# Probing right handed neutrinos at the LHeC and lepton colliders using fat
jet signatures

Arindam Das<EMAIL_ADDRESS>School of Physics, KIAS, Seoul 02455, Korea
Sudip Jana<EMAIL_ADDRESS>Department of Physics and Oklahoma Center
for High Energy Physics, Oklahoma State University, Stillwater, OK 74078-3072,
USA  Sanjoy Mandal<EMAIL_ADDRESS>The Institute of Mathematical
Sciences, C.I.T Campus, Taramani, Chennai 600 113, India  Homi Bhabha National
Institute, BARC Training School Complex, Anushakti Nagar, Mumbai 400085, India
S. Nandi<EMAIL_ADDRESS>Department of Physics and Oklahoma Center for
High Energy Physics, Oklahoma State University, Stillwater, OK 74078-3072, USA

###### Abstract

The inclusion of heavy neutral leptons (right-handed neutrinos) to the
Standard Model (SM) particle content is one of the best motivated ways to
account for the observed neutrino masses and flavor mixing. The modification
of the charged and neutral currents from active-sterile mixing of the neutral
leptons can provide novel signatures which can be tested at the future
collider experiments. In this article, we explore the discovery prospect of a
very heavy right handed neutrino to probe such extensions at the future
collider experiments like Large Hadron electron Collider (LHeC) and linear
collider. We consider the production of the heavy neutrino via the $t$ and
$s$-channel processes and its subsequent decays into the semi-leptonic final
states. We specifically focus on the scenario where the gauge boson produced
from heavy neutrino decay is highly boosted, leading to a fat-jet. We study
the bounds on the sterile neutrino properties from several past experiments
and compare with our results.

††preprint: OSU-HEP-18-06

##  I Introduction

One of the most robust evidence that points out to an important inadequacy of
the SM is the existence of the tiny but non-zero neutrino masses. It seems
unlikely that the very small neutrino masses are generated by the same Higgs
mechanism responsible for the masses of the other SM fermions due to the
absence of right-handed neutrinos. Even then, extremely small Yukawa
couplings, of the order of $\lesssim 10^{-12}$, must be invoked. There are
various BSM extensions which have been proposed to explain small neutrino
masses. Among those, one of the most appealing framework of light neutrino
mass generation is the addition of new states that, once integrated out,
generate the lepton number violating dimension five Weinberg operator
$\mathcal{O}_{5}=\frac{c}{\Lambda}LLHH$ Weinberg:1979sa  . This is embodied by
the so-called seesaw mechanisms. There can be a few different variations of
seesaw, Type-I typ1 , Type-II typ2 , Type-III typ3 , inverse inverse  and
radiative radiative  seesaw.

Most of the UV completed seesaw models contain Standard Model (SM) gauge
singlet heavy right handed neutrino $N$. Through the seesaw mechanism, the
Majorana type Right Handed Neutrinos (RHNs) impart masses to the SM light
neutrinos and hence establishes the fact that SM neutrinos have masses which
have been experimentally observed in a several neutrino oscillation
experiments numass . These RHNs can have masses from eV scale to 1014 GeV
scale depending upon the models. For instance, the sterile neutrinos lightnu
with masses in the eV range could lead to effects in short distance neutrino
oscillation experiments by introducing an additional mass squared difference,
keV mass sterile neutrinos are potential candidates for “warm” dark matter,
MeV scale sterile neutrinos can be possible explanation for MiniBoone mev  and
there can be very heavy sterile neutrinos with masses $M_{GUT}\sim 10^{14}$
GeV, close to $M_{GUT}\sim 10^{16}$ GeV in model of grand unified theories
(GUTs). These RHNs, originally Standard Model (SM) gauge singlet, being mixed
with the SM light neutrinos to interact with the SM gauge bosons. Depending on
the mass of the gauge singlet RHNs and their mixings with the active neutrino
states, seesaw mechanism can be tested at colliders delAguila:2008cj ;
Atre:2009rg ; Datta:1993nm ; Deppisch:2015qwa ; Bhardwaj:2018lma ; Das:2017gke
; Abada:2018sfh ; Jana:2018 ; Cottin:2018nms ; Helo:2018qej ;
Accomando:2017qcs ; Deppisch:2018eth ; Das:2017pvt ; Das:2017flq ; Das:2017deo
; Das:2016akd ; Das:2017nvm ; BhupalDev:2012zg ; Das:2017zjc ; Das:2017rsu ;
Chen:2013fna ; Dev:2018upe ; Dev:2017ftk ; Cely:2012bz ; Ibarra:2011xn ;
Ibarra:2010xw ; Nemevsek:2018bbt ; Nemevsek:2016enw ; Maiezza:2015lza ;
Maiezza:2016bzp ; Buchmuller:1990vh ; Buchmuller:1991wn ; Buchmuller:1992wm ;
Buchmuller:1991ce ; Buchmuller:1991tu ; Mondal:2016kof ; Das:2014jxa ;
Chen:2011hc ; Dev:2013wba ; Dev:2015kca ; Biswal:2017nfl ; Das:2016hof ;
Das:2015toa , as well as, in other non-collider experiments, such as,
neutrinoless double beta decay Dev:2013vxa ; Mitra:2011qr ; Dev:2014xea ;
Rodejohann:2012xd ; Pas:2015eia ; Gonzalez:2017mcg ; Das:2017hmg ; Das:2017nvm
, neutrino experiments lightnu ; mev ; nuex , rare-meson decays Ali:2001gsa ;
Mandal:2016hpr ; Mandal:2017tab , muon $g-2$ g2 , lepton flavor violating
processes $l_{i}\rightarrow l_{j}\gamma,\mu\rightarrow 3e,\mu\rightarrow e$
conversion in nuclei Abada:2007ux ; Abada:2008ea ; Dinh:2012bp , non-unitarity
Antusch:2014woa ; Antusch:2006vwa ; Blennow:2016jkn ; Fernandez-
Martinez:2016lgt ; Fernandez-Martinez:2015hxa , etc.

We are specifically interested in the RHNs at the TeV scale so that they can
be tested at the high energy colliders. At the LHC, the production cross
section of the RHN decreases as the mass of RHN increases as a result of the
properties of the constituent quarks of the proton beams. In the linear
collider the electron and positron are collided to produce the RHN in
association with a light neutrino through the dominant t-channel process. A
subdominant s-channel process also contributes ar1 ; ar2 . Otherwise a variety
of RHN productions at the linear collider have been discussed in ar3  followed
by the bounds on the light heavy mixing angles for the electron flavor at the
linear collider with $500$ GeV and $1$ TeV collider energies. The low mass
range of the RHN has been studied in ar4  which also predicts the limit on the
light heavy mixing and the mass of the RHN up to a mass of $250$ GeV. The
sterile neutrinos at the circular lepton colliders have been studied in ar5
which deals with a comprehensive discussion on the detectors from experimental
point of view. Higgs searches from RHN has been studied in ar6  where the RHN
has been produced from the $W$ and $Z$ mediated processes. Such a RHN decays
into a Higgs and SM light neutrino and the Higgs can dominantly decay into a
pair of $b$-quarks. Hence a $2b$ plus missing momentum will be a signal from
this process. In this paper the RHN up to a $500$ GeV mass have been tested
where the maximum center of mass energy is also taken up to $500$ GeV. The
distinct and interesting signature of the RHN can be displaced vertex search
if the mixing between the light and heavy neutrinos become extremely small.
Such a scenario has been tested in ar7  for the colliders $240$ GeV, $350$ GeV
and $500$ GeV. Another interesting work on the RHNs has been found in the form
of ar8  where a variety of the colliders have been considered to test the
observability of the RHN production. They have discussed several production
modes of the RHNs at the LHC, lepton-Hadron collider (LHeC) 111 In such a
collider we can also nicely study the long lived particles in Curtin:2017bxr ,
beyond the SM physics in Azuelos:2018syu , leptoquarks
AbelleiraFernandez:2012cc , left-right model Lindner:2016lxq , charged Higgs
Azuelos:2017dqw  and heavy Majorana neutrinos Duarte:2018xst . The LHeC design
report can be found in AbelleiraFernandez:2012cc . Mandal:2018qpg  and linear
collider. They have studied all possible modes of the RHN production in these
colliders and compared the bounds on the light-heavy neutrino mixing angles.
In the linear collider, the references ar4 ; ar5 ; ar6 ; ar7 ; ar8  did not go
further than $500$ GeV as they constrained themselves within the center of
mass energy of $500$ GeV. However, none of these papers studied the boosted
object at the LHeC and linear collider respectively.

In our analysis we consider the following things:

  1. 1.

We study the prospect of discovery of RHNs at LHeC considering the boosted
objects for the first time. In the LHeC we concentrate on the lepton number
violating (LNV) and lepton number conserving (LNC) channels to produce the RHN
in association with a jet $(j_{1})$. Hence the RHN will decay into the
dominant $\ell W$ and the $W$ will decay into a pair of jets. The daughter $W$
coming from the heavy RHN will be boosted and its hadronic decay products,
jets, of the $W$ will be collimated such that they can form a fat jet
$(J)$.Hence a signal sample of $\ell+j_{1}+J$ can be studied thoroughly at
this collider. In this process people have mostly studied the lepton number
conserving channel where as the lepton number violating will be potentially
background free. However, for clarity we study the combined channel and the
corresponding SM backgrounds. We consider two scenarios at the LHeC where the
electron and proton beams will have $60$ GeV and $7$ TeV energies where the
center of mass energy becomes $\sqrt{s}=1.3$ TeV. We have also considered
another center of mass energy at the $\sqrt{s}=1.8$ TeV where the proton beam
energy is raised up to the $13.5$ TeV. For both of the colliders we consider
the luminosity at $1$ ab-1. Here the RHN is a first generation RHN $(N_{1})$
and $\ell$ is electron $(e)$. Finally we study up to $3$ ab-1 luminosity.

  2. 2.

At the linear collider the production of the RHNs is occurring from the $s$\-
and $t$\- channel processes in association with a SM light neutrino $(\nu)$.
We consider the linear collider at two different center of mass energies, such
as $\sqrt{s}=1$ TeV and $\sqrt{s}=3$ TeV which can probe up to a high mass of
the RHNs such as $900$ GeV (at the $1$ TeV linear collider) and $2.9$ TeV (at
the $3$ TeV linear collider) due to the almost constant cross section for the
$N\nu$ production. For both of the center of mass energies we consider $1$
ab-1 luminosity. Finally we study up to $3$ $(5)$ ab-1 luminosity for the $1$
$(3)$ TeV linear collider.

At this mass scale, the RHNs will be produced at rest, however, the daughter
particles can be sufficiently boosted. We consider $N\rightarrow\ell
W,W\rightarrow jj$ and $N\rightarrow h\nu,h\rightarrow b\overline{b}$ modes at
the linear collider where $h$ is the SM Higgs boson. If the RHN is
sufficiently heavy, such the, $M_{N}\geq 400$ GeV, the $W$ and $h$ can be
boosted because $M_{W}$ and $M_{h}<<\frac{M_{N}}{2}$. As a result $W$ and $h$
will produce a fat jet $(J)$ and a fat $b$ jet $(J_{b})$ respectively.
Therefore the signal will be $\ell+J$ plus missing momentum and $J_{b}$ plus
missing momentum in the $W$ and $h$ modes respectively at the linear collider.
Therefore studying the signals and the backgrounds for each process we put the
bounds in the mass- mixing plane of the RHNs.

  3. 3.

We want to comment that studying $e^{-}e^{+}\rightarrow
N_{2}\nu_{\mu}/N_{3}\nu_{\tau}$ mode in the $Z$ mediated $s$-channel will be
interesting where $N_{2}(N_{3})$ will be the second (third) generation RHN.
Studying the signal events and the corresponding SM backgrounds one can also
calculate the limits on the mixing angles involved in these processes. Such a
process will be proportional to $\mid V_{\mu N}\mid^{2}(\mid V_{\tau
N}\mid^{2})$. In these processes the signal will be $\mu(\tau)+jj$ plus
missing momentum followed by the decay of $N_{2}(N_{3})\rightarrow\mu jj(\tau
jj)$. One can also calculate the bounds on the mass-mixing plane for different
significances. A boosted analysis could be interesting, however, a non-boosted
study might be more useful as the cross-section goes down with the rise in
collider energy in these processes. Such signals can also be studied if the
RHNs can decay through the LFV modes, such as $e^{-}e^{+}\rightarrow
N\nu_{e},N\rightarrow\mu W,W\rightarrow jj$, however, $\mu\rightarrow e\gamma$
process will make this process highly constrained due to the strong limit
$Br(\mu^{+}\rightarrow e^{+}\gamma)<4.2\times 10^{-13}$ at the $90\%$ C. L.
TheMEG:2016wtm . The corresponding limits on $\tau$ are weaker Aubert:2009ag ;
OLeary:2010hau . Such final states have been studied in ar1  for $M_{N}=150$
GeV, a high mass test with using boosted object will be interesting in future.
A comprehensive LHC study has been performed in Antusch:2018bgr .

  4. 4.

The RHN produced at the linear collider may decay in to another interesting
mode, namely, $N\rightarrow Z\nu,Z\rightarrow b\bar{b}$. Which can be another
interesting channel where boosted objects can be stated. However, precision
measurements at the $Z$-boson resonance using electron-positron colliding
beams at LEP experiment strongly constrains Z boson current, and hence,
$Zb\bar{b}$ coupling. This channel also suffers from larger QCD background
compared to the leptonic decay of Z boson, and hence, leptonic decay of Z
boson has better discovery prospect for this particular mode of RHN decay. On
the other hand, SM Higgs , $h$, mostly decays ($\sim 60\%$) to $b\bar{b}$ due
to large $hb\bar{b}$ coupling. Due to this, we focus on the Higgs decay mode
of RHN, $N\rightarrow h\nu,h\rightarrow b\bar{b}$ to study the fat jet
signature. For the time being, we mainly focus on the first two items. The
investigation of the mode, $N\rightarrow Z\nu,Z\rightarrow b\bar{b}$ is beyond
the scope of this article and shall be presented in future work in detail.

The paper is organised as follows. in Sec. II, we discuss the model and the
interactions of the heavy neutrino with SM particles and also calculate the
production cross sections at different colliders. In Sec.III we discuss the
complete collider study. In Sec. IV we calculate the bounds on the mixing
angles and compare them with the existing results. Finally, we conclude in
Sec. V.

##  II Model and the production mode

In type-I seesaw typ1 , SM gauge-singlet right handed Majorana neutrinos
$N_{R}^{\beta}$ are introduced, where $\beta$ is the flavor index.
$N_{R}^{\beta}$ have direct coupling with SM lepton doublets
$\ell_{L}^{\alpha}$ and the SM Higgs doublet $H$. The relevant part of the
Lagrangian can be written as :

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

After the spontaneous EW symmetry breaking by getting the vacuum expectation
value (VEV) of the Higgs field, $H=\begin{pmatrix}\frac{v}{\sqrt{2}}\\\
0\end{pmatrix}$, we obtain the Dirac mass matrix as
$M_{D}=\frac{Y_{D}v}{\sqrt{2}}$. Using the Dirac and Majorana mass matrices,
the neutrino mass matrix can be written as

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

After diagonalizing this matrix, 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 , Dirac Yukawa term $Y_{D}$ can be as large as $1$, and this
scenario is considered in this paper.

There is another seesaw mechanism, so-called inverse seesaw inverse , where
the light Majorana neutrino mass is generated through tiny lepton number
violation. 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 $M_{N}$ is the Dirac mass matrix, $N_{R}^{\alpha}$ and $S_{L}^{\beta}$
are two SM-singlet heavy neutrinos with the same lepton numbers, and $\mu$ is
a small lepton number violating Majorana mass matrix. After the electroweak
symmetry breaking the neutrino mass matrix is obtained as

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

After 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 small lepton number violating term $\mu$ is responsible for the
tiny neutrino mass generation. The smallness of $\mu$ allows the
$M_{D}M_{N}^{-1}$ parameter to be order one even for an EW scale heavy
neutrino. Since the scale of $\mu$ is much smaller than the scale of $M_{N}$,
the heavy neutrinos become the pseudo-Dirac particles. This is the main
difference between the type-I and the inverse seesaw.

Assuming $M_{D}M_{N}^{-1}\ll 1$, the flavor eigenstates ($\nu$) of the light
Majorana neutrinos can be expressed in terms of the mass eigenstates of the
light ($\nu_{m}$) and heavy ($N_{m}$) Majorana neutrinos such as

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

where

$\displaystyle{\cal R}=M_{D}M_{N}^{-1},\;{\cal
N}=\left(1-\frac{1}{2}\epsilon\right)U_{\rm MNS}$ (8)

with $\epsilon={\cal R}^{*}{\cal R}^{T}$, and $U_{MNS}$ is the usual neutrino
mixing matrix by which the mass matrix $m_{\nu}$ is diagonalized as

$\displaystyle U_{MNS}^{T}m_{\nu}U_{MNS}={\rm diag}(m_{1},m_{2},m_{3}).$ (9)

In the presence of $\epsilon$, the mixing matrix ${\cal N}$ is not unitary
Antusch:2006vwa ; Abada:2007ux ; Antusch:2014woa ; Antusch:2016brq .
Considering the mass eigenstates, the charged current interaction in the
Standard Model is given by

$\displaystyle\mathcal{L}_{CC}=-\frac{g}{\sqrt{2}}W_{\mu}\bar{e}\gamma^{\mu}P_{L}\left({\cal
N}\nu_{m}+{\cal R}N_{m}\right)+h.c.,$ (10)

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 interaction is given by

$\displaystyle\mathcal{L}_{NC}=-\frac{g}{2c_{w}}Z_{\mu}\left[\overline{\nu_{m}}\gamma^{\mu}P_{L}({\cal
N}^{\dagger}{\cal N})\nu_{m}+\overline{N_{m}}\gamma^{\mu}P_{L}({\cal
R}^{\dagger}{\cal R})N_{m}+\left\\{\overline{\nu_{m}}\gamma^{\mu}P_{L}({\cal
N}^{\dagger}{\cal R})N_{m}+h.c.\right\\}\right],$ (11)

where $c_{w}=\cos\theta_{w}$ is the weak mixing angle. Because of non-
unitarity of the matrix ${\cal N}$, ${\cal N}^{\dagger}{\cal N}\neq 1$ and the
flavor-changing neutral current occurs.

The dominant decay modes of the heavy neutrino are $N\rightarrow\ell W$,
$\nu_{\ell}Z$, $\nu_{\ell}h$ and the corresponding partial decay widths are
respectively given by

$\displaystyle\Gamma(N\rightarrow\ell W)$ $\displaystyle=$
$\displaystyle\frac{g^{2}|V_{\ell
N}|^{2}}{64\pi}\frac{(M_{N}^{2}-M_{W}^{2})^{2}(M_{N}^{2}+2M_{W}^{2})}{M_{N}^{3}M_{W}^{2}},$
$\displaystyle\Gamma(N\rightarrow\nu_{\ell}Z)$ $\displaystyle=$
$\displaystyle\frac{g^{2}|V_{\ell N}|^{2}}{128\pi
c_{w}^{2}}\frac{(M_{N}^{2}-M_{Z}^{2})^{2}(M_{N}^{2}+2M_{Z}^{2})}{M_{N}^{3}M_{Z}^{2}},$
$\displaystyle\Gamma(N\rightarrow\nu_{\ell}h)$ $\displaystyle=$
$\displaystyle\frac{|V_{\ell N}|^{2}(M_{N}^{2}-M_{h}^{2})^{2}}{32\pi
M_{N}}\left(\frac{1}{v}\right)^{2}.$ (12)

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
$\left(M_{N}\right)$. Note that for larger values of $M_{N}$, the branching
ratios can be obtained as

$\displaystyle BR\left(N\rightarrow\ell W\right):BR\left(N\rightarrow\nu
Z\right):BR\left(N\rightarrow\nu H\right)\simeq 2:1:1.$ (13)

Figure 1: Heavy neutrino branching ratios ($BR_{i}$) for different decay modes
are shown with respect to the heavy neutrino mass $\left(M_{N}\right)$.

###  II.1 Production cross section at LHeC

The LHeC can produce the RHN in the process $e~{}p\rightarrow N_{1}j_{1}$
through the $t$\- channel exchanging the $W$ boson. In this case the first
generation RHN ($N_{1}$) will be produced. The corresponding Feynman diagram
is given in Fig. 2. The total differential production cross section for this
process is calculated as

Figure 2: Production process, $ep\rightarrow N_{1}j_{1}$ ,of the RHN at the
LHeC through a $t$ channel $W$ boson exchange

$\displaystyle\frac{d\hat{\sigma}_{LHeC}}{d\cos\theta}$ $\displaystyle=$
$\displaystyle\frac{3.89\times
10^{8}}{32\pi}3\times\frac{1}{3}\Big{(}\frac{1}{2}\Big{)}^{2}\Big{(}\frac{M_{inv}^{2}-M_{N}^{2}}{M_{N}^{2}}\Big{)}\times$
(14)
$\displaystyle\frac{256C_{\ell}^{2}C_{q}^{2}\Big{(}\frac{M_{inv}^{2}-M_{N}^{2}}{4}\Big{)}}{\Big{[}M_{N}^{2}-2\Big{\\{}\frac{M_{inv}^{2}}{4}\Big{(}1-\cos\theta\Big{)}\Big{\\}}+\frac{M_{inv}^{2}}{4}\Big{(}1+\cos\theta\Big{)}\Big{]}^{2}+\Gamma_{W}^{2}M_{W}^{2}}$

where $C_{\ell}=C_{q}=\frac{g}{2^{\frac{3}{2}}}$.

Figure 3: RHN production cross section at the LHeC considering
$e~{}p\rightarrow N_{1}j$ process for the $e~{}p$ collider at $\sqrt{s}=318$
GeV (HERA, top left panel), $\sqrt{s}=1.3$ TeV (LHeC, top right panel) and
$\sqrt{s}=1.8$ TeV (HE-LHeC, bottom panel).

Performing the integration over $\cos\theta$ between $[-1,1]$ we find the
cross section as $\hat{\sigma}_{LHeC}$ and finally convoluting the PDF
(CTEQ5M) Pumplin:2002vw  we get the total cross section as

$\displaystyle\sigma=\sum_{i}\int^{1}_{\frac{M^{2}}{E_{CM}^{2}}}dx~{}q_{i}(x,\sqrt{x}E_{CM})~{}\hat{\sigma}_{LHeC}(\sqrt{x}E_{CM})$
(15)

where $E_{CM}$ is the center of mass energy of the LHeC and $i$ runs over the
quark flavors. For different center of mass energies $E$ will be different. In
Fig. 3 we plot the total production cross sections of $N_{1}$ at the three
different collider energies such as $\sqrt{s}=318$ GeV (HERA), $\sqrt{s}=1.3$
TeV (LHeC) and $\sqrt{s}=1.8$ TeV (High Energy LHeC (HE-LHeC)) respectively.
The cross section in Fig. 3 is normalized by the square of the mixing to
correspond the maximum value for a fixed $M_{N}$ according to the relevant
part of the charged current interaction in Eq. 10.

###  II.2 Production cross section at linear collider

Figure 4: RHN production processes at the linear collider. The left panel is
the dominant $t$ channel process and the right panel is $s$ channel process to
produce the $e^{+}e^{-}\rightarrow N_{1}\nu_{1}$. To produce $N_{2}\nu_{2}$
and $N_{3}\nu_{3}$, the $Z$ mediated $s$ channel process will act.

The linear collider can produce the heavy neutrino in the process
$e^{+}e^{-}\rightarrow\overline{\nu_{1}}N_{1}$ through $t$ and $s$-channels
exchanging the $W$ and $Z$ bosons, respectively. The corresponding Feynman
diagrams are given in Fig. 4. The total differential production cross section
for this process is calculated as

$\displaystyle\frac{d\sigma_{ILC}}{d\cos\theta}$ $\displaystyle=$
$\displaystyle(3.89\times 10^{8}\;{\rm pb})\times\frac{\beta}{32\pi
s}\frac{s+M_{N}^{2}}{s}\left(\frac{1}{2}\right)^{2}$ (16)
$\displaystyle\times$
$\displaystyle\left[\frac{16C_{1}^{2}C_{2}^{2}\left(s^{2}-M_{N}^{4}\right)(1+\cos\theta)(1+\beta\cos\theta)}{(M_{N}^{2}-\frac{s-M_{N}^{2}}{2}(1-\beta\cos\theta)-M_{W}^{2})^{2}+M_{W}^{2}\Gamma_{W}^{2}}\right.$
$\displaystyle+$
$\displaystyle\frac{\left(4(C^{2}_{A_{e}}+C^{2}_{V_{e}})(C^{2}_{A_{\nu}}+C^{2}_{V_{\nu}})(1+\beta\cos^{2}\theta)+16C_{A_{e}}C_{V_{e}}C_{A_{\nu}}C_{V_{\nu}}(1+\beta)\cos\theta\right)(s^{2}-M_{N}^{4})}{(s-M_{Z}^{2})^{2}+M_{Z}^{2}\Gamma_{Z}^{2}}$
$\displaystyle-$ $\displaystyle
32C^{2}_{1}C^{2}_{A_{e}}(s^{2}-M_{N}^{4})(1+\cos\theta)(1+\beta\cos\theta)$
$\displaystyle\left.\times\frac{\left(M_{N}^{2}-\frac{s-M_{N}^{2}}{2}(1-\beta\cos\theta)-M_{W}^{2}\right)(s-M_{Z}^{2})+M_{W}M_{Z}\Gamma_{W}\Gamma_{Z}}{((M_{N}^{2}-\frac{s-M_{N}^{2}}{2}(1-\beta\cos\theta)-M_{W}^{2})^{2}+M_{W}^{2}\Gamma_{W}^{2})((s-M_{Z}^{2})^{2}+M_{Z}^{2}\Gamma_{Z}^{2})}\right],$

where $\beta=(s-M_{N}^{2})/(s+M_{N}^{2})$,

$\displaystyle
C_{1}=-C_{2}=\frac{g}{2\sqrt{2}},\;C_{A_{\nu}}=C_{V_{\nu}}=\frac{g}{4\cos\theta_{W}},$
$\displaystyle
C_{A_{e}}=\frac{g}{2\cos\theta_{w}}\left(-\frac{1}{2}+2\sin^{2}\theta_{w}\right),\;C_{V_{e}}=-\frac{g}{4\cos\theta_{w}}.$
(17)

The total production cross section for the process
$e^{+}e^{-}\rightarrow\overline{\nu_{1}}N_{1}$ from the $t$ and $s$ channel
processes at the linear collider at different center of mass energies are
shown in Fig. 5.

Figure 5: RHN production cross section at the linear collider considering
$e^{+}e^{-}\rightarrow N_{1}\nu_{1}$ process at the different center of mass
energies.

The $s$ channel $Z$ mediated process can produce the second (third) generation
of RHNs, $N_{2}(N_{3})$ in association with $\nu_{2}(\nu_{3})$. The cross
sections for different center of mass energies have been given in Fig. 6. The
cross section in this mode decreases with the increase in the center of mass
energy. Such modes can reach up to a cross section of $1$ pb for $M_{N}=100$
GeV at $\sqrt{s}=250$ GeV. Consider the leading decay mode of the RHN into $W$
and $\ell~{}(\mu,\tau)$ followed by the hadronic decay of the $W$ could be
interesting to probe the corresponding mixing angles.

Figure 6: RHN production cross section at the linear collider considering
$e^{+}e^{-}\rightarrow N_{2}\nu_{2}~{}(N_{3}\nu_{3})$ process at the different
center of mass energies from the $s$ channel $Z$ boson exchange.

The cross sections in Figs. 5 and 6 are normalized by the square of the mixing
to correspond the maximum value for a fixed $M_{N}$ according to the relevant
part of the charged current and neutral current interactions in Eqs. 10 and 11
respectively.

##  III Collider Analysis

We implement our model in FeynRules Alloul:2013bka , generate the UFO file of
the model for MadGraph5-aMC@NLO Alwall:2014hca  to calculate the signals and
the backgrounds. Further we use PYTHIA6 Sjostrand:2006za  for LHeC as used in
Mandal:2018qpg  and PYTHIA8 Sjostrand:2014zea  for the linear colliders, where
subsequent decay, initial state radiation, final state radiation and
hadronisation have been carried out. We have indicated in Das:2017gke ;
Bhardwaj:2018lma  that if the RHNs are sufficiently heavy, the daughter
particles can be boosted. We prefer the hadronic decay mode of the $W$ where
the jets can be collimated so that we can call it a fat-jet $(J)$. Such a
topology is very powerful to discriminate the signal from the SM backgrounds.
We perform the detector simulation using DELPHES version 3.4.1
deFavereau:2013fsa . The detector card for the LHeC has been used from LHeC-
card . We use the ILD card for the linear collider. In our analysis the jets
are reconstructed by Cambridge-Achen algorithm Dokshitzer:1997in ;
Wobisch:1998wt  implemented in Fastjet package Cacciari:2011ma ;
Cacciari:2005hq  with the radius parameter as $R=0.8$.

We study the production of the first generation RHN $(N_{1})$ and its
subsequent leading decay mode $(e~{}p\rightarrow
N_{1}~{}j_{1},N_{1}\rightarrow We,W\rightarrow J)$ at the LHeC with
$\sqrt{s}=1.3$ TeV and $1.8$ TeV center of mass energies. The corresponding
Feynman diagram is given in Fig. 7.

Figure 7: $e+J+j_{1}$ final state at the LHeC and HE-LHeC.

We also study the RHN production at the linear collider (International Linear
Collider, ILC) at $\sqrt{s}=1$ TeV and CLIC at $\sqrt{s}=3$ TeV collider
energies. However, for simplicity we will use the term linear collider
unanimously. At the linear collider we consider two sets of signals after the
production of the RHN, such that, $e^{+}~{}e^{-}\rightarrow
N_{1}~{}\nu,N_{1}\rightarrow We,W\rightarrow J$ and $e^{+}~{}e^{-}\rightarrow
N_{1}~{}\nu,N_{1}\rightarrow h\nu,h\rightarrow J_{b}$ where $J_{b}$ is a fat
$b$-jet coming from the boosted SM Higgs decay in the dominant mode. For the
two types of colliders we consider $1000$ fb-1 luminosity. The corresponding
Feynman diagrams are given in Fig. 8.

Figure 8: $e+J+p_{T}^{miss}$ and $J_{b}+p_{T}^{miss}$ final states at the
linear colliders.

For the analysis of signal and background events we use the following set of
basic cuts,

  1. 1.

Electrons in the final state should have the following transverse momentum
$(p_{T}^{e})$ and pseudo-rapidity $(|\eta^{e}|)$ as $p_{T}^{e}>10$ GeV,
$|\eta^{e}|<2.5.$

  2. 2.

Jets are ordered in $p_{T}$, jets should have $p_{T}^{j}>10$ GeV and
$|\eta^{j}|<2.5$.

  3. 3.

Photons are counted if $p_{T}^{\gamma}>10$ GeV and $|\eta^{\gamma}|<2.5$.

  4. 4.

Leptons should be separated by $\Delta R_{\ell\ell}>0.2$.

  5. 5.

The leptons and photons are separated by $\Delta R_{\ell\gamma}>0.3$.

  6. 6.

The jets and leptons should be separated by $\Delta R_{\ell j}>0.3$.

  7. 7.

Fat Jet is constructed with radius parameter $R=0.8$.

Figure 9: Transverse momentum distribution of the associated jet
$(p_{T}^{j_{1}})$ from the signal and background events for $M_{N}=600$ GeV
and $700$ GeV at the $\sqrt{s}=1.3$ TeV LHeC (left panel) and $M_{N}=900$ GeV
and $1$ TeV at the $\sqrt{s}=1.8$ TeV LHeC (right panel)

###  III.1 LHeC analysis for the signal $e^{-}p\rightarrow jN_{1}\rightarrow
e^{\pm}+J+j_{1}$

Producing $N_{1}$ at the LHeC and followed by its decay into leading mode to
study the boosted objects, we consider the final state $e^{\pm}+J+j_{1}$. In
this case we have two different processes, one is them is the $e^{+}+J+j_{1}$
and the other one is $e^{-}+J+j_{1}$. The first one is the Lepton Number
Violating (LNV) channel and the second one is the Lepton Number Conserving
(LNC). At the time of showing the results we combine LNV and LNC channels to
obtain the final state as $e^{\pm}+J+j_{1}$.

The LNV signal is almost background free until some $e^{+}+$jets events appear
from some radiations, however, that effect will be negligible. Therefore for
completeness we include the LNC channel where the leading SM backgrounds will
come from $e^{-}jjj$, $e^{-}jj$ and $e^{-}j$ including initial state and final
state radiations. For completeness we include both of the LNV and LNC
channels. Further we use the fat-jet algorithm to reduce the SM backgrounds.
We have shown the distributions of the transverse momentum of the leading jet
$(p_{T}^{j_{1}})$, lepton $(p_{T}^{e})$ and fat-jet $(p_{T}^{J})$ in
Figs.9-11. The fat-jet mass distribution $(M_{J})$ has been shown in Figs.12.
The invariant mass distribution of the lepton and fat-jet system $(M_{eJ})$
has been shown in Fig. 13. We have also compared the signals with the
corresponding SM backgrounds. As a sample we consider $M_{N}=600$ GeV and
$700$ GeV for $\sqrt{s}=1.3$ TeV LHeC and $M_{N}=900$ GeV, $1$ TeV at
$\sqrt{s}=1.8$ TeV HE-LHeC as shown in Figs.9-13.

Figure 10: Transverse momentum distribution of the electron $(p_{T}^{e})$ from
the signal and background events for $M_{N}=600$ GeV and $700$ GeV at the
$\sqrt{s}=1.3$ TeV LHeC (left panel) and $M_{N}=900$ GeV and $1$ TeV at the
$\sqrt{s}=1.8$ TeV HE-LHeC (right panel)

Figure 11: Transverse momentum distribution of the fat jet $(p_{T}^{J})$ from
the signal and background events for $M_{N}=600$ GeV and $700$ GeV at the
$\sqrt{s}=1.3$ TeV LHeC (left panel) and $M_{N}=900$ GeV and $1$ TeV at the
$\sqrt{s}=1.8$ TeV HE-LHeC (right panel).

Figure 12: Jet mass $(M_{J})$ distribution of the fat jet from the signal and
background events for $M_{N}=600$ GeV and $700$ GeV at the $\sqrt{s}=1.3$ TeV
LHeC (left panel) and $M_{N}=900$ GeV and $1$ TeV at the $\sqrt{s}=1.8$ TeV
HE-LHeC (right panel).

Figure 13: Invariant mass distribution of the fat jet and electron system
$(M_{eJ})$ from the signal and background events for $M_{N}=600$ GeV and $700$
GeV at the $\sqrt{s}=1.3$ TeV LHeC (left panel) and $M_{N}=900$ GeV and $1$
TeV at the $\sqrt{s}=1.8$ TeV HE-LHeC (right panel).

We have chosen $M_{N}=400$ GeV- $900$ GeV for the $1.3$ TeV LHeC and
$M_{N}=800$ GeV- $1.5$ TeV for the $1.8$ TeV HE-LHeC. As benchmark points we
have chosen $M_{N}=600$ GeV, $700$ GeV at the $1.3$ TeV LHeC and $M_{N}=900$
GeV, $1.0$ TeV at the $1.8$ TeV HE-LHeC after the basic cuts. In view of the
distributions in Figs.9-13, we have used the following advanced selection cuts
to reduce the backgrounds:

####  III.1.1 Advanced cuts for $M_{N}=400$ GeV -$900$ GeV at the
$\sqrt{s}=1.3$ TeV LHeC after the detector simulation

  * •

Transverse momentum for lepton and jet, $p_{T}^{e^{\pm}}>50$ GeV.

  * •

Transverse momentum for fat-jet $p_{T}^{J}>175$ GeV.

  * •

Fat-jet mass $M_{J}>70$ GeV.

  * •

Invariant mass window of $e^{\pm}$ and fat-jet $J$, $|M_{eJ}-M_{N}|\leq 20$
GeV.

We have noticed that $M_{J}>70$ GeV cuts out the low energy peaks $(M_{J}\leq
25$ GeV$)$ which come from the hadronic activity of the low energy jets.
Similarly, the $p_{T}^{J}$ and $p_{T}^{e}$ cuts are also very effective. Due
to the presence of the RHN, these distributions from the signal will be in the
high values than the SM backgrounds. Therefore selecting such cuts at high
values, as we have done here, will be extremely useful to reduce the SM
backgrounds.

We have noticed that $ej$ background can completely be reduced with the
application of the kinematic cuts on $p_{T}^{e}$, $p_{T}^{J}$ and $M_{J}$. It
is difficult to obtain a fat jet from this process because the $t$ channel
exchange of the $Z$ boson and photon will contribute to this process, however,
the other low-energy jets may come from the radiations at the initial and
final states. These jets do not help to make the fat jets sufficiently
energetic. Therefore $p_{T}^{J}>175$ GeV $(p_{T}^{J}>400$ GeV$)$ at the LHeC
(HE-LHeC) are very useful. Similarly the $ejjj$ is the irreducible background
in this case which will contribute most among the backgrounds. Whereas $ejj$
is the second leading background in this case. However, both of these
backgrounds can be reduced using the invariant mass cut of the RHN. As the RHN
will decay according to $N\rightarrow eJ$, therefore the invariant mass of the
$eJ$ system with an window of $20$ GeV $(|M_{eJ}-M_{N}|\leq 20$ GeV$)$ will be
extremely useful to reduce the backgrounds further in these colliders. In Tab.
1 we have given the two benchmark scenarios at the $1.3$ TeV LHeC where the
signal events are normalized by the square of the mixing.

Cuts | Signal | Background | Total
---|---|---|---
|  $M_{N_{1}}=600$ GeV |  $M_{N_{1}}=700$ GeV | $ejjj$ | $ejj$ |
Basic Cuts | 645,860 | 261,254 | 70,029,800 | 189,689,000 | 259,718,800
$p^{J}_{T}>175$ GeV | 476,640 | 214,520 | 295,658 | 338,720 | 634,378
$M_{J}>70$ GeV | 356,350 | 160,017 | 35,244 | 17,520 | 52,764
$p_{T}^{e}>50$ GeV | 356,126 | 159,918 | 33,286 | 17,520 | 50,806
$|M_{eJ}-M_{N}|\leq 20$ GeV | 304,457 | 129,690 | 7 | 1 | 8
Table 1: Cut flow of the signal and background events for the final state
$e^{\pm}+J+j_{1}$ for $M_{N}=600$ GeV and $700$ GeV with $\sqrt{s}=1.3$ TeV
LHeC where the signal events are normalized by the square of the mixing.

####  III.1.2 Advanced cuts for $M_{N}=800\,\text{GeV}-1.5$ TeV at the
$\sqrt{s}=1.8$ TeV HE-LHeC after the detector simulation

  * •

Transverse momentum for lepton, $p_{T}^{e^{\pm}}>250$ GeV.

  * •

Transverse momentum for fat-jet $p_{T}^{J}>400$ GeV.

  * •

Fat-jet mass $M_{J}>70$ GeV.

  * •

Invariant mass window of $e^{\pm}$ and fat-jet $J$, $|M_{eJ}-M_{N}|\leq 20$
GeV.

Cuts | Signal | Background | Total
---|---|---|---
|  $M_{N_{1}}=900$ GeV |  $M_{N_{1}}=1$ TeV | $ejjj$ | $ejj$ |
Basic Cuts | 427,311 | 207,015 | 108,243,000 | 273,410,000 | 381,653,000
$p^{J}_{T}>400$ GeV | 158,694 | 110,289 | 12,225 | 12,450 | 24,675
$M_{J}>70$ GeV | 145,558 | 96,787 | 4,596 | 4,150 | 8,746
$p_{T}^{e}>250$ GeV | 144,997 | 96,487 | 4,596 | 4,150 | 8,746
$|M_{eJ}-M_{N}|\leq 20$ GeV | 119,659 | 71,490 | 3 | 1 | 4
Table 2: Cut flow of the signal and background events for the final state
$e^{\pm}+J+j_{1}$ for $M_{N}=900$ GeV and $1.0$ TeV with $\sqrt{s}=1.8$ TeV
HE-LHeC where the signal events are normalized by the square of the mixing.

We have chosen $M_{N}=900~{}\text{GeV}\text{and}\,1$ TeV at the $\sqrt{s}=1.8$
TeV HE-LHeC. The corresponding signals normalized by the square of the mixing
and the SM backgrounds are listed in Tab. 2. Due to the heavier mass range of
the RHN, we have chosen stronger cuts for the transverse momenta of the
electron and fat-jet which became useful to reduce the backgrounds.

###  III.2 Linear collider analysis for the signal $e^{\pm}+J+p_{T}^{miss}$

In linear collider we study $e^{\pm}+J+p_{T}^{miss}$ signal from the leading
decay mode of the RHN at the $1$ TeV and $3$ TeV center of mass energy. The
corresponding distributions for two benchmark points for $M_{N}=500$ GeV,
$800$ GeV at$\sqrt{s}=1$ TeV and $M_{N}=800$ GeV, $2$ TeV at $\sqrt{s}=3$ TeV
linear colliders are given in Figs.14-18 after the basic cuts. We perform a
complete cut based analysis for the signal and the SM backgrounds. In this
process we have $\nu_{e}eW$ as the leading background where as $WW$, $ZZ$ and
$t\bar{t}$ are other important backgrounds.

Figure 14: Missing momentum distribution of the signal and background events
for $M_{N}=500$ GeV and $800$ GeV at the $\sqrt{s}=1$ TeV (left panel) and
$M_{N}=800$ GeV and $2$ TeV at the $\sqrt{s}=3$ TeV (right panel) linear
colliders.

Figure 15: Transverse momentum distribution of the electron $(p_{T}^{e})$ from
the signal and background events for $M_{N}=500$ GeV and $800$ GeV at the
$\sqrt{s}=1$ TeV (left panel) and $M_{N}=800$ GeV and $2$ TeV at the
$\sqrt{s}=3$ TeV (right panel) linear colliders.

Figure 16: Transverse momentum distribution of the fat jet $(p_{T}^{J})$ from
the signal and background events for $M_{N}=500$ GeV and $800$ GeV at the
$\sqrt{s}=1$ TeV (left panel) and $M_{N}=800$ GeV and $2$ TeV at the
$\sqrt{s}=3$ TeV linear colliders.

Figure 17: Jet mass $(M_{J})$ distribution of the fat jet from the signal and
background events for $M_{N}=500$ GeV and $800$ GeV at the $\sqrt{s}=1$ TeV
(left panel) and $M_{N}=800$ GeV and $2$ TeV at the $\sqrt{s}=3$ TeV (right
panel) linear colliders.

Figure 18: $\cos\theta_{J(e)}$ distributions for the $J(e)$ in the first row
(second row) for the $1$ TeV (left column) and $3$ TeV (right column) linear
colliders.

We have shown the missing momentum $(p_{T}^{miss})$, transverse momenta of the
electron $p_{T}^{e}$ and fat-jet $p_{T}^{J}$ in Figs. 14-16 for the linear
colliders. The fat-jet mass $M_{J}$ distribution has been shown in Fig. 17. We
construct the polar angle variable in Fig. 18 for the electron (fat jet),
$\cos\theta_{e}(\cos\theta_{J})$ where
$\theta_{e(J)}=\tan^{-1}\Big{[}\frac{p_{T}^{e(J)}}{p_{z}^{e(J)}}\Big{]}$,
where $p_{z}^{e(J)}$ is the $z$ component of the three momentum of the
electron (fat jet). This is a very effective cut which reduces the SM
background significantly. In view of these distributions, we have used the
following advanced selection cuts to reduce the backgrounds:

####  III.2.1 Advanced cuts for $M_{N}=400$ GeV-$900$ GeV at the $\sqrt{s}=1$
TeV linear collider after the detector simulation

  * •

Transverse momentum for fat-jet $p_{T}^{J}>150$ GeV for $M_{N}$ mass range
$400$ GeV-$600$ GeV and $p_{T}^{J}>250$ GeV for $M_{N}$ mass range $700$
GeV-$900$ GeV.

  * •

Transverse momentum for leading lepton $p_{T}^{e^{\pm}}>100$ GeV for $M_{N}$
mass range $400$ GeV-$600$ GeV and $p_{T}^{e^{\pm}}>200$ GeV for $M_{N}$ mass
range $700$ GeV-$900$ GeV.

  * •

Polar angle of lepton and fat-jet $|\text{cos}~{}\theta_{e}|<0.85$,
$|\text{cos}~{}\theta_{J}|<0.85$.

  * •

Fat-jet mass $M_{J}>70$ GeV.

We have tested $M_{N}=400$ GeV to $900$ GeV at the $\sqrt{s}=1$ TeV at the
linear collider. Hence we consider two benchmark points at the $\sqrt{s}=1$
TeV linear collider such as $M_{N}=500$ GeV and $800$ GeV. The cut flow for
the $\sqrt{s}=1$ TeV are given in the Tabs. 3 and 4 respectively. We have
noticed that $\cos\theta_{e(J)}$ is a very important kinematic variable and
setting $|\cos\theta_{e(J)}|<0.85$ puts a very strong cut for the SM
backgrounds. The $M_{J}>70$ GeV is also effective to cut out the low mass
peaks $(1$ GeV $\leq M_{J}\leq 25$ GeV $)$ from the low energy jets.

Cuts | Signal | Background | Total
---|---|---|---
|  | $\nu_{e}eW$ | $WW$ | $ZZ$ | $t\bar{t}$ |
Basic Cuts | 12,996,200 | 201,586 | 72,244 | 7,200 | 4,300 | 285,330
$|\text{cos}~{}\theta_{J}|\leq 0.85$ | 12,789,800 | 148,802 | 44,910 | 3,800 | 4,100 | 201,600
$|\text{cos}~{}\theta_{e}|\leq\,0.85$ | 12,671,800 | 79,008 | 40,574 | 2,800 | 3,900 | 126,280
$p^{J}_{T}>150$ GeV | 12,308,300 | 70,669 | 40,490 | 2,300 | 3,200 | 116,660
$M_{J}>70$ GeV | 10,923,100 | 62,303 | 37,043 | 2,100 | 2,300 | 103,700
$p_{T}^{\ell}>100\,\text{GeV}$ | 10,714,500 | 57,076 | 33,488 | 1,400 | 1,530 | 93,400
Table 3: Cut flow for the signal and background events for the final state $e^{\pm}+J+p_{T}^{miss}$ for $M_{N}=500$ GeV at the $\sqrt{s}=1$ TeV linear collider. The signal events are normalized by the square of the mixing. Cuts | Signal | Background | Total
---|---|---|---
|  | $\nu_{e}eW$ | $WW$ | $ZZ$ | $t\bar{t}$ |
Basic Cuts | 8,684,990 | 201,586 | 72,244 | 7,200 | 4,300 | 285,330
$|\text{cos}~{}\theta_{J}|\leq 0.85$ | 8,649,570 | 148,802 | 44,910 | 3,800 | 4,100 | 201,600
$|\text{cos}~{}\theta_{e}|\leq\,0.85$ | 8,618,420 | 79,008 | 40,574 | 2,800 | 3,900 | 126,280
$p^{J}_{T}>250$ GeV | 7,681,440 | 59,001 | 40,329 | 2,303 | 2,720 | 104,354
$M_{J}>70$ GeV | 7,176,280 | 53,990 | 36,997 | 2,187 | 2,282 | 95,437
$p_{T}^{\ell}>200\,\text{GeV}$ | 7,080,200 | 38,729 | 26,208 | 942 | 613 | 66,493
Table 4: Cut flow for the signal and background events for the final state
$e^{\pm}+J+p_{T}^{miss}$ for $M_{N}=800$ GeV at the $\sqrt{s}=1$ TeV linear
collider. The signal events are normalized by the square of the mixing.

####  III.2.2 Advanced cuts for $M_{N}=700$ GeV-$2.9$ TeV at the $\sqrt{s}=3$
TeV linear collider after the detector simulation

  * •

Transverse momentum for fat-jet $p_{T}^{J}>250$ GeV for the $M_{N}$ mass range
$700$ GeV-$900$ GeV and $p_{T}^{J}>400$ GeV for $M_{N}$ mass range $1-2.9$
TeV.

  * •

Transverse momentum for leading lepton $p_{T}^{e^{\pm}}>200$ GeV for $M_{N}$
mass range $700-900$ GeV and $p_{T}^{e^{\pm}}>250$ GeV for $M_{N}$ mass range
$1-2.9$ TeV.

  * •

Polar angle of lepton and fat-jet $|\text{cos}~{}\theta_{e}|<0.85$,
$|\text{cos}~{}\theta_{J}|<0.85$.

  * •

Fat-jet mass $M_{J}>70$ GeV.

We have tested $M_{N}=700$ GeV to $2.9$ TeV at the $\sqrt{s}=3$ TeV at the
linear collider. Hence we consider two benchmark points at the $\sqrt{s}=3$
TeV linear collider such as $M_{N}=800$ GeV and $2$ TeV. The cut flow for the
benchmark points at the $\sqrt{s}=3$ TeV are given in the Tabs. 5 and 6
respectively. At the $3$ TeV we see almost the same behavior for the kinematic
variables as we noticed at the $1$ TeV case except the $p_{T}$ distributions
of the electron and fat jet.

Cuts | Signal | Background | Total
---|---|---|---
|  | $\nu_{e}eW$ | $WW$ | $ZZ$ | $t\bar{t}$ |
Basic Cuts | 21,789,900 | 193,533 | 12,135 | 1,361 | 271 | 207,301
$|\text{cos}~{}\theta_{J}|\leq 0.85$ | 13,599,300 | 126,980 | 4,766 | 406 | 215 | 132,367
$|\text{cos}~{}\theta_{e}|\leq\,0.85$ | 12,163,300 | 21,110 | 4,609 | 390 | 195 | 26,304
$p^{J}_{T}>250$ GeV | 12,083,500 | 18,619 | 4,607 | 390 | 189 | 23,807
$M_{J}>70$ GeV | 11,287,000 | 17,442 | 4,411 | 385 | 176 | 22,416
$p_{T}^{\ell}>200\,\text{GeV}$ | 11,094,300 | 16,915 | 4,108 | 343 | 104 | 21,470
Table 5: Cut flow for the signal and background events for the final state $e^{\pm}+J+p_{T}^{miss}$ for $M_{N}=800$ GeV at the $\sqrt{s}=3$ TeV linear collider. The signal events are normalized by the square of the mixing.  Cuts | Signal | Background | Total
---|---|---|---
|  | $\nu_{e}eW$ | $WW$ | $ZZ$ | $t\bar{t}$ |
Basic Cuts | 13,822,500 | 193,533 | 12,135 | 1,382 | 271 | 207,322
$|\text{cos}~{}\theta_{J}|\leq 0.85$ | 12,701,600 | 126,980 | 4,766 | 412 | 215 | 132,374
$|\text{cos}~{}\theta_{e}|\leq\,0.85$ | 12,647,200 | 21,110 | 4,609 | 396 | 195 | 26,310
$p^{J}_{T}>400$ GeV | 12,611,000 | 15,737 | 4,605 | 396 | 184 | 20,923
$M_{J}>70$ GeV | 12,015,600 | 14,889 | 4,410 | 391 | 175 | 19,865
$p_{T}^{\ell}>250\,\text{GeV}$ | 11,987,000 | 14,184 | 4,010 | 336 | 10 | 18,630
Table 6: Cut flow for the signal and background events for the final state
$e^{\pm}+J+p_{T}^{miss}$ for $M_{N}=2$ TeV at the $\sqrt{s}=3$ TeV linear
collider. The signal events are normalized by the square of the mixing.

At this point we must mention that the backgrounds like $ZZ$ and $t\bar{t}$
can have more than one lepton in the final state which has been efficiently
vetoed to reduce the effect.

###  III.3 Linear collider analysis for the signal $J_{b}+p_{T}^{miss}$

Considering the $N\rightarrow h\nu,h\rightarrow J_{b}$ mode at the linear
collider we obtain the $J_{b}+p_{T}^{miss}$ final state. For this final state
the dominant SM backgrounds come from the processes
$h\nu_{\ell}\bar{\nu}_{\ell}$ and $Z\nu_{\ell}\bar{\nu}_{\ell}$. Backgrounds
can also come from the intermediate processes $ZZ$ and $ZH$. We have generated
the background events combining all these processes in MadGraph for our
analysis.

Figure 19: $p_{T}^{miss}$ distribution of the signal and background events for
$M_{N}=700$ GeV and $800$ GeV at the $\sqrt{s}=1$ TeV (left panel) and
$M_{N}=1.5$ TeV and $2$ TeV at the $\sqrt{s}=3$ TeV (right panel) linear
colliders.

Figure 20: Transverse momentum distribution of $J_{b}$ $(p_{T}^{J_{b}})$ from
the signal and background events for $M_{N}=700$ GeV and $800$ GeV at the
$\sqrt{s}=1$ TeV (left panel) and $M_{N}=1.5$ TeV and $2$ TeV at the
$\sqrt{s}=3$ TeV (right panel) linear colliders.

Figure 21: Fat b-Jet mass $(M_{J_{b}})$ distribution from the signal and
background events for $M_{N}=700$ GeV and $800$ GeV at the $\sqrt{s}=1$ TeV
(left panel) and $M_{N}=1.5$ TeV and $2$ TeV at the $\sqrt{s}=3$ TeV (right
panel) linear colliders.

In Figs.19, 20 and 21, we plot the missing momentum $(p_{T}^{miss})$,
transverse momentum of the fat-b jet $p_{T}^{J_{b}}$ and jet mass of the fat-b
jet $(M_{J_{b}})$ distributions for $M_{N}=700$ GeV and $800$ GeV at the
$\sqrt{s}=1$ TeV linear collider and $M_{N}=1.5$ TeV and $2$ TeV at the
$\sqrt{s}=3$ TeV linear collider. In view of these distributions, we have used
the following advanced selection cuts to reduce the SM background:

####  III.3.1 Advanced cuts for $M_{N}=400$ GeV- $900$ GeV at the $\sqrt{s}=1$
TeV linear collider after the detector simulation

  * •

Transverse momentum for $J_{b}$, $p_{T}^{J_{b}}>250$ GeV.

  * •

Fat-b mass, $M_{J_{b}}>115$ GeV.

  * •

Missing energy, $p_{T}^{miss}>150$ GeV.

We consider two benchmark points such as $M_{N}=700$ GeV and $800$ GeV at the
$1$ TeV linear collider to produce the boosted Higgs from RHNs. The cut flow
has been shown in Tab. 7. The b-jets are coming from the SM $h$ as the
$M_{J_{b}}$ distribution peaks at the Higgs mass for the signal at the linear
colliders. As a result $M_{J_{b}}>115$ GeV sets a strong cut on the SM
backgrounds.

Cuts | Signal | Background
---|---|---
|  $M_{N}=700$ GeV |  $M_{N}=800$ GeV |
Basic Cuts | 1,288,150 | 1,248,340 | 19,300
$p_{T}^{miss}>150$ GeV | 1,239,440 | 1,223,480 | 8,373
$p_{T}^{J_{b}}>250$ GeV | 1,100,790 | 1,153,650 | 4,239
$M_{J_{b}}>115$ GeV | 609,330 | 661,258 | 855
Table 7: Cut flow for the signal and background events for the final state
$J_{b}+p_{T}^{miss}$ for $M_{N}=700$ GeV and $800$ GeV at the $\sqrt{s}=1$ TeV
linear collider. The signal events are normalized by the square of the mixing.

####  III.3.2 Advanced cuts for the $M_{N}=1$ TeV -$2.9$ TeV for the
$\sqrt{s}=3$ TeV linear collider after the detector simulation

  * •

Transverse momentum for fat-b $(J_{b})$, $p_{T}^{J_{b}}>350$ GeV.

  * •

Fat-b mass, $M_{J_{b}}>115$ GeV.

  * •

Missing energy, $p_{T}^{miss}>175$ GeV.

We consider two benchmark points such as $M_{N}=1.5$ TeV and $2$ TeV at the
$3$ TeV linear collider for the boosted Higgs production from the RHN. The cut
flow has been shown in Tab. 8. The b-jets are coming from the SM $h$ as the
$M_{J_{b}}$ distribution peaks at the Higgs mass for the signal at the linear
colliders. As a result $M_{J_{b}}>115$ GeV sets a strong cut on the SM
backgrounds. We also consider a strong $p_{T}^{J_{b}}>350$ GeV cut for the
high mass RHNs at the $3$ TeV collider.

Cuts | Signal | Background
---|---|---
|  $M_{N}=1.5$ TeV |  $M_{N}=2$ TeV |
Basic Cuts | 5,077,160 | 4,043,130 | 74,245
$p_{T}^{miss}>175$ GeV | 5,005,240 | 4,011,420 | 39,231
$p_{T}^{J_{b}}>350$ GeV | 4,731,550 | 3,902,490 | 15,327
$M_{J_{b}}>115$ GeV | 2,961,620 | 2,479,960 | 3,740
Table 8: Cut flow for the signal and background events for the final state
$J_{b}+p_{T}^{miss}$ for $M_{N}=1.5$ TeV and $2$ TeV at the $\sqrt{s}=3$ TeV
linear collider. The signal events are normalized by the square of the mixing.

In this work, we adopt a minimalistic approach and consider a flat $70\%$
tagging efficiency for each of the daughter b jets coming from the Higgs
decay.

##  IV Current bounds

The bounds on the light-heavy neutrino mixing for the electron flavor comes
from a variety of searches. As we are interested on the RHN of mass $M_{N}\geq
100$ GeV, therefore we will compare our results with such bounds which are
important for that mass range. The Electroweak Precision Data (EWPD) bounds
have been calculated in delAguila:2008pw ; Akhmedov:2013hec ; deBlas:2013gla
which obtains the bound on $|V_{eN}|^{2}$ as $1.681\times 10^{-3}$ at the
$95\%$ C. L., the LEP2Achard:2001qv , calculated at the $95\%$ C.L., bounds
are rather weaker except $M_{N}=108$ GeV where it touches the EWPD line. The
strongest bounds are coming from the GERDA Agostini:2013mzu  $0\nu 2\beta$
study where the limits as calculated in Deppisch:2015qwa  up to $M_{N}=959$
GeV. The lepton universality limits from deGouvea:2015euy  set bounds on
$|V_{eN}|^{2}$ at $6.232\times 10^{-4}$ up to $M_{N}=1$ TeV at the $95\%$ C.
L. These bounds are plotted in Figs. 22 -27.

Apart from the above mentioned indirect searches, the recent collider searches
for the LHC also set bounds $|V_{eN}|^{2}$ at the $\sqrt{s}=8$ TeV at $95\%$
C. L. from same sign dilepton plus dijet search. The bounds on $|V_{eN}|^{2}$
from ATLAS (ATLAS$8$-$ee$) Aad:2015xaa  and CMS (CMS$8-ee$)
Khachatryan:2016olu  are obtained at $23.3$ fb-1 and $19.7$ fb-1 luminosities
respectively for the $e^{\pm}e^{\pm}+2j$ sample. The ATLAS limit is weaker
than the CMS limits for $100$ GeV $\leq M_{N}\leq 500$ GeV. The LHC has also
published the recent results at $\sqrt{s}=13$ TeV with $35.9$ fb-1 luminosity
which set stronger bounds on $|V_{eN}|^{2}$ than the previous direct searches
for $100$ GeV $\leq M_{N}\leq 500$ GeV. The bounds on $|V_{eN}|^{2}$ from the
$e^{\pm}e^{\pm}+2j$ signal in CMS (CMS$13$-$ee$) Sirunyan:2018xiv  and from
trilepton search at CMS (CMS$13$-$3\ell$) Sirunyan:2018mtv  are also
competitive, however, weaker than the EWPD for $100$ GeV $\leq M_{N}\leq 1.2$
TeV. These limits are also plotted in Figs. 22 -27.

We have explored that at the LHeC with $\sqrt{s}=1.3$ TeV collider energy and
$1$ ab-1 luminosity, the bound on $|V_{eN}|^{2}$ for $MN_{N}=600$ GeV with
$1$-$\sigma$ C.L. is better than the $0\nu 2\beta$ limit from GERDA-low where
as $M_{N}\geq 959$ GeV at $1$-$\sigma$ limit can be probed better than the
GERDA-low and high limit Agostini:2013mzu ; Deppisch:2015qwa . The GERDA
limits are stronger for the $M_{N}$ benchmarks we have studied. The results
have been shown in Fig. 22. In the same figure we show the bounds obtained
from the HE-LHeC with $\sqrt{s}=1.8$ TeV collider energy and $1$ ab -1
luminosity. In this case the current GERDA bounds are stronger up to
$M_{N}=959$ GeV Agostini:2013mzu ; Deppisch:2015qwa . At the HE-LHeC RHN up to
$M_{N}=1.2$ TeV can be probed at $5$-$\sigma$ and these bounds could be
stronger than the limits obtained from the EWPD-e limit delAguila:2008pw ;
Akhmedov:2013hec ; deBlas:2013gla . The improved scenario at the $3$ ab-1
luminosity for the LHeC and HE-LHeC are shown in Figs. 23.

At the linear collider we have explored two sets of signals. one is the
$e+J+p_{T}^{miss}$ and the other one is $J_{b}+p_{T}^{miss}$. Using
$e+J+p_{T}^{miss}$ signal at the $1$ TeV linear collider we have probed RHNs
between $400$ GeV $\leq M_{N}\leq 900$ GeV at $5$-$\sigma$ but the $0\nu
2\beta$ limit from GERDA Deppisch:2015qwa  is stronger than this result for
$M_{N}\leq 959$ GeV, however, the bounds on $|V_{eN}|^{2}$ for the RHNs
heavier than $1$ TeV can be probed at $5$-$\sigma$ significance or more at the
linear collider with the $3$ TeV center of mass energy. All the results are.
In this case apart from the fat jet properties, the polar angle cut for the
leptons worked nicely. The results are shown in Fig. 24. We have also studied
the linear colliders at $1(3)$ TeV center of mass energy with $3(5)$ ab-1
luminosity. We can find the improved results in Fig. 25. Using the
$J_{b}+p_{T}^{miss}$ signal we did a complementarity check where $M_{N}\geq 1$
TeV can be probed better than GERDA Deppisch:2015qwa  at $5$-$\sigma$
significance or more at the $3$ TeV linear collider. The linear collider can
probe $|V_{eN}|^{2}$ down to $\mathcal{O}(10^{-5})$ for $M_{N}=1.35$ TeV at
$3$ TeV, however, compared to this the bounds obtained at the $1$ TeV linear
collider are weaker. The corresponding bounds at the $\sqrt{s}=1$ TeV and $3$
TeV linear collider are plotted in Figs. 24 and 26. The red (blue) band
represents the bounds on $|V_{eN}|^{2}$ at $1$ TeV ($3$ TeV) linear collider
at different confidence levels. Comparing the bounds between the final states
$e+J+p_{T}^{miss}$ and $J_{b}+p_{T}^{miss}$ we find that the former one puts
slightly stronger limits on $|V_{eN}|^{2}$. The results are shown in Fig. 26.
We have also studied the linear colliders at $1(3)$ TeV center of mass energy
with $3(5)$ ab-1 luminosity. We can find the improved results in Fig. 27.
Finally we comment that our results at the linear collider are stronger than
the limits obtained from the EWPD-e delAguila:2008pw ; Akhmedov:2013hec ;
deBlas:2013gla  throughout the study.

Figure 22: The prospective upper limits on $|V_{eN}|^{2}$ at the $1.3$ TeV
LHeC (blue band) and $1.8$ TeV HE-LHeC (red band) at the $1$ ab-1 luminosity
compared to EWPD delAguila:2008pw ; Akhmedov:2013hec ; deBlas:2013gla ,
LEP2Achard:2001qv , GERDA Agostini:2013mzu  $0\nu 2\beta$ study from
Deppisch:2015qwa , ATLAS (ATLAS$8$-$ee$) Aad:2015xaa , CMS (CMS$8-ee$)
Khachatryan:2016olu  at the $8$ TeV LHC, $13$ TeV CMS search for
$e^{\pm}e^{\pm}+2j$ (CMS$13$-$ee$) Sirunyan:2018xiv  and $13$ TeV CMS search
for $3\ell$ (CMS$13$-$ee$) Sirunyan:2018xiv  respectively. Figure 23: Same as
Fig. 22 with $3$ ab-1 luminosity at the $1.3$ TeV LHeC and $1.8$ TeV HE-LHeC.
Figure 24: The prospective upper limits on $|V_{eN}|^{2}$ at the $1$ TeV (red
band) and $3$ TeV (blue band) linear colliders at the $1$ ab-1 luminosity for
$e+J+p_{T}^{miss}$ signal compared to EWPD delAguila:2008pw ; Akhmedov:2013hec
; deBlas:2013gla , LEP2Achard:2001qv , GERDA Agostini:2013mzu  $0\nu 2\beta$
study from Deppisch:2015qwa , ATLAS (ATLAS$8$-$ee$) Aad:2015xaa , CMS
(CMS$8-ee$) Khachatryan:2016olu  at the $8$ TeV LHC, $13$ TeV CMS search for
$e^{\pm}e^{\pm}+2j$ (CMS$13$-$ee$) Sirunyan:2018xiv  and $13$ TeV CMS search
for $3\ell$ (CMS$13$-$ee$) Sirunyan:2018xiv  respectively. Figure 25: Same as
Fig. 24 with $3(5)$ ab-1 luminosity at the $1(3)$ TeV linear collider. Figure
26: The prospective upper limits on $|V_{eN}|^{2}$ at the $1$ TeV (red band)
and $3$ TeV (blue band) linear colliders at the $1$ ab-1 luminosity for
$J_{b}+p_{T}^{miss}$ signal compared to EWPD delAguila:2008pw ;
Akhmedov:2013hec ; deBlas:2013gla , LEP2Achard:2001qv , GERDA Agostini:2013mzu
$0\nu 2\beta$ study from Deppisch:2015qwa , ATLAS (ATLAS$8$-$ee$) Aad:2015xaa
, CMS (CMS$8-ee$) Khachatryan:2016olu  at the $8$ TeV LHC, $13$ TeV CMS search
for $e^{\pm}e^{\pm}+2j$ (CMS$13$-$ee$) Sirunyan:2018xiv  and $13$ TeV CMS
search for $3\ell$ (CMS$13$-$ee$) Sirunyan:2018xiv  respectively. Figure 27:
Same as Fig. 26 with $3(5)$ ab-1 luminosity at the $1(3)$ TeV linear collider.

##  V Conclusion

We have studied the RHNs which can be responsible for the generation of the
tiny light neutrino masses. We have calculated the production cross sections
for the RHNs at the LHeC and linear collider at various center of mass
energies and followed by that we have tested the discovery prospects of this
RHNs. We have chosen $\sqrt{s}=1.3$ TeV and $1.8$ TeV for the LHeC and
$\sqrt{s}=1$ TeV and $3$ TeV for the linear collider. We have considered the
sufficiently heavy mass range of the RHNs. These RHNs can decay dominantly
into $\ell W$ mode. A massive RHN can sufficiently boost the $W$ such that its
hadronic decay modes can form a fat-jet. Therefore we study $e+j_{1}+J$ and
$e+J+p_{T}^{miss}$ at the LHeC and linear collider respectively. Similarly we
consider another interesting mode $N\rightarrow h\nu,h\rightarrow
b\overline{b}$ where a boosted SM Higgs can produce a fat b-jet and test the
$J_{b}+p_{T}^{miss}$ final state at the linear collider. Simulating the events
and passing through the selection cuts for the different colliders we
calculate the bounds on $|V_{eN}|^{2}$ at different luminosities and compare
with the existing bounds. Hence we conclude that $M_{N}\geq 959$ GeV can be
successfully probed at the $1.8$ TeV at the at $5$-$\sigma$ C. L. with $1$
ab-1 and $3$ ab-1 luminosities respectively. Whereas $M_{N}\leq 2.9$ TeV can
be probed at the $3$ TeV linear collider with more than $5$-$\sigma$ C.L using
the $e+J+p_{T}^{miss}$ signal. A complementary signal of $J_{b}+p_{T}^{miss}$
can be useful, too but this is weaker than the bounds obtained by the
$e+J+p_{T}^{miss}$ final state.

Note added:  While in final drafting phase, we noticed Ref.mani  appeared in
arXiv which also studied fat jet signatures from RHNs at the linear colliders.
We have studied LHeC and linear collider at different center of mass energies
using detailed cut based analyses. We have compared our results with all the
existing bounds using the decay modes of the RHNs to $W$ and SM $h$ bosons.
The $0\nu 2\beta$ bound became very strong up to $M_{N}=959$ GeV. At the
linear collider the polar angle variable for the lepton became very useful for
us. In our analysis we have showed that high mass RHNs can be observed at
$5$-$\sigma$ significance or more in these colliders.

## Acknowledgement

The work of SN and SJ was in part supported by US Department of Energy Grant
Number DE-SC 0016013. AD would like to thank Oliver Fischer for useful
discussions on the LHeC and sharing the DELPHES card for that detector. AD
would also like to thank Daniel Jeans (KEK), Kentarou Mawatari (Osaka
University) and Junping Tian (KEK) for useful information regarding the linear
collider card in DELPHES.

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