,” Phys. Rev. D 84, 013005 (2011) doi:10.1103/PhysRevD.84.013005 [arXiv:1103.6217 [hep-ph]].
  * (26) D. N. Dinh, A. Ibarra, E. Molinaro and S. T. Petcov, “The $\mu-e$ Conversion in Nuclei, $\mu\rightarrow e\gamma,\mu\rightarrow 3e$ Decays and TeV Scale See-Saw Scenarios of Neutrino Mass Generation,” JHEP 1208, 125 (2012) Erratum: [JHEP 1309, 023 (2013)] doi:10.1007/JHEP09(2013)023, 10.1007/JHEP08(2012)125 [arXiv:1205.4671 [hep-ph]].
  * (27) J. T. Penedo, S. T. Petcov and T. Yanagida, “Low-Scale Seesaw and the CP Violation in Neutrino Oscillations,” arXiv:1712.09922 [hep-ph].
  * (28) R. W. Rasmussen and W. Winter, “Perspectives for tests of neutrino mass generation at the GeV scale: Experimental reach versus theoretical predictions,” Phys. Rev. D 94, no. 7, 073004 (2016) doi:10.1103/PhysRevD.94.073004 [arXiv:1607.07880 [hep-ph]].
  * (29) P. S. Bhupal Dev, R. Franceschini and R. N. Mohapatra, “Bounds on TeV Seesaw Models from LHC Higgs Data,” Phys. Rev. D 86, 093010 (2012) doi:10.1103/PhysRevD.86.093010 [arXiv:1207.2756 [hep-ph]].
  * (30) G. Aad et al. [ATLAS Collaboration], “Search for heavy Majorana neutrinos with the ATLAS detector in pp collisions at $\sqrt{s}=8$ TeV,” JHEP 1507, 162 (2015) doi:10.1007/JHEP07(2015)162 [arXiv:1506.06020 [hep-ex]].
  * (31) V. Khachatryan et. al. [CMS Collaboration], “Search for heavy Majorana neutrinos in e±e±+jets and e${}^{\pm}\mu^{\pm}$+jets events in proton-proton collisions at $\sqrt{s}=8$ TeV,” JHEP 1604, 169 (2016) doi: 10.1007/JHEP04 (2016)169 [arxiv:1603.02248 [hep-ex]].
  * (32) A. Das, P. S. B. Dev and C. S. Kim, “Constraining Sterile Neutrinos from Precision Higgs Data,” Phys. Rev. D 95, no. 11, 115013 (2017) doi:10.1103/PhysRevD.95.115013 [arXiv:1704.00880 [hep-ph]].
  * (33) R. Ruiz, M. Spannowsky and P. Waite, “Heavy neutrinos from gluon fusion,” Phys. Rev. D 96, no. 5, 055042 (2017) doi:10.1103/PhysRevD.96.055042 [arXiv:1706.02298 [hep-ph]].
  * (34) A. Das, Y. Gao and T. Kamon, “Heavy Neutrino Search via the Higgs boson at the LHC,” arXiv:1704.00881 [hep-ph].
  * (35) J. Kersten and A. Y. Smirnov, “Right-Handed Neutrinos at CERN LHC and the Mechanism of Neutrino Mass Generation,” Phys. Rev. D 76, 073005 (2007) doi:10.1103/PhysRevD.76.073005 [arXiv:0705.3221 [hep-ph]].
  * (36) Z. z. Xing, “Naturalness and Testability of TeV Seesaw Mechanisms,” Prog. Theor. Phys. Suppl. 180, 112 (2009) doi:10.1143/PTPS.180.112 [arXiv:0905.3903 [hep-ph]].
  * (37) X. G. He, S. Oh, J. Tandean and C. C. Wen, “Large Mixing of Light and Heavy Neutrinos in Seesaw Models and the LHC,” Phys. Rev. D 80, 073012 (2009) doi:10.1103/PhysRevD.80.073012 [arXiv:0907.1607 [hep-ph]].
  * (38) F. F. Deppisch and A. Pilaftsis, “Lepton Flavour Violation and theta(13) in Minimal Resonant Leptogenesis,” Phys. Rev. D 83, 076007 (2011) doi:10.1103/PhysRevD.83.076007 [arXiv:1012.1834 [hep-ph]].
  * (39) C. H. Lee, P. S. Bhupal Dev and R. N. Mohapatra, “Natural TeV-scale left-right Seesaw mechanism for neutrinos and experimental tests,” Phys. Rev. D 88, no. 9, 093010 (2013) doi:10.1103/PhysRevD.88.093010 [arXiv:1309.0774 [hep-ph]].
  * (40) R. N. Mohapatra, “Mechanism for Understanding Small Neutrino Mass in Superstring Theories,” Phys. Rev. Lett. 56, 561 (1986). doi:10.1103/PhysRevLett.56.561
  * (41) R. N. Mohapatra and J. W. F. Valle, “Neutrino Mass and Baryon Number Nonconservation in Superstring Models,” Phys. Rev. D 34, 1642 (1986). doi:10.1103/PhysRevD.34.1642
  * (42) A. Das and N. Okada, “Inverse Seesaw neutrino signatures at the LHC and ILC,” Phys. Rev. D 88, 113001 (2013) doi:10.1103/PhysRevD.88.113001 [arXiv:1207.3734 [hep-ph]].
  * (43) A. Das, P. S. Bhupal Dev and N. Okada, “Direct bounds on electroweak scale pseudo-Dirac neutrinos from $\sqrt{s}=8$ TeV LHC data,” Phys. Lett. B 735, 364 (2014) doi:10.1016/j.physletb.2014.06.058 [arXiv:1405.0177 [hep-ph]].
  * (44) A. Das and N. Okada, “Improved bounds on the heavy neutrino productions at the LHC,” Phys. Rev. D 93, no. 3, 033003 (2016) doi:10.1103/PhysRevD.93.033003 [arXiv:1510.04790 [hep-ph]].
  * (45) A. Das, N. Nagata and N. Okada, “Testing the 2-TeV Resonance with Trileptons,” JHEP 1603, 049 (2016) doi:10.1007/JHEP03(2016)049 [arXiv:1601.05079 [hep-ph]].
  * (46) A. Das, P. Konar and S. Majhi, “Production of Heavy neutrino in next-to-leading order QCD at the LHC and beyond,” JHEP 1606, 019 (2016) doi:10.1007/JHEP06(2016)019 [arXiv:1604.00608 [hep-ph]].
  * (47) N. Arkani-Hamed, T. Han, M. Mangano and L. T. Wang, “Physics opportunities of a 100 TeV proton?proton collider,” Phys. Rept. 652, 1 (2016) doi:10.1016/j.physrep.2016.07.004 [arXiv:1511.06495 [hep-ph]].
  * (48) S. Catani, F. Krauss, R. Kuhn and B. R. Webber, “QCD matrix elements + parton showers,” JHEP 0111, 063 (2001) doi:10.1088/1126-6708/2001/11/063 [hep-ph/0109231].
  * (49) A. Das, “Pair production of heavy neutrinos in next-to-leading order QCD at the hadron colliders in the inverse Seesaw framework,” arXiv:1701.04946 [hep-ph].
  * (50) J. Alwal, MadGraph Wiki, Matching of Jets between MadFevent and Pythia, 2011-10-05, https://cp3.irmp.ucl.ac.be/projects/madgraph/wiki/Matching
  * (51) J. Alwall, S. D. Visscher,MadGraph Wiki, Introduction to jet-parton matching in MG/ME, 2011-03-16, https://cp3.irmp.ucl.ac.be/projects/madgraph/wiki/IntroMatching
  * (52) J. Alwall, IPMU, Focus week, Tokyo, Japan, 12 November 2009, http://www.ipmu.jp/sites/default/files/webfm/pdfs/FocusWeek_10
  * (53) J. Alwall, Parton Showers and MLM matching with MadGraph and Pythia, NTU-MadGraph School, May 25-27, 2012. https://cp3.irmp.ucl.ac.be/projects/madgraph/attachment/.../NTU-MLM-lectures.pdf
  * (54) J. Alwall, HELAS/MadGraph Workshop, KEK 18-27 October 2006, http://phya.snu.ac.kr/~particle/wp-content/uploads/2007/03/lectureb-1.pdf
  * (55) J. Alwall, HELAS/MadGraph Workshop, KEK 18-27 October 2006, http://phya.snu.ac.kr/~particle/wp-content/uploads/2007/03/lectureb-2.pdf
  * (56) J. Alwall, HELAS/MadGraph Workshop, KEK 18-27 October 2006, http://phya.snu.ac.kr/~particle/wp-content/uploads/2007/03/lectureb-3.pdf
  * (57) M. L. Mangano, M. Moretti, F. Piccinini, R. Pittau, ALPGEN, http://mlm.web.cern.ch/mlm/talks/kek-alpgen.pdf, http://mlm.home.cern.ch/mlm/alpgen
  * (58) M. L. Mangano, M. Moretti, F. Piccinini, R. Pittau and A. D. Polosa, “ALPGEN, a generator for hard multiparton processes in hadronic collisions,” JHEP 0307, 001 (2003) doi:10.1088/1126-6708/2003/07/001 [hep-ph/0206293].
  * (59) M. L. Mangano, M. Moretti, F. Piccinini and M. Treccani, “Matching matrix elements and shower evolution for top-quark production in hadronic collisions,” JHEP 0701, 013 (2007) doi:10.1088/1126-6708/2007/01/013 [hep-ph/0611129].
  * (60) B. Cooper, J. Katzy, M. L. Mangano, A. Messina, L. Mijovic and P. Skands, “Importance of a consistent choice of alpha(s) in the matching of AlpGen and Pythia,” Eur. Phys. J. C 72, 2078 (2012) doi:10.1140/epjc/s10052-012-2078-y [arXiv:1109.5295 [hep-ph]].
  * (61) J. Alwall et al., “Comparative study of various algorithms for the merging of parton showers and matrix elements in hadronic collisions,” Eur. Phys. J. C 53, 473 (2008) doi:10.1140/epjc/s10052-007-0490-5 [arXiv:0706.2569 [hep-ph]].
  * (62) J. Alwall, M. Herquet, F. Maltoni, O. Mattelaer and T. Stelzer, “MadGraph 5 : Going Beyond,” JHEP 1106, 128 (2011) doi:10.1007/JHEP06(2011)128 [arXiv:1106.0522 [hep-ph]].
  * (63) J. Alwall, S. de Visscher and F. Maltoni, “QCD radiation in the production of heavy colored particles at the LHC,” JHEP 0902, 017 (2009) doi:10.1088/1126-6708/2009/02/017 [arXiv:0810.5350 [hep-ph]].
  * (64) http://home.thep.lu.se/~torbjorn/pythia81html/JetMatching.html
  * (65) http://www.tifr.res.in/~indiacms/indiacms-meetings/march-2009/Matching$%$20Parton$%$
20Showers$%$20and$%$20Matrix$%$20elements.pdf

  * (66) S. Hoeche, F. Krauss, N. Lavesson, L. Lonnblad, M. Mangano, A. Schalicke and S. Schumann, “Matching parton showers and matrix elements,” doi:10.5170/CERN-2005-014.288 hep-ph/0602031.
  * (67) T. Sjostrand, S. Mrenna and P. Z. Skands, “PYTHIA 6.4 Physics and Manual,” JHEP 0605, 026 (2006) doi:10.1088/1126-6708/2006/05/026 [hep-ph/0603175].
  * (68) A. G. Hessler, A. Ibarra, E. Molinaro and S. Vogl, “Impact of the Higgs boson on the production of exotic particles at the LHC,” Phys. Rev. D 91, no. 11, 115004 (2015) doi:10.1103/PhysRevD.91.115004 [arXiv:1408.0983 [hep-ph]].
  * (69) E. J. Williams, “Nature of the high-energy particles of penetrating radiation and status of ionization and radiation formulae,” Phys. Rev. 45, 729 (1934). doi:10.1103/PhysRev.45.729.
  * (70) C. F. von Weizsacker, “Radiation emitted in collisions of very fast electrons,” Z. Phys. 88, 612 (1934). doi:10.1007/BF01333110.
  * (71) V. M. Budnev, I. F. Ginzburg, G. V. Meledin and V. G. Serbo, “The Two photon particle production mechanism. Physical problems. Applications. Equivalent photon approximation,” Phys. Rept. 15, 181 (1975). doi:10.1016/0370-1573(75)90009-5
  * (72) P. S. B. Dev, A. Pilaftsis and U. k. Yang, “New Production Mechanism for Heavy Neutrinos at the LHC,” Phys. Rev. Lett. 112, no. 8, 081801 (2014) doi:10.1103/PhysRevLett.112.081801 [arXiv:1308.2209 [hep-ph]].
  * (73) D. Alva, T. Han and R. Ruiz, “Heavy Majorana neutrinos from $W\gamma$ fusion at hadron colliders,” JHEP 1502, 072 (2015) doi:10.1007/JHEP02(2015)072 [arXiv:1411.7305 [hep-ph]].
  * (74) F. F. Deppisch, P. S. Bhupal Dev and A. Pilaftsis, “Neutrinos and Collider Physics,” New J. Phys. 17, no. 7, 075019 (2015) doi:10.1088/1367-2630/17/7/075019 [arXiv:1502.06541 [hep-ph]].
  * (75) J. Pumplin, D. R. Stump, J. Huston, H. L. Lai, P. M. Nadolsky, W. K. Tung, J. High Energy Phys. 0207 (2002) 012, arXiv: hep-ph/0201195.
  * (76) J. de Favereau et al. [DELPHES 3 Collaboration], “DELPHES 3, A modular framework for fast simulation of a generic collider experiment,” JHEP 1402, 057 (2014) doi:10.1007/JHEP02(2014)057 [arXiv:1307.6346 [hep-ex]].
  * (77) M. Drewes, “The Phenomenology of Right Handed Neutrinos,” Int. J. Mod. Phys. E 22, 1330019 (2013) doi:10.1142/S0218301313300191 [arXiv:1303.6912 [hep-ph]].
  * (78) S. Dube, D. Gadkari and A. M. Thalapillil, “Lepton-Jets and Low-Mass Sterile Neutrinos at Hadron Colliders,” Phys. Rev. D 96, no. 5, 055031 (2017) doi:10.1103/PhysRevD.96.055031 [arXiv:1707.00008 [hep-ph]].
  * (79) A. Das, P. S. B. Dev and R. N. Mohapatra, “Same Sign vs Opposite Sign Dileptons as a Probe of Low Scale Seesaw Mechanisms,” arXiv:1709.06553 [hep-ph].
  * (80) A. Das, P. Konar and A. Thalapillil, “Jet substructure shedding light on heavy Majorana neutrinos at the LHC,” arXiv:1709.09712 [hep-ph].
  * (81) A. Bhardwaj, A. Das, P. Konar and A. Thalapillil, “Challenging Sterile Neutrino Searches at the LHC Complemented by Jet Substructure Techniques,” arXiv:1801.00797 [hep-ph].
  * (82) A. Das, N. Okada and D. Raut, “Enhanced pair production of heavy Majorana neutrinos at LHC,” arXiv:1710.03377 [hep-ph].
  * (83) A. Das, N. Okada and D. Raut, “Heavy Majorana neutrino pair productions at the LHC in minimal U(1) extended Standard Model,” arXiv:1711.09896 [hep-ph].
  * (84) M. Mitra, R. Ruiz, D. J. Scott and M. Spannowsky, “Neutrino Jets from High-Mass $W_{R}$ Gauge Bosons in TeV-Scale Left-Right Symmetric Models,” Phys. Rev. D 94, no. 9, 095016 (2016) doi:10.1103/PhysRevD.94.095016 [arXiv:1607.03504 [hep-ph]].
  * (85) D. Das, K. Ghosh, M. Mitra and S. Mondal, “Probing sterile neutrinos in the framework of inverse seesaw mechanism through leptoquark productions,” Phys. Rev. D 97, no. 1, 015024 (2018) doi:10.1103/PhysRevD.97.015024 [arXiv:1708.06206 [hep-ph]].
  * (86) M. Drewes, B. Garbrecht, D. Gueter and J. Klaric, “Testing the low scale seesaw and leptogenesis,” JHEP 1708, 018 (2017) doi:10.1007/JHEP08(2017)018 [arXiv:1609.09069 [hep-ph]].
  * (87) Y. L. Tang, “Probing the Light Sterile Neutrino Through the Heavy Charged Higgs Decay on the LHC,” arXiv:1712.10108 [hep-ph].
  * (88) V. Khachatryan et al. [CMS Collaboration], “Search for heavy Majorana neutrinos in $\mu^{\pm}\mu^{\pm}$+jets events in proton-proton collisions at $\sqrt{s}$ = 8 TeV,”arXiv:1501.05566 [hep-ex].
  * (89) CMS Collaboration, arXiv:1404.5801 [hep-ex]; For an earlier analysis with $\sqrt{s}=$ 7 TeV LHC data, see S. Chatrchyan, et al., CMS Collaboration, J. High Energy Phys. 1206 (2012) 169, arXiv: 1204.5341 [hep-ex].
  * (90) J. de Blas, “Electroweak limits on physics beyond the Standard Model,” EPJ Web Conf. 60, 19008 (2013) doi:10.1051/epjconf/20136019008 [arXiv:1307.6173 [hep-ph]].
  * (91) O. Adriani et al. [L3 Collaboration], “Search for isosinglet neutral heavy leptons in Z0 decays,” Phys. Lett. B 295, 371 (1992). doi:10.1016/0370-2693(92)91579-X
  * (92) M. Acciarri et al. [L3 Collaboration], “Search for heavy isosinglet neutrinos in $e^{+}e^{-}$ annihilation at 130-GeV less than $S^{(1/2)}$ less than 189-GeV,” Phys. Lett. B 461, 397 (1999) doi:10.1016/S0370-2693(99)00852-7 [hep-ex/9909006].
  * (93) P. Achard et al. [L3 Collaboration], “Search for heavy isosinglet neutrino in $e^{+}e^{-}$ annihilation at LEP,” Phys. Lett. B 517, 67 (2001) doi:10.1016/S0370-2693(01)00993-5 [hep-ex/0107014].
  * (94) P. Abreu et al. [DELPHI Collaboration], “Search for neutral heavy leptons produced in Z decays,” Z. Phys. C 74, 57 (1997) Erratum: [Z. Phys. C 75, 580 (1997)]. doi:10.1007/s002880050370
  * (95) A. Blondel et al. [FCC-ee study Team], “Search for Heavy Right Handed Neutrinos at the FCC-ee,” Nucl. Part. Phys. Proc. 273-275, 1883 (2016) doi:10.1016/j.nuclphysbps.2015.09.304 [arXiv:1411.5230 [hep-ex]].
  * (96) F. del Aguila, J. de Blas and M. Perez-Victoria, “Effects of new leptons in Electroweak Precision Data,” Phys. Rev. D 78, 013010 (2008) doi:10.1103/PhysRevD.78.013010 [arXiv:0803.4008 [hep-ph]].
  * (97) E. Akhmedov, A. Kartavtsev, M. Lindner, L. Michaels and J. Smirnov, “Improving Electro-Weak Fits with TeV-scale Sterile Neutrinos,” JHEP 1305, 081 (2013) doi:10.1007/JHEP05(2013)081 [arXiv:1302.1872 [hep-ph]].
  * (98) S. Chatrchyan et al. [CMS Collaboration], “Search for the standard model Higgs boson decaying to $W^{+}W^{-}$ in the fully leptonic final state in pp collisions at $\sqrt{s}=7$ TeV,” Phys. Lett. B 710, 91 (2012) doi:10.1016/j.physletb.2012.02.076 [arXiv:1202.1489 [hep-ex]].
  * (99) CMS Collaboration [CMS Collaboration], “Search for the standard model Higgs boson decaying to a W pair in the fully leptonic final state in pp collisions at $\sqrt{(}s)=8$ TeV,” CMS-PAS-HIG-12-017.
  * (100) G. Aad et al. [ATLAS Collaboration], “Search for the Standard Model Higgs boson in the $H\rightarrow$ WW(*) $\rightarrow\ell\nu\ell\nu$ decay mode with 4.7 /fb of ATLAS data at $\sqrt{s}=7$ TeV,” Phys. Lett. B 716 (2012) 62 doi:10.1016/j.physletb.2012.08.010 [arXiv:1206.0756 [hep-ex]].
  * (101) S. Chatrchyan et al. [CMS Collaboration], “Search for the standard model Higgs boson in the $H$ to $ZZ$ to 2 $\ell 2\nu$ channel in $pp$ collisions at $\sqrt{s}=7$ TeV,” JHEP 1203, 040 (2012) doi:10.1007/JHEP03(2012)040 [arXiv:1202.3478 [hep-ex]].
  * (102) G. Aad et al. [ATLAS Collaboration], “Search for a standard sodel Higgs boson in the H $\rightarrow$ ZZ $\rightarrow\ell^{+}\ell^{-}\nu\bar{\nu}$ decay channel using 4.7 fb-1 of $\sqrt{s}=$ 7 TeV data with the ATLAS detector,” Phys. Lett. B 717, 29 (2012) doi:10.1016/j.physletb.2012.09.016 [arXiv:1205.6744 [hep-ex]].